A Note on Portfolio Selection under Various Risk Measures

This work gives a brief overview of the portfolio selection problem following the mean-risk approach first proposed by Markowitz (1952). We consider various risk measures, i.e. variance, value-at-risk and expected-shortfall and we study the efficient frontiers obtained by solving the portfolio selection problem under these measures. We show that under the assumption that returns are normally distributed, the efficient frontiers obtained by taking value-at-risk or expected-shortfall are subsets of the mean-variance efficient frontier. We generalize this result for all risk measures having the form gamma*sigma-m for some positive parameter gamma ,here m is the mean and sigma the variance and we show that for these measures Tobin separation holds under some restriction on the choice of gamma.