when a portfolio of investments is put together it is assumed that the distribution of returns will follow a normal distribution under this assumption the probability that returns will move between the mean and three standard deviations either positive or negative is approximately 99 7 this means that the probability of returns moving more than three standard deviations beyond the mean is 0 3 
the assumption that market returns follow a normal distribution is key to many financial models such as harry markowitz s modern portfolio theory mpt and the black scholes merton option pricing model 12 however this assumption does not properly reflect market returns and tail events have a large effect on market returns tail risk is highlighted in nassim taleb s bestselling financial book the black swan 3other distributions and their tailsstock market returns tend to follow a normal distribution that has excess kurtosis kurtosis is a statistical measure that indicates whether observed data follow a heavy or light tailed distribution in relation to the normal distribution the normal distribution curve has a kurtosis equal to three and therefore if a security follows a distribution with kurtosis greater than three it is said to have fat tails a leptokurtic distribution or heavy fat tailed distribution depicts situations in which extreme outcomes have occurred more than expected compared to the normal distribution these curves have excess kurtosis therefore securities that follow this distribution have experienced returns that have exceeded three standard deviations beyond the mean more than 0 3 of the observed outcomes the graph below depicts the normal distribution in green as well as increasingly leptokurtic curves in red and blue which exhibit fat tails thoughtcohedging against tail riskalthough tail events that negatively impact portfolios are rare they may have large negative returns therefore investors should hedge against these events hedging against tail risk aims to enhance returns over the long term but investors must assume short term costs investors may look to diversify their portfolios to hedge against tail risk for example if an investor is long exchange traded funds etfs that track the standard poor s 500 index s p 500 the investor could hedge against tail risk by purchasing derivatives on the cboe volatility index which is inversely correlated to the s p 500 4