# Modified Value-at-Risk (MVaR)

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February 11, 2009
February 11, 2009
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The concept and models of the MVaR

The Modified VaR (MVaR) methodology was suggested by Favre and Galeano (2000), and further introduction to that approach is based on their research. First, they analyze the use of mean-variance analysis against the Sharpe approach for ranking of hedge funds. As discussed in previous chapter, the CAPM framework assumes that returns are normally distributed and investors’ preferences are quadratic meaning that all investors are equally risk averse. They pursue the only interest of maximizing the expected utility of their end of period wealth.

However, in real life, ranking of hedge funds fully depends on the preferences and the degree of risk aversion of investors. The concept of utility functions presents the way obtaining such a ranking. Favre and Galeano compared rankings of hedge fund using power and exponential utility functions with rankings by quadratic approximation.

The results show that the applicability of the analyzed methods is subject to an investor’s degree of the risk aversion. An investor with a low risk aversion may use the mean-variance methodology as it provides an acceptable level of the approximation. However, a high risk averse investor should use the Sharpe ratio.

To overcome the normality assumption of the traditional Sharpe ratio, the suggested approach deploys the Cornish-Fisher Asymptotic Expansion. The Normal VaR is expressed then through the standard deviation and mean as where Zc = critical value for the percentile (1-c).

The Cornish-Fisher Asymptotic Expansion adjusts the standard normal deviates to take into account skewness and kurtosis of the distribution where
S = skewness
K = kurtosis excess.
Therefore, the MVaR is expressed as For the normal distribution S and K are equal to zero, thus Zcf collapses to a standard normal deviate. In fact, the MVaR presents a parametric method of the VaR calculation considering irregularities of the distribution as asymmetry and long tail.

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