Dynamic Mean-Downside Risk Portfolio Selection Problem with Stochastic Interest Rate in Continuous-Time

Abstract

Even though a consensus has been reached long time ago to formulate the interest rate as a stochastic process, most of existing research works on dynamic mean-downside risk portfolio selection problem are still confined to the framework of deterministic interest rate. This work studies a dynamic mean-downside risk portfolio selection problem with a stochastic interest rate in continuous-time financial market. More specifically, we choose the lower-partial moments(LPM), value-at-risk(VaR) and conditional value-at-risk(CVaR) to model our downside risk measures and utilize the Vasicek model to formulate the stochastic interest rate. However, it is not easy to solve these problems analytically since the downside risk measure together with the stochastic interest rate greatly complicates the solution scheme for such a class of portfolio selection problems. By using the martingale method and the inverse Fourier Transformation approach, we successfully derive the semi-analytical optimal portfolio policies and the optimal wealth processes of our models. Our results suggest that the optimal portfolio policies and the optimal wealth processes are strongly correlated to the stochastic interest rate. Finally, we provide some illustrative examples to demonstrate that the portfolio selection model with a stochastic interest rate has an advantage over the model with a deterministic interest rate. Moreover, our illustrative example also shows that the optimal portfolio policy and optimal wealth process of our dynamic mean-downside risk portfolio selection model with a stochastic interest rate differ significantly from the ones generated from the expected utility maximization (EUM) portfolio selection model with a stochastic interest rate.

Keywords: Dynamic portfolio selection, stochastic interest rate, Vasicek model, lower-partial moments, value-at-risk, conditional value-at-risk