stochastic control of funding systems

Insurance: Mathematics and Economics 30 (2002) 323–350

Stochastic control of funding systems

Greg Taylora,b,∗a Taylor Fry Consulting Actuaries, Level 4, 5 Elizabeth Street, Sydney, NSW 2000, Australia

b Centre for Actuarial Studies, Faculty of Economics and Commerce, University of Melbourne, Parkville, Vic. 3052, Australia

Received 1 September 2000; received in revised form 1 September 2001; accepted 14 February 2002

Abstract

This paper is concerned with funding systems, i.e. systems which accumulate funds for the future payment of financialobligations. Commonly, such funding requires a balance between (1) the desire to minimise the contributions that need to bediverted from other use to the support of the Fund, and (2) the need to maintain reasonable solvency in the Fund.

Such funding is discussed here in a general framework. Applications are numerous. The specific applications mentionedin the paper are:

• Defined benefit retirement funding,• Maintenance of a prudential margin by a non-life insurer,• Dividend payment strategy.

The paper applies stochastic optimal control theory to determine how rates of contribution to the Fund and allocation of itsassets by asset sector should respond to changing solvency. These results are obtainable from a particular differential equation,which may be solved numerically. Detailed numerical examples are provided.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Funding; Stochastic control; Dividend strategy

1. Introduction

Consider a financial system subject to stochastic obligations. These are to be funded by contributions whosemagnitude may vary from time to time in a manner which is, to some extent at least, within the discretion of thecontributor.

The contributions accumulate in a Fund of assets to the extent that they are surplus to the requirements tosettle obligations. The choice of assets for investment of the Fund, with particular reference to the riskiness ofthese assets, also lies within the discretion of the contributor, but the Fund is subject to solvency considerations ofsome sort.

The contributor naturally wishes to minimise contributions to the Fund in some long term sense, but subjectof course to solvency considerations. The magnitude of contributions required will be reduced as the investmentreturns of the Fund increase. This suggests a risky investment profile, but again this can be pursued only to theextent that solvency considerations permit.

∗ Present address: Taylor Fry Consulting Actuaries, Level 4, 5 Elizabeth Street, Sydney, NSW 2000, Australia.

0167-6687/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0167-6687(02)00107-5

324 G. Taylor / Insurance: Mathematics and Economics 30 (2002) 323–350

As a motivating example, consider an employer-sponsored defined benefit retirement fund. The Fund pays retire-ment and other benefits, and is supported by contributions, at a discretionary rate, by the employer. The Fund maybe invested in a discretionary manner, but is subject to a regulatory requirement that market value of assets shouldalways exceed vested benefits. The employer wishes to minimise contributions (in a sense still to be defined), butalso seeks stability in contribution rate.

The employer has two variables under his control at each point of time:

(a) the contribution rate,(b) the disposition of assets by sector.

What is required is an algorithm which selects the values of these variables at each point of time in a mannerwhich is in some sense “optimal”.

This retirement funding case is but one of many examples of the general framework described in the first part ofthis section. Other examples include:

(a) the strategy for payment of dividends (negative contributions) by an insurer to its shareholders;(b) the maintenance by a non-life insurer of a prudential margin in excess of the minimum statutory requirement.

There is a reasonably lengthy literature dealing with the retirement funding case. For example,O’Brien (1986,1987), Vandebroek (1990), Haberman and Sung (1994)andHaberman et al. (2000)all use optimal control theoryof one sort or another, sometimes stochastic, sometimes deterministic.

All of these references incorporate the notion of a target fund value, e.g. value of Fund equals present value offuture benefits. Some include the notion of a target contribution rate, such as a steady-state rate.

A different strand of the literature uses stochastic control theory.Taylor (1994)laid out a framework for this witha loss function expressed explicitly in terms of fund contribution rate and fund solvency. Similar frameworks occurin Boulier et al. (1995, 1996)andCairns (2000).

Taylor (1994)andCairns (2000)both include contribution rate and a solvency measure in their loss functions.Taylor and the two Boulier papers drop the concepts of contribution and solvency targets. This will be discussedfurther inSection 2.6.

The present paper seeks to combine the following characteristics:

• stochastic control theory,• both contribution rate and fund solvency manifest in the loss function,• both contribution rate and asset allocation as controls,• no targets for contribution and solvency.

This combination of characteristics does not seem to appear in the prior literature. Moreover, the form in whichsolvency appears in the loss function is somewhat more general here than previously.

A somewhat related series of papers byDufresne (1988, 1989), Haberman (1993, 1994), Gerrard and Haberman(1996)andOwardally and Haberman (1999)deals with the rate of funding a retirement fund, but not using controltheory.

2. Notation and terminology

2.1. General

The development of the optimal control response discussed inSection 1will be made in generality, free of anyof the specific contexts suggested there. Thus, there exists aFundsubject to inflows (contributions) and outflows(drawdowns) from time to time.

The Fund will holdassetsand may have contractual or semi-contractualliabilities. For example, in the retirementfunding case set out inSection 1, the liabilities would comprise the benefits (retirement and other) payable in

G. Taylor / Insurance: Mathematics and Economics 30 (2002) 323–350 325

future years. In this case, the Fund dynamics developed below are very similar to those inCairns (2000). For thisreason, some of the development is banished toAppendix A.

In the non-life prudential margin example, the Fund would consist of the insurer’s total assets, and liabilities allclaims incurred but not settled. The difference between the two would constitute the prudential margin.

2.2. Movements in the Fund

The progress of the Fund will be tracked incontinuous time. Let the Fund be subject to the following at timet(0 ≤ t < ∞), wheret = 0 denotes the present:

c(t) rate of contributionsper unit of the Fund’s liabilities,π(t) rate of drawdownper unit of the Fund’s liabilities,r(t) rate of investment returnon Fund assets.

Each of these quantities is anintensitythat operates over an infinitesimal interval. For example,c(t) dt denotesthe amount of contributions per unit of fund liabilities over the interval(t, t + dt).

Quantities with tildes over them are random variables, whereas those without tildes are deterministic. Assumethat

π(t)dt = π(t)dt + σπ dWπ(t), (2.1)

whereπ (t) is the expectation ofπ(t) andWπ (t) a standard Wiener process.Similarly, assume that

r(t)dt = r(t)dt + σr dWr(t). (2.2)

2.3. Accumulation of assets and liabilities

The Fund’s assets, denotedAt at timet, represent the excess of past contributions and investment returns overdrawdowns. LetLt denote Fund liabilities at timet, andλ(t) the rate of increase of liabilities before drawdowns.

This last type of change in liabilities may be due to recognition of newly accrued liabilities, or fluctuation in thevalue of old (e.g. as a result of inflation). Assume that

λ(t)dt = λ(t)dt + σλ dWλ(t), (2.3)

whereWλ(t) is the standard Wiener process. Assume thatWr , Wπ andW� are stochastically independent.

2.4. Asset allocation

Suppose that there aren + 1 asset sectors, labelled 0,1,2, . . . , n, with 0 denoting the risk-free sector, andndenoting the share market.

Let rk(t) be the rate of investment return in thekth asset sector.Denote

i(t) = [r0(t), r1(t), . . . , rn(t)]T (2.4)

and assume thati(t) is a diffusion process

i(t)dt = i(t)dt + H (t)dW i (t) (2.5)

with W i(t) ad-dimensional Wiener process(1 ≤ d ≤ n+ 1) andH (t) a matrix of dimension(n+ 1)× d such that

V [ i(t)dt ] = H (t)HT(t)dt. (2.6)

Here and throughout the paper vectors and matrices are represented by bold symbols.


Assume thatW i(t) is stochastically independent ofWπ (t) andWλ(t).Denote

C(t) = H (t)HT(t). (2.7)

Assume thatC(t) depends ont only up to a scalar multiplier, i.e. ratios of pairs of elements ofC(t) are independentof t.

Now impose a CAPM structure on (2.5)

rk(t) = E[rk(t)] + r0(t)βk∆, (2.8)

where

βk = ckn(t)

cnn(t), independent oft, (2.9)

and∆ > 0 is a constant.Putk = n in (2.8) and (2.9) to see that

share market rate of return= rn(t) = r0(t)+∆. (2.10)

Also, putk = 0 in (2.8) and (2.9) to see that

c0n(t) = 0. (2.11)

It is assumed that

c00(t) = 0

and hence

c0k(t) = ck0(t) = 0 for k = 0,1, . . . , n. (2.12)

Now (2.8) may be written in vector form

i(t) = [r0(t), r1(t), . . . , rn(t)]T = r0(t)1 + β∆, (2.13)

where1 is the(n+ 1)-vector with each component unity and

β = [β0,β1, . . . ,βn]T. (2.14)

Let pk(t) denote the proportion of assets invested in asset sectork at timet. Write

p(t) = [p0(t), p1(t), . . . , pn(t)]T, (2.15)

wheren∑

k=0

pk(t) = 1. (2.16)

Write p(t) in the form

p(t) =[p0(t)

p¯(t)

], (2.17)

wherep¯(t) is ann-vector.


Similarly, write

β =[

0

β¯

]. (2.18)

Also, letC¯(t) be then × n matrix obtained by deleting the first row and column (all zeros) ofC(t). Assume that

C¯(t) is of full rank (positive definite).

2.5. Vector process

Write

Xt =[At

Lt

]. (2.19)

The controls may be written in an(n+ 1)-vector

u(t) =[c(t)

p¯(t)

]. (2.20)

Then Appendix A.4 shows that

dXt = M(t)Xt dt + N(t)u(t)dt + G(t)dW (t), (2.21)

where

M(t) =[r0(t) −π(t)

0 λ(t)− π(t)

], (2.22)

N(t) =[Lt β

¯TAt∆

0 0

], (2.23)

G(t) =[

[p¯

T(t)C¯(t)p

¯(t)]1/2At −σπLt 0

0 −σπLt σλLt

], (2.24)

W (t) =

WR(t)

Wπ(t)

Wλ(t)

, (2.25)

whereWR(t) is a new standard Wiener process, and the three components ofW (t) are stochastically independent.

2.6. Loss function

Suppose that the Fund process is subjected to control over the time interval [0,T], 0 < T < ≡, with the objectiveof minimising some loss function.

Let V(s, x) denote the value of the loss function over the interval [s, T] whenXs = x. Suppose that it takes theform

V (s, x) = E

[∫ T

s

k(t,Xt , c(t,Xt ))dt +K(T ,XT )

], (2.26)


wherek andK are suitable real-valued and non-negative functions, and it is recognised explicitly thatc(t) is to beused as a control variable and therefore depends onXt .

A reasonable choice fork might be

k(t,Xt , c) = e−αt

[1

2c2(t)+ f

(At

Lt

)], (2.27)

wheref is a non-negative, strictly decreasing function, andα > 0 is a suitable discount rate.The choice ofK(·,·) will be left in abeyance untilSection 3.2.Note thatYt = At/Lt is asolvency ratio, a measure of the solvency of the Fund. It is desirable (all other things

equal) that this be high. Hence the decreasing loss associated with increasingYt by the functionf (·).Note also that the square-bracketed quantity in (2.27) balances the desire for high solvency against the high

contribution rates required to achieve it. Asc(t) increases,k(t,·,·) increases, butk(s,·,·) for s > t decreases becausethe higher value ofc(t) increases solvency at these later times,s.

The discount factor e−αt reflects the usual time weighting of the future, sometimes referred to in the literature asthepsychological discount factor.

Section 1mentioned that much of the existing literature is based on the concepts of target contribution rate andsolvency ratio. It also foreshadowed that these concepts wouldnot be used in the present paper. They are indeedseen to be absent from (2.27).

It is worth pausing briefly to consider this point. For comparison one might consider the loss function defined byCairns (2000)through his Eq. (11). Expressed in the present notation, this is

k(t,Xt , c) = e−αt {[c(t)− cm]2 + f [At − ap]2 + other terms}, (2.27a)

wherecm is the target contribution rate,ap the target solvency ratio andf ≥ 0 is a constant. There is nothing specialabout this example. It is simply chosen as representative of the literature involving these targets.

A loss function of the form (2.27a) will causec(t) andXt to regress to their target values over time. This may beregarded as “normal” behaviour of the system in some sense. However, the raw objectives of the Fund owner mayhave little to do with such behaviour, and may rather consist of the two mutually incompatible objectives of:

• keeping contribution rates (i.e. the costs borne by the owner) as low as possible,• keeping Fund solvency as high as possible.

These motivations are reflected in the alternative form of loss function (2.27). This loss function will also reflectthe compromise between the two motivations, this being represented by the general magnitude off (·). Scalingf (·)up increases the emphasis of solvency; scaling down increases the emphasis on contribution rates.

Although target contribution rates and solvency are not explicit in (2.27), any long term stable behaviour ofthe Fund must reflect these. Effectively, the tension between the two components of (2.27) creates this long termstability (with suitable choice of control parameters).

The Fund steady-state will be identified inSection 3.3and its role in Fund stability is illustrated in the numericalexamples ofSection 4.

Loss functions of a form like (2.27) are not new to the literature.Boulier et al. (1995)used

k(t,Xt , c) = e−αt c2(t) (2.27b)

with solvency subject to some constraints rather than used as a control variable.Cairns (2000)dealt with one example (his Eq. (68)) in which

k(t,Xt , c) = e−αt exp[γ c(t)− θAt ] (2.27c)

which, though not identical to (2.27a), is motivated in the same way.


The generality off (·) renders (2.27) more general than loss functions that have appeared in earlier literature. Itis difficult to retain complete generality in the solution of the above control problem. However,Section 3retains areasonable degree of generality (see (3.11)).

The solvency value functionf has already been assigned the properties

f ≥ 0, f ′ < 0 (2.28)

and would usually have the additional property

f ′′ > 0 (2.29)

which represents a diminishing marginal value of increased solvency.

3. Control problem and its solution

3.1. Control problem

The problem to be solved can be stated as follows:

Problem. Find the control functionsc(t,Xt ) andp(t,Xt ) that minimise loss function (2.26) when the system’sevolution is described by (2.19)–(2.25), supplemented by (2.16).

3.2. Solution

The solution to the control problem is obtained inAppendix C, where it is assumed (see (C.11)) that the functionV(·,·) factorises as follows:

V (s,Xs) = e−αs U(Ys). (3.1)

Comparison of this with (2.26) indicates that the functionK(·,·) also factorises

K(T ,XT ) = e−αT U(YT ). (3.2)

Thus, assumption (3.1) means that the choice ofK(·,·) is not free. The choice (3.2) is reasonable, however, and willbe adopted here. Then the solution to the control problem is found in (C.17)

u(y) = u∗(y) =[c∗(y)p¯∗(y)

]=[ −Uy

−y−1UyU−1yy (C

¯−1β

¯)∆

], (3.3)

wherey = Yt andU(y) is the solution of the ordinary differential equation

αU + a(y)Uy + b(y)Uyy + γ

(U2y

Uyy

)+ U2

y − f = 0 (3.4)

with Uy = dU/dy,Uyy = d2U/dy2 and

a(y) = (π + σ 2π )− (ν + σ 2)y, (3.5)

b(y) = −12σ

2π + σ 2

πy − 12σ

2y2, (3.6)

ν = r0 + π − λ, (3.7)


σ 2 = σ 2π + σ 2

λ , (3.8)

γ = ∆2(β¯

TC¯

−1β¯). (3.9)

By (3.3), the controlp∗ may be expressed in terms ofc∗

p¯∗(y) = y−1

[d

dylogc∗(y)

]−1

(C¯

−1β¯)∆. (3.10)

The governing equation (3.4) is straightforward, and may be solved by finite difference methods.Appendix Econsiders the case in whichf (·) can be expanded as follows:

f (y) =∞∑j=1

fjy−j . (3.11)

It is shown that (3.4) is solved by

U(y) =∞∑j=1

ujy−j , (3.12)

where the coefficientsuj are obtainable from recursion (E.11).The precise details of (3.12) are of little interest as it is of limited practical use. Asymptotic results apply to

unrealistically large values ofy, and the series converges rather slowly.The results which will be useful for computational purposes are that

U(y) ∼ u1

yfor largey (3.13)

Uy(y) ∼ −u1

y2for largey (3.14)

with

u1 = f1

α + ν + (1/2)θ∆2, (3.15)

θ = β¯

TC¯

−1β¯. (3.16)

3.3. Steady-state

In any application of solution (3.3) forc∗(y), it will be of interest to compare the result with the steady-statecontribution rate. The steady-state can be defined as that in which

E[Xt+dt |Xt ] ∝ Xt , (3.17)

i.e.

dXt = qXt dt (3.18)

for some constantq > 0.By (2.21), this is the same as

[M(t)− q1]Xt + N(t)u(t) = 0, (3.19)

where1 is the identity matrix of same dimension asM(t).


With Xt , u(t), M(t), N (t) given by (2.19) and (2.20), (2.22) and (2.23), the solution of (3.19) is straightforward

q = λ(t)− π(t), (3.20)

[r0(t)+ π(t)− λ(t)+ β¯

Tp(t)∆]At = [π(t)− c(t)]Lt , (3.21)

i.e.

c = π − (r + π − λ)Y, (3.22)

where the time argument has been suppressed, (A.13) used to definer, andc written for the steady-state value ofc.This result shows that, for any particular solvency ratioY, there is a corresponding steady-state contribution ratec.

4. Numerical examples

4.1. Employee benefit funding

Consider the case of an employee benefit fund with two types of asset available:

• short risk-free debt (known returnr0),• share market index (β = 1, expected returnr0 +∆).

Suppose that the fund is subject to the basic parameters set out inTable 4.1. These lead to the derived parametersappearing inTable 4.2. The only parameter calling for specific comment is the coefficient involved inf (·).

Table 4.1Fund parameters

Parameter Value

Liabilitiesπ 6%σ 2π 0.0010λ 9%σ 2λ 0.0005

Assetsr0 5%∆ 5%C¯

(20%)2

Controlα 15%f(y) 0.002/y

Table 4.2Derived parameters

Parameter Value

ν 2%σ 2 0.0015γ 0.0625θ 25


Fig. 4.1. Optimal controls.

Typical values ofy might lie between 30 and 70%. These values ofy maintain reasonable balance between thevalue of assets and the actuarial value (value of accrued liabilities less value of future contributions for existing fundmembers).

These values ofyyield values off (y) between about 0.003 and 0.007. The corresponding steady-state contributionrates, given by (3.22), lie between 3.0 and 5.2%. Hence, the values of (1/2)c2 entering the loss function (2.27) liein the range 0.0004–0.0013.

Thus, the choice of coefficient inf (·) renders the two contributions to (2.27) of similar orders of magnitude (oneof them between 1 and 10 times the other). The procedure for numerical solution of (3.4) is described inAppendixF. Fig. 4.1displays the results of those calculations. The functionsc∗(·) andp

¯∗(·) are plotted on the left and right

axes, respectively. The steady-state contribution ratec(·) is plotted against the left axis.Note that both contribution rate and solvency ratio are expressed in terms of liabilities. The fact that contribution

rate is expressed this way, rather than in terms of salary roll, gives it rather lower values than one is accustomed tosee.

In the formulation of the control problem used here,p¯∗(y) has been left unconstrained. Hence, it takes values in

excess of 100% (i.e. the asset portfolio becomes leveraged) for the larger values ofy (>127%).There are twopoints of equilibriumwith respect to contribution rate, vizy = 39%,c(y) = 4.83% andy = 88%,

c(y) = 1.66%. These are equilibrium points in the sense thatc(y) = c(y).Note that the first is astablepoint of equilibrium in that for initial values in the vicinity ofy = 40%, the system

will tend to evolve towards, and remain close to, this point. For example, an upward perturbation of the solvencyratio will force the optimal contribution rate below its steady-state value, causing the solvency ratio to return towardsits equilibrium value.

Though the system always tends to evolve towards this equilibrium, it will not remain precisely there because ofstochastic variation in the system.


Fig. 4.2. Effect of varying solvency loss function on contribution rates.

The casey = 80%, on the other hand, is anunstablepoint of equilibrium. Initial values ofy < 80% will tend toevolve downward, while values ofy > 80% will tend to evolve upward (in fact, without limit).

Figs. 4.2 and 4.3illustrate the effect of varying the solvency componentf (y) of the loss function. In the aboveexample,f (y) = f1/y, with f1 = 0.0020. The figures illustrate alternative values of 0.0005, 0.0010, 0.0025 and0.0050.

In Fig. 4.2, contribution rates increase asf1 increases, reflecting the increasing concern with solvency. InFig. 4.3,the risky asset proportionincreaseswith increasingf1. One might have expected increasing concern with solvency toyield the opposite investment strategy. It should be remembered, however, that the increase in risky asset proportionis also accompanied by increase in contribution rates (Fig. 4.2).

Figs. 4.4 and 4.5illustrate the effects on contribution rate and asset allocation, respectively, of variations in otherparameters. In each case, one parameter is varied from the example illustrated inFig. 4.1. The variations (compareTable 4.1) are:

• steeperf (·)• r0 = 3%• As in Table 4.1(no variation)• C

¯= (15%)2

• α = 20%

They appear in this descending order at the left side ofFig. 4.4. In Fig. 4.5they appear in the following descendingorder on the right side:

• C¯

= (15%)2

• As in Table 4.1(no variation)• r0 = 3%


Fig. 4.3. Effect of varying solvency loss function on asset allocation.

Fig. 4.4. Contribution rates: variation in parameters.


Fig. 4.5. Asset allocation: variation in parameters.

• steeperf (·)• α = 20%

The “steeperf (·)” in Figs. 4.4 and 4.5is

f (y) = 0.00175

y+ 0.0001

y2(4.1)

which

• has a steeper gradient thanf (y) = 0.002/y for y < 80%;• gives the same value at the stable equilibrium point [f (39%) = 0.0051] of the earlierf (·).

The steeper loss function is seen inFig. 4.4to produce a steeper responsec∗(·). Other features ofFig. 4.4are asexpected. The required contribution rates are:

• increased in response to lower expected asset returns,• decreased in response to lower share market volatility,• decreased in response to a higher discount factorα.

The most obvious feature ofFig. 4.5is the sharp increase in optimal risky asset proportion in response to lowervolatility in that sector. In general, however, the optimal values of the risky asset proportion, taken over the 30–70%range of solvency ratio, are considerably below those observed in practice, which might be of the order of 60%.This might be due to the choice of loss function, particularly its unboundedness in the vicinity of zero solvency.


Fig. 4.6. Simulated solvency ratios.

Alternatively, it might be partly due to the definition of liabilities. The volume of liabilities used in this numericalexample is representative of accrued liabilitieswithout any deduction in respect of future contributions. If thisdeduction were made, the volume of liabilities would change substantially, with consequential effects on otheraspects of the example.

Fig. 4.6displays simulated paths of the solvency ratioYt generated by the parameters set out inTable 4.1. Thereare eight paths

• four commencing atY0 = 75%,• four commencing atY0 = 25%.

The general tendency towards the stable equilibrium pointy = 39% can be observed.Fig. 4.7reproduces oneof the solvency ratio paths fromFig. 4.6, and superimposes the antithetic movement in contribution rate.Fig. 4.8superimposes asset allocation.

4.2. Dividend equalisation

Suppose an insurer is in a stationary state in the sense that expected assets, liabilities and annual profit areconstant over time. Expected annual profit is 21

2% of liabilities before allowance for investment earnings on surplus(net assets). The insurer underwrites only short tail business, and so losses will be assumed paid instantly. Theytherefore have no bearing on the accumulation of liabilities.

The insurer operates a dividend equalisation fund (DEF) which can supplement dividend when profit is low.When profits are sufficient, part may be paid as a contribution to the Fund, reducing dividend to less than profit.

Since the insurer is in a stationary state (apart from the DEF), surplus will consist of a constant amount(in expectation at least) plus the DEF. Management of the latter is equivalent to management of total surplus.


Fig. 4.7. Simulated solvency ratio and contribution rate.

Fig. 4.8. Simulated solvency ratio and asset allocation.


There is a prior actuarial literature on optimal dividend strategy, e.g.Bühlmann (1970), Pechlivanides (1978),Waldmann (1988), all of which may be traced ultimately tode Finetti (1957). The approaches taken there are ratherdifferent from the DEF approach described in the present section. All of the given references are concerned withthe maximisation of the expected utility of future dividends. Often the utility is taken as the discounted sum.

For the approach taken here, the evolution of surplus may be described by (A.1) and (A.2) with the appropriateinterpretation. In fact, the second of these will be modified slightly, so that the system develops according to

dAt = r(t)At dt + [c(t)− π(t)]Lt dt (4.2)

dLt = ξ (t)Lt dt, (4.3)

whereAt is the surplus,Lt the total liabilities of the insurer,c(t) the contribution to DEF,π(t) the drawdown on DEFwhich is equal to the budgeted dividend (21

2% of liabilities) less profit andξ(t) the rate of variation of liabilities.Because of the insurer’s steady-state

E[π(t)] = E[ξ (t)] = 0. (4.4)

Thus (4.13) becomes (cf. (2.1))

dLt = σξLt dWξ(t) (4.5)

with Wξ (t) a standard Weiner process independent ofWπ (t).Within this framework,c(t) takes both positive and negative values. In this case, the loss function (2.27) is not

monotone increasing inc(t) as is required from it. Therefore, define the shifted contribution rate

ch(t) = c(t)+ h (4.6)

for some constanth > 0.Also define

πh(t) = π(t)+ h. (4.7)

Note that the asset process (4.2) may be written in the alternative form

dAt = r(t)At dt + [ch(t)− πh(t)]Lt dt. (4.8)

The process has not changed, but ifch(t) replacesc(t) in the loss function, the latter becomes monotone inch(t)(and hence inc(t)) provided thatch(t) > 0, i.e.c(t) > −h.

The processes (4.8) and (4.5) is now optimised with respect toch(t). The former of these takes the same formas (A.1), but the latter is slightly different from (A.4), and the development of control equation (3.4) changes as aresult.

The details of this appear inAppendix G. The result is that (3.4) continues to hold subject to (3.7)–(3.9), but withc andπ replaced bych andπh, and (3.5) and (3.6) modified as follows:

a(y) = πh − (νh + σ 2ξ )y, (4.9)

b(y) = −12σ

2π − 1

2σ2ξ y

2, (4.10)

νh = r0 + πh. (4.11)

Suppose the relevant parameters are as set out inTable 4.3.The steady-state equation corresponding to (3.22) may be derived simply from (4.2) and (4.3) as:

c = π − rY, (4.12)

c = −rY (4.13)

for π = 0 as inTable 4.3.


Table 4.3Fund parameters

Parameter Value

Liabilitiesπ 0%σ 2π (5%)2

σ 2ξ (1%)2

Assetsr0 5%∆ 5%C¯

(20%)2

Controlα 15%f (y) 0.0050/y

Numerical solution of the control equation (3.4) withh = 10% yields the optimal controls illustrated inFig. 4.9.Stable equilibrium occurs aty = 48%.Figs. 4.10 and 4.11show simulated paths of contribution rate and risky

asset proportion, respectively, with an initial solvency ratio of 30%.From the definition ofπ(t)

profit = 212% − π(t). (4.14)

Then dividend paid is

dividend= profit − c(t). (4.15)

Fig. 4.12plots simulated tracks of profit and dividend. The smoothing effect of the DEF can be seen. Note thatdividend generally exceeds 21

2% because of investment earnings of the DEF.Fig. 4.13illustrates the fact that thevariation in dividend apparent inFig. 4.12represents a response to the insurer’s fluctuating solvency.

Fig. 4.9. Optimal controls.


Fig. 4.10. Simulated solvency ratio and contribution rate.

Fig. 4.11. Simulated solvency ratio and asset allocation.


Fig. 4.12. Profit and dividend.

Fig. 4.13. Solvency ratio and dividend.

Appendix A. Fund dynamics

A.1. Accumulation of assets and liabilities

With the notation ofSection 2, dynamics of asset and liability accumulations are as follows:

dAt = r(t)At dt + [c(t)− π(t)]Lt dt, (A.1)

dLt = [λ(t)− π(t)]Lt dt. (A.2)

Eqs. (2.1)–(2.3)may be used to re-express (A.1) and (A.2) as follows:

dAt = r(t)At dt + [c(t)− π(t)]Lt dt + σrAt dWr(t)− σπLt dWπ(t), (A.3)


dLt = [λ(t)− π(t)]Lt dt + σλLt dWλ(t)− σπLt dWπ(t). (A.4)

A.2. Vector process

The two dynamic equations (A.3) and (A.4) can conveniently be abbreviated to vector form by writing

Xt =[At

Lt

]. (A.5)

Then

dXt = M(t)Xt dt + c(t)N(t)dt + G(t)dW (t), (A.6)

where

M(t) =[r(t) −π(t)

0 λ(t)− π(t)

], (A.7)

N(t) =[Lt

0

], (A.8)

G(t) =[σrAt −σπLt 0

0 −σπLt σλLt

], (A.9)

W (t) =

Wr(t)

Wπ(t)

Wλ(t)

. (A.10)

A.3. Asset allocation

With p(t) representing the disposition of assets as defined in (2.15)

r(t)dt = pT(t)i(t)dt,

= pT(t) [i(t)dt + H (t)dW i (t)] [by (2.5)],

= pT(t)[r0(t)1 dt + β∆dt + H (t)dW i (t)] [by (2.13)],

(A.11)

= [r0(t)+ pT(t)β∆] dt + pT(t)H (t)dW i (t) (A.12)

by (2.16).Compare (2.2) with (A.12) to find that

r(t) = r0(t)+ pT(t)β∆. (A.13)

The same two relations yield, respectively

V [r(t)dt ] = σ 2r dt, (A.14)

V [r(t)dt ] = pT(t)H (t)HT(t)p(t)dt = pT(t)C(t)p(t)dt (A.15)

by (2.7). By (A.14) and (A.15)

σ 2r = pT(t)C(t)p(t). (A.16)


Because of (2.16), there are onlyn degrees of freedom in the choice ofp(t). It will prove useful to recognise thisby writing p(t) in the form:

p(t) =[p0(t)

p¯(t)

], (A.17)

wherep¯(t) is ann-vector.

Similarly, write

β =[

0

β¯

]. (A.18)

Substitute (A.17) and (A.18) in (A.13) to obtain

r(t) = r0(t)+ p¯

T(t)β¯∆. (A.19)

By (2.12), (A.16) becomes

σ 2r = p

¯T(t)C

¯(t)p

¯(t), (A.20)

whereC¯(t) is then× n matrix obtained by deleting the first row and column (all zeros) ofC(t). Assume thatC

¯(t)

is of full rank (positive definite).

A.4. Extended vector process

The vector process (A.6) is defined in terms ofc(·), which will be regarded as a control variable. In the contextof Section 2.4, the vectorp

¯(·) is also a control variable. It affects (A.6) through (A.7), (A.9), (A.19) and (A.20).

It will be convenient to place all controls in a single(n+ 1)-vector

u(t) =[c(t)

p¯(t)

]. (A.21)

Then the vector process (A.6) may be expressed in the same form, but with the control vectoru(t) in place ofc(t)

dXt = M(t)Xt dt + N(t)u(t)dt + G(t)dW (t), (A.22)

where

M(t) =[r0(t) −π(t)

0 λ(t)− π(t)

], (A.23)

N(t) =[Lt β

¯TAt∆

0 0

](A.24)

G(t) =[

[p¯

T(t)C¯(t)p

¯(t)]1/2At −σπLt 0

0 −σπLt σλLt

], (A.25)

W (t) =

WR(t)

Wπ(t)

Wλ(t)

, (A.26)

whereWR(t) is a new standard Wiener process, and the three components ofW (t) are stochastically independent.


Appendix B. Bellman’s optimality principle

It is required to minimise the loss function (2.26) by means of control vectoru(t), with Xt given by (2.21).According to Bellman’s optimality principle (see e.g.Arnold (1974, pp. 211–214)), this is done by setting

minu

[DV (s, x)+ k(s, x, c)] = 0, 0 ≤ s ≤ T (B.1)

with end condition

V (T , x) = K(T , x) (B.2)

and whereD is thetotal derivativeoperator

D = ∂

∂s+ [M(t)x + N(t)u(t)]T

∂

∂x+ 1

2Tr

[G(t)GT(t)

∂2

∂x2

], (B.3)

where∂2/∂x2 is 2× 2 matrix of operators.

Appendix C. Optimal control

The square-bracketed term in (B.1) is

Vs + xTMTVx + uTNTVx + 12Tr(GGTVxx)+ k, (C.1)

where arguments of the functions appearing here are temporarily suppressed for conciseness, and a subscriptrepresents partial differentiation with respect to the variable appearing as the subscript.

Recall from (B.1) that this last expression is to be minimised with respect tou, and so consider the terms whichdepend onu. Also, by (2.27), the termk includes the quantity

e−αs uT[

12 00 0

]u. (C.2)

Finally, (2.24) shows that the second last member of (C.1) depends onu throughG. Specifically,

12Tr(GGTVxx) = 1

2p¯

TC¯p¯A2VAA + other terms independent ofu. (C.3)

The total of these three contributions to (C.1) is

uTNTVx + 12uTDu, (C.4)

where

D =[

e−αs 00 A2VAAC

¯

]. (C.5)

Now (C.4) is to be minimised with respect tou. The value ofu at which the minimum is attained is

u = u∗ = −D−1NTVx. (C.6)

The minimum value of (C.4) is

−12V

Tx ND−1NTVx. (C.7)

When this is substituted in (C.1), the following version of (B.1) is obtained

Vs + xTMTVx − 12V

Tx ND−1NTVx + 1

2L2Tr(G∗G∗TVxx)+ e−αsf = 0, (C.8)


whereG∗ is G with the effect ofu removed, since this effect is now included in the third member of (C.8), and withthe factorLt taken outside, i.e.

G∗(t) =[

0 −σπ 00 −σπ σλ

]. (C.9)

By (2.23) and (C.5)

ND−1NT = [eαsL2 +∆2V −1AA β

¯TC

¯−1β

¯]

[1 00 0

]. (C.10)

To this point, the representation (2.19) ofXt has been used, i.e. in terms ofAt andLt . All other quantities have alsobeen represented this way.

It will be beneficial, in fact, to express all these quantities in terms of the solvency ratioAt /Lt . This is done inAppendix D.

It is assumed that

V (s,Xs) = e−αs U(Ys) (C.11)

for some functionU(·). The restriction implied by (C.11) for the functionK(·,·) introduced in (2.26) is discussed inSection 3.2.

Apply the substitutions (D.3)–(D.8) in (C.8) to obtain

−αU + [(r0 + π − λ)y − π ]Uy − [1 +∆2(β¯

TC¯

−1β¯)U−1

yy ]U2y + 1

2[(y − 1)2σ 2π + y2σ 2

λ ]Uyy

+[(y − 1)σ 2π + yσ 2

λ ]Uy + f = 0, (C.12)

i.e.


(U2y

Uyy

)+ (Uy)

2 − f = 0 (C.13)

with

a(y) = (π + σ 2π )− (r0 + π − λ+ σ 2

π + σ 2λ )y, (C.14)

b(y) = −12σ

2π + σ 2

πy − 12(σ

2π + σ 2

λ )y2, (C.15)

γ = ∆2(β¯

TC¯

−1β¯). (C.16)

Note that the result (C.13), from which dependency ons has vanished, confirms the choice of solution (C.11).It is now possible to calculate the optimal control (C.6) in terms ofU(·). Substitute (C.5), (D.7), (2.23) and (D.4)

into (C.6) to obtain

u∗ =[ −Uy

−y−1UyU−1yy (C

¯−1β

¯)∆

]. (C.17)

Appendix D. Expression of quantities in terms of solvency ratio

Define thesolvency ratio

Yt = At

Lt

. (D.1)

Suppose thatV (s,Xs) = e−αs U(Ys) (D.2)


for some functionU(·). Then, forYs = y,

Vs = −α e−αs U(y) (D.3)

V x = Vyyx = e−αs UyL−1[

1−y

], (D.4)

V xx = V yxyTx + Vyyxx = Vyyyxy

Tx + Vyyxx = e−αsL−2[UyyP (y)+ UyQ(y)], (D.5)

where

P (y) =[

1 −y

−y y2

], Q(y) =

[0 −1

−1 2y

]. (D.6)

Note that

VAA = e−αs L−2Uyy, (D.7)

L2Tr(G∗G∗TV xx) = e−αs{Uyy[(y − 1)2σ 2π + y2σ 2

λ ] + 2Uy [(y − 1)σ 2π + yσ 2

λ ]}. (D.8)

Appendix E. Solution of differential equation (C.13)

Eq. (C.13)is repeated here for convenience


(U2y

Uyy

)+ (Uy)

2 − f = 0.

Write (C.14) and (C.15) in the forms

a(y) = a0 + a1y, (E.1)

b(y) = b0 + b1y + b2y2. (E.2)

Try a solution to (C.13) of the form

U(y) =∞∑j=1

ujy−j . (E.3)

In this case,

Uy = −∞∑j=1

juj y−j−1, (E.4)

Uyy =∞∑j=1

j (j + 1)ujy−j−2. (E.5)

Also let

U2y

Uyy=

∞∑j=1

gjy−j , (E.6)

U2y =

∞∑j=4

hjy−j , (E.7)


f (y) =∞∑j=1

fjy−j . (E.8)

Substitute (E.1)–(E.8) in (C.13) to obtain

∞∑j=4

y−j {uj [α − a1j + b2j (j + 1)] + uj−1[−a0(j − 1)+ b1j (j − 1)]

+ uj−2b0(j − 1)(j − 2)+ γgj + hj − fj } + similar terms forj = 1,2,3 = 0. (E.9)

The “similar terms” may in fact be obtained by simply settingj = 1,2,3 in {. . .} and adopting the convention that

u0 = u−1 = u−2 = · · · = 0. (E.10)

The solution of (E.9) is obtained by setting the coefficient of each powery−j to 0. The solution is therefore

uj [α − a1j + b2j (j+1)]+uj−1[−a0(j − 1)+ b1j (j − 1)]+uj−2b0(j − 1)(j − 2)+ γgj + hj − fj = 0,

j = 1,2, . . . (E.11)

subject to (E.10).Values ofu1, u2, . . . , may be calculated recursively from (E.11), but first it is necessary to expressgj andhj in

terms of theu coefficients.Consider thegj . The power series forU2y /Uyy is obtained from (E.4) and (E.5)

U2y

Uyy= y−4[u1 + 2u2y

−1 + 3u3y−2 + · · · ]2

y3

2u1

[1 + 3u2

u1y−1 + 6u3

u1y−2 + · · ·

]−1

= 1

2u1y

−1 + 1

2u2y

−2 + 1

2

(u2

2

u1

)y−3 + · · · . (E.12)

Thus

g1 = 1

2u1, g2 = 1

2u2, g3 = 1

2

u22

u1. (E.13)

Similarly,

h1 = h2 = h3 = 0, h4 = u21, h5 = 4u1u2, h6 = 4u2

2 + 6u1u3. (E.14)

Eq. (E.11), with substitutions (E.13) and (E.14), yields the following:

j = 1 : u1(α − a1 + 2b2 + 12γ )− f1 = 0, (E.15)

j = 2 : u2(α − 2a1 + 6b2 + 12γ )+ u1(−a0 + 2b1)− f2 = 0, (E.16)

j = 3 : u3(α − 3a1 + 12b2)+ u2(−2a0 + 6b1)+ 2b0u1 + 1

2γu2

2

u1− f3 = 0, j = 4, etc. (E.17)

Recursive solution of (E.15) and (E.16) in that order yields

u1 = f1

α − a1 + 2b2 + 12γ

(E.18)

u2 = f2

α − 2a1 + 6b2 + 12γ

+ f1(a0 − 2b1)

(α − a1 + 2b2 + 12γ )(α − 2a1 + 6b2 + 1

2γ ). (E.19)

The solution foru3 is similarly obtained from (E.17) using the values ofu1 andu2 from (E.18) and (E.19).


The values of the coefficientsaj andbj are obtained by comparing (E.1) and (E.2) with (C.14) and (C.15)

a0 = π + σ 2π , a1 = −(r0 + π − λ+ σ 2

π + σ 2λ ), (E.20)

b0 = −12σ

2π , b1 = σ 2

π , b2 = −12(σ

2π + σ 2

λ ). (E.21)

Substitution of (E.20) and (E.21) into (E.18) and (E.19) yields

u1 = f1

α + ν + 12θ∆

2, (E.22)

u2 = f2

α + 2ν − σ 2 + 12θ∆

2+ f1(π − σ 2

π )

(α + ν + 12θ∆

2)(α + 2ν − σ 2 + 12θ∆

2), (E.23)

where

ν = r0 + π − λ, (E.24)

σ 2 = σ 2π + σ 2

λ , (E.25)

θ = β¯

TC¯

−1β¯. (E.26)

Appendix F. Numerical solution of Eq. (3.4)

Eq. (3.4)is solved numerically as follows. Convert it to its finite analogue

αU + aδU + bδ2U + γ (δU)2

δ2U+ (δU)2 − f = 0 (F.1)

equivalently

b(δ2U)2 + [αU + aδU + (δU)2 − f ](δ2U)+ γ (δU)2 = 0, (F.2)

whereδ denotes the finite difference operator with step sized

δU(y) = [U(y + d)− U(y)]

d. (F.3)

Eq. (F.2)is a quadratic inδ2U and so may be solved in terms ofU andδU. It may be shown that the solution alwaysexists. Note thatγ > 0 by the positive definiteness ofC

¯. Also, by (3.6) and (3.8)

b(y) = −12σ

2π (y − 1)2 − σ 2

λ y2 < 0. (F.4)

The opposite signs ofγ andb are sufficient to guarantee a positive discriminant in (F.2).Choose a large valueyL of y (yL = 5 in the present example) and, on the basis of (3.13) and (3.14), assume that

U(yL) = u1

yL, (F.5)

δU(yL − d) = −u1

y2L

, (F.6)

with u1 a parameter yet to be determined.Solve (F.2) for δ2U and take this to be the value ofδ2U(yL − 2d). There are now values ofU(yL),

δU(yL − d)andδ2U(yL − 2d). Application of (F.3) yieldsδU(yL − 2d) andU(yL − d), so (F.2) may be re-appliedwith yL replaced byyL − d. This yields a solution forδ2U(yL − 3d).


In this manner values are obtained forU(yL − jd), j = 0,1, etc., ending atU(d). These values depend on thevalue ofu1 used to initialise the computations in (F.5) and (F.6).

To select this value note that, by (3.12)

U(y) → ∞ asy ↓ 0. (F.7)

Therefore, selectu1 by trial in such a way thatU(d) is finite but large. Guidance in this choice is provided by (3.15).Finally, the solution to the control problem is taken as the finite form of (3.3):

c∗(y) = −δU(y − d), (F.8)

p¯∗(y) = −1.25δU(y − d)

yδ2U(y − 2d), (F.9)

where 1.25 is the value of(C¯

−1β¯)∆.

Appendix G. Control of DEF

The dynamics of the Fund are given by (4.8) and (4.5). Then (A.6) continues to hold but with (A.9) replaced by:

M(t) =[r0(t) −π(t)

0 ξ(t)

], G(t) =

[σrAt −σπLt 0

0 0 σξLt

]. (G.1)

It may then be checked that, in (D.8),

G∗G∗TP (y) =[

σ 2π −yσ 2

π

−yσ 2ξ y2σ 2

ξ

], (G.2)

G∗G∗TQ(y) =[

0 −σ 2π

−σ 2ξ 2yσ 2

ξ

], (G.3)

leading to the replacement of (D.8) by the following:

L2Tr(G∗G∗TV xx) = e−αs{Uyy[σ2π + y2σ 2

ξ ] + 2Uyyσ2ξ }. (G.4)

This leads once again to (3.4) but withc andπ replaced bych andπh, and the definitions ofa(y) andb(y) in (3.5)and (3.6) modified as follows:

a(y) = πh − (νh + σ 2ξ )y, (G.5)

b(y) = −12σ

2π − 1

2σ2ξ y

2, (G.6)

νh = r0 + πh. (G.7)

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