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Computerized Investing > Fourth Quarter 2012

# Interpreting the Sharpe Ratio

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## by CI Staff

The relationship between risk and return is an essential concept in finance, which argues that riskier investments should compensate investors with higher returns and safer investments should not experience exorbitant price fluctuations.

When comparing the performance of two securities, funds or portfolios, investors must consider risk-adjusted returns to see if they are being adequately compensated for the risk they are assuming. The goal is to achieve the largest return per unit of risk.

William Sharpe devised the Sharpe ratio in 1966 to measure this risk/return relationship, and it has been one of the most-used investment ratios ever since. Here, we discuss how to calculate and interpret the Sharpe ratio.

## Components of the Ratio

Much of the ratio’s fame is attributable to its simplicity, as it comprises only three components. The formula is as follows:

Sharpe Ratio = (Rx – Rf) ÷ StdDev(Rx)

Where:

Rx = average rate of return from investment X

Rf = risk-free rate

StdDev(Rx) = standard deviation of Rx

When analyzing the Sharpe ratio, the higher the value, the more excess return investors can expect to receive for the extra volatility they are exposed to by holding a riskier asset. Similarly, a risk-free asset or a portfolio with no excess return would have a Sharpe ratio of zero.

### Average Return

The Sharpe ratio was originally developed as a forecasting tool, but it can also be used to calculate a historical risk-adjusted return. Expected average returns are used to calculate the forward-looking ratio, whereas actual returns are used in the historical ratio.

The expected return is also known as the required rate of return because it represents the minimum return investors require to compensate them for the added risk, which includes both the riskiness of the investment and the time value of money.

### Risk-Free Rate

The risk-free rate is the return investors require to compensate for the time value of money alone. Typically, investors use the return on U.S. Treasury bills for the risk-free rate because it is reasonable to assume the U.S. government will not default on its debt obligations, and thus investors need only be compensated for the time their capital is tied up in the security.

The Sharpe ratio requires that Rf represents the average return of the risk-free rate over the time period under evaluation. When analyzing a three-year period, investors must average the rate of return on T-bills over the same three-year period.

Traditionally, the shortest-dated bill is used since it is the least volatile. However, some argue the risk-free security should match the duration of the investment. Since equities theoretically have an infinite duration, one could argue that the longest-dated bill should be used.

### Standard Deviation

The standard deviation of a security measures how far returns deviate on average from its mean (or average) return. Standard deviation is a common indicator used to measure the volatility, and thus the riskiness, of an investment. For instance, an investment that deviates only 3% from its mean on average is judged as less risky than an investment with a 20% average deviation.

## Using the Sharpe Ratio

As an example of how to calculate and interpret the Sharpe ratio, we downloaded monthly data on the S&P 500 index, the S&P MidCap 400 index, the S&P SmallCap 600 index and 90-day T-bills from January of 2008 to July of 2012 into an Excel spreadsheet.

Since the S&P 500 consists of 500 large-cap U.S. companies, it should theoretically be the least volatile and also the least rewarding. Similarly, the S&P SmallCap 600 should be the most volatile and also the most rewarding of these three indexes, as small-cap stocks are generally considered to be riskier.

When computing the Sharpe ratio, investors must first make sure they have an abundance of consistent and comparable data points. The more data points we use, the more likely distribution is normal and thus the more accurate our results will be. Although we use monthly data here, shorter intervals can be used but are more volatile and thus may require a longer time period to compensate for the added volatility.

In Excel, we used the function “=Average” to calculate the average monthly return and the function “=StDevP” to calculate standard deviation based on the entire population. The function “=StDevP” is used when the entire population is present or when an individual is only interested in the sample and does not want to generalize the data to represent the entire population. For sampling, the function “=StDev” can be used. Basing the standard deviation on the entire population may be preferable when comparing historical performance, whereas sampling may be preferable when forecasting. We then apply the components to the aforementioned formula to get the Sharpe ratio.

The resulting Sharpe ratios shown in Table 1 indicate that the S&P SmallCap 600, with a Sharpe ratio of 0.06, provided the highest monthly return per unit of risk out of the three indexes over the 4½-year period. As expected, the S&P 500, with a Sharpe ratio of 0.003, had the lowest volatility (standard deviation of 5.67%) and produced the lowest average return (0.05%). Meanwhile, the S&P SmallCap 600 experienced the most volatility, with a standard deviation of 7.12%, and the largest returns, averaging 0.46%.

 S&P 500 S&P MidCap 400 S&P SmallCap 600 T-Bills Average Monthly Return (%) 0.05 0.40 0.46 0.03 Monthly St. Deviation (%) 5.67 6.78 7.12 Sharpe Ratio 0.0031 0.0544 0.0601 Monthly Returns (%) 1/31/2008 -6.12 -6.24 -4.97 0.27 2/29/2008 -3.48 -2.00 -3.15 0.18 3/31/2008 -0.60 -1.14 0.25 0.11 4/30/2008 4.75 7.61 3.92 0.11 ~ ~ ~ ~ ~ 4/30/2012 -0.75 -0.52 -1.32 0.01 5/31/2012 -6.27 -6.42 -6.38 0.01 6/30/2012 3.96 1.73 4.04 0.01 7/31/2012 1.26 -0.12 -0.84 0.01 Source: standardandpoors.com

Therefore, the S&P SmallCap 600 earned an average excess return of 6% per unit of risk, whereas the S&P 500 earned an average excess return of 0.3% per unit of risk. Although the S&P SmallCap 600 is more volatile and thus riskier, holders of the index were much better compensated for the risk compared to holders of the S&P 500 during that period. If investors expect this to continue in the future, they should favor the S&P SmallCap 600 over the S&P 500, as it would offer a higher expected return per unit of risk than the S&P 500.

Note that since all three indexes constitute diversified portfolios and that since diversification reduces volatility, their standard deviations are quite similar. Furthermore, historical performance is not a guarantee of future results. In general, small caps will outperform large caps when the market is improving, but small caps will severely underperform when the market is deteriorating. Although choosing the investment with the highest Sharpe ratio is logical, diversification and risk aversion should be considered first.

## The Ratio’s Weaknesses

### Relative Value

The Sharpe ratio provides valuable information only when compared with another investment. To illustrate, if Company A has a Sharpe ratio of 1.0, does that make it a good investment? What if its competitor, Company B, has a Sharpe ratio of 3.0? All else equal, Company B is more attractive because, although Company A appears to have a high ratio, Company B’s ratio is better.

Moreover, negative Sharpe ratios, which are quite common during bear markets, do not provide useful information because the risk-free asset is then outperforming the investment on a risk-adjusted basis. In that case, investors often flood the bond market in search of the highest risk-adjusted returns available.

### Total Risk

Since standard deviation measures total risk, the Sharpe ratio does not determine what investment is best for a diversified portfolio, rather it shows which investment is better of the two being compared. The total risk of an investment comprises both firm-specific and systemic risk, whereas a well-diversified portfolio should contain virtually no firm-specific risk because it is offset by the other securities. Therefore, it may be appropriate to choose an investment with a lower Sharpe ratio in the interest of maintaining a well-diversified portfolio.

### Normal Distribution

Standard deviation requires that an investment’s returns are normally distributed. That is, they must take the shape of a bell curve. The Sharpe ratio is not a suitable measurement for investments with asymmetric expected returns.

Even if returns are normally distributed, bell curves have real limitations. For instance, they do not take big market moves into account, which can impact long-term returns and affect leveraged investments.

Furthermore, the time period used in the calculation will affect results. Going too far back may not provide an accurate representation of the current situation.

### Volatility

Standard deviation includes movement in every direction, which many consider a weakness because it does not differentiate between upside and downside volatility.

However, because standard deviation and volatility measure the predictability of an investment, which is then translated into risk, high volatility means returns are inconsistent. Strong upside performance of a highly volatile stock can turn severely negative in an instant; thus it is still a risky investment.

## Discussion

Richard Oehlberg from CA posted about 1 year ago:

William F. Sharpe has posted his 1994 paper describing the Sharpe Ratio on line at http://www.stanford.edu/~wfsharpe/art/sr/sr.htm.

That paper and this article present a different formula. Specifically, the standard deviation in Sharpe's paper is for the difference between the Asset less the risk free return (Rx-Rf), and not for the asset only (Rx) as indicated above.

Wayne Thorp from IL posted about 1 year ago:

@Richard,

Thank you for pointing out the variation. Sharpe revised the ratio in 1994. We are using his original formula for this discussion. Wayne A. Thorp, CFA, editor, Computerized Investing

Daniel Pry from TX posted about 1 year ago:

Seems that the formula and calculations in this article should be fixed to reflect 1994 formula. Not really interested in the pre-1994 versions.

James Bier from NM posted about 1 year ago:

I don't think that standard deviation requires a normal distribution. All standard distributions have a standard deviation, a measure of scatter. I don't see where the Sharpe's Ratio requires it either but I don't think you should compare Sharpe Ratios from different distributions.

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