How Sharp Are Sharpe Ratios?

Most investors begin with an investment goal. The risk-free asset should be linked to the investment goal. Because not all investors have the same investment goal, the risk-free asset must vary among investors. For example, the risk-free asset may be very different for an investor saving to purchase a home next year vs. an investor saving for retirement.

Once investment goals and risk management assets are defined, many investors choose to allocate some of their savings to assets that increase the uncertainty of meeting their goal but have higher expected returns. Most investors would like to maximize their expected return relative to the risk taken, which is always defined relative to their goal. The most efficient risk-return tradeoff should allow investors to potentially achieve higher returns with minimal risk.

To understand how efficiently an investment is expected to increase returns relative to the risk taken, investors commonly use a simple model called the Sharpe ratio. Sharpe ratios model how much additional return an asset provides over a risk-free rate per unit of volatility. Like all models, the Sharpe ratio is a simplification of reality and is incomplete. The Sharpe ratio requires two key simplifying assumptions: (1) The risk-free asset is a one-month Treasury bill or a similar short-term nominal bond and (2) volatility is a sufficient statistic to summarize risk. Neither assumption can be true for all investors.

Models can be useful for gaining insights that help guide decisions. In this respect, Sharpe ratios are no different; they may contain some useful information for investors. However, the danger lies in applying a model without understanding its limitations. Sharpe ratios alone may not lead to the most efficient risk-return tradeoff because they only account for volatility risk. Reality, however, is far more complex. The risks that matter to investors go beyond volatility and may differ depending on the investment goal. Proper risk management is multifaceted. Shouldn’t risk measurement and assessment be as well?

Illustrating the Problem

To understand why the Sharpe ratio does not properly account for all the risks investors may care about, consider the following examples, summarized in Exhibit 1. An investor can invest in Project A, which has a 99.9% probability of a 10% return but has a small 0.1% chance of losing everything. Alternatively, he could invest in Project B, in which the outcome is a 10% return or better with 99.9% probability. Project B also has a 0.1% chance of loss but that potential loss is limited to 5%. Project C is the third option. Like Project B, the possible outcomes are a loss of 5%, gain of 10%, or gain of 50%. However, the good state of a 50% return is more likely for Project C.

Sharpe ratios alone would indicate that Project A is better than B and B is better than C. Looking at the payoffs, however, it is clear that any rational investor should prefer project C over B and B over A.

The reason the Sharpe ratio falls short is that it only accounts for volatility risk. The risk of a rare disaster, such as for Project A, involves the skewness of the distribution. The potential for good returns also shows up as skewness. Although most investors prefer the opportunity for really good returns (positive skew), they would like to avoid the risk of very bad outcomes (negative skew). The Sharpe ratio does not distinguish between the two. And it fails to capture other features of the distribution, such as fat tails.

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