research article time consistent strategies for mean
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 709129 16 pageshttpdxdoiorg1011552013709129
Research ArticleTime Consistent Strategies for Mean-VarianceAsset-Liability Management Problems
Hui-qiang Ma12 Meng Wu3 and Nan-jing Huang2
1 School of Economics Southwest University for Nationalities Chengdu Sichuan 610041 China2Department of Mathematics Sichuan University Chengdu Sichuan 610064 China3 Business School Sichuan University Chengdu Sichuan 610064 China
Correspondence should be addressed to Meng Wu shancherishhotmailcom
Received 28 May 2013 Accepted 6 August 2013
Academic Editor Cheng Shao
Copyright copy 2013 Hui-qiang Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper studies the optimal time consistent investment strategies in multiperiod asset-liability management problems undermean-variance criterion By applying time consistent model of Chen et al (2013) and employing dynamic programming techniquewe derive two-time consistent policies for asset-liability management problems in a market with and without a riskless assetrespectively We show that the presence of liability does affect the optimal strategy More specifically liability leads a parallel shiftof optimal time-consistent investment policy Moreover for an arbitrarily risk averse investor (under the variance criterion) withliability the time-diversification effects could be ignored in a market with a riskless asset however it should be considered in amarket without any riskless asset
1 Introduction
By using variance as a risk measure Markowitz [1] proposedthe classic mean-variance portfolio selection model whichhas become the theoretical foundation of modern financetheory and has been extended in several directions Oneof the main extensions for portfolio selection is to studythe optimal policy in a multiperiod setting For exampleLi and Ng [2] and Zhou and Li [3] employed an embed-ding technique to derive analytical solutions to multiperiodand continuous-time mean-variance models respectively Inmost of the studies in the multiperiod environment therehas been a common assumption that an investor has nolong-term liability However in reality many investmentinstitutions (eg pension funds insurance company andbanks) have paid great attention to their portfolios whiletaking into account their liabilities Further it has beenshown by Kell and Muller [4] and Sharpe and Tint [5] thatliability does affect the optimal policy More specifically in asingle-period setting liability leads a parallel shift of mean-variance optimal investment policy and affects the mean-variance efficient frontier Due to both theoretical interestand practical importance of asset-liability management the
research on mean-variance asset-liability management hasattracted recent attentions For example among othersLeippold et al [6] considered the multiperiod asset-liabilitymanagement problem where the liability is exogenous andfixed and derived an analytical optimal policy and an efficientfrontier Further they extended their research to the casewhere the liability is endogenous and controllable in [7] Chiuand Li [8] and Xie et al [9] studied continuous-time mean-variance asset-liability management problems respectivelyFurthermore Xie [10] studied mean-variance model withstochastic liability in a Markovian regime switching financialmarket and Zeng and Li [11] investigated asset-liabilitymanagement problem in a jump diffusion market
In most of the literature the popular approaches of deal-ing with dynamic mean-variance asset-liability managementproblem are embedding technique which was developed byLi and Ng [2] and dual method However since the iterated-expectation property does not hold for the variance operatorthe optimal asset-liability management policy (called pre-commitment strategy) derived by both the approaches doesnot satisfy Bellmanrsquos optimality principle and is time incon-sistentThemain reason is that the precommitment strategyfor time interval [119905 119879] computed at time 119905will not necessarily
2 Mathematical Problems in Engineering
coincide with the strategy which computed at time 119905 + Δ119905As a result at time 119905 + Δ119905 the strategy computed at time 119905
will not be implemented by the investor unless there existssome commitment mechanism Strotz [12] first formalizedtime inconsistency and pointed out that the conflict couldbe solved by a time-consistent strategy Very recently muchmore scholars have paid their attentions on constructinga time-consistent mean-variance portfolio choice Amongothers Basak and Chabakauri [13] provided a fully analyt-ical characterization of the optimal time-consistent mean-variance portfolio within a general incomplete market econ-omy Wang and Forsyth [14] developed a numerical schemefor determining the optimal asset allocation strategy for time-consistent continuous time mean-variance optimization Byallowing the trade-off between the mean and the variance ofthe terminal wealth varying over time Cui et al [15] proposedaweak time consistency to comparewith Bellmanrsquos optimalityprinciple and derived an optimal mean-variance portfoliostrategy In all the literature mentioned above the studiesmainly referred to Bellmanrsquos optimality principle Howeverthe requirement (for short REQ) that local optimum isalso globally optimum is not necessarily satisfied whichis an essential requirement in solving the relevant optimalportfolio problem by the dynamic programming techniqueIn order to make up this shortfall Chen et al [16] by using atime consistent dynamic risk measure proposed a separabledynamic mean-variance model and showed that the relevantoptimal investment policy satisfies not only the Bellmanrsquosoptimality principle but also the REQ
Although asset-liability management is an importantissue in modern finance theory the time-consistent asset-liability management problem has not attracted enoughattention Recently Li et al [17] reported the time-consistentasset-liability management problem in the continuous-timesetting They employed Basak and Chabakaurirsquos [13] modelto study the continuous-time asset-liability managementproblem They derived the time-consistent optimal strategyand showed that the time-consistent efficient frontier withliability is below that without liability However the derivedtime-consistent policy does not satisfy the REQ As the afore-mentioned importance of the REQ in this paper we employthemodel of Chen et al [16] to analysis themultiperiod asset-liability management problem We derive time-consistentoptimal investment policies in a market with and withouta riskless asset respectively After comparing the optimaltime-consistent policies withmyopic strategies we show thatfor an arbitrarily risk averse investor if there is a risklessasset in the market the time-diversification effects arisingfrommultiperiod optimization can be ignored otherwise theeffects should be considered
This paper proceeds as follows In the next sectionwe formulate a time-consistent asset-liability managementmodel In Section 3 we derive the time-consistent optimalpolicy for a market without riskless asset Section 4 derivesthe time-consistent optimal policy for a market with bothriskless and risky assets Section 5 performs numericalexamples to illustrate our results The paper is concluded inSection 6
2 Model Formulation
Throughout this paper we assume that 119879 isin N is a fixed andfinite time horizon and trading only takes place at time 119896 =
0 1 119879 Let (ΩF 119901) be a probability space and let a 120590-fieldF
119905be the available information at time 119905
Consider a securitymarket consisting of one riskless assetand 119899 risky assets The return of the riskless asset at the 119905thinvestment period is assumed to be 119904
119905 The return of the 119894th
risky asset at the 119905th investment period is denoted by 119877119894
119905
and the relevant random return vector is denoted by 119877119905=
(1198771
119905 119877
119899
119905)1015840 which isF
119905+1-measurable
Consider an investor with an initial endowment 1199090and a
liability 1198970 We assume that the liability cannot be controlled
and denote by 119897119905the accumulative liability at time 119905 Let119876
119905be
the return of the liability at the 119905th investment period whichis F119905+1
-measurable It is clear that 119897119905+1
= 119876119905119897119905 Assume that
(1198771015840
119905 119876119905)1015840 119905 = 0 119879minus1 are statistically independentDenote
the expected return vector by 120583119905= 119864(119877
119905) and the variance-
covariancematrix by Σ119905= (Cov(119877119894
119905 119877119895
119905))119899times119899
which is assumedto be positive definite throughout this paper It is clear that119864(1198771199051198771015840
119905) = Σ119905+ 1205831199051205831015840
119905is also positive definite Denote by Σ
0
119905=
(Cov(119876119905 1198771
119905) Cov(119876
119905 119877119899
119905))1015840 the covariance vector The
investor begins hisher investment at time 0 and invests thecash amount119906119894
119905in the 119894th risky asset at the beginning of the 119905th
investment period where 119905 = 0 119879 minus 1 119906119905= (1199061
119905 119906
119899
119905)1015840
is called an investment decision at the 119905th investment periodand 119906 = (119906
0 119906
119879minus1) is an investment policy during the
entire investment horizonWe denote the wealth and the sur-plus of the investor at time 119905 by 119909
119905and 119878119905= 119909119905minus119897119905 respectively
A single-period conditional risk mapping is defined as
120588119905(119878119905+1
| F119905) = Var
119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
) (1)
where 119864119905(119878119905+1
) = 119864(119878119905+1
| F119905) and Var
119905(119878119905+1
) = Var(119878119905+1
|
F119905) are the single-period conditional expectation of 119878
119905+1and
its conditional variance respectively Tradeoff parameter 120582119905
as a constant which is defined on [0infin] is a weight whichpresents the relative importance of expected profit comparedto the risk Note that if 120582
119905= 0 then the investor is arbitrary
risk averse who only focus on the risk if 120582119905= infin then the
investor is risk neutral who only concerns maximizing theirexpected profit if 120582
119905isin (0infin) the investor is risk averse who
considers both expected profit and the risk in his decisionIn multiperiod portfolio selection two main optimal
investment policies are the myopic and time-consistentstrategiesMyopic Strategy is a strategy whereby at each time119905 the investor determines their optimal investment decisionassuming the instantaneous moments of assets returns willremain fixed at their current values for the remainder of theinvestment horizon More specifically for any 119905 isin 0 119879 minus
1 the myopic strategy 119906119898119910
119905is a solution to the following
problem
min119906119905isinΠ119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
) (2)
where Π119905is a set of all permit policies at time 119905 Intuitively
a myopic investor only cares about the mean and variance ofthe surplus at the current period
Mathematical Problems in Engineering 3
To measure the total risk of an investor amongmultiperi-ods (after time 119905) we employ a separable expected conditionalmapping which is defined as (see Chen et al [16])
119879
sum119895=119905+1
119864119905(120588119895minus1
(119878119895| F119895minus1
)) (3)
which reflects all the risk in the future Following thisassumption a separable dynamic mean-variance problem isdefined as
min119906isinΠ
119879
sum119895=1
119864 (120588119895minus1
(119878119895| F119895minus1
)) (4)
where Π is a set of all permit policies The optimal policyof problem (4) which satisfies both Bellmanrsquos optimalityprinciple and requirement REQ is called Time-ConsistentStrategy Note that both Bellmanrsquos optimality principle andrequirement REQ could be proved by following the method-ology of Chen et al [16]Thus problem (4) can be recursivelysolved by the dynamic programming technique Applying theiterated-expectation property of the expected operator thatis 119864(119864(sdot | F
119895) | F119896) = 119864(sdot | F
119896) for 119895 gt 119896 we have
119879
sum119895=1
119864 (120588119895minus1
(119878119895| F119895minus1
))
= 1205880(1198781) + 119864 (120588
1(1198782)
+ 1198641(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
(5)(for more details see Chen et al [16]) Then problem (4) isequivalent tomin119906isinΠ
1205880(1198781)
+ 119864 (1205881(1198782) + 1198641(1205882(1198783)
+ sdot sdot sdot + 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
(6)It follows from Bellmanrsquos optimality principle and (6) that(4) is equivalent to find an optimal strategy to satisfy thefollowing problem
min1199060
(1205880(1198781)
+ 119864min1199061
(1205881(1198782)
+ 1198641min1199062
(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
min119906119879minus2
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
min119906119879minus1
(120588119879minus1
(119878119879))) )))
(7)
We solve this problem in the following sections In order todiscuss the impact of riskless asset the market is consideredin two cases with and without riskless asset We demonstratethese results in Sections 3 and 4 respectively
3 Time Consistent Optimal Strategy withoutRiskless Asset
Consider a market consisting of only 119899 risky assets andassume that the wealth process 119909
119905is in a self-financing
fashion We list the notations of this section in Table 1 Thewealth process 119909
119905could be described as follows
119909119905+1
= 1198771015840
119905119906119905 1198681015840119906119905= 119909119905 119905 = 0 1 119879 minus 1 (8)
where 119868 = (1 1)1015840isin R119899 In this setting problem (6) can
be written as follows
min 1205880(1198781) + 1198640(1205881(1198782)
+ 1198641(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(9)
Since Σ1
119879minus1is positive definite we have 119886
119879minus1gt 0 Further
it follows from the nonnegative definiteness of 119864(119877119879minus1
1198771015840
119879minus1)
that Σ1
119879minus2is also positive definite and 119886
119879minus2gt 0 By using
mathematical induction we conclude that for any 119905 =
0 119879 minus 1 Σ1119905is also positive definite and 119886
119905gt 0
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (9) is givenin the following theorem
Theorem 1 The time-consistent optimal investment policy ofproblem (9) is given by
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868)
+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
for 119905 = 0 119879 minus 1
(10)
4 Mathematical Problems in Engineering
Table 1 Model notations of Section 3
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119902119905+1
Σ1
119879minus1= Σ119879minus1
Σ1
119905= Σ119905+ 119864 (119877
1199051198771015840
119905)119886119905+1
119886119879minus1
= 1198681015840(Σ1
119879minus1)minus1
119868 119886119905= 1198681015840(Σ1
119905)minus1
119868
119887119879minus1
= 1198681015840(Σ1
119879minus1)minus1
120583119879minus1
119887119905= 1198681015840(Σ1
119905)minus1
120583119905
119888119879minus1
= 1205831015840
119879minus1(Σ1
119879minus1)minus1
120583119879minus1
119888119905= 1205831015840
119905(Σ1
119905)minus1
120583119905
119889119879minus1
= 1198681015840(Σ1
119879minus1)minus1
Σ0
119879minus1119889119905= 1198681015840(Σ1
119905)minus1
Σ0
119905
119902119879minus1
= 119887119879minus1
119886119879minus1
119902119905= 119887119905119886119905
120572119879minus1
= 119888119879minus1
minus 119902119879minus1
119887119879minus1
120572119905= 119888119905minus 119902119905119887119905
120574119879minus1
= 119889119879minus1
119886119879minus1
120574119905= 119889119905119886119905
120574119879minus1
= 120574119879minus1
120574119905= 120574119905+ 120574119905+1
119864(1198761199051198771015840
119905)(Σ1
119905)minus1
119868119886119905
1205731119879minus1
= 120582119879minus1
119864(119876119879minus1
) minus 120582119879minus1
(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
1205731119905
= minus1205821
1199051205831015840
119905(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868) minus 120582
1
119905120574119905+1
1205831015840
119905(Σ1
119905)minus1
119864(119876119905119877119905)
+ 120582119879minus1
119887119879minus1
120574119879minus1
+ (120582119905+ 1205731119905+1
)119864(119876119905)
1205732119879minus1
= Var(119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1+ 120574119879minus1
119889119879minus1
1205732119905
= [Var(119876119905) + 1205732119905+1
119864(1198762
119905)] + 119886
1199051205742
119905minus [Σ0
119905+ 120574119905+1
119864(119876119905119877119905)]1015840
sdot (Σ1
119905)minus1[Σ0
119905+ 120574119905+1
119864(119876119905119877119905)]
Proof When 119905 = 119879minus1 for given wealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879 minus 1)th period problem (9) can beexpressed as follows
min Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 1198771015840
119879minus1119906119879minus1
= 119909119879
1198681015840119906119879minus1
= 119909119879minus1
119897119879= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(11)
Substituting the binding constraints into the objective func-tion we have
min 119881119879minus1
(119909119879minus1
119897119879minus1
)
= 1199061015840
119879minus1Σ1
119879minus1119906119879minus1
minus 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21199061015840
119879minus1] 119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
st 1198681015840119906119879minus1
= 119909119879minus1
(12)
which is a linear-quadratic program By using the Lagrangemultiplier technique and letting 120596
119879be the Lagrange multi-
plier the Lagrange function is defined as
119871 (119906119879minus1
) = 1199061015840
119879minus1Σ1
119879minus1119906119879minus1
minus 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
minus 120596119879(119909119879minus1
minus 1198681015840119906119879minus1
)
(13)
By using the first-order necessary optimality condition wehave
2Σ1
119879minus1119906119879minus1
minus 2(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
) + 120596119879119868 = 0 (14)
119909119879minus1
minus 1198681015840119906119879minus1
= 0 (15)
From (14) we can easily have
119906lowast
119879minus1= (Σ1
119879minus1)minus1
(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
) minus1
2120596119879(Σ1
119879minus1)minus1
119868
(16)
1198681015840119906lowast
119879minus1= 1198681015840(Σ1
119879minus1)minus1
(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
)
minus1
21205961198791198681015840(Σ1
119879minus1)minus1
119868
= 119889119879minus1
119897119879minus1
+120582119879minus1
2119887119879minus1
minus1
2119886119879minus1
120596119879
= 119909119879minus1
(17)
which implies that the Lagrange multiplier 120596119879is
120596119879= 2
119889119879minus1
119886119879minus1
119897119879minus1
+ 120582119879minus1
119887119879minus1
119886119879minus1
minus 21
119886119879minus1
119909119879minus1
= 2120574119879minus1
119897119879minus1
+ 120582119879minus1
119902119879minus1
minus 21
119886119879minus1
119909119879minus1
(18)
Substituting 120596119879into (16) we have
119906lowast
119879minus1=
(Σ1
119879minus1)minus1
119868
119886119879minus1
119909119879minus1
+ (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2(Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
(19)
Mathematical Problems in Engineering 5
Further by substituting 119906lowast
119879minus1into the objective function of
problem (12) (see Appendix A for more details) we have
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) =1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(20)
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem is given as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 1198771015840
119879minus2119906119879minus2
= 119909119879minus1
1198681015840119906119879minus2
= 119909119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(21)
It follows from (20) that
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
=1
119886119879minus1
119864119879minus2
(1199092
119879minus1) minus 120582119879minus1
119902119879minus1
119864119879minus2
(119909119879minus1
)
minus 2120574119879minus1
119864119879minus2
(119909119879minus1
119897119879minus1
) + 1205731119879minus1
119864119879minus2
(119897119879minus1
)
+ 1205732119879minus1
119864119879minus2
(1198972
119879minus1) minus
(120582119879minus1
)2
4120572119879minus1
(22)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (21) we have
min 119881119879minus2
(119909119879minus2
119897119879minus2
)
st 1198681015840119906119879minus2
= 119909119879minus2
(23)
where
119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2 [(Σ0
119879minus2)1015840
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2)] 119897119879minus2
+1205821
119879minus2
21205831015840
119879minus2119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(24)
By setting 120596119879minus1
be the Lagrange multiplier the Lagrangefunction for problem (23) is
119871 (119906119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2Φ1015840
119879minus2119906119879minus2
+ Ψ119879minus2
minus 120596119879minus1
(119909119879minus2
minus 1198681015840119906119879minus2
)
(25)
where
Φ119879minus2
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] 119897119879minus2
+1205821
119879minus2
2120583119879minus2
Ψ119879minus2
= [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(26)
From the first-order necessary optimality condition we have
2Σ1
119879minus2119906119879minus2
minus 2Φ119879minus2
+ 120596119879minus1
119868 = 0 (27)
119909119879minus2
minus 1198681015840119906119879minus2
= 0 (28)
From (27) we have
119906lowast
119879minus2= (Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
(Σ1
119879minus2)minus1
119868 (29)
1198681015840119906lowast
119879minus2= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
1198681015840(Σ1
119879minus2)minus1
119868
= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
119886119879minus2
= 119909119879minus2
(30)
which implies
120596119879minus1
= 21198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
minus 21
119886119879minus2
119909119879minus2
(31)
Substituting 120596119879minus1
into (29) we get
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+ (Σ1
119879minus2)minus1
Φ119879minus2
minus1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
(Σ1
119879minus2)minus1
119868
(32)
Taking Φ119879minus2
into account we have
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+1205821
119879minus2
2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+ (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868] 119897119879minus2
(33)
6 Mathematical Problems in Engineering
Substituting 119906lowast
119879minus2into (23) gives (see Appendix B for more
details)
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2
minus(120582119879minus1
)2
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(34)
Next by using mathematical induction we show thatboth (10) and
119881lowast
119905(119909119905 119897119905) =
1
119886119905
1199092
119905minus 1205821
119905119902119905119909119905minus 2120574119905119909119905119897119905
+ 1205731119905119897119905+ 12057321199051198972
119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(35)
hold Suppose that (10) and (35) hold for time 119905 119905+1 119879minus1At the beginning of (119905minus1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905)
+ 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 1198771015840
119905minus1119906119905minus1
= 119909119905
1198681015840119906119905minus1
= 119909119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(36)
It follows from (35) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) =
1
119886119905
119864119905minus1
(1199092
119905) minus 1205821
119905119902119905119864119905minus1
(119909119905)
minus 2120574119905119864119905minus1
(119909119905119897119905) + 1205731119905119864119905minus1
(119897119905)
+ 1205732119905119864119905minus1
(1198972
119905) minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(37)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (36) we have
min 119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
st 1198681015840119906119905minus1
= 119909119905minus1
(38)
where
Φ119905minus1
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
+1205821
119905minus1
2120583119905minus1
Ψ119905minus1
= [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
+ 1205731119905] 119864 (119876
119905minus1) 119897119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(39)
Letting 120596119905be the Lagrange multiplier the Lagrange function
for problem (38) is given by
119871 (119906119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
minus 120596119905(119909119905minus1
minus 1198681015840119906119905minus1
)
(40)
It follows from the first-order necessary optimality conditionthat
2Σ1
119905minus1119906119905minus1
minus 2Φ119905minus1
+ 120596119905119868 = 0
119909119905minus1
minus 1198681015840119906119905minus1
= 0
(41)
Thus we have
119906lowast
119905minus1= (Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905(Σ1
119905minus1)minus1
119868 (42)
1198681015840119906lowast
119905minus1= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
21205961199051198681015840(Σ1
119905minus1)minus1
119868
= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905119886119905minus1
= 119909119905minus1
(43)
which implies
120596119905= 2
1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
minus 21
119886119905minus1
119909119905minus1
(44)
It follows from (42) that
119906lowast
119905minus1=
(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
minus1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
(Σ1
119905minus1)minus1
119868
+ (Σ1
119905minus1)minus1
Φ119905minus1
=(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
+1205821
119905minus1
2(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868)
+ (Σ1
119905minus1)minus1
[Σ0
119905minus1minus 120574119905minus1
119868 + 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
(45)
Substituting 119906lowast
119905minus1into the objective function of problem (38)
(see Appendix C for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4120572119894
(46)This completes the proof
Mathematical Problems in Engineering 7
Remark 2 If an investor does not have any liability that is119897119905equiv 0 for any 119905 isin 0 1 119879 minus 1 then the optimal time-
consistent investment strategy can be simplified as follows
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(47)
which is exactly the same as that in [16] This implies thatthe result of Chen et al [16] is a special case of Theorem 1Therefore Theorem 1 generalizes their result
Corollary 3 If the returns of liability and risky assets areuncorrelated that is Σ0
119905= 0 for any 119905 isin 0 1 119879 minus 1 then
the optimal investment policy for problem (9) is
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(48)
Proof Since Σ0119905= 0 it is easy to verify that 119889
119905= 1198681015840(Σ1
119905)minus1Σ0
119905=
0 120574119905= 119889119905119886119905= 0 and 120574
119905= 0 Substituting them into (10) gives
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(49)
This completes the proof
Remark 4 After comparing Corollary 3 and Remark 2 it isquite clear that if the return of liability is uncorrelated withthat of risky asset then the liability does not affect the time-consistent optimal policy in a market without riskless asset
Remark 5 If the return of liability is correlated to those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905 (50)
which depends on the current value of liability 119897119905 and the
covariance between the returns of liability and risky assetsΣ0
119905
Next we compare the time-consistent strategy with themyopic strategy in a market without riskless asset In such amarket problem (2) can be expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(51)
By using the same method in the proof ofTheorem 1 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+
120582119905
2(Σ119905)minus1
(120583119905minus 119902119905119868)
+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(52)
where 119886119905= 1198681015840(Σ119905)minus1119868 119902119905= 1198681015840(Σ119905)minus1120583119905119886119905 120574119905= 1198681015840(Σ119905)minus1Σ0
119905119886119905
It is clear that the difference between two strategies enters intoall of the three parts More specifically the following featureholds if the investor is arbitrarily risk averse that is 120582
119905rarr 0
then both the time consistent optimal strategy and myopicstrategy reduce to
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
119905 = 0 119879 minus 1
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(53)
respectively After comparing these two strategies we findthat if an investor is arbitrarily risk averse then heshe shouldconcern about the time-diversification effects arising frommultiperiod optimization
4 Time Consistent Optimal Strategy withRiskless Asset
In this section we consider a market which is consistingof one riskless asset and 119899 risky assets and assume that thewealth process 119909
119905is also in a self-financing fashion We list
the notations of this section in Table 2 The wealth process 119909119905
can be described as follows
119909119905= 1198751015840
119905minus1119906119905minus1
+ 119904119905minus1
119909119905minus1
(54)
where119875119905= 119877119905minus119904119905119868 In this setting problem (6) can be written
as followsmin 120588
0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot
+119864119879minus3
(120588119879minus2
(119878119879minus1
) + 119864119879minus2
(120588119879minus1
(119878119879))) ))
st 119909119905+1
= 1198751015840
119905119906119905+ 119904119905119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(55)
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (55) is givenin the following theorem
Theorem 6 The optimal investment strategy of problem (55)is given by
119906lowast
119905= Σminus1
119905120593119905119897119905+
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (56)
where 120593119905= 119864(119875
119905119876119905) minus 119864(119875
119905)119864(119876119905) and 120598
119905= 119864(119875
119905)
8 Mathematical Problems in Engineering
Table 2 Model notations of Section 4
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119904119905+1
1205731119879minus1
= 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1205731119905
= (120582119905+ 1205731119905+1
) 119864(119876119905) minus 1205821
1199051205981015840
119905(Σ119905)minus1120593119905
1205732119879minus1
= Var(119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1205732119905
= Var(119876119905) + 1205732119905+1
119864(1198762
119905) minus 1205931015840
119905(Σ119905)minus1120593119905
Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879minus1)th period problem (55) reducesto
min119906119879minus1
Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 119909119879
= 119904119879minus1
119909119879minus1
+ 1198751015840
119879minus1119906119879minus1
119897119879
= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(57)
Substituting the binding constraints into the objective func-tion we have
min119906119879minus1
119881119879minus1
(119909119879minus1
119897119879minus1
) (58)
where
119881119879minus1
(119909119879minus1
119897119879minus1
) = 1199061015840
119879minus1Σ119879minus1
119906119879minus1
minus 2(1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
(59)
It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have
119889 (119881119879minus1
(119909119879minus1
119897119879minus1
))
119889 (119906119879minus1
)= 2Σ119879minus1
119906119879minus1
minus 2(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 0
(60)
which implies
119906lowast
119879minus1= (Σ119879minus1
)minus1
[120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
] (61)
Substituting 119906lowast
119879minus1into the objective function of problem (58)
gives (see Appendix D for more details)
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(62)
where119872119879minus1
= 1205981015840
119879minus1Σminus1
119879minus1120598119879minus1
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 119909119879minus1
= 119904119879minus2
119909119879minus2
+ 1198751015840
119879minus2119906119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(63)
From (62) we can easily have
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) = 1205732119879minus1
119864 (1198762
119879minus2) 1198972
119879minus2
+ 1205731119879minus1
119864 (119876119879minus2
) 119897119879minus2
minus 120582119879minus1
119904119879minus1
119904119879minus2
119909119879minus2
minus 120582119879minus1
119904119879minus1
1205981015840
119879minus2119906119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(64)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (63) we have
min119906119879minus2
119881119879minus2
(119909119879minus2
119897119879minus2
) (65)
where119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ119879minus2
119906119879minus2
minus 2(1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(66)
The first-order necessary optimality condition implies
119889 (119881119879minus2
(119909119879minus2
119897119879minus2
))
119889 (119906119879minus2
)= 2Σ119879minus2
119906119879minus2
minus 2(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 0
(67)
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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2 Mathematical Problems in Engineering
coincide with the strategy which computed at time 119905 + Δ119905As a result at time 119905 + Δ119905 the strategy computed at time 119905
will not be implemented by the investor unless there existssome commitment mechanism Strotz [12] first formalizedtime inconsistency and pointed out that the conflict couldbe solved by a time-consistent strategy Very recently muchmore scholars have paid their attentions on constructinga time-consistent mean-variance portfolio choice Amongothers Basak and Chabakauri [13] provided a fully analyt-ical characterization of the optimal time-consistent mean-variance portfolio within a general incomplete market econ-omy Wang and Forsyth [14] developed a numerical schemefor determining the optimal asset allocation strategy for time-consistent continuous time mean-variance optimization Byallowing the trade-off between the mean and the variance ofthe terminal wealth varying over time Cui et al [15] proposedaweak time consistency to comparewith Bellmanrsquos optimalityprinciple and derived an optimal mean-variance portfoliostrategy In all the literature mentioned above the studiesmainly referred to Bellmanrsquos optimality principle Howeverthe requirement (for short REQ) that local optimum isalso globally optimum is not necessarily satisfied whichis an essential requirement in solving the relevant optimalportfolio problem by the dynamic programming techniqueIn order to make up this shortfall Chen et al [16] by using atime consistent dynamic risk measure proposed a separabledynamic mean-variance model and showed that the relevantoptimal investment policy satisfies not only the Bellmanrsquosoptimality principle but also the REQ
Although asset-liability management is an importantissue in modern finance theory the time-consistent asset-liability management problem has not attracted enoughattention Recently Li et al [17] reported the time-consistentasset-liability management problem in the continuous-timesetting They employed Basak and Chabakaurirsquos [13] modelto study the continuous-time asset-liability managementproblem They derived the time-consistent optimal strategyand showed that the time-consistent efficient frontier withliability is below that without liability However the derivedtime-consistent policy does not satisfy the REQ As the afore-mentioned importance of the REQ in this paper we employthemodel of Chen et al [16] to analysis themultiperiod asset-liability management problem We derive time-consistentoptimal investment policies in a market with and withouta riskless asset respectively After comparing the optimaltime-consistent policies withmyopic strategies we show thatfor an arbitrarily risk averse investor if there is a risklessasset in the market the time-diversification effects arisingfrommultiperiod optimization can be ignored otherwise theeffects should be considered
This paper proceeds as follows In the next sectionwe formulate a time-consistent asset-liability managementmodel In Section 3 we derive the time-consistent optimalpolicy for a market without riskless asset Section 4 derivesthe time-consistent optimal policy for a market with bothriskless and risky assets Section 5 performs numericalexamples to illustrate our results The paper is concluded inSection 6
2 Model Formulation
Throughout this paper we assume that 119879 isin N is a fixed andfinite time horizon and trading only takes place at time 119896 =
0 1 119879 Let (ΩF 119901) be a probability space and let a 120590-fieldF
119905be the available information at time 119905
Consider a securitymarket consisting of one riskless assetand 119899 risky assets The return of the riskless asset at the 119905thinvestment period is assumed to be 119904
119905 The return of the 119894th
risky asset at the 119905th investment period is denoted by 119877119894
119905
and the relevant random return vector is denoted by 119877119905=
(1198771
119905 119877
119899
119905)1015840 which isF
119905+1-measurable
Consider an investor with an initial endowment 1199090and a
liability 1198970 We assume that the liability cannot be controlled
and denote by 119897119905the accumulative liability at time 119905 Let119876
119905be
the return of the liability at the 119905th investment period whichis F119905+1
-measurable It is clear that 119897119905+1
= 119876119905119897119905 Assume that
(1198771015840
119905 119876119905)1015840 119905 = 0 119879minus1 are statistically independentDenote
the expected return vector by 120583119905= 119864(119877
119905) and the variance-
covariancematrix by Σ119905= (Cov(119877119894
119905 119877119895
119905))119899times119899
which is assumedto be positive definite throughout this paper It is clear that119864(1198771199051198771015840
119905) = Σ119905+ 1205831199051205831015840
119905is also positive definite Denote by Σ
0
119905=
(Cov(119876119905 1198771
119905) Cov(119876
119905 119877119899
119905))1015840 the covariance vector The
investor begins hisher investment at time 0 and invests thecash amount119906119894
119905in the 119894th risky asset at the beginning of the 119905th
investment period where 119905 = 0 119879 minus 1 119906119905= (1199061
119905 119906
119899
119905)1015840
is called an investment decision at the 119905th investment periodand 119906 = (119906
0 119906
119879minus1) is an investment policy during the
entire investment horizonWe denote the wealth and the sur-plus of the investor at time 119905 by 119909
119905and 119878119905= 119909119905minus119897119905 respectively
A single-period conditional risk mapping is defined as
120588119905(119878119905+1
| F119905) = Var
119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
) (1)
where 119864119905(119878119905+1
) = 119864(119878119905+1
| F119905) and Var
119905(119878119905+1
) = Var(119878119905+1
|
F119905) are the single-period conditional expectation of 119878
119905+1and
its conditional variance respectively Tradeoff parameter 120582119905
as a constant which is defined on [0infin] is a weight whichpresents the relative importance of expected profit comparedto the risk Note that if 120582
119905= 0 then the investor is arbitrary
risk averse who only focus on the risk if 120582119905= infin then the
investor is risk neutral who only concerns maximizing theirexpected profit if 120582
119905isin (0infin) the investor is risk averse who
considers both expected profit and the risk in his decisionIn multiperiod portfolio selection two main optimal
investment policies are the myopic and time-consistentstrategiesMyopic Strategy is a strategy whereby at each time119905 the investor determines their optimal investment decisionassuming the instantaneous moments of assets returns willremain fixed at their current values for the remainder of theinvestment horizon More specifically for any 119905 isin 0 119879 minus
1 the myopic strategy 119906119898119910
119905is a solution to the following
problem
min119906119905isinΠ119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
) (2)
where Π119905is a set of all permit policies at time 119905 Intuitively
a myopic investor only cares about the mean and variance ofthe surplus at the current period
Mathematical Problems in Engineering 3
To measure the total risk of an investor amongmultiperi-ods (after time 119905) we employ a separable expected conditionalmapping which is defined as (see Chen et al [16])
119879
sum119895=119905+1
119864119905(120588119895minus1
(119878119895| F119895minus1
)) (3)
which reflects all the risk in the future Following thisassumption a separable dynamic mean-variance problem isdefined as
min119906isinΠ
119879
sum119895=1
119864 (120588119895minus1
(119878119895| F119895minus1
)) (4)
where Π is a set of all permit policies The optimal policyof problem (4) which satisfies both Bellmanrsquos optimalityprinciple and requirement REQ is called Time-ConsistentStrategy Note that both Bellmanrsquos optimality principle andrequirement REQ could be proved by following the method-ology of Chen et al [16]Thus problem (4) can be recursivelysolved by the dynamic programming technique Applying theiterated-expectation property of the expected operator thatis 119864(119864(sdot | F
119895) | F119896) = 119864(sdot | F
119896) for 119895 gt 119896 we have
119879
sum119895=1
119864 (120588119895minus1
(119878119895| F119895minus1
))
= 1205880(1198781) + 119864 (120588
1(1198782)
+ 1198641(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
(5)(for more details see Chen et al [16]) Then problem (4) isequivalent tomin119906isinΠ
1205880(1198781)
+ 119864 (1205881(1198782) + 1198641(1205882(1198783)
+ sdot sdot sdot + 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
(6)It follows from Bellmanrsquos optimality principle and (6) that(4) is equivalent to find an optimal strategy to satisfy thefollowing problem
min1199060
(1205880(1198781)
+ 119864min1199061
(1205881(1198782)
+ 1198641min1199062
(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
min119906119879minus2
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
min119906119879minus1
(120588119879minus1
(119878119879))) )))
(7)
We solve this problem in the following sections In order todiscuss the impact of riskless asset the market is consideredin two cases with and without riskless asset We demonstratethese results in Sections 3 and 4 respectively
3 Time Consistent Optimal Strategy withoutRiskless Asset
Consider a market consisting of only 119899 risky assets andassume that the wealth process 119909
119905is in a self-financing
fashion We list the notations of this section in Table 1 Thewealth process 119909
119905could be described as follows
119909119905+1
= 1198771015840
119905119906119905 1198681015840119906119905= 119909119905 119905 = 0 1 119879 minus 1 (8)
where 119868 = (1 1)1015840isin R119899 In this setting problem (6) can
be written as follows
min 1205880(1198781) + 1198640(1205881(1198782)
+ 1198641(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(9)
Since Σ1
119879minus1is positive definite we have 119886
119879minus1gt 0 Further
it follows from the nonnegative definiteness of 119864(119877119879minus1
1198771015840
119879minus1)
that Σ1
119879minus2is also positive definite and 119886
119879minus2gt 0 By using
mathematical induction we conclude that for any 119905 =
0 119879 minus 1 Σ1119905is also positive definite and 119886
119905gt 0
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (9) is givenin the following theorem
Theorem 1 The time-consistent optimal investment policy ofproblem (9) is given by
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868)
+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
for 119905 = 0 119879 minus 1
(10)
4 Mathematical Problems in Engineering
Table 1 Model notations of Section 3
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119902119905+1
Σ1
119879minus1= Σ119879minus1
Σ1
119905= Σ119905+ 119864 (119877
1199051198771015840
119905)119886119905+1
119886119879minus1
= 1198681015840(Σ1
119879minus1)minus1
119868 119886119905= 1198681015840(Σ1
119905)minus1
119868
119887119879minus1
= 1198681015840(Σ1
119879minus1)minus1
120583119879minus1
119887119905= 1198681015840(Σ1
119905)minus1
120583119905
119888119879minus1
= 1205831015840
119879minus1(Σ1
119879minus1)minus1
120583119879minus1
119888119905= 1205831015840
119905(Σ1
119905)minus1
120583119905
119889119879minus1
= 1198681015840(Σ1
119879minus1)minus1
Σ0
119879minus1119889119905= 1198681015840(Σ1
119905)minus1
Σ0
119905
119902119879minus1
= 119887119879minus1
119886119879minus1
119902119905= 119887119905119886119905
120572119879minus1
= 119888119879minus1
minus 119902119879minus1
119887119879minus1
120572119905= 119888119905minus 119902119905119887119905
120574119879minus1
= 119889119879minus1
119886119879minus1
120574119905= 119889119905119886119905
120574119879minus1
= 120574119879minus1
120574119905= 120574119905+ 120574119905+1
119864(1198761199051198771015840
119905)(Σ1
119905)minus1
119868119886119905
1205731119879minus1
= 120582119879minus1
119864(119876119879minus1
) minus 120582119879minus1
(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
1205731119905
= minus1205821
1199051205831015840
119905(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868) minus 120582
1
119905120574119905+1
1205831015840
119905(Σ1
119905)minus1
119864(119876119905119877119905)
+ 120582119879minus1
119887119879minus1
120574119879minus1
+ (120582119905+ 1205731119905+1
)119864(119876119905)
1205732119879minus1
= Var(119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1+ 120574119879minus1
119889119879minus1
1205732119905
= [Var(119876119905) + 1205732119905+1
119864(1198762
119905)] + 119886
1199051205742
119905minus [Σ0
119905+ 120574119905+1
119864(119876119905119877119905)]1015840
sdot (Σ1
119905)minus1[Σ0
119905+ 120574119905+1
119864(119876119905119877119905)]
Proof When 119905 = 119879minus1 for given wealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879 minus 1)th period problem (9) can beexpressed as follows
min Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 1198771015840
119879minus1119906119879minus1
= 119909119879
1198681015840119906119879minus1
= 119909119879minus1
119897119879= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(11)
Substituting the binding constraints into the objective func-tion we have
min 119881119879minus1
(119909119879minus1
119897119879minus1
)
= 1199061015840
119879minus1Σ1
119879minus1119906119879minus1
minus 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21199061015840
119879minus1] 119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
st 1198681015840119906119879minus1
= 119909119879minus1
(12)
which is a linear-quadratic program By using the Lagrangemultiplier technique and letting 120596
119879be the Lagrange multi-
plier the Lagrange function is defined as
119871 (119906119879minus1
) = 1199061015840
119879minus1Σ1
119879minus1119906119879minus1
minus 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
minus 120596119879(119909119879minus1
minus 1198681015840119906119879minus1
)
(13)
By using the first-order necessary optimality condition wehave
2Σ1
119879minus1119906119879minus1
minus 2(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
) + 120596119879119868 = 0 (14)
119909119879minus1
minus 1198681015840119906119879minus1
= 0 (15)
From (14) we can easily have
119906lowast
119879minus1= (Σ1
119879minus1)minus1
(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
) minus1
2120596119879(Σ1
119879minus1)minus1
119868
(16)
1198681015840119906lowast
119879minus1= 1198681015840(Σ1
119879minus1)minus1
(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
)
minus1
21205961198791198681015840(Σ1
119879minus1)minus1
119868
= 119889119879minus1
119897119879minus1
+120582119879minus1
2119887119879minus1
minus1
2119886119879minus1
120596119879
= 119909119879minus1
(17)
which implies that the Lagrange multiplier 120596119879is
120596119879= 2
119889119879minus1
119886119879minus1
119897119879minus1
+ 120582119879minus1
119887119879minus1
119886119879minus1
minus 21
119886119879minus1
119909119879minus1
= 2120574119879minus1
119897119879minus1
+ 120582119879minus1
119902119879minus1
minus 21
119886119879minus1
119909119879minus1
(18)
Substituting 120596119879into (16) we have
119906lowast
119879minus1=
(Σ1
119879minus1)minus1
119868
119886119879minus1
119909119879minus1
+ (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2(Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
(19)
Mathematical Problems in Engineering 5
Further by substituting 119906lowast
119879minus1into the objective function of
problem (12) (see Appendix A for more details) we have
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) =1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(20)
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem is given as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 1198771015840
119879minus2119906119879minus2
= 119909119879minus1
1198681015840119906119879minus2
= 119909119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(21)
It follows from (20) that
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
=1
119886119879minus1
119864119879minus2
(1199092
119879minus1) minus 120582119879minus1
119902119879minus1
119864119879minus2
(119909119879minus1
)
minus 2120574119879minus1
119864119879minus2
(119909119879minus1
119897119879minus1
) + 1205731119879minus1
119864119879minus2
(119897119879minus1
)
+ 1205732119879minus1
119864119879minus2
(1198972
119879minus1) minus
(120582119879minus1
)2
4120572119879minus1
(22)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (21) we have
min 119881119879minus2
(119909119879minus2
119897119879minus2
)
st 1198681015840119906119879minus2
= 119909119879minus2
(23)
where
119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2 [(Σ0
119879minus2)1015840
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2)] 119897119879minus2
+1205821
119879minus2
21205831015840
119879minus2119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(24)
By setting 120596119879minus1
be the Lagrange multiplier the Lagrangefunction for problem (23) is
119871 (119906119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2Φ1015840
119879minus2119906119879minus2
+ Ψ119879minus2
minus 120596119879minus1
(119909119879minus2
minus 1198681015840119906119879minus2
)
(25)
where
Φ119879minus2
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] 119897119879minus2
+1205821
119879minus2
2120583119879minus2
Ψ119879minus2
= [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(26)
From the first-order necessary optimality condition we have
2Σ1
119879minus2119906119879minus2
minus 2Φ119879minus2
+ 120596119879minus1
119868 = 0 (27)
119909119879minus2
minus 1198681015840119906119879minus2
= 0 (28)
From (27) we have
119906lowast
119879minus2= (Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
(Σ1
119879minus2)minus1
119868 (29)
1198681015840119906lowast
119879minus2= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
1198681015840(Σ1
119879minus2)minus1
119868
= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
119886119879minus2
= 119909119879minus2
(30)
which implies
120596119879minus1
= 21198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
minus 21
119886119879minus2
119909119879minus2
(31)
Substituting 120596119879minus1
into (29) we get
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+ (Σ1
119879minus2)minus1
Φ119879minus2
minus1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
(Σ1
119879minus2)minus1
119868
(32)
Taking Φ119879minus2
into account we have
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+1205821
119879minus2
2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+ (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868] 119897119879minus2
(33)
6 Mathematical Problems in Engineering
Substituting 119906lowast
119879minus2into (23) gives (see Appendix B for more
details)
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2
minus(120582119879minus1
)2
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(34)
Next by using mathematical induction we show thatboth (10) and
119881lowast
119905(119909119905 119897119905) =
1
119886119905
1199092
119905minus 1205821
119905119902119905119909119905minus 2120574119905119909119905119897119905
+ 1205731119905119897119905+ 12057321199051198972
119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(35)
hold Suppose that (10) and (35) hold for time 119905 119905+1 119879minus1At the beginning of (119905minus1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905)
+ 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 1198771015840
119905minus1119906119905minus1
= 119909119905
1198681015840119906119905minus1
= 119909119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(36)
It follows from (35) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) =
1
119886119905
119864119905minus1
(1199092
119905) minus 1205821
119905119902119905119864119905minus1
(119909119905)
minus 2120574119905119864119905minus1
(119909119905119897119905) + 1205731119905119864119905minus1
(119897119905)
+ 1205732119905119864119905minus1
(1198972
119905) minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(37)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (36) we have
min 119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
st 1198681015840119906119905minus1
= 119909119905minus1
(38)
where
Φ119905minus1
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
+1205821
119905minus1
2120583119905minus1
Ψ119905minus1
= [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
+ 1205731119905] 119864 (119876
119905minus1) 119897119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(39)
Letting 120596119905be the Lagrange multiplier the Lagrange function
for problem (38) is given by
119871 (119906119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
minus 120596119905(119909119905minus1
minus 1198681015840119906119905minus1
)
(40)
It follows from the first-order necessary optimality conditionthat
2Σ1
119905minus1119906119905minus1
minus 2Φ119905minus1
+ 120596119905119868 = 0
119909119905minus1
minus 1198681015840119906119905minus1
= 0
(41)
Thus we have
119906lowast
119905minus1= (Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905(Σ1
119905minus1)minus1
119868 (42)
1198681015840119906lowast
119905minus1= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
21205961199051198681015840(Σ1
119905minus1)minus1
119868
= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905119886119905minus1
= 119909119905minus1
(43)
which implies
120596119905= 2
1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
minus 21
119886119905minus1
119909119905minus1
(44)
It follows from (42) that
119906lowast
119905minus1=
(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
minus1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
(Σ1
119905minus1)minus1
119868
+ (Σ1
119905minus1)minus1
Φ119905minus1
=(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
+1205821
119905minus1
2(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868)
+ (Σ1
119905minus1)minus1
[Σ0
119905minus1minus 120574119905minus1
119868 + 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
(45)
Substituting 119906lowast
119905minus1into the objective function of problem (38)
(see Appendix C for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4120572119894
(46)This completes the proof
Mathematical Problems in Engineering 7
Remark 2 If an investor does not have any liability that is119897119905equiv 0 for any 119905 isin 0 1 119879 minus 1 then the optimal time-
consistent investment strategy can be simplified as follows
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(47)
which is exactly the same as that in [16] This implies thatthe result of Chen et al [16] is a special case of Theorem 1Therefore Theorem 1 generalizes their result
Corollary 3 If the returns of liability and risky assets areuncorrelated that is Σ0
119905= 0 for any 119905 isin 0 1 119879 minus 1 then
the optimal investment policy for problem (9) is
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(48)
Proof Since Σ0119905= 0 it is easy to verify that 119889
119905= 1198681015840(Σ1
119905)minus1Σ0
119905=
0 120574119905= 119889119905119886119905= 0 and 120574
119905= 0 Substituting them into (10) gives
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(49)
This completes the proof
Remark 4 After comparing Corollary 3 and Remark 2 it isquite clear that if the return of liability is uncorrelated withthat of risky asset then the liability does not affect the time-consistent optimal policy in a market without riskless asset
Remark 5 If the return of liability is correlated to those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905 (50)
which depends on the current value of liability 119897119905 and the
covariance between the returns of liability and risky assetsΣ0
119905
Next we compare the time-consistent strategy with themyopic strategy in a market without riskless asset In such amarket problem (2) can be expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(51)
By using the same method in the proof ofTheorem 1 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+
120582119905
2(Σ119905)minus1
(120583119905minus 119902119905119868)
+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(52)
where 119886119905= 1198681015840(Σ119905)minus1119868 119902119905= 1198681015840(Σ119905)minus1120583119905119886119905 120574119905= 1198681015840(Σ119905)minus1Σ0
119905119886119905
It is clear that the difference between two strategies enters intoall of the three parts More specifically the following featureholds if the investor is arbitrarily risk averse that is 120582
119905rarr 0
then both the time consistent optimal strategy and myopicstrategy reduce to
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
119905 = 0 119879 minus 1
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(53)
respectively After comparing these two strategies we findthat if an investor is arbitrarily risk averse then heshe shouldconcern about the time-diversification effects arising frommultiperiod optimization
4 Time Consistent Optimal Strategy withRiskless Asset
In this section we consider a market which is consistingof one riskless asset and 119899 risky assets and assume that thewealth process 119909
119905is also in a self-financing fashion We list
the notations of this section in Table 2 The wealth process 119909119905
can be described as follows
119909119905= 1198751015840
119905minus1119906119905minus1
+ 119904119905minus1
119909119905minus1
(54)
where119875119905= 119877119905minus119904119905119868 In this setting problem (6) can be written
as followsmin 120588
0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot
+119864119879minus3
(120588119879minus2
(119878119879minus1
) + 119864119879minus2
(120588119879minus1
(119878119879))) ))
st 119909119905+1
= 1198751015840
119905119906119905+ 119904119905119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(55)
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (55) is givenin the following theorem
Theorem 6 The optimal investment strategy of problem (55)is given by
119906lowast
119905= Σminus1
119905120593119905119897119905+
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (56)
where 120593119905= 119864(119875
119905119876119905) minus 119864(119875
119905)119864(119876119905) and 120598
119905= 119864(119875
119905)
8 Mathematical Problems in Engineering
Table 2 Model notations of Section 4
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119904119905+1
1205731119879minus1
= 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1205731119905
= (120582119905+ 1205731119905+1
) 119864(119876119905) minus 1205821
1199051205981015840
119905(Σ119905)minus1120593119905
1205732119879minus1
= Var(119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1205732119905
= Var(119876119905) + 1205732119905+1
119864(1198762
119905) minus 1205931015840
119905(Σ119905)minus1120593119905
Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879minus1)th period problem (55) reducesto
min119906119879minus1
Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 119909119879
= 119904119879minus1
119909119879minus1
+ 1198751015840
119879minus1119906119879minus1
119897119879
= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(57)
Substituting the binding constraints into the objective func-tion we have
min119906119879minus1
119881119879minus1
(119909119879minus1
119897119879minus1
) (58)
where
119881119879minus1
(119909119879minus1
119897119879minus1
) = 1199061015840
119879minus1Σ119879minus1
119906119879minus1
minus 2(1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
(59)
It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have
119889 (119881119879minus1
(119909119879minus1
119897119879minus1
))
119889 (119906119879minus1
)= 2Σ119879minus1
119906119879minus1
minus 2(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 0
(60)
which implies
119906lowast
119879minus1= (Σ119879minus1
)minus1
[120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
] (61)
Substituting 119906lowast
119879minus1into the objective function of problem (58)
gives (see Appendix D for more details)
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(62)
where119872119879minus1
= 1205981015840
119879minus1Σminus1
119879minus1120598119879minus1
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 119909119879minus1
= 119904119879minus2
119909119879minus2
+ 1198751015840
119879minus2119906119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(63)
From (62) we can easily have
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) = 1205732119879minus1
119864 (1198762
119879minus2) 1198972
119879minus2
+ 1205731119879minus1
119864 (119876119879minus2
) 119897119879minus2
minus 120582119879minus1
119904119879minus1
119904119879minus2
119909119879minus2
minus 120582119879minus1
119904119879minus1
1205981015840
119879minus2119906119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(64)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (63) we have
min119906119879minus2
119881119879minus2
(119909119879minus2
119897119879minus2
) (65)
where119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ119879minus2
119906119879minus2
minus 2(1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(66)
The first-order necessary optimality condition implies
119889 (119881119879minus2
(119909119879minus2
119897119879minus2
))
119889 (119906119879minus2
)= 2Σ119879minus2
119906119879minus2
minus 2(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 0
(67)
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
To measure the total risk of an investor amongmultiperi-ods (after time 119905) we employ a separable expected conditionalmapping which is defined as (see Chen et al [16])
119879
sum119895=119905+1
119864119905(120588119895minus1
(119878119895| F119895minus1
)) (3)
which reflects all the risk in the future Following thisassumption a separable dynamic mean-variance problem isdefined as
min119906isinΠ
119879
sum119895=1
119864 (120588119895minus1
(119878119895| F119895minus1
)) (4)
where Π is a set of all permit policies The optimal policyof problem (4) which satisfies both Bellmanrsquos optimalityprinciple and requirement REQ is called Time-ConsistentStrategy Note that both Bellmanrsquos optimality principle andrequirement REQ could be proved by following the method-ology of Chen et al [16]Thus problem (4) can be recursivelysolved by the dynamic programming technique Applying theiterated-expectation property of the expected operator thatis 119864(119864(sdot | F
119895) | F119896) = 119864(sdot | F
119896) for 119895 gt 119896 we have
119879
sum119895=1
119864 (120588119895minus1
(119878119895| F119895minus1
))
= 1205880(1198781) + 119864 (120588
1(1198782)
+ 1198641(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
(5)(for more details see Chen et al [16]) Then problem (4) isequivalent tomin119906isinΠ
1205880(1198781)
+ 119864 (1205881(1198782) + 1198641(1205882(1198783)
+ sdot sdot sdot + 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
(6)It follows from Bellmanrsquos optimality principle and (6) that(4) is equivalent to find an optimal strategy to satisfy thefollowing problem
min1199060
(1205880(1198781)
+ 119864min1199061
(1205881(1198782)
+ 1198641min1199062
(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
min119906119879minus2
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
min119906119879minus1
(120588119879minus1
(119878119879))) )))
(7)
We solve this problem in the following sections In order todiscuss the impact of riskless asset the market is consideredin two cases with and without riskless asset We demonstratethese results in Sections 3 and 4 respectively
3 Time Consistent Optimal Strategy withoutRiskless Asset
Consider a market consisting of only 119899 risky assets andassume that the wealth process 119909
119905is in a self-financing
fashion We list the notations of this section in Table 1 Thewealth process 119909
119905could be described as follows
119909119905+1
= 1198771015840
119905119906119905 1198681015840119906119905= 119909119905 119905 = 0 1 119879 minus 1 (8)
where 119868 = (1 1)1015840isin R119899 In this setting problem (6) can
be written as follows
min 1205880(1198781) + 1198640(1205881(1198782)
+ 1198641(1205882(1198783) + sdot sdot sdot
+ 119864119879minus3
(120588119879minus2
(119878119879minus1
)
+119864119879minus2
(120588119879minus1
(119878119879))) ))
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(9)
Since Σ1
119879minus1is positive definite we have 119886
119879minus1gt 0 Further
it follows from the nonnegative definiteness of 119864(119877119879minus1
1198771015840
119879minus1)
that Σ1
119879minus2is also positive definite and 119886
119879minus2gt 0 By using
mathematical induction we conclude that for any 119905 =
0 119879 minus 1 Σ1119905is also positive definite and 119886
119905gt 0
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (9) is givenin the following theorem
Theorem 1 The time-consistent optimal investment policy ofproblem (9) is given by
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868)
+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
for 119905 = 0 119879 minus 1
(10)
4 Mathematical Problems in Engineering
Table 1 Model notations of Section 3
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119902119905+1
Σ1
119879minus1= Σ119879minus1
Σ1
119905= Σ119905+ 119864 (119877
1199051198771015840
119905)119886119905+1
119886119879minus1
= 1198681015840(Σ1
119879minus1)minus1
119868 119886119905= 1198681015840(Σ1
119905)minus1
119868
119887119879minus1
= 1198681015840(Σ1
119879minus1)minus1
120583119879minus1
119887119905= 1198681015840(Σ1
119905)minus1
120583119905
119888119879minus1
= 1205831015840
119879minus1(Σ1
119879minus1)minus1
120583119879minus1
119888119905= 1205831015840
119905(Σ1
119905)minus1
120583119905
119889119879minus1
= 1198681015840(Σ1
119879minus1)minus1
Σ0
119879minus1119889119905= 1198681015840(Σ1
119905)minus1
Σ0
119905
119902119879minus1
= 119887119879minus1
119886119879minus1
119902119905= 119887119905119886119905
120572119879minus1
= 119888119879minus1
minus 119902119879minus1
119887119879minus1
120572119905= 119888119905minus 119902119905119887119905
120574119879minus1
= 119889119879minus1
119886119879minus1
120574119905= 119889119905119886119905
120574119879minus1
= 120574119879minus1
120574119905= 120574119905+ 120574119905+1
119864(1198761199051198771015840
119905)(Σ1
119905)minus1
119868119886119905
1205731119879minus1
= 120582119879minus1
119864(119876119879minus1
) minus 120582119879minus1
(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
1205731119905
= minus1205821
1199051205831015840
119905(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868) minus 120582
1
119905120574119905+1
1205831015840
119905(Σ1
119905)minus1
119864(119876119905119877119905)
+ 120582119879minus1
119887119879minus1
120574119879minus1
+ (120582119905+ 1205731119905+1
)119864(119876119905)
1205732119879minus1
= Var(119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1+ 120574119879minus1
119889119879minus1
1205732119905
= [Var(119876119905) + 1205732119905+1
119864(1198762
119905)] + 119886
1199051205742
119905minus [Σ0
119905+ 120574119905+1
119864(119876119905119877119905)]1015840
sdot (Σ1
119905)minus1[Σ0
119905+ 120574119905+1
119864(119876119905119877119905)]
Proof When 119905 = 119879minus1 for given wealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879 minus 1)th period problem (9) can beexpressed as follows
min Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 1198771015840
119879minus1119906119879minus1
= 119909119879
1198681015840119906119879minus1
= 119909119879minus1
119897119879= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(11)
Substituting the binding constraints into the objective func-tion we have
min 119881119879minus1
(119909119879minus1
119897119879minus1
)
= 1199061015840
119879minus1Σ1
119879minus1119906119879minus1
minus 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21199061015840
119879minus1] 119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
st 1198681015840119906119879minus1
= 119909119879minus1
(12)
which is a linear-quadratic program By using the Lagrangemultiplier technique and letting 120596
119879be the Lagrange multi-
plier the Lagrange function is defined as
119871 (119906119879minus1
) = 1199061015840
119879minus1Σ1
119879minus1119906119879minus1
minus 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
minus 120596119879(119909119879minus1
minus 1198681015840119906119879minus1
)
(13)
By using the first-order necessary optimality condition wehave
2Σ1
119879minus1119906119879minus1
minus 2(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
) + 120596119879119868 = 0 (14)
119909119879minus1
minus 1198681015840119906119879minus1
= 0 (15)
From (14) we can easily have
119906lowast
119879minus1= (Σ1
119879minus1)minus1
(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
) minus1
2120596119879(Σ1
119879minus1)minus1
119868
(16)
1198681015840119906lowast
119879minus1= 1198681015840(Σ1
119879minus1)minus1
(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
)
minus1
21205961198791198681015840(Σ1
119879minus1)minus1
119868
= 119889119879minus1
119897119879minus1
+120582119879minus1
2119887119879minus1
minus1
2119886119879minus1
120596119879
= 119909119879minus1
(17)
which implies that the Lagrange multiplier 120596119879is
120596119879= 2
119889119879minus1
119886119879minus1
119897119879minus1
+ 120582119879minus1
119887119879minus1
119886119879minus1
minus 21
119886119879minus1
119909119879minus1
= 2120574119879minus1
119897119879minus1
+ 120582119879minus1
119902119879minus1
minus 21
119886119879minus1
119909119879minus1
(18)
Substituting 120596119879into (16) we have
119906lowast
119879minus1=
(Σ1
119879minus1)minus1
119868
119886119879minus1
119909119879minus1
+ (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2(Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
(19)
Mathematical Problems in Engineering 5
Further by substituting 119906lowast
119879minus1into the objective function of
problem (12) (see Appendix A for more details) we have
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) =1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(20)
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem is given as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 1198771015840
119879minus2119906119879minus2
= 119909119879minus1
1198681015840119906119879minus2
= 119909119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(21)
It follows from (20) that
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
=1
119886119879minus1
119864119879minus2
(1199092
119879minus1) minus 120582119879minus1
119902119879minus1
119864119879minus2
(119909119879minus1
)
minus 2120574119879minus1
119864119879minus2
(119909119879minus1
119897119879minus1
) + 1205731119879minus1
119864119879minus2
(119897119879minus1
)
+ 1205732119879minus1
119864119879minus2
(1198972
119879minus1) minus
(120582119879minus1
)2
4120572119879minus1
(22)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (21) we have
min 119881119879minus2
(119909119879minus2
119897119879minus2
)
st 1198681015840119906119879minus2
= 119909119879minus2
(23)
where
119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2 [(Σ0
119879minus2)1015840
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2)] 119897119879minus2
+1205821
119879minus2
21205831015840
119879minus2119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(24)
By setting 120596119879minus1
be the Lagrange multiplier the Lagrangefunction for problem (23) is
119871 (119906119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2Φ1015840
119879minus2119906119879minus2
+ Ψ119879minus2
minus 120596119879minus1
(119909119879minus2
minus 1198681015840119906119879minus2
)
(25)
where
Φ119879minus2
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] 119897119879minus2
+1205821
119879minus2
2120583119879minus2
Ψ119879minus2
= [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(26)
From the first-order necessary optimality condition we have
2Σ1
119879minus2119906119879minus2
minus 2Φ119879minus2
+ 120596119879minus1
119868 = 0 (27)
119909119879minus2
minus 1198681015840119906119879minus2
= 0 (28)
From (27) we have
119906lowast
119879minus2= (Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
(Σ1
119879minus2)minus1
119868 (29)
1198681015840119906lowast
119879minus2= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
1198681015840(Σ1
119879minus2)minus1
119868
= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
119886119879minus2
= 119909119879minus2
(30)
which implies
120596119879minus1
= 21198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
minus 21
119886119879minus2
119909119879minus2
(31)
Substituting 120596119879minus1
into (29) we get
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+ (Σ1
119879minus2)minus1
Φ119879minus2
minus1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
(Σ1
119879minus2)minus1
119868
(32)
Taking Φ119879minus2
into account we have
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+1205821
119879minus2
2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+ (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868] 119897119879minus2
(33)
6 Mathematical Problems in Engineering
Substituting 119906lowast
119879minus2into (23) gives (see Appendix B for more
details)
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2
minus(120582119879minus1
)2
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(34)
Next by using mathematical induction we show thatboth (10) and
119881lowast
119905(119909119905 119897119905) =
1
119886119905
1199092
119905minus 1205821
119905119902119905119909119905minus 2120574119905119909119905119897119905
+ 1205731119905119897119905+ 12057321199051198972
119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(35)
hold Suppose that (10) and (35) hold for time 119905 119905+1 119879minus1At the beginning of (119905minus1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905)
+ 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 1198771015840
119905minus1119906119905minus1
= 119909119905
1198681015840119906119905minus1
= 119909119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(36)
It follows from (35) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) =
1
119886119905
119864119905minus1
(1199092
119905) minus 1205821
119905119902119905119864119905minus1
(119909119905)
minus 2120574119905119864119905minus1
(119909119905119897119905) + 1205731119905119864119905minus1
(119897119905)
+ 1205732119905119864119905minus1
(1198972
119905) minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(37)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (36) we have
min 119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
st 1198681015840119906119905minus1
= 119909119905minus1
(38)
where
Φ119905minus1
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
+1205821
119905minus1
2120583119905minus1
Ψ119905minus1
= [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
+ 1205731119905] 119864 (119876
119905minus1) 119897119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(39)
Letting 120596119905be the Lagrange multiplier the Lagrange function
for problem (38) is given by
119871 (119906119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
minus 120596119905(119909119905minus1
minus 1198681015840119906119905minus1
)
(40)
It follows from the first-order necessary optimality conditionthat
2Σ1
119905minus1119906119905minus1
minus 2Φ119905minus1
+ 120596119905119868 = 0
119909119905minus1
minus 1198681015840119906119905minus1
= 0
(41)
Thus we have
119906lowast
119905minus1= (Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905(Σ1
119905minus1)minus1
119868 (42)
1198681015840119906lowast
119905minus1= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
21205961199051198681015840(Σ1
119905minus1)minus1
119868
= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905119886119905minus1
= 119909119905minus1
(43)
which implies
120596119905= 2
1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
minus 21
119886119905minus1
119909119905minus1
(44)
It follows from (42) that
119906lowast
119905minus1=
(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
minus1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
(Σ1
119905minus1)minus1
119868
+ (Σ1
119905minus1)minus1
Φ119905minus1
=(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
+1205821
119905minus1
2(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868)
+ (Σ1
119905minus1)minus1
[Σ0
119905minus1minus 120574119905minus1
119868 + 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
(45)
Substituting 119906lowast
119905minus1into the objective function of problem (38)
(see Appendix C for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4120572119894
(46)This completes the proof
Mathematical Problems in Engineering 7
Remark 2 If an investor does not have any liability that is119897119905equiv 0 for any 119905 isin 0 1 119879 minus 1 then the optimal time-
consistent investment strategy can be simplified as follows
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(47)
which is exactly the same as that in [16] This implies thatthe result of Chen et al [16] is a special case of Theorem 1Therefore Theorem 1 generalizes their result
Corollary 3 If the returns of liability and risky assets areuncorrelated that is Σ0
119905= 0 for any 119905 isin 0 1 119879 minus 1 then
the optimal investment policy for problem (9) is
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(48)
Proof Since Σ0119905= 0 it is easy to verify that 119889
119905= 1198681015840(Σ1
119905)minus1Σ0
119905=
0 120574119905= 119889119905119886119905= 0 and 120574
119905= 0 Substituting them into (10) gives
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(49)
This completes the proof
Remark 4 After comparing Corollary 3 and Remark 2 it isquite clear that if the return of liability is uncorrelated withthat of risky asset then the liability does not affect the time-consistent optimal policy in a market without riskless asset
Remark 5 If the return of liability is correlated to those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905 (50)
which depends on the current value of liability 119897119905 and the
covariance between the returns of liability and risky assetsΣ0
119905
Next we compare the time-consistent strategy with themyopic strategy in a market without riskless asset In such amarket problem (2) can be expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(51)
By using the same method in the proof ofTheorem 1 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+
120582119905
2(Σ119905)minus1
(120583119905minus 119902119905119868)
+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(52)
where 119886119905= 1198681015840(Σ119905)minus1119868 119902119905= 1198681015840(Σ119905)minus1120583119905119886119905 120574119905= 1198681015840(Σ119905)minus1Σ0
119905119886119905
It is clear that the difference between two strategies enters intoall of the three parts More specifically the following featureholds if the investor is arbitrarily risk averse that is 120582
119905rarr 0
then both the time consistent optimal strategy and myopicstrategy reduce to
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
119905 = 0 119879 minus 1
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(53)
respectively After comparing these two strategies we findthat if an investor is arbitrarily risk averse then heshe shouldconcern about the time-diversification effects arising frommultiperiod optimization
4 Time Consistent Optimal Strategy withRiskless Asset
In this section we consider a market which is consistingof one riskless asset and 119899 risky assets and assume that thewealth process 119909
119905is also in a self-financing fashion We list
the notations of this section in Table 2 The wealth process 119909119905
can be described as follows
119909119905= 1198751015840
119905minus1119906119905minus1
+ 119904119905minus1
119909119905minus1
(54)
where119875119905= 119877119905minus119904119905119868 In this setting problem (6) can be written
as followsmin 120588
0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot
+119864119879minus3
(120588119879minus2
(119878119879minus1
) + 119864119879minus2
(120588119879minus1
(119878119879))) ))
st 119909119905+1
= 1198751015840
119905119906119905+ 119904119905119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(55)
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (55) is givenin the following theorem
Theorem 6 The optimal investment strategy of problem (55)is given by
119906lowast
119905= Σminus1
119905120593119905119897119905+
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (56)
where 120593119905= 119864(119875
119905119876119905) minus 119864(119875
119905)119864(119876119905) and 120598
119905= 119864(119875
119905)
8 Mathematical Problems in Engineering
Table 2 Model notations of Section 4
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119904119905+1
1205731119879minus1
= 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1205731119905
= (120582119905+ 1205731119905+1
) 119864(119876119905) minus 1205821
1199051205981015840
119905(Σ119905)minus1120593119905
1205732119879minus1
= Var(119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1205732119905
= Var(119876119905) + 1205732119905+1
119864(1198762
119905) minus 1205931015840
119905(Σ119905)minus1120593119905
Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879minus1)th period problem (55) reducesto
min119906119879minus1
Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 119909119879
= 119904119879minus1
119909119879minus1
+ 1198751015840
119879minus1119906119879minus1
119897119879
= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(57)
Substituting the binding constraints into the objective func-tion we have
min119906119879minus1
119881119879minus1
(119909119879minus1
119897119879minus1
) (58)
where
119881119879minus1
(119909119879minus1
119897119879minus1
) = 1199061015840
119879minus1Σ119879minus1
119906119879minus1
minus 2(1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
(59)
It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have
119889 (119881119879minus1
(119909119879minus1
119897119879minus1
))
119889 (119906119879minus1
)= 2Σ119879minus1
119906119879minus1
minus 2(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 0
(60)
which implies
119906lowast
119879minus1= (Σ119879minus1
)minus1
[120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
] (61)
Substituting 119906lowast
119879minus1into the objective function of problem (58)
gives (see Appendix D for more details)
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(62)
where119872119879minus1
= 1205981015840
119879minus1Σminus1
119879minus1120598119879minus1
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 119909119879minus1
= 119904119879minus2
119909119879minus2
+ 1198751015840
119879minus2119906119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(63)
From (62) we can easily have
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) = 1205732119879minus1
119864 (1198762
119879minus2) 1198972
119879minus2
+ 1205731119879minus1
119864 (119876119879minus2
) 119897119879minus2
minus 120582119879minus1
119904119879minus1
119904119879minus2
119909119879minus2
minus 120582119879minus1
119904119879minus1
1205981015840
119879minus2119906119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(64)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (63) we have
min119906119879minus2
119881119879minus2
(119909119879minus2
119897119879minus2
) (65)
where119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ119879minus2
119906119879minus2
minus 2(1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(66)
The first-order necessary optimality condition implies
119889 (119881119879minus2
(119909119879minus2
119897119879minus2
))
119889 (119906119879minus2
)= 2Σ119879minus2
119906119879minus2
minus 2(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 0
(67)
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Table 1 Model notations of Section 3
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119902119905+1
Σ1
119879minus1= Σ119879minus1
Σ1
119905= Σ119905+ 119864 (119877
1199051198771015840
119905)119886119905+1
119886119879minus1
= 1198681015840(Σ1
119879minus1)minus1
119868 119886119905= 1198681015840(Σ1
119905)minus1
119868
119887119879minus1
= 1198681015840(Σ1
119879minus1)minus1
120583119879minus1
119887119905= 1198681015840(Σ1
119905)minus1
120583119905
119888119879minus1
= 1205831015840
119879minus1(Σ1
119879minus1)minus1
120583119879minus1
119888119905= 1205831015840
119905(Σ1
119905)minus1
120583119905
119889119879minus1
= 1198681015840(Σ1
119879minus1)minus1
Σ0
119879minus1119889119905= 1198681015840(Σ1
119905)minus1
Σ0
119905
119902119879minus1
= 119887119879minus1
119886119879minus1
119902119905= 119887119905119886119905
120572119879minus1
= 119888119879minus1
minus 119902119879minus1
119887119879minus1
120572119905= 119888119905minus 119902119905119887119905
120574119879minus1
= 119889119879minus1
119886119879minus1
120574119905= 119889119905119886119905
120574119879minus1
= 120574119879minus1
120574119905= 120574119905+ 120574119905+1
119864(1198761199051198771015840
119905)(Σ1
119905)minus1
119868119886119905
1205731119879minus1
= 120582119879minus1
119864(119876119879minus1
) minus 120582119879minus1
(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
1205731119905
= minus1205821
1199051205831015840
119905(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868) minus 120582
1
119905120574119905+1
1205831015840
119905(Σ1
119905)minus1
119864(119876119905119877119905)
+ 120582119879minus1
119887119879minus1
120574119879minus1
+ (120582119905+ 1205731119905+1
)119864(119876119905)
1205732119879minus1
= Var(119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1+ 120574119879minus1
119889119879minus1
1205732119905
= [Var(119876119905) + 1205732119905+1
119864(1198762
119905)] + 119886
1199051205742
119905minus [Σ0
119905+ 120574119905+1
119864(119876119905119877119905)]1015840
sdot (Σ1
119905)minus1[Σ0
119905+ 120574119905+1
119864(119876119905119877119905)]
Proof When 119905 = 119879minus1 for given wealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879 minus 1)th period problem (9) can beexpressed as follows
min Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 1198771015840
119879minus1119906119879minus1
= 119909119879
1198681015840119906119879minus1
= 119909119879minus1
119897119879= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(11)
Substituting the binding constraints into the objective func-tion we have
min 119881119879minus1
(119909119879minus1
119897119879minus1
)
= 1199061015840
119879minus1Σ1
119879minus1119906119879minus1
minus 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21199061015840
119879minus1] 119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
st 1198681015840119906119879minus1
= 119909119879minus1
(12)
which is a linear-quadratic program By using the Lagrangemultiplier technique and letting 120596
119879be the Lagrange multi-
plier the Lagrange function is defined as
119871 (119906119879minus1
) = 1199061015840
119879minus1Σ1
119879minus1119906119879minus1
minus 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
minus 120596119879(119909119879minus1
minus 1198681015840119906119879minus1
)
(13)
By using the first-order necessary optimality condition wehave
2Σ1
119879minus1119906119879minus1
minus 2(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
) + 120596119879119868 = 0 (14)
119909119879minus1
minus 1198681015840119906119879minus1
= 0 (15)
From (14) we can easily have
119906lowast
119879minus1= (Σ1
119879minus1)minus1
(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
) minus1
2120596119879(Σ1
119879minus1)minus1
119868
(16)
1198681015840119906lowast
119879minus1= 1198681015840(Σ1
119879minus1)minus1
(Σ0
119879minus1119897119879minus1
+120582119879minus1
2120583119879minus1
)
minus1
21205961198791198681015840(Σ1
119879minus1)minus1
119868
= 119889119879minus1
119897119879minus1
+120582119879minus1
2119887119879minus1
minus1
2119886119879minus1
120596119879
= 119909119879minus1
(17)
which implies that the Lagrange multiplier 120596119879is
120596119879= 2
119889119879minus1
119886119879minus1
119897119879minus1
+ 120582119879minus1
119887119879minus1
119886119879minus1
minus 21
119886119879minus1
119909119879minus1
= 2120574119879minus1
119897119879minus1
+ 120582119879minus1
119902119879minus1
minus 21
119886119879minus1
119909119879minus1
(18)
Substituting 120596119879into (16) we have
119906lowast
119879minus1=
(Σ1
119879minus1)minus1
119868
119886119879minus1
119909119879minus1
+ (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2(Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
(19)
Mathematical Problems in Engineering 5
Further by substituting 119906lowast
119879minus1into the objective function of
problem (12) (see Appendix A for more details) we have
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) =1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(20)
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem is given as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 1198771015840
119879minus2119906119879minus2
= 119909119879minus1
1198681015840119906119879minus2
= 119909119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(21)
It follows from (20) that
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
=1
119886119879minus1
119864119879minus2
(1199092
119879minus1) minus 120582119879minus1
119902119879minus1
119864119879minus2
(119909119879minus1
)
minus 2120574119879minus1
119864119879minus2
(119909119879minus1
119897119879minus1
) + 1205731119879minus1
119864119879minus2
(119897119879minus1
)
+ 1205732119879minus1
119864119879minus2
(1198972
119879minus1) minus
(120582119879minus1
)2
4120572119879minus1
(22)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (21) we have
min 119881119879minus2
(119909119879minus2
119897119879minus2
)
st 1198681015840119906119879minus2
= 119909119879minus2
(23)
where
119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2 [(Σ0
119879minus2)1015840
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2)] 119897119879minus2
+1205821
119879minus2
21205831015840
119879minus2119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(24)
By setting 120596119879minus1
be the Lagrange multiplier the Lagrangefunction for problem (23) is
119871 (119906119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2Φ1015840
119879minus2119906119879minus2
+ Ψ119879minus2
minus 120596119879minus1
(119909119879minus2
minus 1198681015840119906119879minus2
)
(25)
where
Φ119879minus2
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] 119897119879minus2
+1205821
119879minus2
2120583119879minus2
Ψ119879minus2
= [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(26)
From the first-order necessary optimality condition we have
2Σ1
119879minus2119906119879minus2
minus 2Φ119879minus2
+ 120596119879minus1
119868 = 0 (27)
119909119879minus2
minus 1198681015840119906119879minus2
= 0 (28)
From (27) we have
119906lowast
119879minus2= (Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
(Σ1
119879minus2)minus1
119868 (29)
1198681015840119906lowast
119879minus2= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
1198681015840(Σ1
119879minus2)minus1
119868
= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
119886119879minus2
= 119909119879minus2
(30)
which implies
120596119879minus1
= 21198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
minus 21
119886119879minus2
119909119879minus2
(31)
Substituting 120596119879minus1
into (29) we get
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+ (Σ1
119879minus2)minus1
Φ119879minus2
minus1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
(Σ1
119879minus2)minus1
119868
(32)
Taking Φ119879minus2
into account we have
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+1205821
119879minus2
2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+ (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868] 119897119879minus2
(33)
6 Mathematical Problems in Engineering
Substituting 119906lowast
119879minus2into (23) gives (see Appendix B for more
details)
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2
minus(120582119879minus1
)2
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(34)
Next by using mathematical induction we show thatboth (10) and
119881lowast
119905(119909119905 119897119905) =
1
119886119905
1199092
119905minus 1205821
119905119902119905119909119905minus 2120574119905119909119905119897119905
+ 1205731119905119897119905+ 12057321199051198972
119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(35)
hold Suppose that (10) and (35) hold for time 119905 119905+1 119879minus1At the beginning of (119905minus1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905)
+ 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 1198771015840
119905minus1119906119905minus1
= 119909119905
1198681015840119906119905minus1
= 119909119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(36)
It follows from (35) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) =
1
119886119905
119864119905minus1
(1199092
119905) minus 1205821
119905119902119905119864119905minus1
(119909119905)
minus 2120574119905119864119905minus1
(119909119905119897119905) + 1205731119905119864119905minus1
(119897119905)
+ 1205732119905119864119905minus1
(1198972
119905) minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(37)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (36) we have
min 119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
st 1198681015840119906119905minus1
= 119909119905minus1
(38)
where
Φ119905minus1
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
+1205821
119905minus1
2120583119905minus1
Ψ119905minus1
= [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
+ 1205731119905] 119864 (119876
119905minus1) 119897119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(39)
Letting 120596119905be the Lagrange multiplier the Lagrange function
for problem (38) is given by
119871 (119906119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
minus 120596119905(119909119905minus1
minus 1198681015840119906119905minus1
)
(40)
It follows from the first-order necessary optimality conditionthat
2Σ1
119905minus1119906119905minus1
minus 2Φ119905minus1
+ 120596119905119868 = 0
119909119905minus1
minus 1198681015840119906119905minus1
= 0
(41)
Thus we have
119906lowast
119905minus1= (Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905(Σ1
119905minus1)minus1
119868 (42)
1198681015840119906lowast
119905minus1= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
21205961199051198681015840(Σ1
119905minus1)minus1
119868
= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905119886119905minus1
= 119909119905minus1
(43)
which implies
120596119905= 2
1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
minus 21
119886119905minus1
119909119905minus1
(44)
It follows from (42) that
119906lowast
119905minus1=
(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
minus1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
(Σ1
119905minus1)minus1
119868
+ (Σ1
119905minus1)minus1
Φ119905minus1
=(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
+1205821
119905minus1
2(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868)
+ (Σ1
119905minus1)minus1
[Σ0
119905minus1minus 120574119905minus1
119868 + 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
(45)
Substituting 119906lowast
119905minus1into the objective function of problem (38)
(see Appendix C for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4120572119894
(46)This completes the proof
Mathematical Problems in Engineering 7
Remark 2 If an investor does not have any liability that is119897119905equiv 0 for any 119905 isin 0 1 119879 minus 1 then the optimal time-
consistent investment strategy can be simplified as follows
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(47)
which is exactly the same as that in [16] This implies thatthe result of Chen et al [16] is a special case of Theorem 1Therefore Theorem 1 generalizes their result
Corollary 3 If the returns of liability and risky assets areuncorrelated that is Σ0
119905= 0 for any 119905 isin 0 1 119879 minus 1 then
the optimal investment policy for problem (9) is
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(48)
Proof Since Σ0119905= 0 it is easy to verify that 119889
119905= 1198681015840(Σ1
119905)minus1Σ0
119905=
0 120574119905= 119889119905119886119905= 0 and 120574
119905= 0 Substituting them into (10) gives
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(49)
This completes the proof
Remark 4 After comparing Corollary 3 and Remark 2 it isquite clear that if the return of liability is uncorrelated withthat of risky asset then the liability does not affect the time-consistent optimal policy in a market without riskless asset
Remark 5 If the return of liability is correlated to those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905 (50)
which depends on the current value of liability 119897119905 and the
covariance between the returns of liability and risky assetsΣ0
119905
Next we compare the time-consistent strategy with themyopic strategy in a market without riskless asset In such amarket problem (2) can be expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(51)
By using the same method in the proof ofTheorem 1 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+
120582119905
2(Σ119905)minus1
(120583119905minus 119902119905119868)
+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(52)
where 119886119905= 1198681015840(Σ119905)minus1119868 119902119905= 1198681015840(Σ119905)minus1120583119905119886119905 120574119905= 1198681015840(Σ119905)minus1Σ0
119905119886119905
It is clear that the difference between two strategies enters intoall of the three parts More specifically the following featureholds if the investor is arbitrarily risk averse that is 120582
119905rarr 0
then both the time consistent optimal strategy and myopicstrategy reduce to
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
119905 = 0 119879 minus 1
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(53)
respectively After comparing these two strategies we findthat if an investor is arbitrarily risk averse then heshe shouldconcern about the time-diversification effects arising frommultiperiod optimization
4 Time Consistent Optimal Strategy withRiskless Asset
In this section we consider a market which is consistingof one riskless asset and 119899 risky assets and assume that thewealth process 119909
119905is also in a self-financing fashion We list
the notations of this section in Table 2 The wealth process 119909119905
can be described as follows
119909119905= 1198751015840
119905minus1119906119905minus1
+ 119904119905minus1
119909119905minus1
(54)
where119875119905= 119877119905minus119904119905119868 In this setting problem (6) can be written
as followsmin 120588
0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot
+119864119879minus3
(120588119879minus2
(119878119879minus1
) + 119864119879minus2
(120588119879minus1
(119878119879))) ))
st 119909119905+1
= 1198751015840
119905119906119905+ 119904119905119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(55)
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (55) is givenin the following theorem
Theorem 6 The optimal investment strategy of problem (55)is given by
119906lowast
119905= Σminus1
119905120593119905119897119905+
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (56)
where 120593119905= 119864(119875
119905119876119905) minus 119864(119875
119905)119864(119876119905) and 120598
119905= 119864(119875
119905)
8 Mathematical Problems in Engineering
Table 2 Model notations of Section 4
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119904119905+1
1205731119879minus1
= 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1205731119905
= (120582119905+ 1205731119905+1
) 119864(119876119905) minus 1205821
1199051205981015840
119905(Σ119905)minus1120593119905
1205732119879minus1
= Var(119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1205732119905
= Var(119876119905) + 1205732119905+1
119864(1198762
119905) minus 1205931015840
119905(Σ119905)minus1120593119905
Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879minus1)th period problem (55) reducesto
min119906119879minus1
Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 119909119879
= 119904119879minus1
119909119879minus1
+ 1198751015840
119879minus1119906119879minus1
119897119879
= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(57)
Substituting the binding constraints into the objective func-tion we have
min119906119879minus1
119881119879minus1
(119909119879minus1
119897119879minus1
) (58)
where
119881119879minus1
(119909119879minus1
119897119879minus1
) = 1199061015840
119879minus1Σ119879minus1
119906119879minus1
minus 2(1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
(59)
It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have
119889 (119881119879minus1
(119909119879minus1
119897119879minus1
))
119889 (119906119879minus1
)= 2Σ119879minus1
119906119879minus1
minus 2(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 0
(60)
which implies
119906lowast
119879minus1= (Σ119879minus1
)minus1
[120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
] (61)
Substituting 119906lowast
119879minus1into the objective function of problem (58)
gives (see Appendix D for more details)
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(62)
where119872119879minus1
= 1205981015840
119879minus1Σminus1
119879minus1120598119879minus1
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 119909119879minus1
= 119904119879minus2
119909119879minus2
+ 1198751015840
119879minus2119906119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(63)
From (62) we can easily have
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) = 1205732119879minus1
119864 (1198762
119879minus2) 1198972
119879minus2
+ 1205731119879minus1
119864 (119876119879minus2
) 119897119879minus2
minus 120582119879minus1
119904119879minus1
119904119879minus2
119909119879minus2
minus 120582119879minus1
119904119879minus1
1205981015840
119879minus2119906119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(64)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (63) we have
min119906119879minus2
119881119879minus2
(119909119879minus2
119897119879minus2
) (65)
where119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ119879minus2
119906119879minus2
minus 2(1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(66)
The first-order necessary optimality condition implies
119889 (119881119879minus2
(119909119879minus2
119897119879minus2
))
119889 (119906119879minus2
)= 2Σ119879minus2
119906119879minus2
minus 2(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 0
(67)
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Further by substituting 119906lowast
119879minus1into the objective function of
problem (12) (see Appendix A for more details) we have
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) =1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(20)
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem is given as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 1198771015840
119879minus2119906119879minus2
= 119909119879minus1
1198681015840119906119879minus2
= 119909119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(21)
It follows from (20) that
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
=1
119886119879minus1
119864119879minus2
(1199092
119879minus1) minus 120582119879minus1
119902119879minus1
119864119879minus2
(119909119879minus1
)
minus 2120574119879minus1
119864119879minus2
(119909119879minus1
119897119879minus1
) + 1205731119879minus1
119864119879minus2
(119897119879minus1
)
+ 1205732119879minus1
119864119879minus2
(1198972
119879minus1) minus
(120582119879minus1
)2
4120572119879minus1
(22)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (21) we have
min 119881119879minus2
(119909119879minus2
119897119879minus2
)
st 1198681015840119906119879minus2
= 119909119879minus2
(23)
where
119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2 [(Σ0
119879minus2)1015840
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2)] 119897119879minus2
+1205821
119879minus2
21205831015840
119879minus2119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(24)
By setting 120596119879minus1
be the Lagrange multiplier the Lagrangefunction for problem (23) is
119871 (119906119879minus2
) = 1199061015840
119879minus2Σ1
119879minus2119906119879minus2
minus 2Φ1015840
119879minus2119906119879minus2
+ Ψ119879minus2
minus 120596119879minus1
(119909119879minus2
minus 1198681015840119906119879minus2
)
(25)
where
Φ119879minus2
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] 119897119879minus2
+1205821
119879minus2
2120583119879minus2
Ψ119879minus2
= [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus(120582119879minus1
)2
4120572119879minus1
(26)
From the first-order necessary optimality condition we have
2Σ1
119879minus2119906119879minus2
minus 2Φ119879minus2
+ 120596119879minus1
119868 = 0 (27)
119909119879minus2
minus 1198681015840119906119879minus2
= 0 (28)
From (27) we have
119906lowast
119879minus2= (Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
(Σ1
119879minus2)minus1
119868 (29)
1198681015840119906lowast
119879minus2= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
1198681015840(Σ1
119879minus2)minus1
119868
= 1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
minus1
2120596119879minus1
119886119879minus2
= 119909119879minus2
(30)
which implies
120596119879minus1
= 21198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
minus 21
119886119879minus2
119909119879minus2
(31)
Substituting 120596119879minus1
into (29) we get
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+ (Σ1
119879minus2)minus1
Φ119879minus2
minus1198681015840(Σ1
119879minus2)minus1
Φ119879minus2
119886119879minus2
(Σ1
119879minus2)minus1
119868
(32)
Taking Φ119879minus2
into account we have
119906lowast
119879minus2=
(Σ1
119879minus2)minus1
119868
119886119879minus2
119909119879minus2
+1205821
119879minus2
2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+ (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868] 119897119879minus2
(33)
6 Mathematical Problems in Engineering
Substituting 119906lowast
119879minus2into (23) gives (see Appendix B for more
details)
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2
minus(120582119879minus1
)2
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(34)
Next by using mathematical induction we show thatboth (10) and
119881lowast
119905(119909119905 119897119905) =
1
119886119905
1199092
119905minus 1205821
119905119902119905119909119905minus 2120574119905119909119905119897119905
+ 1205731119905119897119905+ 12057321199051198972
119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(35)
hold Suppose that (10) and (35) hold for time 119905 119905+1 119879minus1At the beginning of (119905minus1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905)
+ 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 1198771015840
119905minus1119906119905minus1
= 119909119905
1198681015840119906119905minus1
= 119909119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(36)
It follows from (35) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) =
1
119886119905
119864119905minus1
(1199092
119905) minus 1205821
119905119902119905119864119905minus1
(119909119905)
minus 2120574119905119864119905minus1
(119909119905119897119905) + 1205731119905119864119905minus1
(119897119905)
+ 1205732119905119864119905minus1
(1198972
119905) minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(37)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (36) we have
min 119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
st 1198681015840119906119905minus1
= 119909119905minus1
(38)
where
Φ119905minus1
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
+1205821
119905minus1
2120583119905minus1
Ψ119905minus1
= [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
+ 1205731119905] 119864 (119876
119905minus1) 119897119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(39)
Letting 120596119905be the Lagrange multiplier the Lagrange function
for problem (38) is given by
119871 (119906119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
minus 120596119905(119909119905minus1
minus 1198681015840119906119905minus1
)
(40)
It follows from the first-order necessary optimality conditionthat
2Σ1
119905minus1119906119905minus1
minus 2Φ119905minus1
+ 120596119905119868 = 0
119909119905minus1
minus 1198681015840119906119905minus1
= 0
(41)
Thus we have
119906lowast
119905minus1= (Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905(Σ1
119905minus1)minus1
119868 (42)
1198681015840119906lowast
119905minus1= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
21205961199051198681015840(Σ1
119905minus1)minus1
119868
= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905119886119905minus1
= 119909119905minus1
(43)
which implies
120596119905= 2
1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
minus 21
119886119905minus1
119909119905minus1
(44)
It follows from (42) that
119906lowast
119905minus1=
(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
minus1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
(Σ1
119905minus1)minus1
119868
+ (Σ1
119905minus1)minus1
Φ119905minus1
=(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
+1205821
119905minus1
2(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868)
+ (Σ1
119905minus1)minus1
[Σ0
119905minus1minus 120574119905minus1
119868 + 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
(45)
Substituting 119906lowast
119905minus1into the objective function of problem (38)
(see Appendix C for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4120572119894
(46)This completes the proof
Mathematical Problems in Engineering 7
Remark 2 If an investor does not have any liability that is119897119905equiv 0 for any 119905 isin 0 1 119879 minus 1 then the optimal time-
consistent investment strategy can be simplified as follows
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(47)
which is exactly the same as that in [16] This implies thatthe result of Chen et al [16] is a special case of Theorem 1Therefore Theorem 1 generalizes their result
Corollary 3 If the returns of liability and risky assets areuncorrelated that is Σ0
119905= 0 for any 119905 isin 0 1 119879 minus 1 then
the optimal investment policy for problem (9) is
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(48)
Proof Since Σ0119905= 0 it is easy to verify that 119889
119905= 1198681015840(Σ1
119905)minus1Σ0
119905=
0 120574119905= 119889119905119886119905= 0 and 120574
119905= 0 Substituting them into (10) gives
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(49)
This completes the proof
Remark 4 After comparing Corollary 3 and Remark 2 it isquite clear that if the return of liability is uncorrelated withthat of risky asset then the liability does not affect the time-consistent optimal policy in a market without riskless asset
Remark 5 If the return of liability is correlated to those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905 (50)
which depends on the current value of liability 119897119905 and the
covariance between the returns of liability and risky assetsΣ0
119905
Next we compare the time-consistent strategy with themyopic strategy in a market without riskless asset In such amarket problem (2) can be expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(51)
By using the same method in the proof ofTheorem 1 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+
120582119905
2(Σ119905)minus1
(120583119905minus 119902119905119868)
+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(52)
where 119886119905= 1198681015840(Σ119905)minus1119868 119902119905= 1198681015840(Σ119905)minus1120583119905119886119905 120574119905= 1198681015840(Σ119905)minus1Σ0
119905119886119905
It is clear that the difference between two strategies enters intoall of the three parts More specifically the following featureholds if the investor is arbitrarily risk averse that is 120582
119905rarr 0
then both the time consistent optimal strategy and myopicstrategy reduce to
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
119905 = 0 119879 minus 1
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(53)
respectively After comparing these two strategies we findthat if an investor is arbitrarily risk averse then heshe shouldconcern about the time-diversification effects arising frommultiperiod optimization
4 Time Consistent Optimal Strategy withRiskless Asset
In this section we consider a market which is consistingof one riskless asset and 119899 risky assets and assume that thewealth process 119909
119905is also in a self-financing fashion We list
the notations of this section in Table 2 The wealth process 119909119905
can be described as follows
119909119905= 1198751015840
119905minus1119906119905minus1
+ 119904119905minus1
119909119905minus1
(54)
where119875119905= 119877119905minus119904119905119868 In this setting problem (6) can be written
as followsmin 120588
0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot
+119864119879minus3
(120588119879minus2
(119878119879minus1
) + 119864119879minus2
(120588119879minus1
(119878119879))) ))
st 119909119905+1
= 1198751015840
119905119906119905+ 119904119905119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(55)
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (55) is givenin the following theorem
Theorem 6 The optimal investment strategy of problem (55)is given by
119906lowast
119905= Σminus1
119905120593119905119897119905+
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (56)
where 120593119905= 119864(119875
119905119876119905) minus 119864(119875
119905)119864(119876119905) and 120598
119905= 119864(119875
119905)
8 Mathematical Problems in Engineering
Table 2 Model notations of Section 4
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119904119905+1
1205731119879minus1
= 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1205731119905
= (120582119905+ 1205731119905+1
) 119864(119876119905) minus 1205821
1199051205981015840
119905(Σ119905)minus1120593119905
1205732119879minus1
= Var(119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1205732119905
= Var(119876119905) + 1205732119905+1
119864(1198762
119905) minus 1205931015840
119905(Σ119905)minus1120593119905
Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879minus1)th period problem (55) reducesto
min119906119879minus1
Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 119909119879
= 119904119879minus1
119909119879minus1
+ 1198751015840
119879minus1119906119879minus1
119897119879
= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(57)
Substituting the binding constraints into the objective func-tion we have
min119906119879minus1
119881119879minus1
(119909119879minus1
119897119879minus1
) (58)
where
119881119879minus1
(119909119879minus1
119897119879minus1
) = 1199061015840
119879minus1Σ119879minus1
119906119879minus1
minus 2(1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
(59)
It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have
119889 (119881119879minus1
(119909119879minus1
119897119879minus1
))
119889 (119906119879minus1
)= 2Σ119879minus1
119906119879minus1
minus 2(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 0
(60)
which implies
119906lowast
119879minus1= (Σ119879minus1
)minus1
[120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
] (61)
Substituting 119906lowast
119879minus1into the objective function of problem (58)
gives (see Appendix D for more details)
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(62)
where119872119879minus1
= 1205981015840
119879minus1Σminus1
119879minus1120598119879minus1
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 119909119879minus1
= 119904119879minus2
119909119879minus2
+ 1198751015840
119879minus2119906119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(63)
From (62) we can easily have
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) = 1205732119879minus1
119864 (1198762
119879minus2) 1198972
119879minus2
+ 1205731119879minus1
119864 (119876119879minus2
) 119897119879minus2
minus 120582119879minus1
119904119879minus1
119904119879minus2
119909119879minus2
minus 120582119879minus1
119904119879minus1
1205981015840
119879minus2119906119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(64)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (63) we have
min119906119879minus2
119881119879minus2
(119909119879minus2
119897119879minus2
) (65)
where119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ119879minus2
119906119879minus2
minus 2(1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(66)
The first-order necessary optimality condition implies
119889 (119881119879minus2
(119909119879minus2
119897119879minus2
))
119889 (119906119879minus2
)= 2Σ119879minus2
119906119879minus2
minus 2(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 0
(67)
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Substituting 119906lowast
119879minus2into (23) gives (see Appendix B for more
details)
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2
minus(120582119879minus1
)2
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(34)
Next by using mathematical induction we show thatboth (10) and
119881lowast
119905(119909119905 119897119905) =
1
119886119905
1199092
119905minus 1205821
119905119902119905119909119905minus 2120574119905119909119905119897119905
+ 1205731119905119897119905+ 12057321199051198972
119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(35)
hold Suppose that (10) and (35) hold for time 119905 119905+1 119879minus1At the beginning of (119905minus1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905)
+ 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 1198771015840
119905minus1119906119905minus1
= 119909119905
1198681015840119906119905minus1
= 119909119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(36)
It follows from (35) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) =
1
119886119905
119864119905minus1
(1199092
119905) minus 1205821
119905119902119905119864119905minus1
(119909119905)
minus 2120574119905119864119905minus1
(119909119905119897119905) + 1205731119905119864119905minus1
(119897119905)
+ 1205732119905119864119905minus1
(1198972
119905) minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(37)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (36) we have
min 119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
st 1198681015840119906119905minus1
= 119909119905minus1
(38)
where
Φ119905minus1
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
+1205821
119905minus1
2120583119905minus1
Ψ119905minus1
= [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
+ 1205731119905] 119864 (119876
119905minus1) 119897119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4120572119894
(39)
Letting 120596119905be the Lagrange multiplier the Lagrange function
for problem (38) is given by
119871 (119906119905minus1
) = 1199061015840
119879minus1Σ1
119905minus1119906119905minus1
minus 2Φ1015840
119905minus1119906119905minus1
+ Ψ119905minus1
minus 120596119905(119909119905minus1
minus 1198681015840119906119905minus1
)
(40)
It follows from the first-order necessary optimality conditionthat
2Σ1
119905minus1119906119905minus1
minus 2Φ119905minus1
+ 120596119905119868 = 0
119909119905minus1
minus 1198681015840119906119905minus1
= 0
(41)
Thus we have
119906lowast
119905minus1= (Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905(Σ1
119905minus1)minus1
119868 (42)
1198681015840119906lowast
119905minus1= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
21205961199051198681015840(Σ1
119905minus1)minus1
119868
= 1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
minus1
2120596119905119886119905minus1
= 119909119905minus1
(43)
which implies
120596119905= 2
1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
minus 21
119886119905minus1
119909119905minus1
(44)
It follows from (42) that
119906lowast
119905minus1=
(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
minus1198681015840(Σ1
119905minus1)minus1
Φ119905minus1
119886119905minus1
(Σ1
119905minus1)minus1
119868
+ (Σ1
119905minus1)minus1
Φ119905minus1
=(Σ1
119905minus1)minus1
119868
119886119905minus1
119909119905minus1
+1205821
119905minus1
2(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868)
+ (Σ1
119905minus1)minus1
[Σ0
119905minus1minus 120574119905minus1
119868 + 120574119905119864 (119876119905minus1
119877119905minus1
)] 119897119905minus1
(45)
Substituting 119906lowast
119905minus1into the objective function of problem (38)
(see Appendix C for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4120572119894
(46)This completes the proof
Mathematical Problems in Engineering 7
Remark 2 If an investor does not have any liability that is119897119905equiv 0 for any 119905 isin 0 1 119879 minus 1 then the optimal time-
consistent investment strategy can be simplified as follows
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(47)
which is exactly the same as that in [16] This implies thatthe result of Chen et al [16] is a special case of Theorem 1Therefore Theorem 1 generalizes their result
Corollary 3 If the returns of liability and risky assets areuncorrelated that is Σ0
119905= 0 for any 119905 isin 0 1 119879 minus 1 then
the optimal investment policy for problem (9) is
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(48)
Proof Since Σ0119905= 0 it is easy to verify that 119889
119905= 1198681015840(Σ1
119905)minus1Σ0
119905=
0 120574119905= 119889119905119886119905= 0 and 120574
119905= 0 Substituting them into (10) gives
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(49)
This completes the proof
Remark 4 After comparing Corollary 3 and Remark 2 it isquite clear that if the return of liability is uncorrelated withthat of risky asset then the liability does not affect the time-consistent optimal policy in a market without riskless asset
Remark 5 If the return of liability is correlated to those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905 (50)
which depends on the current value of liability 119897119905 and the
covariance between the returns of liability and risky assetsΣ0
119905
Next we compare the time-consistent strategy with themyopic strategy in a market without riskless asset In such amarket problem (2) can be expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(51)
By using the same method in the proof ofTheorem 1 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+
120582119905
2(Σ119905)minus1
(120583119905minus 119902119905119868)
+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(52)
where 119886119905= 1198681015840(Σ119905)minus1119868 119902119905= 1198681015840(Σ119905)minus1120583119905119886119905 120574119905= 1198681015840(Σ119905)minus1Σ0
119905119886119905
It is clear that the difference between two strategies enters intoall of the three parts More specifically the following featureholds if the investor is arbitrarily risk averse that is 120582
119905rarr 0
then both the time consistent optimal strategy and myopicstrategy reduce to
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
119905 = 0 119879 minus 1
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(53)
respectively After comparing these two strategies we findthat if an investor is arbitrarily risk averse then heshe shouldconcern about the time-diversification effects arising frommultiperiod optimization
4 Time Consistent Optimal Strategy withRiskless Asset
In this section we consider a market which is consistingof one riskless asset and 119899 risky assets and assume that thewealth process 119909
119905is also in a self-financing fashion We list
the notations of this section in Table 2 The wealth process 119909119905
can be described as follows
119909119905= 1198751015840
119905minus1119906119905minus1
+ 119904119905minus1
119909119905minus1
(54)
where119875119905= 119877119905minus119904119905119868 In this setting problem (6) can be written
as followsmin 120588
0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot
+119864119879minus3
(120588119879minus2
(119878119879minus1
) + 119864119879minus2
(120588119879minus1
(119878119879))) ))
st 119909119905+1
= 1198751015840
119905119906119905+ 119904119905119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(55)
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (55) is givenin the following theorem
Theorem 6 The optimal investment strategy of problem (55)is given by
119906lowast
119905= Σminus1
119905120593119905119897119905+
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (56)
where 120593119905= 119864(119875
119905119876119905) minus 119864(119875
119905)119864(119876119905) and 120598
119905= 119864(119875
119905)
8 Mathematical Problems in Engineering
Table 2 Model notations of Section 4
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119904119905+1
1205731119879minus1
= 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1205731119905
= (120582119905+ 1205731119905+1
) 119864(119876119905) minus 1205821
1199051205981015840
119905(Σ119905)minus1120593119905
1205732119879minus1
= Var(119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1205732119905
= Var(119876119905) + 1205732119905+1
119864(1198762
119905) minus 1205931015840
119905(Σ119905)minus1120593119905
Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879minus1)th period problem (55) reducesto
min119906119879minus1
Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 119909119879
= 119904119879minus1
119909119879minus1
+ 1198751015840
119879minus1119906119879minus1
119897119879
= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(57)
Substituting the binding constraints into the objective func-tion we have
min119906119879minus1
119881119879minus1
(119909119879minus1
119897119879minus1
) (58)
where
119881119879minus1
(119909119879minus1
119897119879minus1
) = 1199061015840
119879minus1Σ119879minus1
119906119879minus1
minus 2(1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
(59)
It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have
119889 (119881119879minus1
(119909119879minus1
119897119879minus1
))
119889 (119906119879minus1
)= 2Σ119879minus1
119906119879minus1
minus 2(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 0
(60)
which implies
119906lowast
119879minus1= (Σ119879minus1
)minus1
[120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
] (61)
Substituting 119906lowast
119879minus1into the objective function of problem (58)
gives (see Appendix D for more details)
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(62)
where119872119879minus1
= 1205981015840
119879minus1Σminus1
119879minus1120598119879minus1
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 119909119879minus1
= 119904119879minus2
119909119879minus2
+ 1198751015840
119879minus2119906119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(63)
From (62) we can easily have
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) = 1205732119879minus1
119864 (1198762
119879minus2) 1198972
119879minus2
+ 1205731119879minus1
119864 (119876119879minus2
) 119897119879minus2
minus 120582119879minus1
119904119879minus1
119904119879minus2
119909119879minus2
minus 120582119879minus1
119904119879minus1
1205981015840
119879minus2119906119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(64)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (63) we have
min119906119879minus2
119881119879minus2
(119909119879minus2
119897119879minus2
) (65)
where119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ119879minus2
119906119879minus2
minus 2(1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(66)
The first-order necessary optimality condition implies
119889 (119881119879minus2
(119909119879minus2
119897119879minus2
))
119889 (119906119879minus2
)= 2Σ119879minus2
119906119879minus2
minus 2(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 0
(67)
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Remark 2 If an investor does not have any liability that is119897119905equiv 0 for any 119905 isin 0 1 119879 minus 1 then the optimal time-
consistent investment strategy can be simplified as follows
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(47)
which is exactly the same as that in [16] This implies thatthe result of Chen et al [16] is a special case of Theorem 1Therefore Theorem 1 generalizes their result
Corollary 3 If the returns of liability and risky assets areuncorrelated that is Σ0
119905= 0 for any 119905 isin 0 1 119879 minus 1 then
the optimal investment policy for problem (9) is
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(48)
Proof Since Σ0119905= 0 it is easy to verify that 119889
119905= 1198681015840(Σ1
119905)minus1Σ0
119905=
0 120574119905= 119889119905119886119905= 0 and 120574
119905= 0 Substituting them into (10) gives
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+
1205821
119905
2(Σ1
119905)minus1
(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1
(49)
This completes the proof
Remark 4 After comparing Corollary 3 and Remark 2 it isquite clear that if the return of liability is uncorrelated withthat of risky asset then the liability does not affect the time-consistent optimal policy in a market without riskless asset
Remark 5 If the return of liability is correlated to those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
(Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905 (50)
which depends on the current value of liability 119897119905 and the
covariance between the returns of liability and risky assetsΣ0
119905
Next we compare the time-consistent strategy with themyopic strategy in a market without riskless asset In such amarket problem (2) can be expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 1198771015840
119905119906119905= 119909119905+1
1198681015840119906119905= 119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(51)
By using the same method in the proof ofTheorem 1 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+
120582119905
2(Σ119905)minus1
(120583119905minus 119902119905119868)
+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(52)
where 119886119905= 1198681015840(Σ119905)minus1119868 119902119905= 1198681015840(Σ119905)minus1120583119905119886119905 120574119905= 1198681015840(Σ119905)minus1Σ0
119905119886119905
It is clear that the difference between two strategies enters intoall of the three parts More specifically the following featureholds if the investor is arbitrarily risk averse that is 120582
119905rarr 0
then both the time consistent optimal strategy and myopicstrategy reduce to
119906lowast
119905=
(Σ1
119905)minus1
119868
119886119905
119909119905+ (Σ1
119905)minus1
(Σ0
119905minus 120574119905119868 + 120574119905+1
119864 (119876119905119877119905)) 119897119905
119905 = 0 119879 minus 1
119906119898119910
119905=
(Σ119905)minus1
119868
119886119905
119909119905+ (Σ119905)minus1
(Σ0
119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1
(53)
respectively After comparing these two strategies we findthat if an investor is arbitrarily risk averse then heshe shouldconcern about the time-diversification effects arising frommultiperiod optimization
4 Time Consistent Optimal Strategy withRiskless Asset
In this section we consider a market which is consistingof one riskless asset and 119899 risky assets and assume that thewealth process 119909
119905is also in a self-financing fashion We list
the notations of this section in Table 2 The wealth process 119909119905
can be described as follows
119909119905= 1198751015840
119905minus1119906119905minus1
+ 119904119905minus1
119909119905minus1
(54)
where119875119905= 119877119905minus119904119905119868 In this setting problem (6) can be written
as followsmin 120588
0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot
+119864119879minus3
(120588119879minus2
(119878119879minus1
) + 119864119879minus2
(120588119879minus1
(119878119879))) ))
st 119909119905+1
= 1198751015840
119905119906119905+ 119904119905119909119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
119905 = 0 1 119879 minus 1
(55)
By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (55) is givenin the following theorem
Theorem 6 The optimal investment strategy of problem (55)is given by
119906lowast
119905= Σminus1
119905120593119905119897119905+
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (56)
where 120593119905= 119864(119875
119905119876119905) minus 119864(119875
119905)119864(119876119905) and 120598
119905= 119864(119875
119905)
8 Mathematical Problems in Engineering
Table 2 Model notations of Section 4
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119904119905+1
1205731119879minus1
= 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1205731119905
= (120582119905+ 1205731119905+1
) 119864(119876119905) minus 1205821
1199051205981015840
119905(Σ119905)minus1120593119905
1205732119879minus1
= Var(119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1205732119905
= Var(119876119905) + 1205732119905+1
119864(1198762
119905) minus 1205931015840
119905(Σ119905)minus1120593119905
Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879minus1)th period problem (55) reducesto
min119906119879minus1
Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 119909119879
= 119904119879minus1
119909119879minus1
+ 1198751015840
119879minus1119906119879minus1
119897119879
= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(57)
Substituting the binding constraints into the objective func-tion we have
min119906119879minus1
119881119879minus1
(119909119879minus1
119897119879minus1
) (58)
where
119881119879minus1
(119909119879minus1
119897119879minus1
) = 1199061015840
119879minus1Σ119879minus1
119906119879minus1
minus 2(1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
(59)
It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have
119889 (119881119879minus1
(119909119879minus1
119897119879minus1
))
119889 (119906119879minus1
)= 2Σ119879minus1
119906119879minus1
minus 2(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 0
(60)
which implies
119906lowast
119879minus1= (Σ119879minus1
)minus1
[120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
] (61)
Substituting 119906lowast
119879minus1into the objective function of problem (58)
gives (see Appendix D for more details)
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(62)
where119872119879minus1
= 1205981015840
119879minus1Σminus1
119879minus1120598119879minus1
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 119909119879minus1
= 119904119879minus2
119909119879minus2
+ 1198751015840
119879minus2119906119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(63)
From (62) we can easily have
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) = 1205732119879minus1
119864 (1198762
119879minus2) 1198972
119879minus2
+ 1205731119879minus1
119864 (119876119879minus2
) 119897119879minus2
minus 120582119879minus1
119904119879minus1
119904119879minus2
119909119879minus2
minus 120582119879minus1
119904119879minus1
1205981015840
119879minus2119906119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(64)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (63) we have
min119906119879minus2
119881119879minus2
(119909119879minus2
119897119879minus2
) (65)
where119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ119879minus2
119906119879minus2
minus 2(1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(66)
The first-order necessary optimality condition implies
119889 (119881119879minus2
(119909119879minus2
119897119879minus2
))
119889 (119906119879minus2
)= 2Σ119879minus2
119906119879minus2
minus 2(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 0
(67)
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 2 Model notations of Section 4
For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821
119879minus1= 120582119879minus1
1205821
119905= 120582119905+ 1205821
119905+1119904119905+1
1205731119879minus1
= 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1205731119905
= (120582119905+ 1205731119905+1
) 119864(119876119905) minus 1205821
1199051205981015840
119905(Σ119905)minus1120593119905
1205732119879minus1
= Var(119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1205732119905
= Var(119876119905) + 1205732119905+1
119864(1198762
119905) minus 1205931015840
119905(Σ119905)minus1120593119905
Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1
and liability 119897119879minus1
at the beginning of the (119879minus1)th period problem (55) reducesto
min119906119879minus1
Var119879minus1
(119878119879) minus 120582119879minus1
119864119879minus1
(119878119879)
st 119909119879
= 119904119879minus1
119909119879minus1
+ 1198751015840
119879minus1119906119879minus1
119897119879
= 119876119879minus1
119897119879minus1
119878119879= 119909119879minus 119897119879
(57)
Substituting the binding constraints into the objective func-tion we have
min119906119879minus1
119881119879minus1
(119909119879minus1
119897119879minus1
) (58)
where
119881119879minus1
(119909119879minus1
119897119879minus1
) = 1199061015840
119879minus1Σ119879minus1
119906119879minus1
minus 2(1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906119879minus1
+ Var (119876119879minus1
) 1198972
119879minus1minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
(59)
It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have
119889 (119881119879minus1
(119909119879minus1
119897119879minus1
))
119889 (119906119879minus1
)= 2Σ119879minus1
119906119879minus1
minus 2(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 0
(60)
which implies
119906lowast
119879minus1= (Σ119879minus1
)minus1
[120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
] (61)
Substituting 119906lowast
119879minus1into the objective function of problem (58)
gives (see Appendix D for more details)
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(62)
where119872119879minus1
= 1205981015840
119879minus1Σminus1
119879minus1120598119879minus1
When 119905 = 119879 minus 2 for given wealth 119909119879minus2
and liability 119897119879minus2
at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows
min Var119879minus2
(119878119879minus1
) minus 120582119879minus2
119864119879minus2
(119878119879minus1
)
+ 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
))
st 119909119879minus1
= 119904119879minus2
119909119879minus2
+ 1198751015840
119879minus2119906119879minus2
119897119879minus1
= 119876119879minus2
119897119879minus2
119878119879minus1
= 119909119879minus1
minus 119897119879minus1
(63)
From (62) we can easily have
119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) = 1205732119879minus1
119864 (1198762
119879minus2) 1198972
119879minus2
+ 1205731119879minus1
119864 (119876119879minus2
) 119897119879minus2
minus 120582119879minus1
119904119879minus1
119904119879minus2
119909119879minus2
minus 120582119879minus1
119904119879minus1
1205981015840
119879minus2119906119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(64)
Substituting 119864119879minus2
(119881lowast
119879minus1(119909119879minus1
119897119879minus1
)) and the binding con-straints into the objective function of problem (63) we have
min119906119879minus2
119881119879minus2
(119909119879minus2
119897119879minus2
) (65)
where119881119879minus2
(119909119879minus2
119897119879minus2
) = 1199061015840
119879minus2Σ119879minus2
119906119879minus2
minus 2(1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)119906119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
(66)
The first-order necessary optimality condition implies
119889 (119881119879minus2
(119909119879minus2
119897119879minus2
))
119889 (119906119879minus2
)= 2Σ119879minus2
119906119879minus2
minus 2(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 0
(67)
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Thus
119906lowast
119879minus2= (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+1205821
119879minus2
2(Σ119879minus2
)minus1
120598119879minus2
(68)
Substituting 119906lowast
119879minus2into the objective function of problem (65)
(see Appendix E for more details) we have
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(69)
where119872119879minus2
= 1205981015840
119879minus2Σminus1
119879minus2120598119879minus2
Next by using mathematical induction we show that
both (56) and
119881lowast
119905(119909119905 119897119905) = 12057321199051198972
119905+ 1205731119905119897119905minus 1205821
119905119904119905119909119905minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(70)
hold where 119872119894= 1205981015840
119894Σminus1
119894120598119894with 119894 = 0 1 119879 minus 1 Suppose
that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909
119905minus1and
liability 119897119905minus1
the corresponding optimal investment problemis
min Var119905minus1
(119878119905) minus 120582119905minus1
119864119905minus1
(119878119905) + 119864119905minus1
(119881lowast
119905(119909119905 119897119905))
st 119909119905= 119904119905minus1
119909119905minus1
+ 1198751015840
119905minus1119906119905minus1
119897119905= 119876119905minus1
119897119905minus1
119878119905= 119909119905minus 119897119905
(71)
It follows from (70) that
119864119905minus1
(119881lowast
119905(119909119905 119897119905)) = 120573
2119905119864 (1198762
119905minus1) 1198972
119905minus1+ 1205731119905119864 (119876119905minus1
) 119897119905
minus 1205821
119905119904119905119904119905minus1
119909119905minus1
minus 1205821
1199051199041199051205981015840
119905minus1119906119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(72)
Substituting 119864119905minus1
(119881lowast
119905(119909119905 119897119905)) and the binding constraints into
the objective function of problem (71) we havemin119906119905minus1
119881119905minus1
(119909119905minus1
119897119905minus1
) (73)
where
119881119905minus1
(119909119905minus1
119897119905minus1
) = 1199061015840
119905minus1Σ119905minus1
119906119905minus1
minus 2(1205931015840
119905minus1119897119905minus1
+1205821
119905minus1
21205981015840
119905minus1)119906119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
(74)
The first-order necessary optimality condition gives
119889 (119881119905minus1
(119909119905minus1
119897119905minus1
))
119889 (119906119905minus1
)= 2Σ119905minus1
119906119905minus1
minus 2(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 0
(75)
which implies
119906lowast
119905minus1= Σminus1
119905minus1120593119905minus1
119897119905minus1
+1205821
119905minus1
2Σminus1
119905minus1120598119905minus1
(76)
Substituting 119906lowast
119905minus1into the objective function of problem (73)
(see Appendix F for more details) we have
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(77)
This completes the proof
Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909
119905 However Theorem 1 gives an opposite conclusion
This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested
Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin
0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (78)
which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result
Corollary 9 If the return of liability 119876119905 is uncorrelated with
those of risky assets 119877119905 that is Σ0
119905= 0 for any 119905 isin 0 1 119879minus
1 then the optimal policy for problem (55) is
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (79)
Proof Since Σ0119905= 0 it is easy to have
120593119905= 119864 (119875
119905119876119905) minus 119864 (119875
119905) 119864 (119876
119905)
= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877
119905minus 119904119905119868) 119864 (119876
119905)
= 119864 (119877119905119876119905) minus 119864 (119877
119905) 119864 (119876
119905)
= Σ0
119905
= 0
(80)
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Substituting 120593119905= 0 into (56) gives
119906lowast
119905=
1205821
119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (81)
This completes the proof
Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset
Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is
Σminus1
119905120593119905119897119905 (82)
which depends on the current value of the liability
Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows
min119906119905
Var119905(119878119905+1
) minus 120582119905119864119905(119878119905+1
)
st 119909119905+1
= 119904119905119909119905+ 1198751015840
119905119906119905
119897119905+1
= 119876119905119897119905
119878119905+1
= 119909119905+1
minus 119897119905+1
(83)
By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by
119906119898119910
119905= Σminus1
119905120593119905119897119905+
120582119905
2Σminus1
119905120598119905 119905 = 0 119879 minus 1 (84)
Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is
arbitrarily risk averse that is 120582119905
rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to
119906lowast
119905= Σminus1
119905120593119905119897119905 119905 = 0 119879 minus 1 (85)
This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market
Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered
Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05
119905Time-consistent strategy with
liability 119906lowast
119905
Time-consistent strategywithout liability
lowast
119905
0 (
11941198970+ 0903119909
0+ 3736
01071198970minus 0066119909
0+ 0203
minus05771198970+ 0163119909
0minus 3939
) (
09031199090+ 3736
minus00661199090+ 0203
01631199090minus 3939
)
1 (
minus03721198971+ 1018119909
1+ 1743
01981198971minus 0060119909
1+ 0095
minus06331198971+ 0042119909
1minus 1838
) (
10181199091+ 1743
minus00601199091+ 0095
00421199091minus 1838
)
Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0+ 3736
minus00661199090+ 0107119897
0+ 0203
01631199090minus 0577119897
0minus 3939
) (
10181199090+ 0449119897
0+ 1743
minus00601199090+ 0150119897
0+ 0095
00421199090minus 0599119897
0minus 1838
)
1 (
10181199091minus 0372119897
1+ 1743
minus00601199091+ 0198119897
1+ 0095
00421199091minus 0633119897
1minus 1838
) (
10181199091+ 0449119897
1+ 1743
minus00601199091+ 0150119897
1+ 0095
00421199091minus 0599119897
1minus 1838
)
5 Numerical Illustration
In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583
119905=
(1162 1246 1228) and
Σ119905= (
00146 00187 00145
00187 00854 00104
00145 00104 00289
) (86)
respectively The expected return of the liability 119864(119876119905) is
1136 the corresponding variance Var(119876119905) is 001 and the
covariance vector Σ0119905is (00006 00149 00050)1015840
Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability
Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582
119905=
05 and 120582119905
= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset
Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 5 Investment strategies in a market without riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
09031199090+ 1194119897
0
minus00661199090+ 0107119897
0
01631199090minus 0577119897
0
) (
10181199090+ 0449119897
0
minus00601199090+ 0150119897
0
00421199090minus 0599119897
0
)
1 (
10181199091minus 0372119897
1
minus00601199091+ 0198119897
1
00421199091minus 0633119897
1
) (
10181199091+ 0449119897
1
minus00601199091+ 0150119897
1
00421199091minus 0599119897
1
)
Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970+ 7889
01501198970minus 0062
minus05991198970minus 3572
) (
04441198970+ 3867
01501198970minus 0031
minus05991198970minus 1751
)
1 (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
) (
04441198971+ 3867
01501198971minus 0031
minus05991198971minus 1751
)
Table 7 Investment strategies in a market with a riskless asset for1205820= 1205821= 0
119905 Time-consistent strategy 119906lowast
119905Myopic strategy 119906
my119905
0 (
04441198970
01501198970
minus05991198970
) (
04441198970
01501198970
minus05991198970
)
1 (
04441198971
01501198971
minus05991198971
) (
04441198971
01501198971
minus05991198971
)
riskless asset 119904119905is 104 Then we have 120598
119905= 120583119905minus 119904119905119868 =
(0122 0206 0188)1015840 and120593
119905= Σ0
119905= (00006 00149 00050)
1015840Tables 6 and 7 show the time-consistent strategy and the
myopic strategy in a market with a riskless asset for 120582119905= 05
and 120582119905= 0 respectively From Table 6 it is clear that the two
strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects
6 Conclusion
In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset
Appendices
A Proof of (20)
Firstly substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ1
119879minus1119906lowast
119879minus1gives
119906lowast
119879minus1
1015840
Σ1
119879minus1119906lowast
119879minus1=
119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
1015840
(Σ1
119879minus1)minus1
times 119868
119886119879minus1
119909119879minus1
+ [Σ0
119879minus1minus 120574119879minus1
119868]
times119897119879minus1
+120582119879minus1
2[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[Σ0
119879minus1minus 120574119879minus1
119868] 1198972
119879minus1
+ 120582119879minus1
[Σ0
119879minus1minus 120574119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868] 119897119879minus1
+(120582119879minus1
)2
4[120583119879minus1
minus 119902119879minus1
119868]1015840
times (Σ1
119879minus1)minus1
[120583119879minus1
minus 119902119879minus1
119868]
=1
119886119879minus1
1199092
119879minus1+ [(Σ
0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
minus119886119879minus1
1205742
119879minus1] 1198972
119879minus1
+ [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
]
times 120582119879minus1
119897119879minus1
+(120582119879minus1
)2
4120572119879minus1
(A1)
Secondly by substituting 119906lowast
119879minus1into 2[(Σ
0
119879minus1)1015840
119897119879minus1
+
(120582119879minus1
2)1205831015840
119879minus1]119906lowast
119879minus1 we have
2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] 119906lowast
119879minus1
=2
119886119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
119868119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
times [Σ0
119879minus1minus 120574119879minus1
119868] 119897119879minus1
+ 120582119879minus1
[(Σ0
119879minus1)1015840
119897119879minus1
+120582119879minus1
21205831015840
119879minus1] (Σ1
119879minus1)minus1
times [120583119879minus1
minus 119902119879minus1
119868]
= 2120574119879minus1
119897119879minus1
119909119879minus1
+ 119902119879minus1
120582119879minus1
119909119879minus1
+ 2 [(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1minus 120574119879minus1
119889119879minus1
] 1198972
119879minus1
+ 2120582119879minus1
[(Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
minus 119887119879minus1
120574119879minus1
] 119897119879minus1
+(120582119879minus1
)2
2120572119879minus1
(A2)
Thus119881lowast
119879minus1(119909119879minus1
119897119879minus1
)
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
120583119879minus1
+119887119879minus1
120574119879minus1
] 119897119879minus1
+ [Var (119876119879minus1
) minus (Σ0
119879minus1)1015840
(Σ1
119879minus1)minus1
Σ0
119879minus1
+120574119879minus1
119889119879minus1
] 1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
=1
119886119879minus1
1199092
119879minus1minus 120582119879minus1
119902119879minus1
119909119879minus1
minus 2120574119879minus1
119909119879minus1
119897119879minus1
+ 1205731119879minus1
119897119879minus1
+ 1205732119879minus1
1198972
119879minus1minus
(120582119879minus1
)2
4120572119879minus1
(A3)
B Proof of (34)It is easy to verify that
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
=119889119879minus2
119886119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
+ 120574119879minus1
1198681015840(Σ1
119879minus2)minus1
119864 (119876119879minus2
119877119879minus2
)
119886119879minus2
minus 120574119879minus2
= 120574119879minus2
minus 120574119879minus2
= 0
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868] =119887119879minus2
119886119879minus2
minus 119902119879minus2
= 0
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
119868
= 119886119879minus2
120574119879minus2
+ 120574119879minus1
119864 (119876119879minus2
1198771015840
119879minus2) (Σ1
119879minus2)minus1
119868
119886119879minus2
= 119886119879minus2
120574119879minus2
(B1)
Firstly we compute 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2 It follows from the
previous equalities that both coefficients of 119909119879minus2
and 119897119879minus2
119909119879minus2
in 119906lowast
119879minus2
1015840Σ1
119879minus2119906lowast
119879minus2are 0 the coefficient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B2)
the coefficient of 119897119879minus2
is
1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
minus 1205821
119879minus2120574119879minus2
1198681015840(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
= 1205821
119879minus2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
(B3)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]1015840
sdot (Σ1
119879minus2)minus1
sdot [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
minus 119886119879minus2
1205742
119879minus2
(B4)
and the constant term is
(1205821
119879minus2)2
4[120583119879minus2
minus 119902119879minus2
119868]1015840
(Σ1
119879minus2)minus1
[120583119879minus2
minus 119902119879minus2
119868]
=(1205821
119879minus2)2
4120572119879minus2
(B5)
Secondly we compute Φ1015840
119879minus2119906lowast
119879minus2 The coefficient of 1199092
119879minus2
is 0 the coefficient of term 119909119879minus2
is
1205821
119879minus2
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
119868
119886119879minus2
=1205821
119879minus2119902119879minus2
2 (B6)
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
the coefficient of 119897119879minus2
119909119879minus2
is 120574119879minus2
the coefficient of 119897119879minus2
is
1205821
119879minus2
2[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
times (Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868)
+1205821
119879minus2
21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
(B7)
the coefficient of 1198972119879minus2
is
[Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
= [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)] minus 119886119879minus2
1205742
119879minus2
(B8)
and the constant term is
(1205821
119879minus2
2)
2
1205831015840
119879minus2(Σ1
119879minus2)minus1
(120583119879minus2
minus 119902119879minus2
119868) = (1205821
119879minus2
2)
2
120572119879minus2
(B9)
Then we can compute 119881lowast
119879minus2(119909119879minus2
119897119879minus2
) easily The coeffi-cient of 1199092
119879minus2is
1198681015840
119886119879minus2
(Σ1
119879minus2)minus1 119868
119886119879minus2
=1
119886119879minus2
(B10)
the coefficient of 119909119879minus2
isminus1205821119879minus2
119902119879minus2
the coefficient of 119897119879minus2
119909119879minus2
is minus2120574119879minus2
the coefficient of 1198972119879minus2
is
[Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] + 119886
119879minus21205742
119879minus2
minus [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]1015840
(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
)]
= 1205732119879minus2
(B11)
the coefficient of 119897119879minus2
is
minus 1205821
119879minus21205831015840
119879minus2(Σ1
119879minus2)minus1
times [Σ0
119879minus2+ 120574119879minus1
119864 (119876119879minus2
119877119879minus2
) minus 120574119879minus2
119868]
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)]
= 1205731119879minus2
(B12)
and the constant term is
minus1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B13)
Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) =1
119886119879minus2
1199092
119879minus2minus 1205821
119879minus2119902119879minus2
119909119879minus2
minus 2120574119879minus2
119909119879minus2
119897119879minus2
+ 1205731119879minus2
119897119879minus2
+ 1205732119879minus2
1198972
119879minus2minus
1205822
119879minus1
4120572119879minus1
minus(1205821
119879minus2)2
4120572119879minus2
(B14)
C Proof of (46)It is easy to verify that
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
=119889119905minus1
119886119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
+ 120574119905
1198681015840(Σ1
119905minus1)minus1
119864 (119876119905minus1
119877119905minus1
)
119886119905minus1
minus 120574119905minus1
= 120574119905minus1
minus 120574119905minus1
= 0
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =119887119905minus1
119886119905minus1
minus 119902119905minus1
= 0
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
119868
= 119886119905minus1
120574119905minus1
+ 120574119905
119864 (119876119905minus1
1198771015840
119905minus1) (Σ1
119905minus1)minus1
119868
119886119905minus1
= 119886119905minus1
120574119905minus1
(C1)
Firstly we compute 119906lowast
119905minus1
1015840Σ1
119905minus1119906lowast
119905minus1 It follows from the
previous equalities that both coefficients of 119909119905minus1
and 119897119905minus1
119909119905minus1
are 0 the coefficient of 1199092119905minus1
is
1198681015840
119886119905minus1
(Σ1
119905minus1)minus1 119868
119886119905minus1
=1
119886119905minus1
(C2)
the coefficient of 119897119905minus1
is
1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
minus 1205821
119905minus1120574119905minus1
1198681015840(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
= 1205821
119905minus1[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
times (Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868]
(C3)
the coefficient of 1198972119879minus2
is
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]1015840
sdot (Σ1
119905minus1)minus1
sdot [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C4)
and the constant term is
(1205821
119905minus1)2
4[120583119905minus1
minus 119902119905minus1
119868]1015840
(Σ1
119905minus1)minus1
[120583119905minus1
minus 119902119905minus1
119868] =(1205821
119905minus1)2
4120572119905minus1
(C5)
Secondly we compute Φ1015840
119905minus1119906lowast
119905minus1 The coefficient of 1199092
119905minus1is
0 the coefficient of 119909119905minus1
is
1205821
119905minus1
2
1205831015840
119905minus1(Σ1
119905minus1)minus1
119868
119886119905minus1
=1205821
119905minus1119902119905minus1
2 (C6)
the coefficient of 119897119905minus1
119909119905minus1
is 120574119905minus1
the coefficient of 119897119905minus1
is
1205821
119905minus1
2[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times (120583119905minus1
minus 119902119905minus1
119868) +1205821
119905minus1
21205831015840
119905minus1(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
(C7)
the coefficient of 1198972119905minus1
is
[Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
= [Σ0
119905minus1+ 120574119905119864(119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)] minus 119886119905minus1
1205742
119905minus1
(C8)
and the constant term is
(1205821
119905minus1
2)
2
120583119905minus1
(Σ1
119905minus1)minus1
(120583119905minus1
minus 119902119905minus1
119868) = (1205821
119905minus1
2)
2
120572119905minus1
(C9)
Then it is easy to compute 119881lowast
119905minus1(119909119905minus1
119897119905minus1
) The coefficientof 1199092
119905minus1is 1119886
119905minus1 the coefficient of 119909
119905minus1is minus120582
1
119905minus1119902119905minus1
thecoefficient of 119897
119905minus1119909119905minus1
is minus2120574119905minus1
the coefficient of 1198972119905minus1
is
[Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] + 119886
119905minus11205742
119905minus1
minus [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]1015840
(Σ1
119905minus1)minus1
times [Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
)]
= 1205732119905minus1
(C10)
the coefficient of 119897119905minus1
is
minus 1205821
119905minus11205831015840
119905minus1(Σ1
119905minus1)minus1
[Σ0
119905minus1+ 120574119905119864 (119876119905minus1
119877119905minus1
) minus 120574119905minus1
119868]
+ (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) = 1205731119905minus1
(C11)
and the constant term is minus(14)sum119879minus1119894=119905minus1
(1205821
119894)2120572119894
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) =1
119886119905minus1
1199092
119905minus1minus 1205821
119905minus1119902119905minus1
119909119905minus1
minus 2120574119905minus1
119909119905minus1
119897119905minus1
+ 1205731119905minus1
119897119905minus1
+ 1205732119905minus1
1198972
119905minus1minus
1
4
119879minus1
sum119894=119905minus1
(1205821
119894)2
120572119894
(C12)
D Proof of (62)
Substituting 119906lowast
119879minus1into 119906
lowast
119879minus1
1015840Σ119879minus1
119906lowast
119879minus1 we have
119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1= (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)
times (Σ119879minus1
)minus1
(120593119879minus1
119897119879minus1
+120582119879minus1
2120598119879minus1
)
= 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
1198972
119879minus1+ 120582119879minus1
1205981015840
119879minus1
times (Σ119879minus1
)minus1
120593119879minus1
119897119879minus1
+(120582119879minus1
)2
4119872119879minus1
2 (1205931015840
119879minus1119897119879minus1
+120582119879minus1
21205981015840
119879minus1)119906lowast
119879minus1= 2119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1
(D1)
Thus
119881lowast
119879minus1(119909119879minus1
119897119879minus1
) = minus119906lowast
119879minus1
1015840
Σ119879minus1
119906lowast
119879minus1+ Var (119876
119879minus1) 1198972
119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
+ 120582119879minus1
119864 (119876119879minus1
) 119897119879minus1
= [Var (119876119879minus1
) minus 1205931015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
] 1198972
119879minus1
+ 120582119879minus1
[119864 (119876119879minus1
) minus 1205981015840
119879minus1(Σ119879minus1
)minus1
120593119879minus1
]
times 119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus1
1198972
119879minus1+ 1205731119879minus1
119897119879minus1
minus 120582119879minus1
119904119879minus1
119909119879minus1
minus(120582119879minus1
)2
4119872119879minus1
(D2)
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
E Proof of (69)Taking 119906
lowast
119879minus2into account we have
119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2= (1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2)
times (Σ119879minus2
)minus1
(120593119879minus2
119897119879minus2
+1205821
119879minus2
2120598119879minus2
)
= 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1198972
119879minus2+ 1205821
119879minus21205981015840
119879minus2
times (Σ119879minus2
)minus1
120593119879minus2
119897119879minus2
+(1205821
119879minus2)2
4119872119879minus2
2 1205931015840
119879minus2119897119879minus2
+1205821
119879minus2
21205981015840
119879minus2119906lowast
119879minus2= 2119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
(E1)Thus
119881lowast
119879minus2(119909119879minus2
119897119879minus2
) = minus119906lowast
119879minus2
1015840
Σ119879minus2
119906lowast
119879minus2
+ [Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2)] 1198972
119879minus2
+ [120582119879minus2
119864 (119876119879minus2
) + 1205731119879minus1
119864 (119876119879minus2
)] 119897119879minus1
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
= 1205732119879minus2
1198972
119879minus2+ 1205731119879minus2
119897119879minus2
minus 1205821
119879minus2119904119879minus2
119909119879minus2
minus(120582119879minus1
)2
4119872119879minus1
minus(1205821
119879minus2)2
4119872119879minus2
(E2)where1205731119879minus2
= (120582119879minus2
+ 1205731119879minus1
) 119864 (119876119879minus2
) minus 1205821
119879minus21205981015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
1205732119879minus2
= Var (119876119879minus2
) + 1205732119879minus1
119864 (1198762
119879minus2) minus 1205931015840
119879minus2(Σ119879minus2
)minus1
120593119879minus2
(E3)
F Proof of (77)It follows from 119906
lowast
119905minus1that
119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1= (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)
times (Σ119905minus1
)minus1
(120593119905minus1
119897119905minus1
+1205821
119905minus1
2120598119905minus1
)
= 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1198972
119905minus1+ 1205821
119905minus11205981015840
119879minus1
times (Σ119905minus1
)minus1
120593119905minus1
119897119905minus1
+(1205821
119905minus1)2
4119872119905minus1
2 (1205931015840
119879minus1119897119905minus1
+1205821
119905minus1
21205981015840
119879minus1)119906lowast
119905minus1= 2119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
(F1)
Thus
119881lowast
119905minus1(119909119905minus1
119897119905minus1
) = minus119906lowast
119905minus1
1015840
Σ119905minus1
119906lowast
119905minus1
+ [Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1)] 1198972
119905minus1
+ [120582119905minus1
119864 (119876119905minus1
) + 1205731119905119864 (119876119905minus1
)] 119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905
(1205821
119894)2
4119872119894
= 1205732119905minus1
1198972
119905minus1+ 1205731119905minus1
119897119905minus1
minus 1205821
119905minus1119904119905minus1
119909119905minus1
minus
119879minus1
sum119894=119905minus1
(1205821
119894)2
4119872119894
(F2)
where
1205731119905minus1
= (120582119905minus1
+ 1205731119905) 119864 (119876
119905minus1) minus 1205821
119905minus11205981015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
1205732119905minus1
= Var (119876119905minus1
) + 1205732119905119864 (1198762
119905minus1) minus 1205931015840
119879minus1(Σ119905minus1
)minus1
120593119905minus1
(F3)
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995
[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990
[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004
[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011
[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006
[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009
[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011
[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955
[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010
[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011
[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012
[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013
[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of