# The Cornish Fisher Expansion and its Limitations.

## Introduction

In Part 2 of the PRIIPs series we experienced (using the SMI) that daily returns on stocks or stock indices are usually not normally distributed, but that extreme events (such as returns outside twice the standard deviation) occur more frequently than it would be the case with a normal distribution . To take into account this phenomenon, the PRIIPs Delegated Regulation requires to use the Cornish Fisher expansion for Category 2 instruments.

In this article we take a closer look at the Cornish Fisher expansion and point out its limitations.

## The Cornish Fisher Expansion and its Limits

### How does the Cornish Fisher expansion work?

The Cornish Fisher expansion uses not only the mean and standard deviation, but also the skewness and the excess kurtosis to obtain a probability distribution for the returns. So, if the time series of the returns yields a skewness of -0.367 and an excess kurtosis of 2.356 (these are the values for the DAX between July 2013 and June 2018), then the distribution would look like this:

The blue curve shows the probability density of the standard normal distribution, the yellow curve the Cornish Fisher density with the same mean (= 0) and the same standard deviation (= 1), but with the specified values for the skewness and the excess kurtosis. To the left, the yellow curve is significantly higher.

For the quantiles of the distributions, the probability distributions, defined as the the integrals of the probability densities, are relevant.

If we want to know how much the 1% quantile deviates from the mean, we look for the intersection of the green horizontal with the blue or yellow curves and get: -2.326 (blue) or -3.096 (yellow). So 1% of the values are below (normal distribution) or below (Cornish Fisher with skewness and excess kurtosis as specified).

How do we obtain these Cornish Fisher values? For each quantile there is exactly one blue and one yellow intersection. If we can specify a function that calculates the yellow (Cornish-Fisher quantile) values from the blue (normal distribution quantile) values, we have solved the problem.

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## What are the limitations of the Cornish Fisher Expansion?

In the Cornish Fisher expansion up to the fourth moment, this transformation is a 3rd degree polynomial. For the skewness and excess kurtosis values as above, it looks like this (If we evaluate this function at -2.326, the function value is -3.096.).

DThe daily Volkswagen returns have a skewness of -1.245 and an excess kurtosis of 14.09 during the same observation period. The Cornish Fisher “transformation” is no longer strictly monotonously increasing, but reverses in the middle area.

The probability “distributions” and the probability “densities” then look like this:

This is certainly no probability distribution. How does this nonsense come about? The Cornish Fisher approach is only valid if the transformation function is monotonic. As a rule of thumb, this is the case for an excess kurtosis of up to about 8 (depending on the skewness). In the period from July 2013 to June 2018, the validity of the Cornish Fisher transformation for the DAX stocks Volkswagen, Linde and Adidas was violated.

It becomes even worse with Bitcoin (excess kurtosis = 87) and for the CHF / EUR exchange rate (excess kurtosis = 589) in the period from 2013 to 2017.

## Conclusion

With a recommended holding period of one day, a Bitcoin investment of USD 10,000 would deliver the following performance scenarios (without costs):

**Pessimistic:** 14,110 USD**Moderate:** 9,681 USD**Optimistic:** 7,435 USD

This is obvious nonsense, but is directly derived from the Delegated Regulation.

We have already reported these extreme cases (short holding time, extremely high kurtosis) to the supervisory authorities.

For longer holding periods (N trading days), the skewness decreases with and the excess kurtosis with 1 / N, so that the following plausible picture emerges for CHF / EUR with 512 trading days.

If the higher statistical moments are extreme and the holding times are short, we recommend a category 3 simulation which, assuming the extrapolation of historical return distributions, always delivers plausible results.

**In the next post: PRIIPs category 3 methodology for equity instruments.**