Merton's portfolio problem
Merton's portfolio problem is a well known problem in continuous-time finance and in particular intertemporal portfolio choice. An investor must choose how much to consume and must allocate his 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.
The investor lives from time 0 to time T; his wealth at time t is denoted Wt. He starts with a known initial wealth W0 (which may include the present value of wage income). At time t he must choose what amount of his 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
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, and ρ is the subjective discount rate.
The wealth evolves according to the
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:
where is a constant which expresses the investor's risk aversion: the higher the gamma, the more reluctance to own stocks.
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