Chebyshev’s Inequality and Tail Risk

“The sacred geometry of chance

The hidden law of a probable outcome”

Sting, “Shape of My Heart”

Tail risk is commonly defined as probability of rare events (technically speaking tail risk is risk of an asset moving more than three standard deviations away from the average).  From the graphical presentation of the returns of most of the financial assets easily can be seen that the tails of the distribution are fatter than the normal distribution. Earlier posts in this blog placed an equal focus on daily returns of certain CEE indices as well as the standardized returns (number of standard deviations from the average return) in an effort to grasp more value-added information from data available.

The assumptions of the underlying distribution predetermine the results and generally this presents problem to analysts (for instance, VaR under normal distribution assumption has significant flaws). In some previous posts in this blog two Extreme Value Theory approaches have been used with relation to Romanian and Bulgarian stock exchanges. For the sake of completeness, another measure needs to be revealed as well – Chebyshev’s inequality. One advantage of Chebyshev’s inequality is that it is valid in any distribution. The drawback is that it is too general and can provide too high probability of extreme values. So, as the normal distribution underestimates extreme values probabilities, Chebyshev’s inequality overestimates the extreme values probabilities.

For a given mean and standard deviation, Chebyshev’s inequality states that:

for any t>0.

Stated in an equivalent form:

However, t needs to be greater than 1.

Chebyshev’s inequality says that no more than 1/t^2 of the values can be more than t standard deviations away from the mean (stated as a maximum), or put in an alternative way, at least 1−1/t^2 of the values are within t standard deviations of the mean (stated as a minimum).

Chebyshev’s inequality results in higher probability of extreme cases compared to the normal distribution. For instance, if we want to know what is the probability of having cases at 3 standard deviations from the mean (t=+/-3 in the Chebyshev’s inequality formula). Chebyshev’s inequality states that at least 88.8% of values must lie within +/-3 standard deviations from the mean (or equivalently, no more than 11% of the values can be more than 3 standard deviations from the mean). The result for normal distribution is 99.7% (this is also known as empirical rule 68–95–99.7 or the three-sigma rule that briefly states that extreme values are barely possible). The chart and table below present four cases for probabilities to have more than t standard deviations from the mean: Chebyshev’s inequality, one-tailed version of Chebyshev’s inequality (also known as Cantelli’s inequality) and Standard normal distribution probabilities – one-tailed and two-tailed. Cantelli’s inequality is helpful in identifying the worst confidence level for heavily skewed or leptokurtic distributions (Lleo and Ziemba, 2014).