Optimal Portfolio and Consumption Rule with a CIR Model Under HARA Utility

Abstract

In the real-world environments, different individuals have different risk preferences. This paper investigates the optimal portfolio and consumption rule with a Cox–Ingersoll–Ross (CIR) model in a more general utility framework. After consumption, an individual invests his wealth into the financial market with one risk-free asset and multiple risky assets, where the short-term rate is driven by the CIR model and stock price dynamics are simultaneously influenced by random sources from both stochastic interest rate and stock market itself. The individual hopes to optimize their portfolios and consumption rules to maximize expected utility of terminal wealth and intermediate consumption. Risk preference of individual is assumed to satisfy hyperbolic absolute risk aversion (HARA) utility, which contains power utility, logarithm utility, and exponential utility as special cases. By using the principle of stochastic optimality and Legendre transform-dual theory, the explicit expressions of the optimal portfolio and consumption rule are obtained. The sensitivity of the optimal strategies to main parameters is analysed by a numerical example. In addition, economic implications are also presented. Our research results show that Legendre transform-dual theory is an effective methodology in dealing with the portfolio selection problems with HARA utility and interest rate risk can be completely hedged by constructing specific portfolios.