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.