Merton's portfolio problem is a problem in continuous-time finance and in particular intertemporal portfolio choice. An investor must choose how much to consume and must allocate their wealth between stocks and a risk-free asset so as to maximize expected utility. The problem was formulated and solved by Robert C. Merton in 1969 both for finite lifetimes and for the infinite case. Research has continued to extend and generalize the model to include factors like transaction costs and bankruptcy.
Problem statement The investor lives from time 0 to time T; their wealth at time T is denoted WT. They start with a known initial wealth W0 (which may include the present value of wage income). At time t they must choose what amount of their wealth to consume, ct, and what fraction of wealth to invest in a stock portfolio, πt (the remaining fraction 1 − πt being invested in the risk-free asset). The objective is
max E [ ∫ 0 T e − ρ t u ( c t ) d t + ϵ γ e − ρ T u ( W T ) ] {\displaystyle \max E\left[\int _{0}^{T}e^{-\rho t}u(c_{t})\,dt+\epsilon ^{\gamma }e^{-\rho T}u(W_{T})\right]}
where E is the expectation operator, u is a known utility function (which applies both to consumption and to the terminal wealth, or bequest, WT), ε parameterizes the desired level of bequest, ρ is the subjective discount rate, and γ {\displaystyle \gamma } is a constant which expresses the investor's risk aversion: the higher the gamma, the more reluctance to own stocks. The wealth evolves according to the stochastic differential equation
d W t = [ ( r + π t ( μ − r ) ) W t − c t ] d t + W t π t σ d B t {\displaystyle dW_{t}=[(r+\pi _{t}(\mu -r))W_{t}-c_{t}]\,dt+W_{t}\pi _{t}\sigma \,dB_{t}}
where r is the risk-free rate, (μ, σ) are the expected return and volatility of the stock market and dBt is the increment of the Wiener process, i.e. the stochastic term of the SDE. The utility function is of the constant relative risk aversion (CRRA) form:
u ( x ) = x 1 − γ 1 − γ . {\displaystyle u(x)={\frac {x^{1-\gamma }}{1-\gamma }}.}
Consumption cannot be negative: ct ≥ 0, while πt is unrestricted (that is borrowing or shorting stocks is allowed). Investment opportunities are assumed constant, that is r, μ, σ are known and constant, in this (1969) version of the model, although Merton allowed them to change in his intertemporal CAPM (1973).
Solution Somewhat surprisingly for an optimal control problem, a closed-form solution exists. The optimal consumption and stock allocation depend on wealth and time as follows:
π ( W , t ) = μ − r σ 2 γ . {\displaystyle \pi (W,t)={\frac {\mu -r}{\sigma ^{2}\gamma }}.}
This expression is commonly referred to as Merton's fraction. Because W and t do not appear on the right-hand side; a constant fraction of wealth is invested in stocks, no matter what the age or prosperity of the investor.
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