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

$$ K = \frac{\frac{1}{n}\sum_{i=1}^{n}{(r_i-\bar{r})^{4}}}{(\frac{1}{n}\sum_{1=0}^{n}{(r_i-\bar{r})^{2}})^2} - 3 $$

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. We've defined excess kurtosis in the equation above but in everything that follows, everything that we call kurtosis is in fact excess kurtosis.

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 |

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 |

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 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 |

The canonical Big Day example is the 19th of October 1987. How could we have hedged this event

Maybe one could have bought put options on the S&P 500? If one can't see into the future that would imply that you had in place an investment process which required you to buy puts on a regular basis. Unfortunately, as is well known, purchasing put options systematically is an almost guaranteed money loser. Implied volatilities are almost always higher than realised volatilities and so although purchasing put options removes the nasty left tail, it moves the mean of the distribution to the left. Hedging one's positions with options loses you money almost all the time and most investors will bail out of an underperforming manager before the put hedge kicks in. This investor preference is probably rational from a career perspective even if it isn't rational from a long term return perspective.

The “put option” hedge becomes even more problematic when one is dealing with a complex dynamic highly diversified futures portfolio such as CTAs would hold. What

$$ \text{Kurt}\left( \sum_{i=1}^{n}{X_i}\right) = {{1} \over {n^2}}\sum_{i=1}^{n}{\text{Kurt}\left(X_i\right)} $$

If we have 100 assets each with a kurtosis of 5 then a moment's work with the trusty HP12C shows that the kurtosis of the final distribution is 0.05. Wow! Our problem is solved…or is it?

Even though CTAs are some of the most highly diversified investment instruments in the world, the much vaunted statements about “100, 150, 250 assets traded” are in fact misleading. There aren't 100 completely independent assets in the world. Oh, that there were! As we mentioned in a previous post there might be at most ten

There is no simple answer to this. Why should diversification increase Sharpe ratio but not reduce kurtosis? Here at Cantab this is a very active area of research. We have developed an interesting theoretical “generative” model of idealised markets which has many of the same features as we see in the real world and may provide insight into this effect.

We can see that there are quite a lot of tail events and indeed some as large as 9 standard deviations, but we are reaching the limit of being able to work out what is going on by eyeballing the graph. So we turn to our standard deviation and kurtosis statistics. In line with the analysis above, let's assume that we have more of these Sharpe ratio 1.0 assets and they're uncorrelated to each other. What happens to the Sharpe and kurtosis as we add these assets together?

Number of Assets | Return | Vol | Sharpe | Kurtosis |
---|---|---|---|---|

1 | 10.0% | 10.4% | 0.96 | 1.7 |

2 | 10.7% | 7.3% | 1.46 | 1.0 |

3 | 10.3% | 6.0% | 1.72 | 0.7 |

4 | 10.0 | 5.2% | 1.92 | 0.5 |

5 | 10.2% | 4.6% | 2.20 | 0.3 |

However, it can get much worse than this. Let us assume that our assets are uncorrelated in the low volatility state, but one random day out of a hundred not only do they have a “big day” at the same time, they're also correlated at 75% to each other on that day.

Number of Assets | Return | Vol | Sharpe | Kurtosis |
---|---|---|---|---|

1 | 10.0% | 10.4% | 0.96 | 1.7 |

2 | 10.7% | 7.6% | 1.37 | 4.63 |

3 | 10.3% | 6.4% | 1.61 | 10.12 |

4 | 10.0 | 5.7% | 1.74 | 13.15 |

5 | 10.2% | 5.2% | 1.98 | 19.28 |

There is a minus 5% day ($17\sigma$) and plus 6% day $(19\sigma$) in that distribution. This is exceptionally punchy for a 5% volatility portfolio. Indeed, we are getting $4\sigma$ days — which we would expect once every 127 years — about once every 2 years on average. This is starting to look very much like real markets, real asset managers, and real risks. It seems as if this simple generative model of assets which look uncorrelated — but suddenly aren't — is generating something which looks quite realistic.

To summarise, the key feature is not that assets can have big days or even that assets can be correlated: it is that

In this post and the previous one we have examined fat tail events and how to deal with them. Investors are rightly unhappy with fat tail events especially when they are fat tailed losses. Investors would like their managers to reduce this risk, but since the kurtosis in markets comes from a variety of sources the approaches have to be multi-layered.

Firstly, we have shown that by scaling positions by recent volatility, one can reduce the kurtosis of most financial securities considerably. Secondly, we presented a theoretical model which would have indicated that by adding together lots of uncorrelated assets with kurtosis, the portfolio kurtosis should have negligable kurtosis. However, in reality for real managers or widely available models, this doesn't seem to be the case. Finally, we presented a more subtle model of multivariate kurtosis. We built a set of uncorrelated Gaussian Mixture models, where the high volatility component of the mixture happens infrequently, but when it does the assets are highly correlated. It was this model which seemed to have the features of diversified asset portfolios.

Clearly this generalised model for asset returns has consequences for risk management and allocation of capital to assets. Optimal solutions under the simpler asset returns models are no longer optimal and new optimal allocations must be calculated. Ideally you would be able to forecast when these random big events are going to happen and suitably adjust your risk profile when a big event is likely to happen tomorrow. Even if that was possible for the known unknowns like FOMC announcements or central bank governors speaking, it is going to be impossible to predict events like 9/11. The difficulty is compounded by the relative infrequency of these events. Events which happen once every 100 days will only have happened around 70 times in the past 30 years. That might seem like a reasonable amount of data, but once you factor in different classifications of events (geopolitical, economic, surprise announcement etc), the number of data points you have to build a forecasting model is very small. It is also critical not to be misled by a generative model which produces plausible returns. It's possible that this correlated Gaussian Mixture model produces very realistic results, but in fact the underlying processes in the real world are completely different. If a model like this does encapsulate some of the features of the real world, then it suggests that the free lunch of diversification is in fact a more complex dining experience. There are trade-offs to be made between increasing Sharpe ratio (something which investors are rightly rather keen on), but there may be an implied and hidden cost associated with this — a higher frequency of unexpectedly large returns.

This is one of those rare cases where the complexities of real world comes to our aid. In our generative model, we assumed that each of the underlying assets was a black box return generating system with a Sharpe ratio of 1.0. In the real world, the constituents of most investors portfolios are managers or, in our case, a diversified basket of models each of which may trade 100 or more assets. We have the ability to look inside these 'black boxes' and analyse the commonality of the positions within the models which can help to determine those times in which fatter tails are more likely and also to manage the trade-off between higher Sharpe and higher kurtosis. For us, this is an area of very active fundamental financial research and is something that we will cover in future posts.

Risk management is a topic which comes up very frequently in our discussions and meetings with investors. It is a complex multifaceted problem involving understanding “risk” on a variety of related dimensions. Volatility, fat tails, skewness, liquidity, VAR, expected shortfall, drawdown…risk of permanent impairment of capital. As we have seen, none of these issues are entirely independent from each other, which makes analysing and understanding them subtle, challenging and ultimately satisfying.