Portfolio Policies with Unobservable
Markov Modulated Drift Process and Bounded Expected Loss.
J. Saß and R. Wunderlich
In: C. Fernandes, H. Schmidli, N.Kolev (eds.): Proceedings of the Third Brazilian Conference: on Statistical Modelling in Insurance and Finance, Maresias, March 25-30, 2007, Institute of Mathematics and Statistics, University of Sao Paulo, 242-247, 2007.
In this paper we consider the computation of optimal portfolio strategies for utility maximizing investors under a shortfall risk constraint based on a financial market model with partial information on the Markov modulated drift process. Without a risk constraint the distribution of the optimal terminal wealth often is quite skew and 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. Since a strict restriction to portfolio values above a benchmark leads to a considerable decrease in the optimal portfolio's expected utility, it seems to be reasonable to allow shortfall and to restrict only some shortfall risk measure. Value at risk (VaR) is a widely used risk measure and 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. Basak and Shapiro (2001) show in Basak and Shapiro (2001) that the distribution of the resulting optimal terminal wealth has more desirable properties. We use a hidden Markov model (HMM) where the drift follows a continuous time Markov chain. Sass and Haussmann (2004) show in that on market data utility maximization strategies based on such a model can outperform strategies based on the assumption of a constant drift parameter. Extending these results we obtain quite explicit representations for the form of the optimal terminal wealth and the trading strategies which we compute by using Monte Carlo methods. The same model was used in Gabih et al. (2005). Here we use a different class of risk constraints, consider non-constant interest rates and can give more explicit results on the existence and uniqueness of an optimal solution.