**Special to AllAboutAlpha.com by:** By Neil Kotecha, CAIA, Vice President, Senior Research Analyst, BNY Mellon Wealth Management

Using risk-adjusted return ratios is a necessary yet difficult task to do when analyzing investment managers. Ranjan Bhaduri points out the weaknesses of the Sharpe ratio in analyzing managed futures products in this July post at AllAboutAlpha.com. However, there are times when market anomalies make using the Sharpe and information ratios difficult even on traditional products. During these times, investors should not use the standard version of these ratios, for they can be misleading and result in ill-informed investment decisions.

Between 1970 and the end of 2008 there have been few periods of extreme losses among US and international equities. The S&P 500 Index’s rolling three-year returns have been positive in all but three periods (1972 – 1975, 1999 – 2003 & 2006 – 2008). Similarly, the MSCI EAFE Index has only had three-year declines in 1972 – 1975, 1989 – 1992, 1999 – 2003 & 2006 – 2008. During these periods, many formulas broke down.

Each of the aforementioned ratios is calculated by dividing a type of excess return by a measurement of risk. As a reminder, the Sharpe ratio uses investment returns in excess of the risk-free rate of return as its numerator, then divides that by the standard deviation of the product (its risk). Similarly, the information ratio uses investment returns in excess of the return of an assigned benchmark as its numerator and then divides that by the tracking error of the product to its benchmark (its risk).

When the investment returns are sufficiently low in both instances, the numerators become negative and the ratios break down. Consider the following examples for the Sharpe ratio, which also apply to the information ratio.

The Sharpe ratio holds when it is positive. Investment A has twice the return and the same volatility so it is preferred over Investment B.

In this case, negative Sharpe ratios hold. Investment B is preferred due to a smaller loss. Its Sharpe ratio is also larger.

In this instance, the Sharpe ratios are identical as the one above. But Investment A is preferred as it has the same return but smaller volatility. Unfortunately, the Sharpe ratio indicates Investment B is preferred.

Two formulas that adjust each ratio for negative return environments should be used in lieu of the traditional approaches. In each instance they adjust the denominator by using the standard deviation raised to the power of excess return divided by the absolute value of the excess return. Craig L. Israelsen, the creator of the two ratios, expressed as such with the formulas below.

An important observation is that the changes to both ratios do not have an impact if excess returns are positive as the exponent becomes one divided by one.

The modified Sharpe ratio holds when it is positive. Investment A has twice the return and the same volatility so it is preferred over Investment B. Note that the modified Sharpe ratio has the same value as the normal Sharpe ratio when positive.

In this case, negative modified Sharpe ratios hold. Investment B is preferred due to a smaller loss. Its Sharpe ratio is also larger.

This is the same scenario used in the normal Sharpe ratio example; however it uses the modified calculation. Unlike the traditional Sharpe ratio, it holds as Investment A is preferred and has a larger modified Sharpe ratio value.

It should be noted that the modified ratios are limited in their use. An impact of the new denominator is that the output range of the ratio is very large and less statistically relevant. So if one modified Sharpe ratio is twice as large as another, it cannot be said that one investment’s risk-adjusted returns are twice as good as another’s. However, it can still be said that one is better on a risk-adjusted basis. Therefore, it is advisable to use these ratios only for ranking purposes.

The modified Sharpe ratio is unnecessary during normal market environments as investment returns will often exceed the risk-free rate over long time periods. It only applies when market declines are significant and sustained. However, both ratios are useful in environments when underperformance (against the risk-free rate or a benchmark) is widespread and sustained. It is very important that investors do not use negative results from these traditional ratios as they can be misleading.

## 4 Comments

September 2, 2009 at 9:44 pm## nick gogerty

September 2, 2009 at 10:28 pm## Peter Urbani

October 2, 2009 at 3:22 am## Carvalho