Portfolio optimization: Volatility constraints versus shortfall constraints

In this paper we examine two models of portfolio optimization. Volatility (standard deviation) constraints as well as shortfall constraints are considered and compared. We present a general condition under which the restriction to a certain risk level concerning volatility can be transformed to a special shortfall constraint and vice versa. We show that under this condition the Value at Risk of a portfolio can be easily calculated and restricted using the variance of the portfolio even if the return distribution is asymmetric. Finally, we give two examples of portfolio optimization where we show how the derived condition can be applied.