Explicit solutions to an optimal portfolio choice problem with stochastic income

This paper solves in closed form the optimal portfolio choice problem for an investor with negative exponential utility over terminal wealth facing imperfectly hedgeable stochastic income. The returns on income and the stock are imperfectly correlated, so the market is incomplete. We identify how an investor adjusts their [Merton, Rev. Econ. Statist. 51 (1969) 247] portfolio of the stock and riskless asset via an intertemporal hedging demand, in reaction to the stochastic income. Under general assumptions on the process governing the income, the sign of the hedging demand is the opposite of the sign of the correlation between the income and stock. The optimal portfolio in the stock is long stock if the risk premium is positive and correlation negative, and short if these signs are reversed. Specializing in turn to normally distributed income and lognormally distributed income with or without mean-reversion, the effect of a number of parameters on the optimal portfolio in the stock can be studied. When the risk premium on the stock and correlation have opposite signs, the optimal portfolio decreases in magnitude with risk aversion, unhedgeable variance of income and time, and increases in magnitude with the magnitude of correlation under both the lognormal (without mean-reversion) and normal models for income. These relationships do not necessarily hold if the signs on the risk premium and correlation are the same, in particular the optimal portfolio is not necessarily monotone in risk aversion, violating well-known static results in models without income. When income follows a lognormal model with mean-reversion, more complicated behavior can occur. For instance, the optimal portfolio does not have to be monotonic in time, regardless of the signs of correlation and the risk premium.