Lifetime investment and consumption using a defined-contribution pension scheme

Journal of Economic Dynamics & Control 36 (2012) 1303–1321

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Lifetime investment and consumption using a defined-contribution pension scheme Paul Emms Department of Mathematics, King’s College London, Strand, London, United Kingdom

a r t i c l e i n f o

abstract

Article history: Received 9 April 2011 Received in revised form 1 November 2011 Accepted 31 December 2011 Available online 7 March 2012

During the accumulation phase of a defined-contribution pension scheme, a scheme member invests part of their stochastic income in a portfolio of a stock and a bond in order to build up sufficient funds for retirement. It is assumed that the remainder of their salary pre-retirement is consumed, an annuity is purchased at retirement, and the stock allocation and consumption pre-retirement maximise the total expected lifetime consumption using a CARA utility function. Perfect correlation between the scheme member’s income and the stock price leads to analytical expressions for the controls for a general income model. If the correlation is imperfect then analytical controls are found for two particular stochastic income models. & 2012 Published by Elsevier B.V.

JEL classification: C61 D31 D52 D91 Keywords: Defined-contribution pension Lifecycle model Stochastic income Replacement ratio

1. Introduction Increased longevity and lower birth rates in many countries have led to an ongoing debate concerning how an individual should save for their retirement (Turner et al., 2006). Recent legislation in the U.K. seeks to encourage individual responsibility through work-related saving plans. Many occupational pension plans are now defined-contribution (DC) schemes, where the scheme member (and often their employer) makes regular tax-free contributions from their salary to an invested savings fund. At retirement, the DC scheme member receives a taxed income from their fund often with restrictions on withdrawal. The characteristic of this scheme is that the financial risk is borne by the scheme member, and so essentially the DC scheme is a deferred investment with tax advantages. The relationship between the consumption of an individual, their income, and the difference between these two quantities, their savings, has occupied economists since Keynes (1936). The Permanent Income Hypothesis was formulated by Friedman (1957) in discrete time, and relates the consumption of an individual to their future (expected) income. This has become known as the lifecycle model in the Economics literature (Campbell and Viceira, 2001). One of the main tools used to tackle the strategic problem of lifetime saving is stochastic optimal control theory. Using this theory, Samuelson (1969) considers the lifetime consumption problem for an individual who can invest their wealth

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in a stock, but there is no explicit income process. Merton (1971) considers the continuous time investment and consumption problem, and also considers an explicit stochastic income process. There are many papers in the Economics and Finance literature that build on these ideas. One way to partition the body of knowledge is into ‘‘large’’ models that solve the lifetime consumption problem through simulation or approximation, and ‘‘small’’ models for which closed-form optimal investment and consumption strategies are available. The work of Campbell and Viceira (1999), Cocco et al. (2005), Kahl et al. (2003), and Zeldes (1989) belong to the first group while Henderson (2005), Tepla´ (2000), Choi and Shim (2006), and Svensson and Werner (1993) are representative of the second. An occupational pension is one means of lifetime saving so the Economics and Actuarial literature on this subject are intertwined. We refer the reader to Blake et al. (2001, 2003) for an extensive set of references on pension modelling. This paper focuses specifically on a DC pension as an instrument for lifetime saving, and it is most closely related to the papers by Henderson (2005), Cairns et al. (2006), and Koo (1998). Henderson (2005) considers a stochastic stock and income model, and maximises the expected terminal wealth of an individual. She finds analytical expressions for the optimal investment strategy for particular income models. Cairns et al. (2006) also maximise the expected terminal wealth of an individual when there are fixed contributions to a pension scheme. They find an analytical expression for the optimal investment strategy when there is no correlation between the income and the stock. Koo (1998) considers the properties of the optimal strategies in a stochastic income model, but closed-form expressions for these strategies are not available. In contrast to these papers, we find analytical results for optimal controls: the optimal investment of a DC pension fund and the optimal contribution to that fund. We consider a DC pension scheme member who consumes an income, contributes to a pension fund, chooses how to invest the pension fund, and then annuitises their accumulated pension fund at retirement. Consequently, we consider the optimal lifetime consumption and investment problem where the two phases of a DC pension plan are modelled explicitly. In the accumulation phase, where the scheme member’s income is stochastic, contributions are variable, and the fund can be invested in a stock. In the distribution phase, we fix consumption and suppose income is derived from the compulsory purchase of an annuity at retirement. This is the main form of pension distribution in Austria, Brazil, France, the Netherlands and Sweden (Lunnon, 2002). Other countries allow a lump sum payment on retirement such as in the U.S., or a mixture of an annuity and a lump sum payment such as in the U.K. The constraint of fixed consumption post-retirement allows us to derive analytical results for the optimal investment and consumption processes. The purpose of a pension fund is to provide an ‘‘adequate’’ income for a pensioner in retirement. We fix this income level as a predetermined fraction of the scheme member’s salary just prior to retirement. This is called the replacement condition. In the language of optimal control theory, this requirement is a terminal constraint. Such a constraint makes it much harder to solve the optimisation problem. As a compromise, we weight the utility of income during employment relative to that during retirement, and fix this weighting so that the unconstrained optimisation problem satisfies the replacement condition (Liu, 2007). The resulting model is an example of an optimal control problem in an incomplete market with two control variables (stock allocation and consumption) that can be analysed without a full numerical solution. Henderson (2005) remarks that it is rare to find such problems in the literature. Section 2 describes the deterministic pension model and gives a canonical example that substantiates the methodology. The stochastic model is formulated in Section 3 for a general utility function. We make analytical progress by adopting a utility function that has constant absolute risk aversion (CARA). In Section 4, we develop the theory for a general income model, where the stock and income are perfectly correlated. Section 5 contains two income models with imperfect correlation between stock and income, where analytical expressions for the feedback controls are available. Conclusions can be found in Section 6.

2. Deterministic lifecycle model First we introduce a continuous time, deterministic pension scheme with variable contributions and no equity investment. This scheme introduces the notation of the paper and is a limiting case of the subsequent stochastic model. Suppose that the scheme member enters continuous employment at time t¼0 and receives a salary Y(t) per unit time. Further, suppose that a fraction f(t) of this money is invested for retirement into a pension fund of size X(t). If the member’s salary evolves according to dY ¼ nðt,YÞ, dt

Yð0Þ ¼ y0

ð1Þ

then the pension fund accumulates at the rate dX ¼ rX þfY, dt

Xð0Þ ¼ 0,

ð2Þ

where the constant risk-free interest rate is r. At the fixed retirement time t ¼ t years (after the start of employment), we suppose that the pensioner purchases a level annuity with the content of the pension fund XðtÞ. The annuity provides a fixed income for the pensioner for the fixed lifetime t ¼T, measured from the start of employment. The price of a level unit annuity for a pensioner of age t is denoted

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by aðtÞ and evolves according to da ¼ ðmðtÞ þrÞa1, dt

aðTÞ ¼ 0,

ð3Þ

where mðtÞ is the force of mortality (see Bowers et al., 1986), and we have ignored the insurer’s loading for simplicity. Traditional actuarial models use a functional parameterisation for the force of mortality, which is then fitted to a life table. Analytical expressions for aðtÞ are only available in special cases. Let us define the scheme member’s consumption as cðtÞ :¼ ð1f ÞYðtÞ

ð4Þ

that is the member contributes fY(t) to the pension fund and consumes the remainder of their salary. Thus, we suppose the pension fund holds the entire savings of the member, and X(t) represents their current wealth if we ignore human capital. At retirement, we assume that the scheme member purchases an annuity, which yields a constant income of size r a ðtÞXðtÞ, where r a ðtÞ ¼ 1=aðtÞ is the rate of return offered by the annuity provider. We consider the following problem. What fraction of salary should the scheme member contribute to the pension fund in order to purchase an annuity at retirement that maintains their standard of living ? Suppose the objective of the pension scheme member is to maximise their lifetime consumption taking into account the annuity purchase at retirement: Z t  Z T max ekt Uðct Þdt þ a ekt Uðr a ðtÞXðtÞÞ dt , ð5Þ f

t

0

where k is the subjective discount rate, U(c) is the utility of consumption, and a is a nondimensional weighting that reflects the member’s preference for consumption in retirement compared to consumption in employment. Since consumption post-retirement is constant we can integrate the salvage function and obtain Sekt Uðr a ðtÞXðtÞÞ, where we introduce   1ekðTtÞ S¼a ,

k

which has units of time and reflects the pensioner’s preference for income in retirement weighted by their remaining lifetime. If r a ðtÞ ¼ r then income post-retirement is provided by a perpetuity since the pensioner receives the interest rXðtÞ per unit time on the pension fund at retirement and the face value of the retirement capital is maintained. If the scheme member purchases an annuity on retirement then typically r a ðtÞ 4r and there is decumulation of the pension fund. This optimisation problem is a lifecycle model with a modified terminal boundary condition (Zeldes, 1989). It is easily solved using Pontryagin’s Maximum Principle. The first-order condition yields the consumption as cðtÞ ¼ IðSr a ðtÞeðrkÞðttÞ U 0 ðr a XðtÞÞÞ,

ð6Þ

01

where the function I  U , whereas the second-order condition requires that the utility function U is concave. Thus we see that if the subjective discount rate is equal to the risk-free rate (k ¼ r) then it is optimal for the member to consume a constant amount of their income, c, during employment, and that this value depends on the preference for consumption in retirement over employment through the parameter S. Substituting (6) into (2) gives the pension fund size using the optimal contribution as Z t XðtÞ ¼ ðYðuÞcðu; aÞÞerðtuÞ du: ð7Þ 0

We suppose the scheme member’s preference for consumption a is such that the income post-retirement is a fraction of salary at retirement, that is r a ðtÞXðtÞ ¼ RYðtÞ,

ð8Þ

where R is the desired replacement ratio. This is the replacement condition. If we substitute (8) into (6) and evaluate at t ¼ t then we obtain

a¼

U 0 ½cðtÞ kaðtÞ  : U 0 ½RYðtÞ 1ekðTtÞ

ð9Þ

Consequently, the retirement preference, a, is the product of two nondimensional terms. The first is the ratio of the marginal utility of consumption whilst employed relative to the marginal utility of retirement consumption. The second term is the value of a unit level annuity relative to self-annuitisation using a money market account that pays interest at the subjective rate k: it is a measure of the value for money of the annuity. If the change in utility per unit consumption is increased during employment or the annuity is made more expensive then the retirement preference increases in order to maintain retirement income.

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The retirement preference a is fixed by the budget equation (7) evaluated at retirement. The scheme member places sufficient income in the pension fund to purchase an annuity at retirement and consumes the remainder over their period of employment: F ð0,cð:; aÞ,r1Þ ¼ F ð0,Y,r1ÞaðtÞRYðtÞ,

ð10Þ

where the function 1 : ½t, t-1 and we have introduced the functional Z t emðuÞðtuÞ aðuÞ du, F ðt,a,mÞ :¼

ð11Þ

t

which represents the projected value at retirement t of the money stream a : ½t, t-R þ using the interest rate function m : ½t, t-R þ . This notation will prove useful for the stochastic model in Section 3. 2.1. Analytical expressions for the optimal consumption If the subjective and risk-free interest rates are identical ðk ¼ rÞ then (10) can be simplified because c is the constant c¼

F ð0,Y,r1ÞaðtÞRYðtÞ , Dr ð0Þ

ð12Þ

where Dm ðtÞ :¼

Z t

emðutÞ du ¼

t

1emðttÞ ¼ F ðt,1,m1Þ m

ð13Þ

represents the value at time t of a unit cash flow over ½t, t with constant interest rate m. The corresponding retirement preference is given by (9) with cðtÞ ¼ c. We can see from (12) that as t-T then the consumption during employment cðaÞ approaches F ðt,Y,r1Þ=Dr ð0Þ, which is the projected income at T relative to a unit income, and so is a measure of the average income value. Consequently, if income increases over the employment period then it is optimal to borrow money from the pension fund early in the member’s career, and later replace that money with future expected income. This optimal consumption strategy seems reasonable because an employee often borrows a substantial amount of money early in their career in the form of a mortgage, whilst at the same time saving for their retirement through an occupational pension. We interpret the pension fund as both a deferred investment and a cash account in this paper. Generalising the model in order to allow investment other than in the pension fund is a subject of future work. The HARA family of utility functions also allow us to determine an explicit expression for the optimal consumption cðt; aÞ and the retirement preference a. Following Merton (1990) these utility functions are given by1  z 1z dc UðcÞ ¼ þZ ð14Þ zd 1z subject to the restrictions

za1, d 4 0,

dc þ Z 40, 1z

Z ¼ 1 if z ¼ 1:

Following Pratt (1964) the absolute risk aversion is defined by   U 00 ðcÞ c Z 1 AðcÞ :¼  0 ¼ þ : 1z d U ðcÞ The open-loop optimal consumption rule given by (6) now becomes     1g dRYðtÞ cðtÞ ¼ ðSr a ðtÞeðrkÞðttÞ Þ1=ðz1Þ þ Z Z 1z d on applying (8), while the retirement preference is 0 1z1 d ðF ð0,Y,r1Það t ÞRYð t ÞÞ þ Z D ð0Þ r C aðtÞk B B1z C  a¼ A dRYðtÞ 1ekðTtÞ @ þ Z DðkrÞ=ðz1Þr ð0Þ 1z using (10). If we take the limit z-1 and set Z ¼ 1 then we obtain the exponential utility function 1 dc

UðcÞ ¼ d

1

e

,

For convenience we have multiplied Merton’s definition of the HARA utility function by d

ð15Þ

1

.

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where d is the constant absolute risk aversion. The optimal consumption written in open-loop form is kr  kr  F ð0,L,r1Þ F ð0,Y,r1ÞRaðtÞYðtÞ d ðttÞ þ cðtÞ ¼ d Dr ð0Þ using the first-order condition (6), the budget equation (10), and where the function LðuÞ :¼ tu. Thus consumption decreases linearly over the employment period if k 4 r and reduces to the constant in (12) if k ¼ r. The corresponding retirement weighting is    aðtÞk aðtÞ ðkrÞF ð0,L,r1ÞdF ð0,Y,r1Þ þ : ð16Þ a¼ exp dRYðtÞ 1 þ  k ðT t Þ Dr ð0Þ Dr ð0Þ 1e

2.2. Mean-reverting income We give an example of a pension scheme using the exponential utility function in order to illustrate that the retirement weighting, a, has a plausible numerical value. Let us choose a mean-reverting income model

nðYÞ ¼ fðym YÞ,

ð17Þ

where f,ym 40, giving F ð0,Y,r1Þ ¼ ðy0 ym Þeft Dðr þ fÞ ð0Þ þ ym Dr ð0Þ. This form of income model allows us to write the optimal consumption in Markov form, which is demonstrated later in Section 5.2. Let us take the income model parameters y0, ym and f from the base parameter set given in Table 1 with monetary units in thousands of pounds and time in years. Thus, we suppose the scheme member starts employment on a salary of ` ` E20; 000 per year at t ¼0 (corresponding to age 20 years) that reverts to E50; 000 as t-1. Next, suppose that the force of mortality m is constant, which yields the annuity price as aðtÞ ¼

1eðm þ rÞðTtÞ m þr

from (3). The remaining values of the parameters required for the deterministic model (r, k, d, T, R) are given in Table 1, and imply that the optimal consumption is constant, and death occurs at age 80. In Fig. 1(a) we vary the employment period t, and plot the income of the scheme member YðtÞ, the consumption of the member during employment cðtÞ, and the income of the pensioner in retirement RYðtÞ. An individual can choose from this graph the constant consumption they require during employment, c, and from that value read off the corresponding retirement year t. Notice that for t \ 25 income at retirement is greater than the initial income rate, that is cðtÞ 4 y0 , so borrowing is optimal early in the member’s career. The retirement weighting of the scheme member a is shown in Fig. 1(b) as a function of the retirement year t and the replacement ratio R. The greater the replacement ratio, R, the higher the weighting required for sufficient retirement income. From (16), a-expðdF ð0,Y,r1Þ=Dr ð0ÞÞ as t-T, so the weighting for retirement income is independent of R and less than one if there are very few years of retirement available. If the member retires much earlier then the weighting is very large because there are insufficient funds available to sustain income over the retirement period. In Fig. 1(a) we can see that for very early retirement, c o0, which means the member must borrow money in order to build the pension fund, and they carry that debt throughout their life. In practice, retirement cannot occur at such an early age. Fig. 1(c) shows the pension fund size as a function of employment period. Since death is specified at age 80 no funds are required thereafter. Table 1 Base parameter set for the deterministic and stochastic models. Time is measured relative to the start of employment fixed at age 20 years. Initial income y0 Mean-reverting income ym Reversion parameter f Force of mortality m Risk free interest rate r Subjective discount rate k Risk aversion d Lifetime T Replacement ratio R Stock drift l Stock volatility s Correlation coefficient r Employment period t Number of time steps N

20 50 0.02 0.05 0.05 0.05 1.0 60 2/3 0.10 0.20 1.0 45 1000

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50

5

40

4 Y(τ)

3 α

30 RY(τ)

20

2

R=1

0.8 10 0

1

c(τ) 0

10

20

30 τ

40

50

60

350

2/3 25

30

35

40 τ

45

50

55

60

R=1

300

0.8

250 X(τ)

0 20

200 2/3

150 100 50 0

0

10

20

30

40

50

60

τ Fig. 1. Optimal solutions of a deterministic DC pension fund model for k ¼ r.

Fig. 1(a) suggests a minimal retirement time such that consumption in employment matches consumption in retirement, that is the solution of cðtmin Þ ¼ RYðtmin Þ: For the example in the figure, this gives a minimal retirement age of just under 50 years with the consumption rate over the ` lifetime at just over the initial income of E20; 000 per annum. Retirement at a later age leads to greater income in retirement which is comparable to that during in employment, but requires greater borrowing early in the member’s career.

3. Stochastic asset and income model Now we generalise the deterministic model in order to allow investment of the pension fund and to model the uncertainty of income. Therefore, we suppose a cash amount yt of the pension fund is invested in a stock with unit price Pt, and this price evolves according to dPt ¼ P t ðl dt þ s dW 1t Þ,

P 0 ¼ p0 ,

where l, s, p0 are constants with s, p0 4 0. The remainder of the fund accumulates at the constant risk-free rate r. Let us suppose that the member’s income evolves according to dY t ¼ nðt,Y t Þ dt þ Zðt,Y t ÞdW 2t ,

Y 0 ¼ y0 ,

ð18Þ

where fW 1t g and fW 2t g are standard Brownian motions under a probability measure P with dW 1t dW 2t ¼ r dt, the constant correlation coefficient satisfies 9r9 r1, and the initial income y0 is a positive constant. The value of the pension fund Xt evolves according to the SDE dX t ¼ yt

dPt þ rðX t yt Þ dt þ ðY t ct Þ dt, Pt

ð19Þ

where ct is the consumption given by (4). Following the deterministic model we suppose the objective of the scheme

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member is to maximise their expected total discounted lifetime consumption, that is Z t  ekt Uðct Þ dt þ Sekt Uðr a X t Þ9X 0 ¼ 0,Y 0 ¼ y0 , supE 0

c, y

where we have written r a ¼ r a ðtÞ for the sake of brevity. Hereafter, the expectation operator E is taken under the measure P unless stated otherwise. The objective is similar in form to the original optimal investment and consumption problem (Merton, 1969). Notice that the integration is up to retirement, t, consumption from the annuity replaces the bequest function in Merton’s model, and the utility function U is a function of a money rate rather than absolute wealth. Thus, we restrict the stock allocation and consumption decisions to the pre-retirement phase of the pension, and the salvage function of the optimisation problem models the post-retirement phase. The objective suggests that we define the value function as Z t  Vðt,x,yÞ ¼ sup E eku Uðcu Þ duþ Sekt Uðr a X t Þ9X t ¼ x,Y t ¼ y t

c, y

so the terminal boundary condition is Vðt,x,yÞ ¼ Sekt Uðr a xÞ: If the value function is sufficiently smooth then it satisfies the HJB equation 2

supfV t þ V x ððlrÞy þ rx þ ycÞ þ nðt,yÞV y þ 12 s2 y V xx þ 12Zðt,yÞ2 V yy þ rysZðt,yÞV xy þ ekt UðcÞg ¼ 0, c, y

where subscripts refer to partial derivatives. The first-order conditions for the HJB equation are   bV x þ rZðt,yÞV xy , c ¼ Iðekt V x Þ, y ¼  sV xx

ð20Þ

where the Sharpe ratio is

b¼

lr

s

:

The second-order conditions require that U 00 ,V xx o 0, and provided that this is true then at retirement the first-order conditions c ¼ Iðr a SU 0 ðr a xÞÞ,

y¼

bU 0 ðra xÞ

ð21Þ

sr a U 00 ðra xÞ

yield a local maximum provided that the value function is sufficiently smooth. Consequently, the stock allocation at retirement is positive since U 0 40 and U 00 o0, and we can adjust the scheme member’s preferences so that the consumption yields the desired fraction of income post-retirement. Further analytical progress requires the specification of the form of the utility function. 3.1. Constant absolute risk aversion We focus on the exponential utility function (15) since there is considerable analytical reduction. Substituting the firstorder controls into the HJB equation yields 1

V t þ ðd

ðlogðekt V x Þ1Þ þ y þrxÞV x þ nðt,yÞV y þ

ðbV x þ rZðt,yÞV xy Þ2 1 Zðt,yÞ2 V yy  ¼0 2 2V xx

ð22Þ

with boundary condition Vðt,x,yÞ ¼ d Sektdra x . Let us look for a solution of the form 1

Vðt,x,yÞ ¼ 

1

dHðtÞ

exp½ktHðtÞdx þGðt,yÞ,

ð23Þ

where HðtÞ ¼ r a , so at retirement Gðt,yÞ ¼ logðr a SÞ. Substituting (23) into the HJB equation (22) and collecting terms of x yields dH ¼ HðHrÞ dt

ð24Þ

Gt HG þ nðt,yÞGy þ 12 Zðt,yÞ2 ðGyy þ G2y Þ12ðb þ rZðt,yÞGy Þ2 dyH þrk ¼ 0:

ð25Þ

and

Integrating (24) and applying the terminal boundary condition gives HðtÞ ¼

rr a erðttÞ : rr a ð1erðttÞ Þ

ð26Þ

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The function H(t) is a modified interest rate, which strictly increases from the risk-free rate r at t ¼ 1 to the annuity rate ra at retirement t ¼ t. The reduced HJB equation (25) is a quasilinear PDE and its solution (should it exist) yields the firstorder controls as

yn ¼

1

sdHðtÞ

cn ¼ HðtÞx

ðb þ rZðt,yÞGy ðt,yÞÞ,

Gðt,yÞ

d

ð27Þ

:

ð28Þ

The candidate solution for the value function is concave in x, so the first-order controls are the optimal controls providing that we can find a sufficiently smooth solution of (25). On the optimal state trajectory     b þ rZðt,Y t ÞGy b G n ðb þ rZðt,Y t ÞGy Þ þ dW 1t : ð29Þ dt þ dX t ¼ ðrHðtÞÞX nt þY t þ d dHðtÞ dH Using this expression and the reduced HJB equation (25), we can write down the evolution of the optimal controls as a function of time t and stochastic income Yt:

sdHðtÞdynt ¼ ðrHðtÞÞðb þ rZðt,Y t ÞGy Þ dt þ rdðZðt,Y t ÞGy Þ and

ddcnt ¼ ðb2 g12ð1r2 ÞZ2 ðt,Y t ÞG2y Þ dt þðb þ rZðt,Y t ÞGÞ dW 1t Zðt,Y t ÞG dW 2t , where we have introduced the constant

g ¼ kr þ 12b2 : 2

If k ¼ r þ 12 b and there is perfect correlation between stock and income (9r9 ¼ 1) then the optimal consumption is a P-martingale under suitable growth conditions on Zðt,Y t ÞGðt,Y t Þ. This might be attractive for a pensioner who prefers steady consumption whilst working. In the limit s-1, Z-0 we recover the deterministic results, that is b-0, there is no stock allocation, and the optimal consumption is constant if the subjective interest rate is the risk-free rate (k ¼ r). In general, there is no analytical solution for G, so we focus on special cases of the income model. We can make some general observations by identifying parts of the controls with those of the lifetime consumption problem studied by Merton (1969) and the terminal wealth problem studied by Henderson (2005). 3.2. Log-Linear value function Let us suppose that G is an affine function of the current income y: Gðt,yÞ ¼ G0 ðtÞ þ G1 ðtÞy,

G0 ðtÞ ¼ logðr a SÞ,

G1 ðtÞ ¼ 0:

ð30Þ

Clearly, whether one can make this assumption depends on the form of the income drift and volatility. Sections 4.1, 5.1 and 5.2 contain income models for which G is affine, so there are income models for which this assumption is valid. If such a reduction is possible then the value function given by (23) is log-linear in the state variables x and y. In addition, we can split up the optimal control terms and write

yn ¼ ynm þ ynH ¼

b

sdHðtÞ

þ

rZðt,yÞ G ðtÞ, sdHðtÞ 1 1

cn ¼ cnm þcny þcns ¼ HðtÞxd

ð31Þ 1

G1 ðtÞyd

G0 ðtÞ:

ð32Þ n

The first term in the optimal stock allocation, ym , is the static Merton hedge adjusted by the interest rate H(t) for the n constrained consumption post-retirement. Since H(t) increases with time, ym decreases over the employment period. The n second term, yH , is a dynamic income hedging term provided that the volatility of income depends on current income and the stock is correlated with income. The first term in the optimal consumption, cnm , is the Merton dynamic control adjusted for the required annuity purchase at retirement. The second term, cny , is a dynamic consumption term, whilst the last term, cns , is static. At retirement, the optimal stock allocation is the Merton stock allocation (with r a ¼ r) and the optimal 1 consumption is cn ¼ r a xd logðr a SÞ. Notice that the optimal consumption is linear in both state variables, whereas the optimal stock allocation depends on the form of the volatility of income. On substituting (30) into (29) we obtain an SDE for the optimal pension fund size:        G ðtÞ b þ rZðt,Y t ÞG1 ðtÞ G1 ðtÞ b

n dW 1t þ ðrHðtÞÞX t þ 1 þ b þ rZðt,Y t ÞG1 ðtÞ þ 0 dX t ¼ Yt þ dt: ð33Þ dHðtÞ d d dHðtÞ If we denote the expected value of a random variable under the P measure using an overbar e.g. X n ðtÞ ¼ E½X nt  then taking

P. Emms / Journal of Economic Dynamics & Control 36 (2012) 1303–1321

the expectation of (33) yields   dX n G1 ðtÞ b G0 ðtÞ ¼ ðrHðtÞÞX n ðtÞ þ 1þ ðb þ rZ ðt,Y t ÞG1 ðtÞÞ þ Y ðtÞ þ , dt d d dH

1311

ð34Þ

where X n ð0Þ ¼ 0. If the volatility of income ZðY t ,tÞ is linear in Yt then both optimal controls are linear in the state variables and (33) is a linear second-order SDE, whose solution can be written in terms of fundamental solutions (Kloeden and Platen, 1999). Therefore, we can find the strong solution for the optimal pension fund size for this class of problem in addition to explicit expressions for the optimal controls. It is also straightforward to calculate evolution equations for higher-order moments of the state and control variables. The evolution of the variance of the optimal controls is described in Appendix A. Closure of the model is now similar to the deterministic case: we choose the retirement weighting a such that sufficient income is generated post-retirement, and this fixes the optimal consumption during employment. Thus we pose the boundary condition X n ðt; aÞ ¼ RaðtÞY ðtÞ:

ð35Þ

In general, we must solve this problem numerically by integrating G0 and G1 backwards in time from retirement and then integrating (34) forwards from t ¼0 to t ¼ t. The retirement preference is fixed by varying a over each forward/backward integration until (35) is satisfied. Notice that we stipulate that the expected pension fund size is sufficient to purchase an annuity at retirement, and that the annuity provides a fraction R of the expected income at retirement. If a pensioner requires a guaranteed minimum annuity rate in retirement then the value of that guarantee can be valued as an option (Ballota and Haberman, 2003), but this is outside the scope of the present paper. The deterministic problem given in Section 2.2, where the income model given by (17), reduces to a problem where G is affine. It is straightforward to verify that (30) with G0 ðtÞ ¼ eH ðtÞðttÞ ðlogðr a SÞF ðt,G,HÞÞ,

ð36Þ

G1 ðtÞ ¼ deðH ðtÞ þ fÞðttÞ F ðt,H,H þ f1Þ

ð37Þ

satisfies (25) for b ¼ Z ¼ 0, where GðsÞ ¼ krfym G1 ðsÞ

ð38Þ

and HðtÞ ¼

1 tt

Z t

HðyÞ dy:

ð39Þ

t

One can simplify this solution by changing the order of integration in F ðt,G,HÞ which yields G0 ðtÞ ¼ eH ðtÞðttÞ ðlogðr a SÞF ðt,ðkrÞ1 þ dHym ð1efð:tÞ ÞÞ,HÞ:

ð40Þ

The function H is a running average interest rate, and lies between the risk-free rate r and the annuity rate ra. In general, H cannot be written in closed-form. A term of the form emðtÞðttÞ F ðt,a,mÞ represents the value of a cash flow, a, invested from time t to retirement with variable interest rate, m, whose value is then discounted using the constant interest rate, m(t), in order to find the value of the cash flow at time t. If m is constant then emðttÞ F ðt,a,mÞ is the value of cash flow, a, at time t using discount interest rate m, and in particular emðttÞ F ðt,1,mÞ ¼ Dm ðtÞ, which is closed-form. If the retirement phase of the pension is provided by a perpetuity then H ¼ H ¼ r a ¼ r and we can find closed-form expressions for the optimal controls in feedback form. 4. Perfect correlation between stock and income Perfect correlation between the stock price and the income of the pension scheme member is unlikely to be observed in practice. However, this is a special case for which we can integrate the HJB equation (25) and find an analytical expression for G. If 9r9 ¼ 1 then one can define the income process by dY t ¼ nðt,Y t Þ dt þ rZðt,Y t Þ dW 1t : Using the stock for replication gives a unique risk-neutral pricing measure Q defined by the Radon–Nikody´m derivative dQ 2 ¼ expfbW 1t 12b tg: dP From Girsanov’s theorem (Karatzas and Shreve, 1991), the income process fY t g has drift nðt,Y t ÞrbZðt,Y t Þ under Q. If 9r9 ¼ 1 then the HJB equation (25) becomes a linear PDE: Gt HG þ ðnðt,yÞrbZðt,yÞÞGy þ 12Zðt,yÞ2 Gyy Hdyg ¼ 0:

ð41Þ

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Let us assume that G is sufficiently smooth, and satisfies a polynomial growth condition, so that we can apply the Feynman–Kac theorem (Karatzas and Shreve, 1991, p. 366). This yields the solution of the PDE in integral form:   Z t   Z u  Z t HðyÞ dy logðr a SÞ exp  HðyÞ dy ðdHðuÞY u þ gÞ du , ð42Þ Gðt,yÞ ¼ EQ t,y exp  t

t

t

Q

where Et,y denotes the expectation under the risk-neutral measure given that the income has value y at time t. By swapping the order of integration in (42) we find Gðt,yÞ ¼ eH ðtÞðttÞ ðlogðr a SÞgF ðt,1,HÞdF ðt,HEQ t,y ½Y : ,HÞÞ,

ð43Þ

where the interest rate H is given by (39). Suppose we can write the conditional expected income (under the risk-neutral measure) in affine form

EQ t,y ½Y s  ¼ Mðs,tÞy þNðs,tÞ

ð44Þ

for s Z t Z0 where Mðs,tÞ is dimensionless, Nðs,tÞ has units of income, and Mðt,tÞ ¼ 1, Nðt,tÞ ¼ 0. The functions M and N are determined explicitly by the income model provided that there is an analytical expression for the conditional expectation. Since the functional F defined in (11) is linear in the function a then G is linear in the current income y so G0 ðtÞ ¼ eH ðtÞðttÞ ðlogðr a SÞgF ðt,1,HÞdF ðt,HNð,tÞ,HÞÞ, G1 ðtÞ ¼ deH ðtÞðttÞ F ðt,HMð,tÞ,HÞ:

ð45Þ

From (31) and (32) the corresponding control terms are

ynH ¼ 

rZðt,yÞeH ðtÞðttÞ F ðt,HMð,tÞ,HÞ , sHðtÞ

cny ¼ eH ðtÞðttÞ F ðt,HMð,tÞ,HÞy,   1 cns ¼ eH ðtÞðttÞ F ðt,HNð,tÞ,HÞ þ ðgF ðt,1,HÞlogðr a SÞÞ :

d

Thus, we have found both optimal controls in feedback form, where in general the function H and the functional F must be calculated numerically. Suppose that the volatility of income Zðt,yÞ 40, the function Mðs,tÞ 40, and the stock is n perfectly positively correlated with income: r ¼ 1. Then the income hedge yH o0, so the optimal stock allocation is less n than the Merton allocation ym , and the consumption income hedge cny 40 provided that the current income is positive and Mðs,tÞ 4 0. Further analytical progress depends on the functional form of the income volatility. 4.1. Stochastic income model with mean-reversion If the income model (18) is represented by a linear SDE2 then the conditional expectation satisfies (44) and we can apply the theory in Section 4. Suppose that

nðt,yÞ ¼ fðym yÞ, Zðt,yÞ ¼ Z0 y,

ð46Þ 3

where f,ym , Z0 4 0. This is one income model studied by Henderson (2005), and the solution of (18) for 0 r t rs is given by   Z s F1 du , Y s ¼ Fs,t Y t þ fym u,t t

where the fundamental solution Fs,t ¼ expððf þ 12Z20 ÞðstÞ þ Z0 ðW 1s W 1t ÞÞ. Income is positive under this model, and it is lognormally distributed if fym ¼ 0. The expected income at time s, given that the income at time t r s is y, is

Et,y ½Y s  ¼ yefðstÞ þ ym ð1efðstÞ Þ, which yields under the risk-neutral measure the functions ^

Mðs,tÞ ¼ ef ðstÞ ,

^

Nðs,tÞ ¼ y^ m ð1ef ðstÞ Þ,

where the modified mean-reversion rate and income level are

f^ ¼ f þ rbZ0 , y^ m ¼

2

fym , 9r9 ¼ 1: f^

The income model does not have to be a linear SDE in order to satisfy (44) . The CIR income model in Section 5.1 also satisfies this relationship. In contrast to Henderson (2005) this income model does not lead to analytical controls if there is imperfect correlation between the stock and income. 3

P. Emms / Journal of Economic Dynamics & Control 36 (2012) 1303–1321

1313

The value function is now given by ^

G0 ðtÞ ¼ eH ðtÞðttÞ ðlogðr a SÞF ðt, g1 þ dHy^ m ð1ef ðtÞ ÞÞ,HÞ, ^ ^ 1Þ, G1 ðtÞ ¼ deðH ðtÞ þ f ÞðttÞ F ðt,H,H þ f

which can be compared with the deterministic counterparts (36) and (37) because the income drift is the same in both models. We see that the deterministic value function is the same as the stochastic value function if we make the substitutions

kr-kr þ 12b2 ¼: g, ym -y^ m , f-f^ : Thus, in the stochastic model, the optimal consumption at each instant is equivalent to a deterministic strategy with modified values of k, ym, and f. In particular, for positive Sharpe ratio and perfect correlation, a scheme member facing income and equity risk follows a deterministic consumption strategy with a decreased mean-reverting income level and an increased speed of mean-reversion of income. We study the parametric sensitivity and risk of the optimal strategies in the case that a perpetuity is purchased at retirement. For H ¼ H ¼ r a ¼ r F ðt,1,r1Þ ¼ Dr ðtÞ,

^

^

F ðt,ref ð:tÞ ,r1Þ ¼ ref ðttÞ Dðr þ f^ Þ ðtÞ,

^

^

F ðt,r y^ m ð1ef ð:tÞ Þ,r1Þ ¼ r y^ m ðDr ðtÞef ðttÞ Dðr þ f^ Þ ðtÞÞ, which yields G0 ðtÞ ¼ erðttÞ logðrSÞðg þ dr y^ m ÞDr ðtÞ þ dr y^ m Dr þ f^ ðtÞ, G1 ðtÞ ¼ drDr þ f^ ðtÞ

ð47Þ

using (45) and emðttÞ Dm ðtÞ ¼ Dm ðtÞ. Using these expressions we obtain the optimal stock allocation terms:

ynm ¼

rZ0 Dr þ f^ ðtÞ b n , yH ¼  y, 9r9 ¼ 1: sdr s n

The Merton stock allocation ym is constant and contains the factor r 1 due to the perpetuity purchase. The income hedge ynH is identical to that in Henderson (2005) and is independent of risk aversion. If the member’s income is positively and perfectly correlated with the stock (r ¼ 1), and the Sharpe ratio b is positive, then the optimal stock allocation is less than n the Merton allocation, ym , because income takes the place of stock. The optimal consumption terms are cnm ¼ rx, 1

cns ¼ ðd

cny ¼ rDr þ f^ ðtÞy,

g þ ry^ m ÞDr ðtÞry^ m Dr þ f^ ðtÞd1 erðttÞ logðrSÞ,

so the optimal consumption is a linear function of both state variables. The overall magnitude of the static consumption profile, cns , is set by the replacement condition (35). On substituting (47) into (34) and then integrating, we obtain the following explicit expression for the retirement preference: ! ! " ( tDr þ f^ ð0Þ k d RY ðtÞ b2 g ðy0 ym ÞDf ð0Þ ym y^ m þ exp a¼ tðrðy^ m ym ÞrbZ0 ym Þ ^ r rð1ekðTtÞ Þ Dr ð0Þ dr r þf )# g  þ y^ m Dr ð0Þ : þðr þ rbZ0 Þðy0 ym ÞðDf ð0Þeft Dr þ rbZ0 ð0ÞÞ dr Both controls are linear in the state variables, so we can calculate the variance of each optimal control, and therefore deduce one risk measure associated with adopting an optimal strategy. The derivation of the variance of the optimal controls for this income model is given in Appendix A. Let us adopt the base parameter set given in Table 1 for the stochastic model. The employment period is now fixed at t ¼ 45 years, so that the retirement age is 65 years. We shall use these values in the numerical results unless stated otherwise in order to illustrate the quantitative features of the optimal controls. In Fig. 2 we show the sensitivity of the mean of the optimal controls to the volatility of the stochastic processes. It is clear from Fig. 2(a) that both the optimal stock allocation and the optimal consumption are very sensitive to the volatility of income Z0 . As future income becomes more uncertain, it is optimal to make a lower stock allocation, and consume less in order to achieve the desired annuity rate. Indeed, for Z0 ¼ 0:02 it is optimal to short sell the stock in order to hedge against the future loss of income. These results are a consequence of the perfect correlation between stock and income: when the stock price decreases, so does income, both of which are unfavourable outcomes for the scheme member. Notice n n also that the optimal stock allocation is just ym at retirement (and so independent of Z0 ) since the income hedge yH ¼ 0.

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38 36 34 η0=0.0

32 c

θ

40 35 30 25 20 15 10 5 0 -5 -10 -15

0.01

5

10

15

28

η0=0.0

26

0.01 0.02

24

0.02 0

30

22 20

25

30

35

40

20

45

0

5

10

15

20

40

38

35

36

30

34

25

32

20

30

15 10 0 -5

26

0.25

24

5

10

15

35

20

40

45

40

45

σ=0.2 0.3

22

0.3 0

30

28

σ=0.2

5

25 t

c

θ

t

25

30

35

40

20

45

0

5

10

15

20

t

25

30

35

t

Fig. 2. Mean optimal controls for a stochastic DC pension fund, with a stock that is perfectly positively correlated with income, and where the retirement product is a perpetuity. For the graphs shown in (a) the stock volatility is fixed at s ¼ 0:20, while in (b) the income volatility is fixed at Z0 ¼ 0:01. The expected income per year of the scheme member is shown as a grey line in each plot.

σX

200 180 160 140

0.02

120 100 80 60 40 20 0

0.01

η0=0.0 0

5

10

15

20

2.5 2

0.02

σc

σθ

1.5 1 0.01 0.5 0

5

10

25

30

35

40

45

10 9 8 7 6 5 4 3

0.02 0.01

η0=0.0

2 1

η0=0.0 0

t

15

20

t

25

30

35

40

45

0

0

5

10

15

20

t

25

30

35

40

45

Fig. 3. Standard deviation of the optimal fund size and controls for fixed stock volatility s ¼ 0:2, varying income volatility Z0 , and perfect positive correlation between the stock and income.

P. Emms / Journal of Economic Dynamics & Control 36 (2012) 1303–1321

1315

From the graphs in Fig. 2(b), we see that as the volatility of the stock increases, it is optimal to decrease the stock allocation, n which can be accounted for by the decrease in the Merton term ym . The optimal consumption also decreases, and it is fairly insensitive to changes in stock volatility. In both sets of results, borrowing early in the scheme member’s career is optimal. The standard deviation of the optimal fund size, sX , and the optimal controls sy , sc , are shown in Fig. 3. The last two graphs of Fig. 3 show the variation about the mean value of the controls shown in Fig. 2(a), since the parameter sets are the same. As one might expect, the pension fund size and optimal consumption both increase as income volatility increases: for Z0 ¼ 0:02, the standard deviation of the fund at retirement is of the same order of magnitude as the maximum deterministic fund in Fig. 1. Notice that from (31), there is no variability in the optimal stock allocation at retirement since G1 ðtÞ ¼ 0. 5. Imperfect correlation between stock and income If there is imperfect correlation between the stock and income then the HJB equation is nonlinear, and we cannot directly apply the Feynman–Kac theorem. However, there are two income models that lead to significant reduction of the optimisation problem. As in the previous section, this analytical reduction relies on the fact that the value function is linear in the current income y. If the retirement income is provided by a perpetuity then there are analytical expressions for the optimal stock allocation and optimal consumption for both income models. 5.1. CIR income model with no correlation between stock and income Suppose the income model is given by pffiffiffi nðyÞ ¼ fðym yÞ, ZðyÞ ¼ Z1 y

ð48Þ

with constant f,ym , Z1 4 0, but where income and the stock are uncorrelated so r ¼ 0. The constant Z1 has different units to Z0 with this parameterisation of income volatility. If we look for a solution of the form (30) and substitute into (25) then G0 ðtÞ and G1 ðtÞ must satisfy G00 HðtÞG0 ¼ gfym G1 ,

ð49Þ

G01 ¼ dHðtÞ þ ðHðtÞ þ fÞG1 12Z21 G21

ð50Þ

with boundary conditions G0 ðtÞ ¼ logðr a SÞ and G1 ðtÞ ¼ 0. The last of these equations is a Riccati equation, which is not integrable analytically, so we examine the ðH,G1 Þ phase plane in Fig. 4 in order to gain some qualitative insight into the optimal controls. From (26) we see that H(t) increases with time from above the risk-free rate r at t ¼0 to the annuity rate ra at retirement t ¼ t. There are two equilibrium points for the system (24) and (50) that lie in the relevant part of phase space, which we denote by ðr,G 7 Þ where G7 ¼

f þ r 7ððf þrÞ2 þ 2rdZ21 Þ1=2

Z21

ð51Þ

so G þ 40, G o 0. If we perturb about these points by setting HðtÞ ¼ r þ hðtÞ, G1 ðtÞ ¼ G 7 þ gðtÞ and then linearise then we

Fig. 4. Phase diagram for the construction of the optimal controls for the CIR income model.

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P. Emms / Journal of Economic Dynamics & Control 36 (2012) 1303–1321

obtain the matrix equation y0 ¼ Ay where y ¼ ðh,gÞT and ! r 0 A¼ : d þ G 7 f þrZ21 G 7 The eigenvalues of A are m ¼ r or ! fþr m¼ G 7 Z21 : 2

Z1

From (51) we deduce that the point ðr,G þ Þ is a saddle point, and the point ðr,G Þ is an unstable node. The eigenvectors for each equilibrium point are 0 1   1 0 e17 ¼ @ d þ G 7 A, e27 ¼ : 2 1 Z1 G 7 f From (51) the eigenvector e1þ has positive gradient in the ðH,G1 Þ plane. Using the RHS of (50) we can write   G 1 2 d ¼ dðG Þ ¼  Z1 G ðf þ rÞ r 2 for then, by considering the gradient of d for G o 0, we have d oG o 0, so e1 has negative gradient in the ðH,G1 Þ plane. Fig. 4 shows a sketch of the phase diagram. The trajectory ending at H ¼ r a , G1 ¼ 0 satisfies the terminal boundary conditions and so yields the optimal controls. At retirement we have from the boundary conditions dG1 d ¼ 4 0: dH r a r Consequently, G1 o0 and G1 -G as t-1. It is more difficult to determine the form of G0 because this requires a backward integration and a forward integration in order to fix the retirement preference a. Thus we study the sensitivity of the controls using our numerical example. 40

38

35

36 34

30

32

25 20

c

θ

η1=0.0-0.2

28

η1=0.0

26

15

24

10 5

0.2

30

22 0

5

10

15

20

25

30

35

40

45

20

0

5

10

15

20

t 40

38

35

36

30

34 σ=0.2,

20

c

θ

30

35

40

45

35

40

45

32

25

0.25

15 10

0

5

10

15

30

σ=0.2

28

0.3

26 24

0.3

5 0

25 t

22 20

25 t

30

35

40

45

20

0

5

10

15

20

25

30

t

Fig. 5. Mean optimal controls for a CIR income process where there is no correlation between the stock and income and the retirement product is an annuity. In graphs (a) the stock volatility is fixed at s ¼ 0:2, while in (b) the income volatility is fixed at Z1 ¼ 0:1.

P. Emms / Journal of Economic Dynamics & Control 36 (2012) 1303–1321

1317 n

Since there is no correlation between the stock and income, the stock allocation is just the static Merton hedge ym , while the optimal consumption terms are cny ¼ dG1 ðtÞy,

cns ¼ dG0 ðtÞ,

where G0 ðtÞ and G1 ðtÞ must be calculated numerically from the ODEs (49) and (50). In Fig. 5 we show the sensitivity of the mean controls to the volatility of the stochastic processes. We use larger values of Z1 compared to Z0 than those shown in Fig. 2 because of the change in the parameterisation of income volatility. Since n n the optimal stock allocation is just y ¼ ym , only the consumption varies with Z1 in Fig. 5, and in contrast to the lognormal income model, increasing Z1 leads to increased consumption near retirement. The decreasing mean optimal stock allocation in both (a) and (b) is due to the increase in the modified interest rate H(t) with age, which is in contrast to the results for lognormal income in Section 4.1. This feature illustrates how the choice of retirement product affects the behaviour of the optimal controls. If there is no correlation between stock and income then the qualitative form of the optimal stock allocation leads to short-selling at an early age: this is a result often found in lifecycle models. As the income volatility is increased then it becomes optimal to consume much less early on in the lifecycle. This means the pension fund is larger, and it is much easier to maintain the required standard of living at retirement, so the retirement income preference is extremely small. 5.2. Ornstein–Uhlenbeck income process Consider the case that income is normally distributed and mean reverting:

nðt,yÞ ¼ fðym yÞ, Zðt,yÞ ¼ Z2 , where f,ym , Z2 4 0. For this model we examine how the correlation between stock and income, r, affects the optimal controls and the preference for retirement income. Again we look for a solution of the form (30) and substitute into (25). The functions G0 ðtÞ and G1 ðtÞ must satisfy the nonlinear ODEs G00 HG ¼ g þ G1 ðrZ2 bfym Þ12ð1r2 ÞZ22 G21 , G01 ðH þ fÞG1 ¼ dH with boundary conditions G0 ðtÞ ¼ logðr a SÞ, G1 ðtÞ ¼ 0. On integrating and applying the boundary conditions, we find the closed-form expressions G0 ðtÞ ¼ eH ðtÞðttÞ ðlogðr a SÞF ðt,G,HÞÞ, G1 ðtÞ ¼ deðH ðtÞ þ fÞðttÞ F ðt,H,H þ f1Þ,

ð52Þ

where GðtÞ ¼ g þG1 ðtÞðbrZ2 fym Þ12ð1r2 ÞZ22 ðG1 ðtÞÞ2 : Notice that G0 and G1 take the same functional form as for the deterministic problem (36), (37), and only differ through the definition of the cash flow G. If b ¼ Z2 ¼ 0 then the problem is deterministic because there is no stock allocation, and G reduces to (38). n The optimal investment strategy y is static:

ynH ¼ 

rZ2 ðH ðtÞ þ fÞðttÞ e F ðt,H,H þ f1Þ sHðtÞ

ð53Þ

and it can be directly compared with the terminal wealth problem in Henderson (2005, Table 1, Normal case). We can deduce the sensitivity of the investment strategy to changes in the parameters explicitly. The main features of the control n are that the income hedge, yH , is independent of risk aversion d, the signs of r and b determine whether the member should go long or short the stock, and the stock allocation is monotonic in the risk aversion, so the magnitude of the stock allocation decreases as risk aversion increases. The optimal consumption cn is dynamic with components cny ¼ eðH ðtÞ þ fÞðttÞ F ðt,H,H þ f1Þy, 1 H ðtÞðttÞ

cns ¼ d

e

ðF ðt,G,HÞlogðr a SÞÞ:

ð54Þ

Again, the controls are linear in the state variables, so we can calculate the integrals in (52) numerically in order to determine the mean optimal controls. Results from these computations are shown in Fig. 6, and reveal the sensitivity of the optimal controls to the volatility, and the correlation coefficient. Retirement income is provided by an annuity. It is clear that consumption is relatively insensitive to changes in s, Z2 or r, whereas increasing these quantities leads to a lower mean optimal stock allocation. As income and stock become more correlated, the stock allocation decreases for similar reasons to the perfectly correlated results in Section 4.

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40

38

35

36

30

34 32

20

c

θ

25 σ=0.2

15 5 0

5

10

15

22

20

t

25

30

35

40

20

45

40

38

35

36

5

10

15

20

t

25

30

35

40

45

35

40

45

35

40

45

32

25

c

θ

η2=0.0

20

0.1

15

0

5

10

15

30 28

η2=0.0

26

0.2

24

0.2

10

22 20

t

25

30

35

40

20

45

40

38

35

36

0

5

10

15

20

t

25

30

34

30

32 ρ=0.25,

0.5

20

c

θ

25

30 ρ=0.25 0.75

28 26

0.75

15

24

10 5

0

34

30

5

0.3

24

0.3 0

σ=0.2

28 26

0.25

10

30

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5

10

15

20

t

25

30

35

40

20

45

0

5

10

15

20

t

25

30

Fig. 6. Mean optimal controls for an OU income model. In each row of graphs we vary one parameter with the other two chosen from stock volatility s ¼ 0:2, income volatility Z2 ¼ 0:1, and correlation between stock and income r ¼ 1:0.

Closed-form expressions (rather than integrals) for the optimal controls can be found if the pensioner purchases a perpetuity on retirement. In this case H ¼ H ¼ r ¼ r a and G0 ðtÞ ¼ erðttÞ ðlogðrSÞF ðt,G,r1ÞÞ,

G1 ðtÞ ¼ r dDr þ f ðtÞ,

ð55Þ

where 2

F ðt,G,r1Þ ¼ gDr ðtÞ þ

1 2 r d Z22 ð1r2 Þ r dðbrZ2 fym Þ ðDf ðtÞDr ðtÞÞ 2 ðDr þ 2f ðtÞ2Df ðtÞ þ Dr ðtÞÞ, r þf ðr þ fÞ2

where the discounted unit cash flow Dm ðtÞ is defined in (13). n Consequently the income hedge yH is static:

ynH ðtÞ ¼ 

rZ2 D ðtÞ s rþf

P. Emms / Journal of Economic Dynamics & Control 36 (2012) 1303–1321

1319

while the optimal consumption terms are cny ¼ rDr þ f ðtÞy,

1 rðttÞ

cns ¼ d

e

ðF ðt,G,r1ÞlogðrSÞÞ:

Again, the magnitude of the static consumption term, cns , is set by the replacement condition (35). On substituting (55) into (34) and then integrating, we obtain the following explicit expression for the retirement preference: " # k dðaðtÞRY ðtÞIðtÞÞ exp a¼ , Dr ð0Þ rð1ekðTtÞ Þ where IðtÞ ¼

Z

t 0

ð1rDr þ f ðuÞÞY ðuÞ þ

b 1 ðbrZ2 r dDr þ f ðuÞÞ erðtuÞ F ðu,G,r1Þ du d dr

is a sum of exponential functions, since Y ðtÞ ¼ ðy0 ym Þeft þ ym : As a result, the retirement preference is a complicated function of the model parameters, and this makes it infeasible to perform a sensitivity analysis via comparative statics. 6. Conclusions We have studied the optimal investment and contribution to a DC pension fund when the scheme member’s income is stochastic. The model is similar to the classical lifecycle model, but the two distinct phases of a DC scheme are modelled explicitly, and the form of income in retirement is prescribed. Specifically, we have introduced a parameter that reflects a scheme member’s preference for consumption in retirement over consumption during employment. This parameter is a function of the income model and the desired replacement ratio at retirement. The model is applicable to countries where annuitisation at retirement is compulsory, or where a pensioner chooses at the outset to receive retirement income in the form of an annuity. The other principal assumptions, which make the model tractable, are constant absolute risk aversion, and fixed retirement age. If the value function corresponding to the optimisation problem is log-linear and the market is complete then one can find analytical expressions for the optimal stock and consumption strategies. We illustrated these controls using a positive income model, where we found explicit results for the optimal strategies in feedback form. For a stochastic, two factor incomplete model, analytical expressions for the optimal control laws have been found for two income models. For a CIR income model with no correlation between stock and income, the optimal stock allocation is given by the Merton model, and the optimal consumption has been described qualitatively by a phase diagram. For an Ornstein–Uhlenbeck income process, we calculated a numerical example where positive correlation between stock and income significantly decreases the mean optimal stock allocation, and slightly decreases the mean optimal consumption of the scheme member. There are a number of areas where we seek to generalise the model. If we incorporate a power law or Epstein and Zin (1989) recursive utility function then we will lose analytical tractability, which makes it difficult to assess the properties of the optimal control. In this case, a discrete time numerical solution is required, and the analytical results contained herein can used to test such a numerical model. If one assumes that the scheme member’s lifetime is stochastic then one also loses tractability. In addition, if we make the retirement time a control then we need to introduce a dis-utility of working, so that there is some incentive other than wealth generation for the member. Early on in the scheme, the model often indicates that it is optimal to borrow funds. In the current climate, it would therefore seem appropriate to maintain a separate borrowing account with constraints related to income or future expected income. Constraints on the controls are therefore also a future line of research. Appendix A. Variance of the optimal controls The optimisation problem simplifies considerably if the value function is log-linear in current income. Under this condition, (31) and (32) give the variance of the optimal controls as   rG1 ðtÞ 2 n Varðy Þ ¼ VarðZðt,Y t ÞÞ, ðA:1Þ sdHðtÞ Varðcn Þ ¼ HðtÞ2 VarðX nt Þ þ d

2

G1 ðtÞ2 VarðY t Þ2d

1

HðtÞG1 ðtÞCovðX nt ,Y t Þ:

ðA:2Þ

Thus, we can assess the consumption and pension fund risk associated with adopting the optimal strategy, provided that we can calculate the raw moments of the optimal state variables. Using Itˆo’s lemma with the SDEs (18), (33) and the Itˆo product rule we find   b 1 1 ðb þ rZðt,Y t ÞG1 ðtÞÞX nt þ d G0 ðtÞX nt dt dðX nt Þ2 ¼ 2 ðrHðtÞÞðX nt Þ2 þ ð1 þ d G1 ðtÞÞX nt Y t þ dHðtÞ

1320

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þ 2X nt







2

b þ rZðt,Y t ÞG1 ðtÞ b þ rZðt,Y t ÞG1 ðtÞ dW 1t þ dHðtÞ dHðtÞ

dt,

dðY t Þ2 ¼ 2Y t ðnðt,Y t Þ dt þ Zðt,Y t Þ dW 2t Þ þ Zðt,Y t Þ2 dt,  b 1 1 ðb þ rZðt,Y t ÞG1 ðtÞÞY t dðX nt Y t Þ ¼ nðt,Y t ÞX nt þ ðrHðtÞÞX nt Y t þð1 þ d G1 ðtÞÞY 2t þ d G0 ðtÞY t þ dHðtÞ     b þ rZðt,Y t ÞG1 ðtÞ b þ rZðt,Y t ÞG1 ðtÞ dt þ Y t dW 1t þ Zðt,Y t ÞX nt dW 2t : þ rZðt,Y t Þ dHðtÞ dHðtÞ Applying the expectation operator to the above SDEs under P, and swapping the order of integration, we find the second-order moment equations   dX 2 b 1 1 1 2 ¼ 2 ðrHðtÞÞX 2 þ ð1 þ d G1 ðtÞÞXY þ ðbX þ rG1 ðtÞZX Þ þ d G0 ðtÞX þ 2 ðb þ 2rbG1 ðtÞZ þ r2 G1 ðtÞ2 Z2 Þ, dt dHðtÞ d HðtÞ2 dY 2 ¼ 2nY þ Z2 , dt dXY b r 1 1 ¼ nX þ ðrHðtÞÞXY þ ð1 þ d G1 ðtÞÞY 2 þ ðbY þ rG1 ðtÞZY Þ þ d G0 ðtÞY þ ðbZ þ rG1 ðtÞZ2 Þ, dt dHðtÞ dHðtÞ where we have dropped the star from the pension fund state variable and used overbars to denote expectation. The firstorder moment equations are dX b 1 1 ¼ ðrHðtÞÞX þ ð1 þ d G1 ðtÞÞY þ ðb þ rG1 ðtÞZ Þ þ d G0 ðtÞ, dt dHðtÞ dY ¼n dt

ðA:3Þ

directly from (33) and (18). Using these equations, the evolution equations for the variances and covariance are dðVarðXÞÞ 2brG1 ðtÞ 1 1 2 ¼ 2ðrHðtÞÞVarðXÞ þ 2ð1 þ d G1 ðtÞÞCovðX,YÞ þ CovðZ,YÞ þ 2 ðb þ 2rbG1 ðtÞZ þ r2 G1 ðtÞ2 Z2 Þ, dt dHðtÞ d HðtÞ2 dðVarðYÞÞ ¼ 2Covðn,YÞ þ Z2 , dt dðCovðX,YÞÞ brG1 ðtÞ r 1 ¼ Covðn,XÞ þ ðrHðtÞÞCovðX,YÞ þ ð1þ d G1 ðtÞÞVarðYÞ þ CovðZ,YÞ þ ðbZ þ rG1 ðtÞZ2 Þ: dt dHðtÞ dHðtÞ

ðA:4Þ

The form of income model dictates whether we can use these equations in order to find the variance of the controls using (A.1) and (A.2). For example, if the income model is lognormal (46) and there is perfect correlation between income and stock then the value function is log-linear, and

ZX ¼ Z0 XY , nX ¼ fym X fXY , n ¼ fðym Y Þ, Z ¼ Z0 Y , Z2 ¼ Z20 Y 2 , nY ¼ fym Y fY 2 , ZY ¼ Z20 Y 2 : Consequently, (A.3) and (A.4) are a coupled system of five linear ODEs, which can be solved by numerical integration using G0 ðtÞ and G1 ðtÞ given by (36) and (37). References Ballota, L., Haberman, S., 2003. Valuation of guaranteed annuity conversion options. Insurance: Mathematics and Economics 33, 87–108. Blake, D., Cairns, A.J.G., Dowd, K., 2001. PensionMetrics: stochastic pension plan design and value-at-risk during the accumulation phase. Insurance: Mathematics and Economics 29, 187–215. Blake, D., Cairns, A.J.G., Dowd, K., 2003. Pensionmetrics 2: stochastic pension plan design during the distribution phase. Insurance: Mathematics and Economics 33, 29–47. Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J., 1986. Actuarial Mathematics. The Society of Actuaries. Cairns, A.J.G., Blake, D., Dowd, K., 2006. Stochastic lifestyling: optimal dynamic asset allocation for defined contribution pension plans. Journal of Economic Dynamics and Control 30 (5), 843–877. Campbell, J.Y., Viceira, L.M., 1999. Consumption and portfolio decisions when expected returns are time varying. Quarterly Journal of Economics 114 (2), 433–495. Campbell, J.Y., Viceira, L.M., 2001. Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. Oxford University Press, New York. Choi, K.J., Shim, G., 2006. Disutility, optimal retirement, and portfolio selection. Mathematical Finance 16 (2), 443–467. Cocco, J.F., Gomes, F.J., Maenhout, P.J., 2005. Consumption and portfolio choice over the life cycle. The Review of Financial Studies 18 (2), 491–533. Epstein, L.G., Zin, S.E., 1989. Substitution, risk aversion, and the temporal behaviour of consumption and asset returns: a theoretical framework. Econometrica 57 (4), 937–969.

P. Emms / Journal of Economic Dynamics & Control 36 (2012) 1303–1321

1321

Friedman, M., 1957. A Theory of the Consumption Function. National Bureau of Economic Research. General Series, No. 63. Princeton University Press. Henderson, V., 2005. Explicit solutions to an optimal portfolio choice problem with stochastic income. Journal of Economic Dynamics and Control 29 (7), 1237–1266. Kahl, M., Liu, J., Longstaff, F.A., 2003. Paper millionaires: how valuable is stock to a stockholder who is restricted from selling it? Journal of Financial Economics 67, 385–410 Karatzas, I., Shreve, S.E., 1991. Brownian Motion and Stochastic Calculus. Springer-Verlag. Keynes, J.M., 1936. The General Theory of Employment, Interest, and Money. Prometheus Books. Kloeden, P.E., Platen, E., 1999. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin. Koo, H.K., 1998. Consumption and portfolio selection with labor income. Mathematical Finance 8 (1), 49–65. Liu, J., 2007. Portfolio selection in stochastic environments. Review of Financial Studies 20 (1), 1–39. Lunnon, M., 2002. Annuitization and alternatives. In: Proceedings of the Transactions of the 27th International Congress of Actuaries, Cacun. Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: the continuous-time case. Review of Economics and Statistics 51 (August), 247–257. Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373–413. Merton, R.C., 1990. Continuous-time Finance. Blackwell. Pratt, J.W., 1964. Risk aversion in the small and in the large. Econometrica 32 (1–2), 122–136. Samuelson, P.A., 1969. Lifetime portfolio selection by dynamic stochastic programming. Review of Economics and Statistics 51 (August), 239–246. Svensson, L.E.O., Werner, I.M., 1993. Nontraded assets in incomplete markets. European Economic Review 37, 1149–1168. Tepla´, L., 2000. Optimal hedging and valuation of nontraded assets. European Finance Review 4, 231–251. Turner, A., Drake, J., Hills, J., 2006 (April). Implementing an Integrated Package of Pension Reforms: The Final Report of the Pensions Commission. Zeldes, S.P., 1989. Optimal consumption with stochastic income: deviations from certainty equivalence. The Quarterly Journal of Economics 104 (2), 275–298.