Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences

This paper investigates the dynamic consumption and portfolio choice of an investor with habit formation in preferences and access to a complete financial market with time-varying investment opportunities. An exact and simple characterization of the optimal behavior under general, possibly non-Markov, dynamics of market prices is derived. Relative to the benchmark case of time-additive power utility, habit formation affects the hedge component of the optimal portfolio differently than the speculative component. The quantitative effects of habit formation are studied in three concrete settings. Firstly, a closed-form solution of the optimal consumption and portfolio choice with mean-reverting stock returns is derived. Secondly, with Cox-Ingersoll-Ross interest rate dynamics the optimal strategies are expressed in terms of the solution to a partial differential equation, which has an explicit solution for time-additive utility, but not with habit formation. Thirdly, a new model with both mean-reverting stock returns and stochastic interest rates is studied. Overall, the numerical examples show that, while hedging demands for various assets are affected differently by habit persistence, the main effect on relative asset allocations stems from the fact that some assets (bonds and cash) are better investment objects than others (stocks) when it comes to ensuring that future consumption will not fall below the habit level.