A Word of Caution on the Modified Distribution

By Peter Urbani*

It has become popular to use the Cornish Fisher modification to the normal distribution to add the impact of skewness and kurtosis to fund distributions and Value at Risk (VaR) calculations. This is generally called mVaR OR ‘modified’ VaR.

Unfortunately the Cornish Fisher modification suffers from two rather serious drawbacks that some users may not be fully aware of. Firstly it is not strictly monotone with respect to the confidence level and skewness being used. This means that it is possible to get a VaR number at a higher confidence (e.g. 99%) level that is lower than one at a lower confidence level (e.g. 95%.)

In the above example the 95th percentile VaR for this fund is higher than the 99th percentile VaR

Secondly the Cornish Fisher modification also suffers from Poor Tail behaviour particularly at high confidence levels where you need it the most. These may cause the Cumulative Distribution (CDF) function to turn either in the body or more seriously in the tail of the distribution.

As you can see from the examples shown these problems occur even at low levels of excess skewness and kurtosis and not only at high levels as some people believe. Moreover, it affects fully 50% of all hedge funds so great caution should be used particularly when using this calculation at confidence levels above 95%.

A simple test for the appropriateness of the Cornish Fisher expansion can be downloaded here.

How Prevalent is this problem ?

VERY


*Peter Urbani – is the former Chief Investment Officer of Infiniti Capital, a now defunct Hong Kong based Fund of Funds group.

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