Understanding The Sharpe Ratio

By Investopedia Staff AAA

Since the Sharpe ratio was derived in 1966 by William Sharpe, it has been one of the most referenced risk/return measures used in finance, and much of this popularity can be attributed to its simplicity. The ratio's credibility was boosted further when Professor Sharpe won a Nobel Memorial Prize in Economic Sciences in 1990 for his work on the capital asset pricing model (CAPM). In this article, we'll show you how this historic thinker can help bring you profits. (To find out more on this subject, see The Capital Asset Pricing Model: An Overview and The Sharpe Ratio Can Oversimplify Risk.)

The Ratio Defined
Most people with a financial background can quickly comprehend how the Sharpe ratio is calculated and what it represents. The ratio describes how much excess return you are receiving for the extra volatility that you endure for holding a riskier asset. Remember, you always need to be properly compensated for the additional risk you take for not holding a risk-free asset.

We will give you a better understanding of how this ratio works, starting with its formula:

Return (rx)
The returns measured can be of any frequency (i.e. daily, weekly, monthly or annually), as long as they are normally distributed, as the returns can always be annualized. Herein lies the underlying weakness of the ratio - not all asset returns are normally distributed.

Abnormalities like kurtosis, fatter tails and higher peaks, or skewness on the distribution can be a problematic for the ratio, as standard deviation doesn't have the same effectiveness when these problems exist. Sometimes it can be downright dangerous to use this formula when returns are not normally distributed.

Risk-Free Rate of Return (rf )
The risk-free rate of return is used to see if you are being properly compensated for the additional risk you are taking on with the risky asset. Traditionally, the risk-free rate of return is the shortest dated government T-bill (i.e. U.S. T-Bill). While this type of security will have the least volatility, some would argue that the risk-free security used should match the duration of the investment it is being compared against.

For example, equities are the longest duration asset available, so shouldn't they be compared with the longest duration risk-free asset available - government issued inflation-protected securities (IPS)?

Using a long-dated IPS would certainly result in a different value for the ratio, because in a normal interest rate environment, IPS should have a higher real return than T-bills. Sometimes that yield on IPS has been extremely high. For instance, in the 1990s, Canada's long-dated real return bonds were trading as high as 5%. That meant that any investor purchasing these bonds would have a guaranteed inflation-adjusted return of 5% per year for the next 30 years.

Given that global equities only returned an arithmetic average of 7.2% over inflation for the twentieth century (according to Dimson, Marsh, and Staunton, in their book "Triumph Of The Optimists: 101 Years Of Global Investment Returns" (2002)), the projected excess in the example above was not much for the additional risk of holding equities. (Learn other ways of evaluating your investments, read Measure Your Portfolio's Performance.)

Standard Deviation (StdDev(x))
Now that we have calculated the excess return from subtracting the risk-free rate of return from the return of the risky asset, we need to divide this by the standard deviation of the risky asset being measured. As mentioned above, the higher the number, the better the investment looks from a risk/return perspective.

How the returns are distributed is the Achilles heel of the Sharpe ratio. Bell curves do not take big moves in the market into account. As Benoit Mandelbrot and Nassim Nicholas Taleb note in their article, "How The Finance Gurus Get Risk All Wrong", which appeared in Fortune in 2005, bell curves were adopted for mathematical convenience, not realism.

However, unless the standard deviation is very large, leverage may not affect the ratio. Both the numerator (return) and denominator (standard deviation) could be doubled with no problems. Only if the standard deviation gets too high do we start to see problems. For example, a stock that is leveraged 10 to 1 could easily see a price drop of 10%, which would translate to a 100% drop in the original capital and an early margin call.

Using the Sharpe Ratio
The Sharpe ratio is a risk-adjusted measure of return that is often used to evaluate the performance of a portfolio. The ratio helps to make the performance of one portfolio comparable to that of another portfolio by making an adjustment for risk.

For example, if manager A generates a return of 15% while manager B generates a return of 12%, it would appear that manager A is a better performer. However, if manager A, who produced the 15% return, took much larger risks than manager B, it may actually be the case that manager B has a better risk-adjusted return.

To continue with the example, say that the risk free-rate is 5%, and manager A's portfolio has a standard deviation of 8%, while manager B's portfolio has a standard deviation of 5%. The Sharpe ratio for manager A would be 1.25 while manager B's ratio would be 1.4, which is better than manager A. Based on these calculations, manager B was able to generate a higher return on a risk-adjusted basis.

To give you some insight, a ratio of 1 or better is considered good, 2 and better is very good, and 3 and better is considered excellent.

Conclusion
The Sharpe ratio is quite simple, which lends to its popularity. It's broken down into just three components: asset return, risk-free return and standard deviation of return. After calculating the excess return, it's divided by the standard deviation of the risky asset to get its Sharpe ratio. The idea of the ratio is to see how much additional return you are receiving for the additional volatility of holding the risky asset over a risk-free asset - the higher the better.

Learn more about this type of analysis in Modern Portfolio Theory Stats Primer.

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