Optimal Portfolios under a Value at Risk Constraint

Abstract

Recently, financial institutions discovered that portfolios with a limited Value at Risk often showed returns that were close to the VaR and had large losses in the exceptional cases where losses exceeded VaR. In this paper we consider the construction of portfolios with options that maximize expected return with a restriction on the Value at Risk. These theoretically optimal portfolios indeed have the properties as experienced by financial institutions and illustrate that maximizing under a VaR-constraint is very dangerous. We also show that if one considers market prices of options there will be an even higher impetus to go for gambling portfolios.