Performance Measures

"Future returns may vary."

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 will done ---for better or worse --- in Singapore or in New York.

However skeptical we may be about one measurement or another, we may be reassured that banks, brokerages, hedge funds, and central banks around the globe have very few secrets.

Their tools and our tools may differ in a few details --- but, just as likely, they will not differ at all.

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.

The Usual Suspects

• 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 theoretical benefit of the Sharpe ratio is that if asset A has higher Sharpe ratio than asset B then you can hold a little of A and a little cash and get an asset that has the same Sharpe ratio as B. In this limited sense the asset B is redundant, but one does not want to take this idea too far. The Sharpe ratio just depends on the marginal distribution of the assets, and as candidates for portfolios, the joint distributions matter a lot.
• Information Ratio. This is a like the Sharpe Ratio, except you replace the returns on your benchmark for the risk-free rate. Here the benchmark could be the market return, or a more specialized return such as the Russell 2000. For small portfolios, say those built from a few ETFs, a very good benchmark is the daily rebalanced portfolio. Historically this has been hard to beat.
• 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.
• This is a favorite measure of commodity traders, and it has great intuitive appeal. Hedge funds typically provide historical drawdown information to their clients, and these are among the most studied numbers. It is strange to see how much attention sophisticated investors will give to a five year track record, even thought when asked in the abstract they may say that such a record has very little information. This is just one of the problems of the business.
• Alpha, or Jensen's alpha. Here you take your portfolio and calculated it's ex-post alpha in the "tautological" CAPM. Many sensible people have this as their favorite measure and use the regression p-values as the Holy Grail of excess performance. There is some elegant theory to back this up (which depends on the normality assumption, which we don't believe --- so you pay your money and take your choice.)
• Treynor Ratio. This is analogous to the Sharpe ratio except you take as the denominator the beta of the CAPM (wo intercept). For some asset choices this works very poorly, but still the Treynor Ratio does have some advocates.
• 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.
• Plots of the Wealth Processes, or the Difference between Wealth Processes. Such plots really do give you the full story. I always want to see these, no matter what tables of summary values are presented.

You can find expositions of these measures in many places. The Wikipedia is starting to get these right, but the Wiki articles were wildly inaccurate for several years.

One source that I like is given in the thesis of Venkat Chandramouli, in section 4.2 (beginning on the pdf page 47, thesis page 39).

The "Tap Root" Criticism

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, was not a strong 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 explain almost everything that does not have a less global explanation . For example, one can argue that it explains the "equity premium" ---the fact that over long periods of time equities have provided greater returns than bonds even when both asset classes are adjusted for observed volatilities.