A note on portfolio selections under various risk measures

Enrico G. De Giorgi

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Risk and Portfolio Optimization Stochastic processes and financial applications Financial Risk and Volatility Modeling

Abstract

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 that can be written as a particular combination of mean and variance and we show that for these measures Tobin separation holds under some restrictions.

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References

1999Coherent Measures of Risk 2000Optimization of conditional value-at-risk 2002Conditional value-at-risk for general loss distributions 1997An Overview of Value at Risk 2001Value at Risk 2000Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk 2002Trust Breeds Trust: How Taxpayers Are Treated 2002Economic implications of using a mean-VaR model for portfolio selection: A comparison with mean-variance analysis

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