Non-normality of the return distribution

1. Non-normality of the return distribution

Up to now, we have been using the portfolio volatility as our measure of risk. Loosely speaking, the underlying assumption

2. Volatility describes "normal" risk

is that the portfolio return has a normal distribution and thus that its density function is bell-shaped. It is symmetric such that gains are equally likely as losses of the same magnitude.

3. Non-normality of return

For many financial assets, this assumption of a normal distribution is wrong. Their distribution tends to be skewed to the left with tails that are fatter

4. Non-normality of return

than those of a normal distribution.

5. Non-normality of return

As a consequence, when zooming in on the left tail of the histogram of financial returns, we often find that there are more extreme negative returns happening than is possible under a normal distribution. When this happens, an investor is no longer satisfied with using only the standard deviation as the risk measure but should be using also a downside risk measure that quantifies the risk of losing money. Such a downside risk measure focuses on the left side of the return distribution, instead of considering the complete distribution.

6. Portfolio return semi-deviation

A straightforward way to turn the standard deviation into a downside risk measure is to remove the higher than average returns. We then obtain the so-called portfolio semi deviation. As can be seen on the slide, it is defined as the square root of the average variability of the lower than average returns around the mean.

7. Value-at-risk & expected shortfall

Besides the semi deviation, also the 5% portfolio value-at-risk and 5% expected shortfall are popular downside risk measures.

8. Value-at-risk & expected shortfall

They quantify the risk of the 5% most extreme losses. To understand their definition, take a look again at the return distribution plot, where I indicate their value.

9. Value-at-risk & expected shortfall

5% value-at-risk is the return that is so extremely negative that there is only a 5% chance of observing a return that is even more negative.

10. Value-at-risk & expected shortfall

But how severe are those 5% most extreme losses? This question is answered by computing the average value of the 5% most negative returns. This number is called the 5% expected shortfall.

11. Shape of the distribution

In terms of performance measures, we have thus seen the mean and volatility, as well as downside risk measures such as the semi deviation and value at risk. Typically, investors also report the skewness and excess kurtosis of their portfolio returns to indicate two types of non-normality in the return distribution, namely asymmetry and fat tails.

12. Skewness

The skewness is designed such that it is approximately zero when the distribution is symmetric.

13. Skewness

When the skewness is negative, it indicates that large negative returns occur more often than large positive returns. A negative skewness thus corresponds to a distribution with a long left tail.

14. Skewness

The opposite is true in case of a positive skewness. Then large positive returns are more likely than large negative returns, and the distribution has a long tail to the right.

15. Kurtosis

Besides skewness, also fat tails are a cause of non-normality. A distribution with fat tails is a distribution in which extremely large positive or negative returns occur more often than a normal distribution would predict. A useful statistic to detect fat tails is the excess kurtosis. Its value is zero for the normal distribution. The larger the excess kurtosis, the fatter the tails are compared to the tails of a normal distribution.

16. Let's practice!

Let’s do some exercises to get more insights about the non-normality of the S&P 500 portfolio returns.

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