**Fat Tail Events – Does My Tail Look Fat In This? Part 2 by Dr Ewan Kirk, Cantab Capital**

Investors and managers are concerned with “fat tails”. In the second part of this post, we look at kurtosis in more detail.

### An apology and a warning

This piece is more technical and longer than I had expected. The problem we’re looking at here is subtle and not easy to distill down to a short, punchy and maths-free post. Sometimes the world isn’t simple.

### Fat Tail Events – Introduction

In Part 1 of Does My Tail Look Fat In This, we saw how simple volatility scaling rules can help to reduce the incidence of fat tails and make market processes look a lot more Gaussian. Whilst this was useful, there is *still* a fat tails problem and it is exemplified by the events of “Black Monday” on the 19th of October 1987. Dealing with this event is going to be much more difficult. To understand the problem better it’s time to formalize what we mean by “fat tails”. When we are measuring distributions, we use a measure called kurtosis to describe how fat or thin tailed a distribution is relative to a standard Gaussian. For a sample of *n*n returns *r**i*ri from a market, we calculate the excess kurtosis using the following equation:

[drizzle]Most graphing or spreadsheet packages will calculate this quantity (and many other quantities of interest) so you don’t need to commit this formula to memory! In financial literature, authors sometimes play fast and loose with the terms excess kurtosis and kurtosis.(1) We’ve defined excess kurtosis in the equation above but in everything that follows, everything that we call kurtosis is in fact excess kurtosis.(2)

### What do real markets look like?

The easiest way to get a feel for kurtosis is to calculate it (and other statistics) for various markets and distributions. In all cases we will be comparing to a 10% volatility Gaussian daily distribution.(3)

Market |
Date |
Vol |
Kurtosis |
Big Days |

Gaussian | 01Jan85 | 10.1% | 0 | < 1 |

Crude Oil | 01Jan85 | 35.8% | 9.9 | 39 |

Crude Oil Scaled | 01Jan85 | 10.4% | 2.6 | 19 |

S&P 500 | 01Jan85 | 19.4% | 54.9 | 44 |

S&P 500 Scaled | 01Jan85 | 10.4% | 21.8 | 19 |

In this table, “Big Days” is defined as the number of days in the 30 year period where there is a return — either positive or negative — which is greater than four standard deviations.(4)

The obvious stand out thing in this table is just how kurtotic (or strictly “lepto-kurtotic”) both markets are and how the equities market has an extremely high kurtosis even after scaling to a 10% volatility process.

It is very common when modelling a market to use this empirical data to construct model distributions to describe the potential future path for each of the markets. Unfortunately these empirical measurements are very sensitive to the start date of one’s measurements. For example, if we decided to start the equities data in 1988 instead of 1985 then the equities data would look like this.

Market |
Date |
Vol |
Kurtosis |
Big Days |

Gaussian | 01Jan88 | 10% | 0 | < 1 |

S&P 500 | 01Jan88 | 16.6% | 11.44 | 61 |

S&P 500 Scaled | 01Jan88 | 10.05% | 5.43 | 14 |

The comparison between this table and the previous one gives us some insight into just how difficult it is to quantify tails. Just removing three years of data, the volatility of the data has dropped a little but the kurtosis has dropped from 55 to 11. Even more oddly, the number of “Big Days” has gone up!(5) The curse of sampling error has struck again. Small sample sizes have large noise around the estimates of the statistical parameters of a distribution.

We showed in Part 1 that scaling the return distribution by recent volatility removes some of the “fat tailyness” but this doesn’t work for all markets: the S&P500 from 1985 has a higher kurtosis after scaling than the WTI market has before scaling. This is all very confusing: we need a better framework to think about returns in financial markets before we can make any progress.

Volatility scaling is the first step in a process. If you scale a distribution by recent volatility you can think of it as equivalent to saying that the market process is a “Gaussian Mixture” process. It is a —possibly unknowable— set of Gaussian distributions with different volatilities and, if you scale the returns by recent volatility, you remove a lot of this effect. So far so good. However, these large outlier events — which we would only expect to happen once every 50 years or so — happen way more often in real financial markets and scaling the returns doesn’t remove all of the outliers.

Let’s attack the problem in a typical scientific way which is to assume for the moment at least that the problem doesn’t exist. We are going to take the Big Days(6) out of the distibution entirely. We are going to look at the distribution of Small Days first and then look at the Big Days separately.

Market |
Date |
Vol |
Kurtosis |
Big Days |

Gaussian | 01Jan88 | 10% | 0 | ~0.5 |

S&P 500 Scaled Small Days | 01Jan88 | 9.99% | 1.15 | 0 |

So, it appears that if we ignore the problem, it goes away. I suppose this might not be considered progress but it is a start. The excess kurtosis is small and not anything would really change our view of how to model risk.

### Dealing with the big days

Whilst it is nice to know that if we ignore the problem then it goes away, it isn’t really an approach designed to encourage career longevity either in managers or investors. There *are* big days and they are going to generate large positive or negative returns. How can we hedge or reduce these risks?

The canonical Big Day example is the 19th of October 1987. How could we have hedged this event *ex ante*? Without postulating some psychic abilities to see the future, it isn’t clear that there was any information on the 18th of October 1987 which would have allowed you to predict that the following day was going to be so cataclysmic.(7) There may have been warning signs but it’s probably fair to say that in every market which suffers big moves, there are always warning signs which become apparent in hindsight.

Maybe one could have bought put options on the S&P 500? If one can’t see into the future