Tuesday, June 9, 2009

Normality of returns...are they?

As I make progress with an upcoming article I'm writing on standard deviation, I will occasionally share some of the information I discover. For the data I'm using the monthly returns for the S&P 500 for the 36-year period ending December 2008. I am first calculating the results for the 36-month period ending this date, to conform with what is proposed for GIPS 2010. I will also evaluate the longer period and probably a few other 36-month periods throughout this time frame.

I haven't tested completely for normality, yet, but did discover something of interest. For the 36-year period I determined that at the 97.5% confidence level there should be 11 returns outside the range of plus-or-minus 1.96 standard deviations: there are 25, more than twice the expected number. Most (15) are at the low end. Other authors (e.g., Anson, Mark (2002), The Handbook of Alternative Assets. (Wiley)) have found similar results.

One reasonable question that might arise is "does it matter?" Well, if your analysis is based on an assumption which is flawed, one would suspect that your conclusions might also be flawed, yes? There have been studies that have shown that in some circumstances the absence of a normal distribution isn't a problem. But I like to hearken back to a standard line that IBM used to use when someone would do something other than what was specifically called for: "unpredictable results may occur." Bottom line: we just don't know. In some cases there may not be a problem, but in other cases there will. Over the 36-year period I'm reviewing there are more results outside the boundary just at the low end than are predicted for the entire distribution, so wouldn't we expect our assumptions to lead to "unpredictable results," which likewise might call into question any of our conclusions?

Standard deviation remains the most criticized risk measure, and probably with much justification. We like things that are easily understood: standard deviation fits this bill. However, we also like things that work properly...unfortunately, it's unclear that we can say this holds.

1 comment:

  1. As you indicate, the S&P 500 is like many financial assets which do not reflect a perfectly normal distribution of returns. Rather, they reflect the characteristics of skewness and kurtosis, meaning a greater likelihood of losses and a greater likelihood of extreme event losses. This does not invalidate the usefulness of standard deviation as one of several useful risk statistics to disclose to clients. After all, the standard deviation does answer the very useful question: "How much different is your actual return in any single period likely to be relative to your expected return?" As a matter of short term uncertainty, this seems like a reasonable (albeit imperfect and limited) statistic.

    Admittedly, standard deviation is an incomplete answer, since it doesn't provide a reliable answer to the question of the likelihood of an extreme return. It seems unreasonable for anyone to apply a simple "mean minus so many standard deviations" approach to answer this question AND THEN BE SURPRISED that the answer is wrong. We already KNOW that this is going to understate the likelihood and severity of extreme losses. So, this is more a case of misuse of the standard deviation statistic, where we expect too much of this little guy. He's OK for getting a good sense of short term return uncertainty. For example, we can say: "Under normal market cycles, we can reasonably expect a return of 12% in any year, plus or minus 20%. But in extreme cases, the losses could be significantly greater." That's a limited, but still useful statement. All this criticism of standard deviation has the potential to simply leave the industry reporting returns to clients in great detail without any mention of risk. This in itself is a violation of the CFA Institute's ethical standards, which require managers to provide information on BOTH the return and risk of any investment. Something is always better than nothing. Something with disclosure about the limitations of the information provided is even better. For example, what's so bad about noting that "extreme losses tend to happen about twice as often as we expect" if you feel the need to qualify the limitations of your statistical analysis? A simple chart of risk and return can provide some useful guidance to clients about the relative risk/return opportunities of an investment relative to other market alternatives. Additional information (not necessarily statistics) about the likelihood of losses (skewness) and the likelihood of extreme events (kurtosis) can be provided to help provide a more fair representation and full disclosure about the return and risk characteristics of the investment. Everyone understands that standard deviation has its limits. That doesn't justify its rejection as a risk statistic.


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