Getting a handle on true performance

By Kestner, Lars N.
Publication: Futures (Cedar Falls, Iowa)
Date: Monday, January 1 1996

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Consider two managers trading the same system and markets, except that manager A trades twice as large as manager B. Consequently manager A's average return is twice as large. Is manager A a better manager because of higher average returns? Not necessarily. Managers have different trading styles. Some trade large positions relative to the size of funds under management, while others prefer to keep returns and risk at lower levels by holding smaller positions. Performance ratios attempt to place all managers on a level playing field by adjusting returns for risk. The return part of the calculation is generally agreed upon, but defining risk is another matter. Each ratio divides return by its own measure of risk to arrive at a standardized value for comparing traders.

Look Sharpe. The most popular ratio for evaluating performance is the Sharpe ratio, a measure created by Nobel-laureate William Sharpe. The Sharpe ratio is composed of two statistical measures: average return and standard deviation. First, monthly returns are adjusted by subtracting the monthly return of a risk-free asset, such as Treasury bills, from the total return. If a manager returned 5% but 1% came from posting T-bills as margin, then the adjusted monthly return would be 4%. (This is logical because the return from T-bills is not dependent on the manager's skill.)

Return in the Sharpe ratio is defined as the average adjusted monthly return; risk is defined as the standard deviation of corrected monthly returns. Standard deviation measures how close or far returns vary from the average. Returns dispersed closer to the average have a lower standard deviation and are considered less risky than returns varying further from the average.
To demonstrate this principle, consider two managers. Manager A has 30 months of + 15% returns and 30 months of -10% returns, while manager B returns +5% for 30 months and 0% the other 30 months. Both managers have identical average returns, but manager A's returns show higher dispersion around the average than manager B's returns. As a result, manager A has a standard deviation of 13% while manager B has a standard deviation of 2.5%. Although both managers return 2.5% per month, manager B accomplishes the feat with much less risk as measured by standard deviation. As a result, manager B will have a higher Sharpe ratio (2.5%/2.5% = 1.0 ) than manager A (2.5%/13% = 0.19). Typically, a Sharpe ratio is calculated using three or five years of monthly data.

The Sharpe ratio is not without faults. Critics have argued that the ratio does not accurately reflect performance when autocorrelation exists in returns. Positive autocorrelation occurs when values are not independent from one period to the next, for example, when high returns one month are generally followed by high returns the next month, or when low returns are followed by low returns. Similarly, if high returns are followed by low returns, and low returns by high, then negative autocorrelation exists.

To put this into perspective, take the example of managers C and D. Both have 30 months of +10% returns and 30 months of -5% returns. The only difference is the timing of returns. Manager C has 30 consecutive returns of +10% followed by 30 returns of -5%. Manager D's returns alternate: One period is +10%, the next is -5%, followed by +10%, and so forth. Manager C possesses significant positive autocorrelation while manager D possesses significant negative correlation. Although both have equal Sharpe ratios, manager D's returns are more consistent than manager C's and therefore more likely to be profitable in the future (see "A tale of two managers," left).

Maximum drawdown is the largest percentage retracement from the previous peak in equity. Suppose a manager returns an average of 20% annually for the past three years. In year one the maximum drawdown was 15%, in year two it was 15% and in year three there was no drawdown. Risk as measured by the Sterling ratio would be ((15+15+0)/3) + 10 = 20%. The corresponding Sterling ratio is 20%/20% = 1.0.
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