Composition of an efficient portfolio in the Bielecki and Pliska market model

Abstract

We study the continuous time portfolio optimization model due to Bielecki and Pliska where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors. We introduce the functional Q _γ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient γ to the variance when we keep the value of the factor levels fixed. The coefficient γ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for Q _γ at any moment of time. The single-factor case is analyzed in more details. We present a simple asset allocation example featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of the risk-sensitive control theory thereby using an infinite horizon objective featuring the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to γ.