The canonical continuous time consumption-portfolio choice model was intro- duced byMerton (1969,1971). In his framework, uncertainty is generated by a d-dimensional Brownian motionW and prices/state variables follow a vector diffu- sion process. The investor has a finite horizonŒ0; T . The presentation, throughout this review, focuses on the special case of complete financial markets.
25.2.1 Financial Market
The financial market consists ofd risky assets and a riskless asset. The riskless asset is a money market account that pays interest at the rater .t; Yt/, whereY is ady- dimensional vector of state variables. Risky assets are dividend-paying stocks, with returns evolving according to
dRt D.r .t; Yt/1ı .t; Yt// dtC .t; Yt/ . .t; Yt/ dtCd Wt/ ; S0given d YtDY .t; Yt/ dtCY.t; Yt/ d Wt; Y0given.
(25.1) The vectorRis thed 1vector of cumulative stock returns, 1 .1; :::; 1/0is the d 1 vector of ones,ı .t; Yt/is thed 1 vector of dividend yields and .t; Yt/ thedd matrix of return volatility coefficients. The volatility matrix is assumed to be invertible, ensuring that all risks are hedgeable (the market is complete). The quantity .t; Yt/is the market price of Brownian motion risk, given by .t; Yt/ .t; Yt/1. .t; Yt/r .t; Yt/1/ where .t; Yt/ is the vector of instantaneous expected stock returns. All the coefficients of the return process depend the vector of state variablesY, that satisfies the stochastic differential equation described on the second line of (25.1). The coefficients of this equation,Y .t; Yt/ ; Y.t; Yt/, are assumed to satisfy standard conditions for the existence of a unique strong solution (seeKaratzas and Shreve 1991, p. 338).
The state price density (SPD) implied by the return process (25.1) is t Dexp
Z t
0
r .s; Ys/ ds Z t
0
.s; Ys/0d Ws1 2
Z t
0
.s; Ys/0 .s; Ys/ ds
: (25.2) The SPDt represents the stochastic discount factor that can be used for valuation at date0of cash flows received at the future datet.
The conditional state price density (CSPD) is defined ast;vv=t. It represents the stochastic discount factor for valuation att of random cash flows received at vt.
25.2.2 Choices and Preferences
An investor operating in the market above will consume, invest and leave a bequest at the terminal date. A consumption policyc is a nonnegative stochastic process, adapted to the Brownian filtration. A bequest policyXTis a measurable nonnegative random variable at the terminal date. A portfolio policy is a d-dimensional adapted stochastic process, representing the fractions of wealth invested in the risky stocks. Portfolio components are allowed to take negative values (short sales are permitted).
A consumption-bequest-portfolio policy.c; X; /generates the wealth process Xgiven by
dXt D.Xtr .t; Yt/ct/ dtCXtt0 .t; Yt/ . .t; Yt/ dtCd Wt/ (25.3) subject to the initial conditionX0Dx, wherexis initial wealth.
Investor preferences are defined over consumption-bequest policies. Preferences are assumed to have the von Neumann-Morgenstern (expected utility) representation
E Z T
0
u.cv;v/ dvCU .XT; T /
; (25.4)
where u.cv;v/is the instantaneous utility of consumption at date v andU .XT; T / is the utility of terminal bequest. Utility functions uW ŒAu;1/Œ0; T !Rand U W ŒAU;1/ ! R, are assumed to be twice continuously differentiable, strictly increasing and strictly concave. Marginal utilities are zero at infinity. They are assumed to be infinite atAu; AU. IfAu; AU > 0, the utility functions are extended over the entire positive domain by setting u.c;v/ D 1; U .X; T / D 1for c2Œ0; Au/,X 2Œ0; AU/.
A standard example of utility function is the Hyperbolic Absolute Risk Aversion (HARA) specification
u.c; t/D 1
1R.cAu/1R;
whereR > 0andAuis a constant (Aucan be positive of negative).
The inversesI W RC Œ0; T ! ŒAu;1/and J W RC ! ŒAU;1/ of the marginal utility functions u0.c; t/ and U0.X; T / play a fundamental role. Given the assumptions above, these inverses exist and are unique. They are also strictly decreasing with limiting values limy!0I .y; t/ D limy!0J .y; T / D 1 and limy!1I .y; t/DAu,limy!1J.y; T /DAU.
Throughout the paper it will be assumed that initial wealth is sufficient to finance the minimum consumption level. This condition isx EhRT
0 vACu dvCTACU i
, whereACmax.0; A/.
25.2.3 The Dynamic Choice Problem
The investor maximizes preferences over consumption, bequest and portfolio policies. The dynamic consumption-portfolio choice problem is
max
.c;XT;/E Z T
0
u.cv;v/dvCU.XT; T /
(25.5) subject to the constraints
dXt D.Xtr .t; Yt/ct/ dtCXtt0 .t; Yt/ . .t; Yt/ dtCd Wt/I X0Dx (25.6)
ct 0; Xt 0 (25.7)
for all t 2 Œ0; T . Equation (25.6) is the dynamic evolution of wealth. The first constraint in (25.7) is the nonnegativity restriction on consumption. The second (25.7) is a no-default condition, imposed to ensure that wealth is nonnegative at all times, including the bequest time.
25.2.4 The Static Choice Problem
Pliska(1986),Karatzas et al. (1987) and Cox and Huang (1989) show that the dynamic problem is equivalent to the following static consumption-portfolio choice problem
max.c;X/E Z T
0
u.cv;v/ dvCU .XT; T /
(25.8) subject to the static budget constraint
E Z T
0 scsCTXT
x (25.9)
and the nonnegativity constraintsc 0andXT 0. Equation (25.9) is a budget constraint. It mandates that the present value of consumption and bequest be less than or equal to initial wealth. The objective in (25.8) is to maximize lifetime utility with respect to consumption and bequest, which satisfy the usual nonnegativity restrictions.
In the static problem (25.8) and (25.9) there is no reference to the portfolio, which is treated as a residual decision. The reason for this is because of market completeness. Once a consumption-bequest policy has been identified, there exists a replicating portfolio that finances it.