Risk measures and portfolio optimization

In this paper we investigate portfolio optimization under Value at Risk, Average Value at Risk and Limited Expected Loss constraints in a continuous time framework, where stocks follow a geometric Brownian motion. Analytic expressions for Value at Risk, Average Value at Risk and Limited Expected Loss are derived. We solve the problem of minimizing risk measures applied to portfolios. Moreover, the portfolio's expected return is maximized subject to the aforementioned risk measures. We illustrate the effect of these risk measures on portfolio optimization by using numerical experiments.