Optimal portfolios under bounded shortfall risk and partial information.

R. Wunderlich, J. Saß and A. Gabih

Operations Research Proceedings 2006, accepted

Abstract :
This paper considers the optimal selection of portfolios for utility maximizing investors under a shortfall risk constraint for a financial market model with partial information on the drift parameter. It is known that without risk constraint the distribution of the optimal terminal wealth often is quite skew. In spite of its maximum expected utility there are high probabilities for values of the terminal wealth falling short a prescribed benchmark. This is an undesirable and unacceptable property e.g. from the viewpoint of a pension fund manager. While imposing a strict restriction to portfolio values above a benchmark leads to considerable decrease in the portfolio's expected utility, it seems to be reasonable to allow shortfall and to restrict only some shortfall risk measure. A very popular risk measure is value at risk (VaR) which takes into account the probability of a shortfall but not the actual size of the loss. Therefore we use the so-called expected loss criterion resulting from averaging the magnitude of the losses. And in fact, e.g. in Basak, Shapiro (2001) it is shown that the distribution of the resulting optimal terminal wealth has more desirable properties. We use a financial market model which allows for a non-constant drift which is not directly observable. In particular, we use a hidden Markov model (HMM) where the drift follows a continuous time Markov chain. In Sass and Haussmann (2004) it was shown that on market data utility maximizing strategies based on such a model can outperform strategies based on the assumption of a constant drift parameter. Extending these results to portfolio optimization problems under risk constraints we obtain quite explicit representations for the form of the optimal terminal wealth and the trading strategies which can be computed using Monte Carlo methods.