The Limitations of Cornish-Fisher

In the MathBox entry from 13 December, we showed the implied distribution of DAX returns when using the Cornish Fisher expansion.

As a rule of thumb, this works nicely as long as the excess kurtosis is smaller than 8. What happens if the implied distribution of returns is really fat-tailed?

The exchange rate of EUR and Swiss Franc between the years 2013 and 2017 showed an excess kurtosis of 590. The Cornish-Fisher transformation between the implied distribution of returns and the standard normal distribution is not bijective any more and leads to weird distributions.

The plots show the implied “distribution” density (that is not a density) for the CHF EUR exchange rate return for a time horizon of 1, 16, and 256 business days, respectively. Applying the Cornish-Fisher formulae for extreme cases like CHFEUR leads to misleading results in the risk reward section of a PRIIPs KID.