Optimal Consumption and Investment Strategies with Stochastic Interest Rates

We study the consumption and investment choice of a time-additive power utility investor and demonstrate how the investor should optimally hedge changes in the opportunity set. The investor is allowed to invest in stocks and interest rate dependent assets in a continuous-time dynamically complete market. In particular, we demonstrate that under stochastic interest rates the investor optimally hedges changes in the term structure of interest rates by investing in a coupon bond, or portfolio of bonds, with a payment schedule that matches the forward-expected (i.e certainty equivalent) consumption pattern. This is of conceptual importance since the hedge portfolio does not depend on the specific term structure dynamics (only through the consequences for the optimal consumption pattern). We consider two explicit examples where the dynamics of the term structure of interest rates are given by the Vasicek-model and a three-factor non-Markovian Heath-Jarrow-Morton model.