# Normal VaR, Modified VaR & Fat-Tailed VaR

The Differences between Normal VaR, Modified VaR and “Fat-Tailed” VaR:

In commercial products, VaR is widely used in combination with the normal distribution. In order to deal with the shortcomings of Normal VaR (as described above), Modified Value-at-Risk (mVaR) is often used. The calculation of mVaR is not tied to a distributional assumption. It can be viewed as a non-parametric estimator of the empirical VaR, employing the first four moments computed from the observed returns (Mean, Standard Deviation, Skewness, and Kurtosis).

Some in the industry consider mVaR to be a “Fat-tailed” VaR. This statement is only true to the extent that it distinguishes mVaR from the Normal VaR methodology based on the normal distribution. However, mVaR is a non-parametric estimator and while it has some of its own merits, it is not much different than any other non-parametric estimator and has many deficiencies including:

a. It becomes less reliable for probabilities close to 0 or 1. That is, the deeper we go into the left or the right tail, the worse the approximation gets. In effect, VaR at high confidence levels get more inaccurate. Basically, VaR at 99% cannot be reliably calculated.

b. It works well only for non-normal distributions which are “close” to the normal distribution and not for those which deviate significantly. As a result, it will not work well for distributions with high degree of skewness or fat tails.

Therefore associating mVaR with the term “Fat-tailed” VaR is inaccurate. Fat-tailed VaR is VaR based on a non-normal distributional assumption for risk factor returns. Recognizing that returns are fat-tailed and skewed, one needs to explicitly assume that an appropriate non-normal distribution is necessary to model these properties. In this report, we use the Skewed Student’s t distribution to compute a true and more accurate fat-tailed VaR number.

Investors who are required to select and monitor investment managers should develop a basic understanding of investment statistics. Quantitative tools can provide you with good insight that you can use in your qualitative interviews with managers and when monitoring your investments.