4.2 Kurtosis

Kurtosis, [latex]K[/latex], is a measure of heavy tails for a probability distribution when compared to the normal distribution. It conveys information about the probability of obtaining extreme values as it examines the mass contained in the tails of a distribution relative to the rest of the distribution. For a theoretical distribution, it can be computed using the following formula:

Kurtosis is 3 for the normal distribution, which is commonly used as a benchmark for comparison. Kurtosis greater than three indicates heavy tails, that is higher likelihood of obtaining extreme positive and extreme negative values. As [latex]K = 3[/latex] for normal distribution, often we use excess kurtosis ([latex]EK[/latex]), which is obtained by subtracting 3 from Equation (8). That is, [latex]EK = K - 3[/latex]. Excess kurtosis is 0 for a normally distributed variable. Positive excess kurtosis indicates fatter tails compared to normal distribution with the same mean and standard deviation.

Any statistical data package or Excel will enable you to compute the skewness and kurtosis using empirical data. We discussed in Section 2.2 some of the pitfalls of assuming that our tomorrow will be very much like our yesterday. In an interview with the New Yorker, Prof. Nassim Nicholas Taleb, a finance professor and former stock market trader, called those who believe that history tends to repeat itself “naïve empiricists.”^[1] In his 2007 best-selling book Black Swan, Prof. Taleb popularized the notion of a “black swan” – an event with catastrophic consequences when it occurs, which is so rare that it is impossible to predict using standard analytical tools and historical data. Some events in the recent past commonly considered as black swans are Brexit, the 2008 Global Financial Crisis, and 9/11.

Many are tempted to assign a black swan status to the global COVID-19 pandemic, but Prof. Taleb argues that the pandemic was entirely predictable – he, as well as others, warned governments of the impending crisis in early 2020. His recommendation is to look beyond the normal distribution to account for distributions with “fat tails.” While we may not always know what is in store for us in the fat tail of a distribution, we cannot expect to observe only values close to the mean and should build resilience in expectation of black swan events. Finally, while a black swan event can be devasting for an economy, remember that the flip side of risk is opportunity. There are many who profited from the drastic fall in stock prices in October 2008 and from the global pandemic.