Performance measures are used to quantify the results of a trading strategy. They are usually adjusted for risk. The most popular is the Sharpe ratio.

Example

One stock has an average growth of 10% per annum, another is 30% per annum. You'd rather invest in the second, right? What if I said that the first had a volatility of only 5%, whereas the second was 20%, does that make a difference?

Long answer

Performance measures are used to determine how successful an investment strategy has been. When a hedge fund or trader is asked about past performance the first question is usually ''What was your return?'' Later maybe ''What was your worst month?'' These are both very simple measures of performance. The more sensible measures make allowance for the risk that has been taken, since a high return with low risk is much better than a high return with a lot of risk.

Sharpe ratio

The Sharpe ratio is probably the most important non-trivial risk-adjusted performance measure. It is calculated as

where x is the return on the strategy over some specified period, r is the risk-free rate over that period and a is the standard deviation of returns. The Sharpe ratio will be quoted in annualized terms. A high Sharpe ratio is intended to be a sign of a good strategy.

If returns are normally distributed then the Sharpe ratio is related to the probability of making a return in excess of the risk-free rate. In the expected return versus risk diagram of Modern Portfolio Theory the Sharpe ratio is the slope of the line joining each investment to the risk-free investment. Choosing the portfolio that maximizes the Sharpe ratio will give you the Market Portfolio. We also know from the Central Limit Theorem that if you have many different investments all that matters is the mean and the standard deviation. So as long as the CLT is valid the Sharpe ratio makes sense.

The Sharpe ratio has been criticized for attaching equal weight to upside 'risk' as downside risk since the standard deviation incorporates both in its calculation. This may be important if returns are very skewed.

Modigliani-Modigliani measure

The Modigliani-Modigliani or M2 measure is a simple linear transformation of the Sharpe ratio:

where v is the standard deviation of returns of the relevant benchmark. This is easily interpreted as the return you would expect from your portfolio is it were (de)leveraged to have the same volatility as the benchmark.

Sortino ratio

The Sortino ratio is calculated in the same way as the Sharpe ratio except that it uses the square root of the semi-variance as the denominator measuring risk. The semi-variance is measured in the same way as the variance except that all data points with positive return are replaced with zero, or with some target value.

This measure ignores upside 'risk' completely. However, if returns are expected to be normally distributed the semi-variance will be statistically noisier than the variance because fewer data points are used in its calculation.

Treynor ratio

The Treynor or Reward-to-variability ratio is another Sharpe-like measure, but now the denominator is the systematic risk, measured by the portfolio's beta, (see Capital Asset Pricing Model), instead of the total risk:

In a well-diversified portfolio Sharpe and Treynor are similar, but Treynor is more relevant for less diversified portfolios or individual stocks.

Information ratio

The Information ratio is a different type of performance measure in that it uses the idea of tracking error. The numerator is the return in excess of a benchmark again, but the denominator is the standard deviation of the differences between the portfolio returns and the benchmark returns, the tracking error.

This ratio gives a measure of the value added by a manager relative to their benchmark.

References and Further Reading

Modigliani, F & Modigliani, L 1997 Risk-adjusted performance. Journal of Portfolio Management 23 (2) 45-54

Sharpe, WF 1966 Mutual Fund Performance. Journal of Business January, 119-138

Sortino FA & van der Meer, R 1991 Downside risk. Journal of Portfolio Management 27-31

Treynor, JL 1966 How to rate management investment funds. Harvard Business Review 43 63-75

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