# Understanding Tail Risk and the Odds of Portfolio Losses

## 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 both ends of a normal distribution curve.

### Key Takeaways

• Tail risk is the chance of a loss occurring due to a rare event, as predicted by a probability distribution.
• Colloquially, a short-term move of more than three standard deviations is considered to instantiate tail risk.
• While tail risk technically refers to both the left and right tails, people are most concerned with losses (the left tail).
• Tail events have had experts questions the true probability distribution of returns for investable assets.

## Understanding 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 larger than otherwise anticipated, 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, for example.

The chart below depicts three curves of increasing right-skewness, with fat tails to the downside—and which differ from the symmetrical bell curve shape of the normal distribution.

## Normal Distributions and Asset Returns

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. 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.

## Other Distributions and Their 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/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.

## 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 Cboe Volatility Index, which is inversely correlated to the S&P 500.

Article Sources
Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.
1. Nassim Nicholas Taleb. "The Black Swan: The Impact of the Highly Improbable." Random House Publishing Group, 2007.

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