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From time to time, contributors to this board have debated the question of the size and accuracy of the observed maximum drawdown of a screen. For example, Len Kogan in post #132333 recently commented that "Drawdown, along with -2 and -3sigma are better indicators [of risk] than the usually observed standard deviation and Sharpe Ratio volatility indicators. Drawdown, not volatility, is what we fear the most, second only to inadequate returns."

In post #132341, Eric Mintz replied, "The data we've seen so far supports the idea that the size of past drawdowns is a very poor predictor of future drawdowns. GSD is a vastly superior predictor of drawdowns."

In post #13359, Elan illustrated his doubts about drawdowns by repeating an experiment 10 times. Each time, he simulated 150 monthly returns of a screen with CAGR = 42 and GSD = 35, and calculated the maximum drawdown. He found considerable variation in the size of the maximum drawdown, and concluded from this that Eric Mintz is right, the maximum drawdown is too unreliable.

I would like to provide some theoretical support for this position, and to show exactly how bad the maximum drawdown statistic really is. Along the way I will show how to estimate the anticipated size of the maximum drawdown, and its sampling variability.

Sources

There is an obscure corner of the theory of probability and statistics which describes the behavior of statistics such as the maximum drawdown. This little niche is known as "Extreme Value Theory." To give you an idea of just how small it is, it occupies a mere 5 pages within the 2100 pages of Kendall's Advanced Theory of Statistics. Here is an internet link to a well-written summary of Extreme Value Theory, which I have used as my basic reference for this MI research note:

http://pauillac.inria.fr/algo/bsolve/constant/extval/extval.html

Basic Results

The size of the maximum drawdown that can be observed in a series of monthly returns of a stock screen is a random variable that has a probability distribution. Extreme Value Theory tells us how to find an approximation for this distribution, and how it depends on the number of months over which the screen is to be observed.

If we make the traditional (but frequently questioned) assumption that the logarithms of the monthly returns are independent and normally distributed, then the probability distribution of the smallest log return converges to the Gumbel distribution as the number of months goes to infinity. We shall use this asymptotic distribution to characterize the size of the maximum drawdown. As an approximation, it works quite well.

Within a series of N monthly returns, the estimated size of the smallest standardized log return is:

(log( log( N ) ) + log( 4 pi )) / ( 2 sqrt( 2 log( N ) ) ) - sqrt( 2 log( N ) ).

To destandardize this value, multiply by log( GSD )/sqrt(12) and add log( CAGR )/12, where CAGR and GSD are expressed in "multiplier" form and all logarithms are natural (i.e. not base 10). Then exponentiate. The result is the estimated size of the smallest return.

For example, Elan simulated 150 monthly returns of a screen with CAGR = 1.42 and GSD = 1.35. Thus N = 150 and the expected smallest standardized log return is 2.511. Destandardizing this value and exponentiating, we obtain a smallest return of 83%, i.e. a predicted maximum drawdown of 17%. This prediction is the most likely value for the maximum drawdown. In other words, it is the mode of the extreme value distribution, which is skewed, rather than its mean or median.

I have prepared a Excel workbook with which to calculate the most likely maximum drawdown for any monthly screen. This workbook contains two sheets, one for the theoretical calculation, and another for a simulation which demonstrates the accuracy of the theory. Here is a link:

http://www.Aetheling.com/MI/docs/Extreme.xls

Accuracy of an Extreme Statistic

The asymptotic distribution of extreme values conveys a lot of information about the accuracy of an empirical maximum drawdown. As the number of monthly observations increases, the variability of the size of the maximum drawdown decreases. If we measure this sampling variability with the standard deviation (SD), then the formula for this dependency upon N is given by the following formula. The sampling variability of the standardized maximum monthly log return is

SD(N) = pi / sqrt( 12 log( N ) ).

In Elan's example, this variability is 0.405. As before, we need to destandardize and exponentiate to convert this result into useful units. The spreadsheet shows how to use this variability to calculate 1- and 2-sigma ranges around the predicted maximum drawdown.

As useful and interesting as these numeric quantities may be, the real message of the theory of extreme values is to be found in the dependency of the variability of the observed maximum drawdown on the number of monthly observations. The formula for SD(N) clearly shows that this variability is inversely proportional to the square root of the logarithm of N. In sharp contrast, the variability of the GSD is inversely proportional to the square root of N itself. From this we conclude that as the sample size increases, the accuracy of an observed GSD becomes far superior to the accuracy of an observed maximum drawdown.

Discussion

I replicated Elan's simulation as he described it, but did not obtain the same observed levels of the maximum drawdown. My simulated results conform to the theory of extreme values, so I am quite certain that they are correct. I don't know where the discrepancy occurred, but it is immaterial because our conclusions are the same: the size of an observed maximum drawdown is highly unreliable when compared to the GSD. Since we can accurately predict the most likely size of the maximum drawdown from the GSD and N, there is very little reason to look at an empirical maximum drawdown.

Conclusion: For the purposes of estimating risk it is far more informative to look at the predicted maximum drawdown, as calculated from the GSD and N, than to look at any empirical maximum drawdown.

It is certainly reasonable to wonder what happens to all this theory if the fundamental assumption quoted above, to the effect that the logarithms of monthly returns are normally distributed, is violated. One could argue, for example, that Extreme Value Theory is sensitive to the size of the tails of the log return distribution, and that therefore a fat-tailed hypothesis will throw everything into doubt. Fortunately, one of the remarkable things about Extreme Value Theory is the robustness of the Gumbel distribution. In order to invalidate the above theory, the tails would have to be so fat as to be asymptotically polynomial rather than exponential in shape. This is quite a drastic condition, which I do not believe applies. If I am wrong then a different part of Extreme Value Theory should be used, with somewhat different formulas. Even so, my ultimate conclusion remains unaltered, because the sampling variability of the maximum drawdown still depends on the logarithm of N.

As usual, I need to mention that I do occasionally make arithmetic mistakes. I have tried hard to avoid them, but if any have crept in then I would greatly appreciate learning what they are.

Loren Cobb
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