Performance Measures

Future returns may vary, and in fact they may vary so hugely and in such disturbingly structural ways that one may worry at times if it is a fool's errand even to try to measure the performance of a fund, a manager, an asset class, or --- perhaps especially --- a strategy.

Nevertheless, the job must be done, however skeptical we may be about the uses to which our measurements might be put.

It is intuitive to most people that a proper analysis of returns should somehow account for risk. Unfortunately, there is no clear-cut prescription for doing this, even though there are some widely reported measure such as "beta" and "VAR." We have already seen some of the limitations of these measures.

Here we will be mainly concerned with four measures:

• The Sharpe Ratio. This is the most widely used measure of "risk adjusted returns," though we saw from the RiskMetrics table that its implications are ludicrous at times.
• The Sortino Ratio. This is a modified version of the Sharp Ratio that attempts to measure just "bad" volatility. It seems reasonable, but it has not achieved a fraction of the popularity of the Sharpe Ratio. In theory, this could be because one can go long or go short, but in practice it may have more to do with familiarity and with coding.
• Drawdown. This is a favorite measure of commodity traders, and it has great intuitive appeal.
• The Modigliani-Modigliani measures These measures use a convex combination of the strategy returns and the risk free returns that is driven by the standard deviations of the strategy returns and the benchmark returns (say the SP500). They have the great advantage of permitting a clear interpretation of the difference between two strategies --- a feature that is lacking in the Sharpe and Sortino Ratios.

You can find expositions of these measures in many places. One that is I like is given in the thesis of Venkat Chandramouli, on pages 39--46.

The most fundamental criticism of all of these measures is that they incorporate only the historical risk. This limitation can be serious. For example, for many years the return on unhedged deposits in Mexican banks would have had an infinite Sharpe ratio for US investors. This was in fact a reasonable and well-performing investment if held during certain periods of time. Not unexpectedly, it turned out to be a very poor investment for those Jonny-come-latelies who piled on when it looked the juiciest.

Investments that face real but unobserved risks are said to suffer from the Peso Problem. The sad fact is that every investment suffers from the Peso Problem to some extent. In particular, the issue of unobserved risks has been used to attempt to explain the so-called equity premium (the fact that over long periods of time equities have provided greater returns than bonds when both asset classes are adjusted for observed volatilities).