Sharpening the Sharpe Ratio

hat happens when "excess returns" head south? Well, investors start bailing out and the Sharpe ratio is harder to interpret.

The Sharpe ratio is a measure which simultaneously considers the risk and return of an investment. Devised by William Sharpe, it is calculated (as shown below) by dividing the excess return of an asset by its standard deviation of return. Excess return is typically calculated by subtracting the T-bill rate of return from the asset being analyzed. The higher the Sharpe ratio the better.

Fundamentally the Sharpe ratio is a measure which attempts to calculate the amount of "reward per unit of risk" of an investment. Equating risk with standard deviation of return is not a perfect approach, inasmuch as returns that deviate above the mean are seldom viewed by investors as a "bad thing." Nevertheless, the Sharpe ratio represents a broadly-accepted and useful statistic in the analysis of investment assets. Many investment research firms calculate the Sharpe ratio for a broad array of investment assets. Morningstar, for example, calculates the Sharpe ratio for mutual funds in its Principia Pro software.

				   Return of Asset minus T-bill rate
Sharpe Ratio	=	   ----------------------------------------------
				   Standard Deviation of Return of Asset

				   Excess Return 
Sharpe Ratio	=	   -----------------------------------
				   Standard Deviation of Return 

Let's consider a simple scenario assuming a T-bill 3 year annualized return of 4%.

Figure 1. Sharpe Ratio Math

As seen in Figure 1 Asset A and B had the same annualized return over the three year period, but Asset A had lower volatility (as measured by standard deviation of return), hence it has the higher Sharpe ratio. The Sharpe ratio operates by very straight-forward rules: 1) the higher the Sharpe ratio the better, and 2) at a given level of return, lower standard deviation of return is better. The Sharpe ratio can be calculated over any period of time. Morningstar has chosen to report the Sharpe ratio as a 3 year figure.

The example in Figure 1 assumes positive excess returns. Over the long term this is a reasonable assumption, but over a short run it may not be. Many domestic equity funds have not produced positive excess return over the past three years. In fact, of the 2,581 distinct domestic equity funds with at least three years of performance history as of October 31, 2002 only 391 (or 15.2%) produced an excess return greater than zero. Thus, nearly 85% of the equity funds had a negative excess return, meaning that an unusually large number of mutual funds currently have negative Sharpe ratios. Negative Sharpe ratios are not inherently problematic, but it turns out that comparing them can be.

Now let's turn to a mirror image example. If, for example, the three year annualized return of Asset A and Asset B turn from positive 5% to negative 5% the excess return changes from +1% to -9%. As shown in Figure 2 we now observe that the Sharpe ratio of Asset B is larger than that of Asset A (recall that a smaller negative is a larger number). This is both counterintuitive and logically backwards. Asset B has the same return as Asset A, but with twice the risk. If higher standard deviation is bad when excess return is positive it should still be bad when excess return is negative. Nevertheless, because of the perverse outcome caused by a negative numerator, Asset B will be viewed as superior to Asset A on the basis of a higher Sharpe ratio.

Figure 2. The Negative "Excess Return" Dilemma

The proposed solution to this dilemma is a modification to the Sharpe ratio, as shown below:

				    ER
Modified Sharpe Ratio1 	=	 --------					     	
				   SD (ER/abs ER)	

	ER = Excess Return     (where:  Excess Return = Asset Return - T Bill Return)
	SD = Standard Deviation
	abs = Absolute value

This modified formula adds an exponent to the denominator of the traditional Sharpe ratio. The exponent is excess return divided by the absolute value of excess return. As a reminder, absolute value simply means that a negative number will be treated as a positive number. In this case, a negative ER will be treated as a positive ER (and a positive ER will remain positive).

Perhaps several "real" examples will help (see Figure 3). Consider three domestic midcap value funds as of October 31, 2002 (using data from Morningstar Principia Pro). As shown by the three funds below, the Sharpe ratio is well behaved when dealing with returns sufficiently large enough to produce a positive excess return. Funds with a higher Sharpe ratio have either a larger return or a smaller standard deviation of return, or both. It is important to note that the highest Sharpe ratio, when comparing funds with similar returns, will belong to the fund with the smaller standard deviation. When excess return is positive the Sharpe ratio and the Modified Sharpe ratio will always be identical. The risk free rate used in Figures 3 and 4 was 4.1%, the three year annualized return for 3 month T-Bills as of October 31, 2002. On a technical note, the Sharpe ratio reported by Morningstar will differ slightly from those shown in Figures 3 and 4 because their calculation uses cumulative returns rather than annualized returns.

Figure 3. Regular and Modified Sharpe Ratio under "Normal" conditions

Now, let's compare three funds that ended up with negative Sharpe ratios as of 10/31/02 (see Figure 4). When ranked by the Regular Sharpe ratio (from highest to lowest) Rydex Basic Materials was ranked higher than both Prudential Utility B and Amana Income because of its smaller negative ratio. This, however, is obviously wrong because Rydex had both a lower return and a larger standard deviation of return than the other two funds. When ranked by the Modified Sharpe ratio, Amana Income has the highest Sharpe ratio (i.e. smallest negative) and Rydex Basic Materials earns the lowest ratio (i.e. largest negative). These results using the Modified Sharpe ratio make sense - both intuitively and logically.

When sorting funds by their Sharpe ratio in descending order (i.e. from highest to lowest) in a program such as Morningstar's Principia Pro, the funds will be correctly ordered when the Sharpe ratio is positive. However, if the Sharpe ratio is negative, the ranking of funds according to their Sharpe ratio will be unreliable. As demonstrated in Figure 4 funds with higher return and lower standard deviation can receive a lower (i.e. a larger negative) Sharpe ratio. Calculating the Sharpe ratio using the modified denominator will avoid this problem.

Morningstar acknowledges the anomaly in the Sharpe ratio in the following statement: "There are some drawbacks to using the Sharpe ratio. If two funds have equal positive average excess returns, the one that has the lower return volatility [i.e. standard deviation] receives a higher Sharpe ratio score. However, if the average excess returns are equal and negative, the fund with the higher volatility receives the higher score. While this result is consistent with portfolio theory, many retail investors find it counterintuitive. Unless advised appropriately, they may be reluctant to accept a fund rating based on the Sharpe ratio, or similar measures, in periods when the majority of the funds have negative excess returns."

The "portfolio theory" which Morningstar refers to suggests that if two funds have equivalent negative Sharpe ratios, the fund with the smaller deviation receives the lower (i.e. larger negative) Sharpe ratio because "it took less risks, but yet its performance was no [better] than the other fund."

I find Morningstar's explanation inconsistent with the basic tenet of risk-adjusted returns, which simply is that higher risk is only preferable if accompanied by higher return. Moreover, if the Sharpe ratio penalizes larger standard deviation when excess returns are positive, the same rule should apply when excess returns are negative.

With a large percentage of equity funds currently experiencing negative excess returns it is critically important to modify the Sharpe ratio in order to correctly rank funds on the basis of their return per unit of risk.

With a simple modification to the denominator, the Sharpe ratio gets sharper.

Figure 4. Regular and Modified Sharpe Ratio with Negative Excess Returns

Notes

¹ Special thanks to Dr. Dennis Sentilles (University of Missouri) for his significant contribution in formulating the Modified Sharpe Ratio.