with μ = expected asset return (daily or monthly) zp = distance between u and the VaR in number of standard deviation. In other terms, number of standard deviation at (1-zc) or -1.96 with p = 95% probability σ = standard deviation (daily or monthly).
The Modified VaR (MVaR) uses not only the volatility as risk, but as well skewness and kurtosis. The Modified VaR is:
where S is the skewness, K is the excess kurtosis, and zc is the distance between the portfolio returns and its mean in terms of standard deviation number. If the risk is measured only with the volatility, the risk is often underestimated. The assets returns are negatively skewed and have fat tails. Consequently, the volatility alone is not able to account for that. The Modified VaR model has been developed by AlternativeSoft and then published in the Journal of Alternative Investments, Modified Value-at-Risk Optimization with Hedge Funds, Fall 2002. The Modified VaR is applied in AlternativeSoft’s platform.
The skewness and the kurtosis effect is high if the VaR is computed at 99%. For example, for a portfolio invested in 50% stocks and 50% government bonds, computing the risk with standard deviation only underestimates the risk by more than 35%. AlternativeSoft’s platform considerably reduces this problem by computing optimal portfolios using the mean, the standard deviation, the skewness, the kurtosis and by minimizing the probability of having extreme portfolio losses.