Semi-analytical solution for consumption and investment problem under quadratic security market model with inflation risk

There exists strong empirical evidence that all inflation rates, interest rates, market price of risk, and return volatilities of assets are stochastic, which is now a stylized fact. However, to the best of our knowledge, existing models providing solutions to consumption–investment problems do not consider all of the aforementioned stochastic processes. We consider a consumption–investment problem for a long-term investor with constant relative risk aversion utility under a quadratic security market model in which all of the processes are stochastic. We solve a nonhomogeneous linear partial differential equation for the indirect utility function and derive a semi-analytical solution. This study obtains an optimal portfolio decomposed into myopic demand, intertemporal hedging demand, and “inflation hedging demand,” and presents that all three types of demand are nonlinear functions of the state vector. Our numerical analysis presents the nonlinearity and significance of the market timing effect. The cause of this result lies mainly with inflation hedging demand in addition to myopic demand. This result highlights the importance of the market timing effect and inflation hedging demand.