Optimal Portfolio Strategies with Stochastic Wage Income : The Case of A defined Contribution Pension Plan

We consider a stochastic model for a defined-contribution pension fund in continuous time. In particular, we focus on the portfolio problem of a fund manager who wants to maximize the expected utility of his terminal wealth in a complete financial market. The fund manager must cope with a set of stochastic investment opportunities and with the uncertainty involved by the labor market. After introducing a stochastic interest rate, we assume a market structure characterized by three assets : a riskless asset, a bond and a stock. Moreover, we introduce a stochastic process for salaries, and develop the model according to the stochastic dynamic programming methodology. We show that the optimal portofolio is formed by three components : a speculative component proportional to the market price of risk of the two risky assets through the relative risk aversion index, an hedging component proportional to the diffusion term of the interest rate, and a preference-free hedging component proportional to the volatilities of the salary process. Finally, after specifying a suitable fucntional form for the drift term of the salary process, we find a close form solution to the asset allocation problem.