## Definition

Tail risk is the additional risk of an asset or portfolio of assets moving more than 3 standard deviations from its current price, above the risk of a normal distribution. Prudent asset managers are typically cautious with tail risk involving losses which could damage or ruin portfolios, and not the beneficial tail risk of outsized gains.

## Tail Risk

## What is ‘Tail Risk’

Tail risk is a form of portfolio risk that arises when the possibility that an investment will move more than three standard deviations from the mean is greater than what is shown by a normal distribution. Tail risks include events that have a small probability of occurring and occur at the ends of a normal distribution curve.

## Explaining ‘Tail Risk’

Traditional portfolio strategies typically follow the idea that market returns follow a normal distribution. However, the concept of tail risk suggests that the distribution of returns is not normal, but skewed, and has fatter tails. The fat tails indicate that there is a probability, which may be small, that an investment will move beyond three standard deviations. Distributions that are characterized by fat tails are often seen when looking at hedge fund returns.

## Normal Distribution

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 99.97%. This means that the probability of returns moving more than three standard deviations beyond the mean is 0.03%. The assumption that market returns follow a normal distribution is key to many financial models, such as Harry Markowitz’s modern portfolio theory and the Black-Scholes Merton option pricing model. However, this assumption does not properly reflect market returns and tail events have a large effect on market returns.

## Distribution Tails

Stock 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 tailed distribution, depicts situations in which extreme outcomes have occurred more than expected. Therefore, securities that follow this distribution have experienced returns that have exceeded three standard deviations beyond the mean more than 0.03% of the observed outcomes.

## Hedging Against Tail Risk

Although 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 Chicago Board Options Exchange (CBOE) Volatility Index, which is inversely correlated to the S&P 500.

## Further Reading

- Tail risk and asset prices – academic.oup.com [PDF]
- Can cryptocurrencies be a safe haven: a tail risk perspective analysis – www.tandfonline.com [PDF]
- Tail risk premia and return predictability – www.sciencedirect.com [PDF]
- Comparative analyses of expected shortfall and value-at-risk (2): expected utility maximization and tail risk – ideas.repec.org [PDF]
- Beyond reasonable doubt: multiple tail risk measures applied to European industries – www.tandfonline.com [PDF]
- Capital regulation and tail risk – papers.ssrn.com [PDF]
- A statistical risk assessment of Bitcoin and its extreme tail behavior – www.worldscientific.com [PDF]
- Who should hedge tail risk? – www.tandfonline.com [PDF]