**By Salil Mehta, Statistical Ideas**

There is more risk in less risky asset classes than one may think. This analysis looks at major equity and fixed income asset classes, both in the U.S., as well as internationally. And the study samples two decades of data, from 1990, to 2010. A period that is fairly representative of a lengthier history of markets, through current.

A higher-order measure of risk, named kurtosis, is designed to look at the relative thickness or thinness of the tail-ends of the distribution. Kurtosis can be used to look at the tail risk of an asset class, versus what we would see if it were normally distributed. Only some market participants know that financial market data do not follow a normal distribution, and even for those that do it is a common mistake to then not throw out a common assumption about the underlying kurtosis of the return distributions.

Kurtosis is calculated by taking the typical (return dispersion)^{4}. By taking the fourth power, both positive and negative deviations become positive, and higher values take on significantly greater weight. Then when we see kurtosis levels of, say four or five, for the four risky assets on the right side of the chart below, we know that there has been very heavy distribution in the tails. And while kurtosis doesn’t distinguish between the upper tail and the lower tail, similar to the standard deviation measure, it should be noted that skew was negative for all of the asset classes shown here but for the non-U.S. bonds (for which skewness was virtually nonexistent). We introduce the name “leptokurtic”, which is defined as distributions with fatter tails than the normal distribution, such as the risky assets shown.

Since we see risky assets having this excess kurtosis in its return distribution, how does this relate to what we see in less risky asset classes (on the left of the chart above)? Here we look at bonds, both in the U.S. as well as internationally. And we see that the typical risk measure of standard deviation is about 1/3 that for risky assets (~5% versus ~17%). We might say this makes sense for bonds to have this lower risk, by the standard deviation measure. But what happens to those bonds on the higher-order, kurtosis statistic?

So to be sure, kurtosis is less for bonds than for stocks, regardless of geography. Though not by a lot. Bonds still have a higher degree of kurtosis than would be proportionally assumed by either the normal distribution, let alone the reduction in standard deviation risk of a non-normal distribution. In other words, there is greater tail risk from these “less risky” instruments, than most investors appreciate until after their downturn. This is likely further evidence that statistical aberrations in the markets, created simultaneous, correlated inefficiencies from multiple asset classes.

### Salil Mehta

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