**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.

Great post. Interesting views. I think this is particularly relevant in the discussion about risk parity portfolios. I often wonder if they aren’t measuring risk incorrectly. That would explain the strange performance in those funds this year.

Taleb talks about it. It often happens as a result of volatility suppression. In fat tailed distributions, the tails occur less often, but the impact of those fewer events is larger.

Interesting post. Just to clarify:

- the kurtosis in your definiition would be 3 if normally distributed (ref:http://en.wikipedia.org/wiki/Kurtosis)?

- US large-valued stocks are the most fat tailed; non-US bonds are the least

Also, it would be interesting to plot the mean return vs. kurtosis for your data set (like the convention plot vs. std deviation). Cheers.

I have generally understood Kurtosis to conceptually be a measure of the sharpness of the peak of a probability distribution function, which is really easier to get by the maximum slope of the cumulative distribution function. The common simple implementation just looks at higher moments of the distribution, which actually is more a measure of the tails than the sharpness of the peak.

In the “real world” today approaches like this are far from state of the art, or even standard practice. The classic 1953 paper by Metropolis et. al (including E. Teller of H-bomb fame) was published in 1953. Today, almost all serious work in statistics of non-normal (or non-simple) systems uses some form of Markov Chain Monte Carlo.

Tail risk is still a difficult problem and always will be because it’s in the tails (and therefor has high statistical uncertainty, however you wish to quantify it).

Another problem people in finance ignore or gloss over is that there is no reason to expect (and many reasons not to expect) the distribution to be stationary. Hence we see people discussing things like reversion to the mean, when in fact there is no mathematical basis for expecting this in developed economies. It’s reasonable to anticipate that the non-stationary part of economic distributions generally moves slowly in the absence of large scale perturbations. But we have recently had a moderate perturbation, and there is some evidence (especially in the still declining (or maybe now flat) consumer loan real origination rate, that this will persist for a considerable time (a generation, perhaps two). We also see a non-ending decline in discretionary government spending (now lower than at any time in decades). These two factors, although mostly independent of each other, put us on a course where growth will be lower in the near term and the long term. Of course that is just one reasoned view, it could be wrong, but it does mean that the fundamentals that shaped for instance asset class distributions will continue to move in the future, so that the statistical past provides relatively little in concrete knowledge for the future absent the inclusion of more general macro-economic reasoning.

You raise another important point about the fact that the dynamics can change over time. Therefore, when you look at the distribution of returns, you cannot tell whether the ‘fat’ tails are due to a distribution with positive kurtosis or to an overlap of, e.g., normal distributions whose parameters change over time. To solve this problem, one also needs information about the structure of the ‘time’ distribution of returns. This is in general very difficult to do because there are only relatively short time series in finance and the results are inevitably noisy.

In my experience, i very much agree with you that the math alone cannot help solve this problem without some good insights about what moves the markets (see e.g. the interesting discussion about the dynamics of monetary policy on this site).

There is a lot of discussion about fat tails and other non-normal effects. However, one thing that is often forgotten is that “normal” risk is very relevant in its own right. There is a misperception that fat tails are responsible for big portfolio drawdowns, but large drawdowns are possible even with a “normal” distribution. In fact, the vast majority of the risk is still attributable to the “normal” component of the distribution. Part of the misconception derives from the idea that under a “normal” distribution returns tend to “reverse” to the mean. This is not so. While returns “regress” to the mean, the do not “revert” to it. This is a long way to say that the volatility of prices increases with time even under a normal distribution.

The source of the misunderstanding often comes from looking at the statistics of the “annualized” returns and not at the statistics of the returns compounded over a period of time. While the statistics of the “annualized” returns appears to support the idea that the risks of stocks decrease over the long time, the relevant statistics is the one for the compounded returns, and that increases over time.

This is the ‘Time Diversification Fallacy” which supporter of Modern Portfolio Theory so strongly oppose.

That the distributions are moving over time and that the past means might be no guidance on even the medium to long term future is the risk that John Hussman has taken, unsuccessfully since 2009.

What if we are in a low inflation, high un/underemployment, low growth long term scenario in developed countries as freer trade lifts hundreds of millions more out of poverty in developing countries, but at the expense of job and real wage growth in developed countries?

1.5% inflation, 1% real growth but with much of it coming from population growth not growth per capita and very low growth per capita for most deciles means very low interest rates and relatively high capitalisation rates/PE’s for maybe 2 or 3 decades.

If that is the case, 2.5% capital growth in the stock market on top of 2% dividends could be a very good return.

Only time will tell.

Hussman has become the poster child for the way valuation metrics help/hurt an investor. He’s relied so heavily on long-term valuations that it’s killed his portfolio.