|Simulation|Modified VaR| Correlation| Regression| Cholesky Matrix| Distribution| Diversification|
|Skewness| Sharpe| Fund of Funds |4 Moment CAPM| Stress Test| Coskewness| Black-Litterman|
Skewness measures the distribution asymmetry. A risk-averse investor does not like negative skewness. A distribution with positive skewness has more returns far away to the right of its mean return, as shown below.




Kurtosis measures the fat-tail degree of a distribution.  A risk-averse investor prefers a distribution with low kurtosis (i.e. returns not far away from the mean).  A distribution with positive kurtosis exhibits a peak in the middle and fat tails versus a normal distribution.  A distribution to be normal should have an excess kurtosis equal to 0.  In case of a positive skewness, it is possible to have a high excess kurtosis and to have no future extreme negative returns. The extreme returns will only be positive. This is only possible when the skewness is positive. As soon as the skewness is negative, the impact of a high excess kurtosis affects the extreme negative returns.

The most interesting thing is when the return distribution has a skewness lower than -1 and an excess kurtosis higher than 1. In this case, the probability to have sudden high negative returns increases. For a distribution with a skewness of -1 and an excess kurtosis of 5 (for example, technology stocks, media stocks, telecom stocks or hedge funds in arbitrage strategies), a claasical approach will conclude that the investor will not lose more than -3.5% in the next 1 day with 99% probability.  An approach, accounting for skewness and kurtosis, shows a -7.4% loss in the next 1 day with 99% probability.  This is exactly what one observes on the equity market.  The difference is large: an underestimation of 111% of the downside risks (i.e. 7.4%/3.5%-1=111%) when using volatility only to measure the risk.

To conclude, for optimization and simulation, the technique must account for volatility, skewness, and kurtosis as soon as they are significant and as soon as the investor is risk averse.