The Sharpe Ratio

Motivation(s)

The reward-to-variability ratio is a measure for the performance of mutual funds. Since its introduction, it has gained considerable popularity under a different name: the Sharpe Ratio. To further exacerbate the confusion around the current jargon, several studies have generalized the measure under various names such as return information ratio. The return information ratio encompasses the ratio of the mean to the standard deviation of the distribution of the return on a single investment. While this statistic may be useful, it could lead to wrong decisions because it does not account for the riskless rate of interest.

Proposed Solution(s)

The author proposes unifying these different terms under the Sharpe Ratio and examines a broad range of applications.

This study builds on the Markowitz mean-variance paradigm, which assumes that the mean and standard deviation of the distribution of one-period return are sufficient statistics for evaluating the prospects of an investment portfolio. The focus is on the use of the ratio for making decisions in situations where the first two moments can be usefully summarized with the ratio. Correlations between historic Sharpe Ratios and unbiased forecasts of the ratio are left for other expositions.

Evaluation(s)

The differential return represents the result of a zero-investment strategy i.e. any strategy that involves a zero outlay of money in the present and returns either a positive, negative or zero amount in the future. To compute the return for a zero-investment strategy, the payoff is divided by a notional value. One of the benefits of the Sharpe Ratio is its scale independence to the notional value.

The author derives that the expected return on assets is directly related to the product of the risk position times the Sharpe Ratio of the strategy. This statistic is sufficient for decisions on the optimal risk/return combination when

  • an investor plans to allocate money between a riskless asset and a single risky fund, or

  • a single fund is to be selected to complement a pre-existing group of risky investments.

The optimal solution for each scenario involves maximizing the Sharpe Ratio of the zero-investment strategy. However, the latter requires that all the funds in the set from have similar correlations with the other holdings.

When an investor has a pre-existing set of investments and is considering taking positions in one or more independent and uncorrelated zero-investment strategies, the Sharpe Ratios of the strategies is proportional to their risk levels.

Future Direction(s)

  • Contrarian and momentum strategies have yielded a modest amount of annual gains. Would applying the Sharpe Ratio in a similar automated fashion yield equivalent or better results?

  • To account for correlation, would simply clustering the funds according to their style factors be appropriate?

Question(s)

  • The author claims that annualizing data that apply to periods other than one year before computing a Sharpe Ratio can provide at least reasonably meaningful comparisons among strategies, even if predictions are initially stated in terms of different measurement periods. Another claim is that it is usually desirable to measure risks and returns using fairly short (e.g. monthly) periods. If the Sharpe Ratio is a measure, would plotting this measure over time instead of aggregating the data be useful?

Analysis

The Sharpe Ratio is designed to measure the expected return per unit of risk for a zero investment strategy. The Sharpe Ratio does not cover cases in which only one investment return is involved, and it does not account for correlations.

While the derivations of the measure are intuitive and useful, the paper would be more interesting if the author reported monthly and annual Sharpe Ratio estimates of existing funds (e.g. mutual, hedge, index). Furthermore, since the author mentioned that the Sharpe Ratio has been extended to handle scenarios beyond the original design, it seems worthwhile to consider how these measures hold up against each other.

Notes

Ex Ante Sharpe Ratio

Let \(R^F\) represent the return on fund \(F\) in the forthcoming period and \(R^B\) the return on a benchmark portfolio or security. Define the differential return as

\[\tilde{d} = \tilde{R}^F - \tilde{R}^B\]

where \(\tilde{\cdot}\) indicates that the exact values may not be known in advance.

Let \(\bar{d}\) be the expected value of \(d\) and \(\sigma_d\) be the predicted standard deviation of \(d\). The ex ante Sharpe Ratio

\[S = \frac{\bar{d}}{\sigma_d}\]

indicates the expected differential return per unit of risk associated with the differential return.

Ex Post Sharpe Ratio

Let \(R^F_t\) be the return on fund \(F\) in period \(t\) and \(R^B_t\) the return on a benchmark portfolio or security in period \(t\), The differential return in period \(t\) is

\[D_t = R^F_t - R^B_t.\]

The ex post Sharpe Ratio

\[S_h = \frac{\bar{D}}{\sigma_D} = \frac{ T^{-1} \sum_{t = 1}^T D_t }{ \sqrt{(T - 1)^{-1} \sum_{t = 1}^T (D_t - \bar{D})^2} }\]

indicates the historic average differential return per unit of historic variability of the differential return.

References

Sha94

William F Sharpe. The sharpe ratio. The journal of portfolio management, 21(1):49–58, 1994.