optimization problem of insurance investment based on...

Research ArticleOptimization Problem of Insurance Investment Based onSpectral Risk Measure and RAROC Criterion

Xia Zhao 1 Hongyan Ji 2 and Yu Shi 1

1School of Statistics and Information Shanghai University of International Business and Economics Shanghai 201620 China2School of Statistics Shandong University of Finance and Economics Jinan Shandong 250014 China

Correspondence should be addressed to Xia Zhao zhaoxia-w163com

Received 7 September 2018 Accepted 15 October 2018 Published 30 October 2018

Academic Editor Xue-Jun Xie

Copyright copy 2018 Xia Zhao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper introduces spectral risk measure (SRM) into optimization problem of insurance investment Spectral risk measurecould describe the degree of risk aversion so the underlying strategy might take the investors risk attitude into account Weestablish an optimizationmodel aiming atmaximizing risk-adjusted return of capital (RAROC) involvedwith spectral riskmeasureThe theoretical result is derived and empirical study is displayed under different risk measures and different confidence levelscomparativelyThe result shows that risk attitude has a significant impact on investment strategyWith the increase of risk aversionfactor the investment ratio of risk asset correspondingly reduces When the aversive level increases to a certain extent the impacton investment strategies disappears because of the marginal effect of risk aversion In the case of VaR and CVaR without regard forrisk aversion the investment ratio of risk asset is increasing significantly

1 Introduction

Underwriting business and investment business are twomain fund sources of an insurance company In recentyears more and more insurers have paid attention to theefficiency of investment business because of increasing com-petition among insurance companies continuing decline inunderwriting profits and gradual relaxation of insuranceinvestment policies

The relationship between return and risk needs to befully balanced in insurance investment in which mean-riskoptimization is the most commonly used criterion For themeasurement of risk variance is a common choice Earlystudies for example Lambert and Hofflander [1] Kahaneand Nye [2] and Briys [3] established optimal portfoliomodel for property insurance undermean-variance criterionLater due to the limitation of variance new risk measureswere proposed constantly and mean-risk models were alsoextended in various backgrounds see [4ndash9] In particularruin probability and some down-side risk measures such asVaR and CaR were introduced into insurance business tofind the optimal investment strategy Guo and Li [10] used

mean-VaR model to analyze the choice of optimal portfoliosfor insurers Chen et al [11] investigated an investment-reinsurance problem under dynamic Value-at-Risk (VaR)constraint Zeng et al [12] established twomean-CaR modelsto study reinsurance-investment problem of insurers andobtained the explicit expressions of the optimal deterministicrebalance reinsurance-investment strategies and mean-CaRefficient frontiers

Risk measures used in the above literatures indeeddescribe different risk characteristics of the assets but theydo not take investorsrsquo risk attitude into account Spectral riskmeasures (SRM) proposed by Acerbi et al [13] characterizeinvestorsrsquo risk aversion and have been applied to fields ofbanks and securities for example Adam et al [14] and Diaoet al [15] and the references therein However to the best ofour knowledge there is no literature which studied optimalinvestment problem in insurance business based on SRMOnthe other hand mean is generally used to describe the returnbut the insurer needs to determine the amount of capitalbased on entire risk situation of company The risk-adjustedreturn on capital (RAROC) takes into account the capitalreturn adjusted by risk whichmakes up for the shortcomings

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 9838437 7 pageshttpsdoiorg10115520189838437

2 Mathematical Problems in Engineering

from average-return principle see [16ndash18] and the referencestherein

This paper will introduce spectral risk measure intooptimal investment model with RAROC as optimizationtarget construct optimization model and give its theoreticaland empirical study The rest of this paper is organizedas follows Section 2 illustrates spectral risk measure andinsurance returnmodel used here Section 3 finds the solutionof the optimization problem The empirical application isdisplayed in Section 4 Section 5 concludes the paper

2 Spectral Risk Measure and InsuranceReturn Model

21 Spectral Risk Measure

Definition 1 (see [19]) Suppose that random variable Xrepresents the loss of assets and its distribution function canbe denoted as 119865(119909) = 119875119903(119883 le 119909) Spectral risk measure withconfidence level 119901 = 1 minus 120572 (120572120598(0 1)) is defined as follows

120588 = int10120601 (119901) 119902119901119889119901 (1)

where 120601(119901) (0 1) 997891997888rarr R is a weight function or riskspectral function and 119902119901 = inf119909 | 119865(119909) ge 119901 is 119901-quantile of distribution function SRM is a coherent riskmeasure when 120601(119901) satisfies nonnegativity normalizationand increasingness

Specially 120588 is Value at Risk (VaR) if 120601(119901) = 0 119901 =1 minus 120572 infin119901 = 1 minus 120572 120588 corresponds to Conditional Valueat Risk (CVaR) if 120601(119901) = (1120572)119868119901ge1minus120572 If 120601(119901) =(120574119890minus(1minus119901)120574120572120572(1minus119890minus120574))1198681minus120572le119901le1 120588 is exponential spectral riskmeasure if 120601(119901) = (120573(120572 minus 1 + 119901)120573minus1120572120573)1198681minus120572le119901le1 120573 gt 1(120573(1 minus 119901)120573minus1120572120573)1198681minus120572le119901le1 0 lt 120573 lt 1 120588 is power spectralrisk measure where 120574 gt 0 is the coefficient of absolute riskaversion and 120573 gt 0 is the coefficient of relative risk aversion

Proposition 2 (see [20]) Suppose that 119877 denotes incomevariable and then119883 = minus119877 denotes the loss variable If119877 followsnormal distribution assumption we can get

119878119877119872(119877) = minus119864 (119877) + 119879 (120572) 120590 (119877) (2)

Specially 119881119886119877 (119877) = minus119864 (119877) + Φminus1 (119901) 120590 (119877) (3)

and 119862119881119886119877 (119877) = minus119864 (119877) + 119891 (Φminus1 (119901))120572 120590 (119877) (4)

where 119879(120572) = int10Φminus1(119901)120601(119901)119889119901 Φminus1(119901) is 119901-quantile of

standard normal distribution and 119891() is probability densityfunction of standard normal distribution

22 Insurance Return Model Suppose that insurers invest inN assets one of which is risk-free asset and others are riskassets Therefore the total profit is given as

119877119901 = 1199031198871198770 + g1198770(1 minus 119873minus1sum119894=1

119896119894)1199030 + g1198770119873minus1sum119894=1

119896119894119903119894 (5)

where R0 rb g denote premium charged by insurers therate of underwriting profit and investment ratio respectivelyConstant r0 denotes the rate of risk-free asset return Andrandomvariable ri (i = 1 2 sdot sdot sdot Nminus1) denotes the rate of riskasset return with N(120583i 1205902i ) assumption ki is the investmentweight of the i-th risk asset and we assume that 0 lt sumNminus1

i=1 ki lt1Let K = (R0 gR0k1 gR0k2 sdot sdot sdot gR0kNminus1)T and r =(rb + gr0 r1 minus r0 sdot sdot sdot rNminus1 minus r0)T with mean 120583 and covariance

matrix Σ Then we have R = rTK and E(R) = 120583TK 120590(R) =radicKTΣK 120588c is the upper limit of risk the insurer can bear thatis SRM(Rp) le 120588c3 Optimal Investment Strategy for InsurersBased on SRM-RAROC Criterion

In this section we establish SRM-RAROC optimizationmodel and derive the optimal solution under normal distri-bution assumption

31 Optimization Model Here the investment performanceevaluation is measured by risk-adjusted return on capital(RAROC) instead of the absolute amount of income asfollows

119877119860119877119874119862 = 119864 (119877119901)119878119877119872(119877119901) (6)

Thus the optimization model can be formulated as

max 119877119860119877119874119862 = 119864 (119877119901)119878119877119872(119877119901)

st 0 lt 119873minus1sum119894=1

119896119894 lt 1119878119877119872(119877119901) = minus119864 (119877119901) + 119879 (120572) 120590 (119877119901) le 120588119888119864 (119877119901) = 120583119879119870120590 (119877119901) = radic119870119879Σ119870

(7)

32 Solution of Optimization Model

Step 1 (simplifying optimization model) Define 120579 as n-dimension vector 120579 = (1205791 1205792 sdot sdot sdot 120579119899)119879 where 1205791 = 1(1 +gsum119873minus1119869=1 119896119895)120579119894 = g119896119894minus1(1 + gsum119873minus1119869=1 119896119895) 119894 = 2 3 sdot sdot sdot 119899 Andthen119870 can be rewritten as119870 = 1198770(1 + gsum119873minus1119869=1 119896119895)120579 119868119879120579 = 1where 119868 is n-dimension vector 119868 = (1 1 sdot sdot sdot 1)119879

Let 119903119901 = 119903119879120579 then 120583119901 = 119864(119903119901) = 120583119879120579 120590119901 = V119886119903(119903119901) =radic120579119879Σ120579 With 1 minus 120572 confidence level SRM(119903119901) = minus120583119879120579 +119879(120572)radic120579119879Σ120579 Then we have

Mathematical Problems in Engineering 3

119864 (119877119901) = 120583119879119870 = 1205831198791198770(1 + g119873minus1sum119869=1

119896119895)120579

= 1198770(1 + g119873minus1sum119869=1

119896119895)119864 (119903119901)(8)

and 119878119877119872(119877119901) = minus119864 (119877119901) + 119879 (120572) 120590 (119877119901)= 1198770(1 + g

119873minus1sum119869=1

119896119895)119878119877119872(119903119901) (9)

So RAROC can be rewritten as

119877119860119877119874119862 = 119864 (119877119901)119878119877119872(119877119901)

= 1198770 (1 + gsum119873minus1119869=1 119896119895) 119864 (119903119901)1198770 (1 + gsum119873minus1119869=1 119896119895) 119878119877119872(119903119901)

= 119864 (119903119901)119878119877119872(119903119901)

(10)

And model (7) can be transformed as

max 119877119860119877119874119862 = 119864 (119903119901)119878119877119872(119903119901)

st 119868119879120579 = 1119878119877119872(119903119901) = minus120583119901 + 119879 (120572) 120590119901 le 120588119888120583119901 = 119864 (119903119901) = 120583119879120579120590119901 = 120590 (119903119901) = radic120579119879Σ120579

(11)

Step 2 (effective frontier curve equation of mean-SRM space)Effective frontier in mean-risk space refers to the portfoliothat maximizes the return at a certain level of risk orminimizes the risk at a certain level of return Thereforemathematical expression of curve equation of effective fron-tier can be given as

min (minus120583T120579 + T (120572)radic120579119879Σ120579)st 120583119901 = 120583119879120579

119868119879120579 = 1(12)

Solving model (12) by Lagrange multiplier method yields

120579 = 1119889Σminus1 ((119888120583119901 minus 119887) 120583 + (119886 minus 119887120583119901) 119868) (13)

where 119886 = 120583119879Σminus1120583 119887 = 120583119879Σminus1119868 = 119868119879Σminus1120583 119888 = 119868119879Σminus1119868 119889 =119886119888 minus 1198872

So then 120590p2 = 120579TΣ120579 = (c120583p2 minus 2b120583p + a)dTherefore the effective frontier curve equation is

119878119877119872(119903119901) = minus120583119901 + 119879 (120572)radic 1119889 (1198881205831199012 minus 2119887120583119901 + 119886) (14)

Assume that 120583119900119901119905 be the optimal return for a given risk 120588based on formula (14) the corresponding optimal portfolioweights on effective frontier curve can be solved as


Step 3 (RAROC maximized portfolio under SRM con-straints) Let 119877119860119877119874119862 = 120583119901119878119877119872(119903119901) = 119906 that is theslope 119906 of line 120583119901 = 119906119878119877119872(119903119901) will be maximized in theprocessing of optimization From portfolio theory in financewe know that maximum value is obtained when the line istangent to the effective leading edge The tangent point is theoptimal portfolio when the tangent point is on the left of theconstraint line while the intersection of the constraint lineand the effective frontier is the optimal portfolio when thetangent point is on the right of the constraint line

Let (SRMT 120583T) denote the intersection portfolio It isobvious that SRMT = 120588c at the intersection point Fromeffective frontier curve equation we can find that

120583T = (bT2 + d120588c) + Tradicd (2b120588c + c120588c2 + a minus T2)cT2 minus d

(16)

Let (SRMtg 120583tg) denote tangent portfolio The formula120597SRM120597120583p = 1u is true for tangent point when the line istangent to the effective frontier So it follows that

minus 1 + T (120572) (c120583p minus b)dradic(1d) (c120583p2 minus 2b120583p + a)

= minus120583p + T (120572)radic(1d) (c120583p2 minus 2b120583p + a)120583p

(17)

which results in the following tangent point portfolio

(SRMtg 120583tg) = (radicab (T minus radica) ab) (18)

Summarily the optimal solution of optimization modelcan be expressed as

(SRMopt 120583opt) = (SRMtg 120583tg) if 120588tg le 120588c(SRMT 120583T) if 120588tg gt 120588c (19)

and the optimal portfolio weight is

120579opt = 1dΣminus1 ((c120583opt minus b) 120583 + (a minus b120583opt) I) (20)

Therefore the optimal investment ratio of each risk asset is

kiminus1 = 120579ig1205791 i = 2 3 sdot sdot sdot N (21)


Table 1 Descriptive statistical analysis of risk assets

Changrsquoan Vehicle (000625) JoinTo Energy (000600) Yili (600887)Median 0196729 0005992 -0070959Maximum 1341006 1124433 1299169Minimum -1631535 -1184555 -1196948Standard deviation 0866075 0632811 0698381Skewness -0545350 -0132884 0049686Kurtosis 2653994 2993299 2709674J-B statistic 0545562 0034779 0039235p value 0761260 0982761 0980574

minus15

minus10

minus05

00

05

10

15

Qua

ntile

s of N

orm

al

minus15

minus10

minus05

00

05

10

15

Qua

ntile

s of N

orm

al

minus12

minus08

minus04

00

04

08

12

Qua

ntile

s of N

orm

al

minus1 0 1 2minus2Quantiles of_________000625

minus10 minus05 00 05 10 15minus15Quantiles of_________00600


Figure 1 QQ chart of each risky assetrsquos return

and the corresponding proportion of investment in risk-freeassets is

1 minus Nminus1sumi=1

ki = 1 minus 1 minus 1205791g1205791 = g1205791 + 1205791 minus 1g1205791 (22)

4 Data Analysis

41 Data Selection In this section we assume that insurersinvest in one risk-free asset and three security risk assetsBank deposit is regarded as risk-free asset and the selectedrisk assets are JoinTo Energy (000600) Changrsquoan Vehicle(000625) and Yili (600887) Yearly data is chosen fromJanuary 1 2006 to December 31 2016 The data for ChinaLife InsuranceCompany Ltd is calculated from the companyrsquosannual report and semi-annual report Data of risk assetsis obtained from Guotai An CSMAR series of researchdatabases

42 Descriptive Statistics of Risk Asset Data We conductdescriptive statistical analysis of risk assets see Table 1 andFigure 1

It can be found fromTable 1 that the distribution of returnfor three risk assets presents a certain degree of skewness andflatter peak than normal distribution FromQQ chart and thefact that JB statistic of each risky asset return rate is less than599 which is 95 quantile of 1205942(2) we can conclude that thereturn of each risky asset is subject to a normal distributionapproximately

43 Calculation of Related Variables

(1) Investment Ratio and Rate of Underwriting Profit Theinvestment ratio is an important indicator to measure thelevel of capital utilization of an insurance company whichcan be calculated by investment assets divided by total assetsThe rate of underwriting profit can be calculated by thedifference between total profit and investment profit dividedby underwriting income Based on investment data andunderwriting data of China Life Insurance Company Ltd weobtain that g = 9408 and rb = minus01776 here(2) Rate of Return for Risk-Free Asset and Risky Asset FromChina Life Insurance Company Ltdrsquos data of the amount ofbank deposit and bank deposit return in 2006ndash2016 we cancalculate the mean of the return rate E(r0) = 00428 as risk-free return rate Each risky asset return is calculated by ri =ln(PitPitminus1) where Pit and Pitminus1 denote i-th assetrsquos price attime t and t-1 respectively So it can be calculated fromGuotaiAn CSMAR Series Research Database that the average returnof JoinTo Energy (000600) Changrsquoan Vehicle (000625) andYili (600887) are 00580 00516 and 00409 respectively44 Optimal Insurance Investment Strategy Based on SRM-RAROC Criterion In this section we conduct empiricalanalysis for optimal insurance investment strategy The con-fidence level is set to 90 and 95 and the upper limit ofrisk is assumed to be 002 Based on formulas in Section 2we calculate the optimal weights of the assets under differentconfidence levels The results are displayed in Table 2


Table2Optim

alinvestm

entstrategyu

nder

confi

dencelevel

120572=005

120588 119888=0

02Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash07136

02538

-00903

01229

CVaR

mdash08062

01812

-00674

00799

Expo

nentialSRM

120574=02

08150

01743

-00652

00758

120574=04

08261

01657

-00625

00707

120574gt06

08400

01548

-00590

006

42

Powe

rSRM

120573=11

08177

01723

-0064

600746

120573=12

08342

01594

-0060

5006

69120573gt

1308400

01548

-00590

006

42120572=

01120588 c=

002Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash06357

03148

-010

9601591

CVaR

mdash07350

02370

-00850

0113

0

Expo

nentialSRM

120574=02

07393

02336

-00840

0111

0120574=

0407437

02302

-00829

01089

120574ge35

08335

01599

-0060

6006

73

Powe

rSRM

120573=11

07405

02327

-00837

0110

4120573=

1207457

02286

-00824

01080

120573ge3

08335

01599

-0060

6006

73Re

markthen

egativev

alue

means

short-s

ellin

g


From Table 2 we can obtain the following conclusion(1) Generally the optimal investment proportion in riskasset is decreasing as the factor of risk aversion increaseswhich reflects the effect of risk attitude on investmentstrategy When the factor of risk aversion reaches some fixvalue the optimal investment proportion stays in a roughlyidentical level which shows the existence of marginal effectfrom risk aversion extent(2) Compared with the results under different risk mea-sures risk aversion attitude shows a significant effect on thechoice and assignment among risk assets and risk-free assetand hence the risk attitude should not be ignored(3) As confidence level becomes bigger the optimalinvestment proportion in risky asset increases and the onein risk-free asset decreases This is a natural conclusion sincebigger confidence level will make the value of risk be smallerwhich results in the increasing trend of investing in riskassets

5 Conclusions

This paper constructed an insurance optimization modelincluding spectral risk measure and risk-adjusted return ofcapital and conducted theoretical and empirical analysis Themain innovation of this paper is introduction of spectral riskmeasure in insurance business which makes the risk attitudeof the investor be considered in decision-making The resulttells us that more risk aversion will decrease the investmentratio in risk asset and increase the interest of investing inrisk-free asset However the impact on investment strategieswill disappear when the level of risk aversion increases to acertain extent Furthermore both confidence level and thethreshold of risk play a significant role on optimal strategyFor convenience here we suppose that the underwritinginsurance return follows a deterministic process and the pol-icy constraints for insurance investment are not involved Itmay be also of interest to extend the research to the case withstochastic insurance surplus process andor the governmentpolicy constraint We will explore these problems in thefollowing studies

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was partially supported by NSFC (71671104)Project of Humanities Social Sciences of MOE China(16YJA910003 18YJC630220) Key Project of National SocialScience of China (16AZD019) Special Funds of TaishanScholar Project (tsqn20161041) and SUIBE PostgraduateInnovative Talents Training Project

References

[1] E W Lambert and A E Hofflander ldquoImpact of New Multi-ple Line Underwriting on Investment Portfolios of Property-Liability Insurersrdquo Journal of Risk and Insurance vol 33 no 2p 209 1966

[2] Y Kahane and D Nye ldquoA Portfolio Approach to the Property-Liability Insurance Industryrdquo Journal of Risk and Insurance vol42 no 4 p 579 1975

[3] E P Briys ldquoInvestment portfolio behavior of non-life insurersa utility analysisrdquo Insurance Mathematics amp Economics vol 4no 2 pp 93ndash98 1985

[4] A J Frost ldquoImplications of modern portfolio theory for lifeassurance companies [J]rdquo Journal of the Institute of Actuariesvol 26 pp 47ndash68 1983

[5] W Guo ldquoOptimal portfolio choice for an insurer with lossaversionrdquo InsuranceMathematics amp Economics vol 58 pp 217ndash222 2014

[6] Y Zeng Z Li and Y Lai ldquoTime-consistent investment andreinsurance strategies for mean-variance insurers with jumpsrdquoInsuranceMathematics amp Economics vol 52 no 3 pp 498ndash5072013

[7] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential LEvy processrdquo Insurance Mathe-matics amp Economics vol 52 no 3 pp 508ndash521 2013

[8] P Artzner F Delbaen and J-M Eber ldquoCoherent measures ofriskrdquoMathematical Finance vol 9 no 3 pp 203ndash228 1999

[9] R G Emanuela ldquoRisk measures via g-expectations [J]rdquo Insur-ance Mathematics amp Economics vol 39 pp 19ndash34 2006

[10] W Guo and X Li ldquoOptimal insurance investment strategyunder VaR restriction [J]rdquo Journal of Systems amp Managementvol 18 no 5 pp 118ndash124 2009

[11] S Chen Z Li and K Li ldquoOptimal investment-reinsurance pol-icy for an insurance company with VaR constraintrdquo InsuranceMathematics amp Economics vol 47 no 2 pp 144ndash153 2010

[12] Y Zeng and Z Li ldquoOptimal reinsurance-investment strategiesfor insurers under mean-CaR criteriardquo Journal of Industrial andManagement Optimization vol 8 no 3 pp 673ndash690 2012

[13] C Acerbi ldquoSpectral measures of risk a coherent representationof subjective risk aversionrdquo Journal of Banking amp Finance vol26 no 7 pp 1505ndash1518 2002

[14] AAdamMHoukari and J-P Laurent ldquoSpectral riskmeasuresand portfolio selectionrdquo Journal of Banking amp Finance vol 32no 9 pp 1870ndash1882 2008

[15] X Diao B Tong and CWu ldquoSpectral risk measurement basedon EVT and its application in risk management [J]rdquo Journal ofSystems Engineering vol 6 no 30 pp 354ndash405 2015

[16] A Milne and M Onorato ldquoRisk-Adjusted Measures of ValueCreation in Financial InstitutionsrdquoEuropean FinancialManage-ment vol 18 no 4 pp 578ndash601 2012

[17] L Wang and J Li ldquoResearch on Insurance Investment StrategyBased on Risk-Adjusted Return Rate under Policy Constraints[J]rdquo Chinese Journal of Management Science vol 20 pp 16ndash222012

[18] X Chen Z Han and K Tang ldquoResearch on the OptimizationModel of Insurance Investment Based on VaR and RAROC [J]rdquoThe Journal of Quantitative amp Technical Economics vol 4 pp111ndash117 2006


[19] K Dowd and D Blake ldquoAfter VaR The theory estimation andinsurance applications of quantile-based riskmeasuresrdquo Journalof Risk and Insurance vol 73 no 2 pp 193ndash229 2006

[20] X Li and X Liu ldquoThe mean-spectral measures of risk efficientfrontier of portfolio and its empirical test [J]rdquo Chinese Journalof Management Science vol 5 no 13 pp 6ndash11 2005

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from average-return principle see [16ndash18] and the referencestherein

This paper will introduce spectral risk measure intooptimal investment model with RAROC as optimizationtarget construct optimization model and give its theoreticaland empirical study The rest of this paper is organizedas follows Section 2 illustrates spectral risk measure andinsurance returnmodel used here Section 3 finds the solutionof the optimization problem The empirical application isdisplayed in Section 4 Section 5 concludes the paper

2 Spectral Risk Measure and InsuranceReturn Model

21 Spectral Risk Measure

Definition 1 (see [19]) Suppose that random variable Xrepresents the loss of assets and its distribution function canbe denoted as 119865(119909) = 119875119903(119883 le 119909) Spectral risk measure withconfidence level 119901 = 1 minus 120572 (120572120598(0 1)) is defined as follows

120588 = int10120601 (119901) 119902119901119889119901 (1)

where 120601(119901) (0 1) 997891997888rarr R is a weight function or riskspectral function and 119902119901 = inf119909 | 119865(119909) ge 119901 is 119901-quantile of distribution function SRM is a coherent riskmeasure when 120601(119901) satisfies nonnegativity normalizationand increasingness

Specially 120588 is Value at Risk (VaR) if 120601(119901) = 0 119901 =1 minus 120572 infin119901 = 1 minus 120572 120588 corresponds to Conditional Valueat Risk (CVaR) if 120601(119901) = (1120572)119868119901ge1minus120572 If 120601(119901) =(120574119890minus(1minus119901)120574120572120572(1minus119890minus120574))1198681minus120572le119901le1 120588 is exponential spectral riskmeasure if 120601(119901) = (120573(120572 minus 1 + 119901)120573minus1120572120573)1198681minus120572le119901le1 120573 gt 1(120573(1 minus 119901)120573minus1120572120573)1198681minus120572le119901le1 0 lt 120573 lt 1 120588 is power spectralrisk measure where 120574 gt 0 is the coefficient of absolute riskaversion and 120573 gt 0 is the coefficient of relative risk aversion

Proposition 2 (see [20]) Suppose that 119877 denotes incomevariable and then119883 = minus119877 denotes the loss variable If119877 followsnormal distribution assumption we can get

119878119877119872(119877) = minus119864 (119877) + 119879 (120572) 120590 (119877) (2)

Specially 119881119886119877 (119877) = minus119864 (119877) + Φminus1 (119901) 120590 (119877) (3)

and 119862119881119886119877 (119877) = minus119864 (119877) + 119891 (Φminus1 (119901))120572 120590 (119877) (4)

where 119879(120572) = int10Φminus1(119901)120601(119901)119889119901 Φminus1(119901) is 119901-quantile of

standard normal distribution and 119891() is probability densityfunction of standard normal distribution

22 Insurance Return Model Suppose that insurers invest inN assets one of which is risk-free asset and others are riskassets Therefore the total profit is given as

119877119901 = 1199031198871198770 + g1198770(1 minus 119873minus1sum119894=1

119896119894)1199030 + g1198770119873minus1sum119894=1

119896119894119903119894 (5)

where R0 rb g denote premium charged by insurers therate of underwriting profit and investment ratio respectivelyConstant r0 denotes the rate of risk-free asset return Andrandomvariable ri (i = 1 2 sdot sdot sdot Nminus1) denotes the rate of riskasset return with N(120583i 1205902i ) assumption ki is the investmentweight of the i-th risk asset and we assume that 0 lt sumNminus1

i=1 ki lt1Let K = (R0 gR0k1 gR0k2 sdot sdot sdot gR0kNminus1)T and r =(rb + gr0 r1 minus r0 sdot sdot sdot rNminus1 minus r0)T with mean 120583 and covariance

matrix Σ Then we have R = rTK and E(R) = 120583TK 120590(R) =radicKTΣK 120588c is the upper limit of risk the insurer can bear thatis SRM(Rp) le 120588c3 Optimal Investment Strategy for InsurersBased on SRM-RAROC Criterion

In this section we establish SRM-RAROC optimizationmodel and derive the optimal solution under normal distri-bution assumption

31 Optimization Model Here the investment performanceevaluation is measured by risk-adjusted return on capital(RAROC) instead of the absolute amount of income asfollows

119877119860119877119874119862 = 119864 (119877119901)119878119877119872(119877119901) (6)

Thus the optimization model can be formulated as

max 119877119860119877119874119862 = 119864 (119877119901)119878119877119872(119877119901)

st 0 lt 119873minus1sum119894=1

119896119894 lt 1119878119877119872(119877119901) = minus119864 (119877119901) + 119879 (120572) 120590 (119877119901) le 120588119888119864 (119877119901) = 120583119879119870120590 (119877119901) = radic119870119879Σ119870

(7)

32 Solution of Optimization Model

Step 1 (simplifying optimization model) Define 120579 as n-dimension vector 120579 = (1205791 1205792 sdot sdot sdot 120579119899)119879 where 1205791 = 1(1 +gsum119873minus1119869=1 119896119895)120579119894 = g119896119894minus1(1 + gsum119873minus1119869=1 119896119895) 119894 = 2 3 sdot sdot sdot 119899 Andthen119870 can be rewritten as119870 = 1198770(1 + gsum119873minus1119869=1 119896119895)120579 119868119879120579 = 1where 119868 is n-dimension vector 119868 = (1 1 sdot sdot sdot 1)119879

Let 119903119901 = 119903119879120579 then 120583119901 = 119864(119903119901) = 120583119879120579 120590119901 = V119886119903(119903119901) =radic120579119879Σ120579 With 1 minus 120572 confidence level SRM(119903119901) = minus120583119879120579 +119879(120572)radic120579119879Σ120579 Then we have


119864 (119877119901) = 120583119879119870 = 1205831198791198770(1 + g119873minus1sum119869=1

119896119895)120579

= 1198770(1 + g119873minus1sum119869=1

119896119895)119864 (119903119901)(8)

and 119878119877119872(119877119901) = minus119864 (119877119901) + 119879 (120572) 120590 (119877119901)= 1198770(1 + g

119873minus1sum119869=1

119896119895)119878119877119872(119903119901) (9)


119877119860119877119874119862 = 119864 (119877119901)119878119877119872(119877119901)


= 119864 (119903119901)119878119877119872(119903119901)

(10)


max 119877119860119877119874119862 = 119864 (119903119901)119878119877119872(119903119901)


(11)



119868119879120579 = 1(12)











(16)




(17)












minus15

minus10

minus05

00

05

10

15

Qua

ntile

s of N

orm

al

minus15

minus10

minus05

00

05

10

15

Qua

ntile

s of N

orm

al

minus12

minus08

minus04

00

04

08

12

Qua

ntile

s of N

orm

al








4 Data Analysis







Table2Optim

alinvestm

entstrategyu

nder

confi

dencelevel

120572=005

120588 119888=0

02Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash07136

02538

-00903

01229

CVaR

mdash08062

01812

-00674

00799

Expo

nentialSRM

120574=02

08150

01743

-00652

00758

120574=04

08261

01657

-00625

00707

120574gt06

08400

01548

-00590

006

42

Powe

rSRM

120573=11

08177

01723

-0064

600746

120573=12

08342

01594

-0060

5006

69120573gt

1308400

01548

-00590

006

42120572=

01120588 c=

002Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash06357

03148

-010

9601591

CVaR

mdash07350

02370

-00850

0113

0

Expo

nentialSRM

120574=02

07393

02336

-00840

0111

0120574=

0407437

02302

-00829

01089

120574ge35

08335

01599

-0060

6006

73

Powe

rSRM

120573=11

07405

02327

-00837

0110

4120573=

1207457

02286

-00824

01080

120573ge3

08335

01599

-0060

6006

73Re

markthen

egativev

alue

means

short-s

ellin

g



5 Conclusions


Data Availability




Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









119864 (119877119901) = 120583119879119870 = 1205831198791198770(1 + g119873minus1sum119869=1

119896119895)120579

= 1198770(1 + g119873minus1sum119869=1

119896119895)119864 (119903119901)(8)

and 119878119877119872(119877119901) = minus119864 (119877119901) + 119879 (120572) 120590 (119877119901)= 1198770(1 + g

119873minus1sum119869=1

119896119895)119878119877119872(119903119901) (9)


119877119860119877119874119862 = 119864 (119877119901)119878119877119872(119877119901)


= 119864 (119903119901)119878119877119872(119903119901)

(10)


max 119877119860119877119874119862 = 119864 (119903119901)119878119877119872(119903119901)


(11)



119868119879120579 = 1(12)











(16)




(17)












minus15

minus10

minus05

00

05

10

15

Qua

ntile

s of N

orm

al

minus15

minus10

minus05

00

05

10

15

Qua

ntile

s of N

orm

al

minus12

minus08

minus04

00

04

08

12

Qua

ntile

s of N

orm

al








4 Data Analysis







Table2Optim

alinvestm

entstrategyu

nder

confi

dencelevel

120572=005

120588 119888=0

02Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash07136

02538

-00903

01229

CVaR

mdash08062

01812

-00674

00799

Expo

nentialSRM

120574=02

08150

01743

-00652

00758

120574=04

08261

01657

-00625

00707

120574gt06

08400

01548

-00590

006

42

Powe

rSRM

120573=11

08177

01723

-0064

600746

120573=12

08342

01594

-0060

5006

69120573gt

1308400

01548

-00590

006

42120572=

01120588 c=

002Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash06357

03148

-010

9601591

CVaR

mdash07350

02370

-00850

0113

0

Expo

nentialSRM

120574=02

07393

02336

-00840

0111

0120574=

0407437

02302

-00829

01089

120574ge35

08335

01599

-0060

6006

73

Powe

rSRM

120573=11

07405

02327

-00837

0110

4120573=

1207457

02286

-00824

01080

120573ge3

08335

01599

-0060

6006

73Re

markthen

egativev

alue

means

short-s

ellin

g



5 Conclusions


Data Availability




Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018











minus15

minus10

minus05

00

05

10

15

Qua

ntile

s of N

orm

al

minus15

minus10

minus05

00

05

10

15

Qua

ntile

s of N

orm

al

minus12

minus08

minus04

00

04

08

12

Qua

ntile

s of N

orm

al








4 Data Analysis







Table2Optim

alinvestm

entstrategyu

nder

confi

dencelevel

120572=005

120588 119888=0

02Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash07136

02538

-00903

01229

CVaR

mdash08062

01812

-00674

00799

Expo

nentialSRM

120574=02

08150

01743

-00652

00758

120574=04

08261

01657

-00625

00707

120574gt06

08400

01548

-00590

006

42

Powe

rSRM

120573=11

08177

01723

-0064

600746

120573=12

08342

01594

-0060

5006

69120573gt

1308400

01548

-00590

006

42120572=

01120588 c=

002Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash06357

03148

-010

9601591

CVaR

mdash07350

02370

-00850

0113

0

Expo

nentialSRM

120574=02

07393

02336

-00840

0111

0120574=

0407437

02302

-00829

01089

120574ge35

08335

01599

-0060

6006

73

Powe

rSRM

120573=11

07405

02327

-00837

0110

4120573=

1207457

02286

-00824

01080

120573ge3

08335

01599

-0060

6006

73Re

markthen

egativev

alue

means

short-s

ellin

g



5 Conclusions


Data Availability




Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018









Table2Optim

alinvestm

entstrategyu

nder

confi

dencelevel

120572=005

120588 119888=0

02Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash07136

02538

-00903

01229

CVaR

mdash08062

01812

-00674

00799

Expo

nentialSRM

120574=02

08150

01743

-00652

00758

120574=04

08261

01657

-00625

00707

120574gt06

08400

01548

-00590

006

42

Powe

rSRM

120573=11

08177

01723

-0064

600746

120573=12

08342

01594

-0060

5006

69120573gt

1308400

01548

-00590

006

42120572=

01120588 c=

002Risk

measure

Factor

ofris

kaversio

nRisk-fr

eeasset

JoinTo

Energy

(000

600)

Changrsquoa

nVe

hicle(00

0625)

Yili(600

887)

VaR

mdash06357

03148

-010

9601591

CVaR

mdash07350

02370

-00850

0113

0

Expo

nentialSRM

120574=02

07393

02336

-00840

0111

0120574=

0407437

02302

-00829

01089

120574ge35

08335

01599

-0060

6006

73

Powe

rSRM

120573=11

07405

02327

-00837

0110

4120573=

1207457

02286

-00824

01080

120573ge3

08335

01599

-0060

6006

73Re

markthen

egativev

alue

means

short-s

ellin

g



5 Conclusions


Data Availability




Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018










5 Conclusions


Data Availability




Acknowledgments


References





























Journal of












Journal of







Volume 2018






Volume 2018


















Journal of












Journal of







Volume 2018






Volume 2018








optimization problem of insurance investment based on...

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