Sharpe Ratio Example - Illustrating How the Equation is Used

One of the biggest benefits to the Sharpe ratio is its simplicity. The equation involves only three terms and using the equation is a simple as placing the correct numbers in the correct places. It can be used to compare two options to each other, to easily determine which one provides a better return for the risk taken.

How to Use the Equation

In the equation S = R – Rf / s, R is a fairly simple to determine, as it is the actual rate of return. Since there is no way to determine for certain how an option will perform, when the Sharpe ratio is used to make a prediction, the rate of return is based on past performance. In other words, the R for predicted and past returns will be the same value; it is just a question of whether it is factual or estimated. The standard deviation used is a measure of the volatility of the option that is being evaluated. The R and s both refer to the portfolio that is being evaluated, and the Rf is an industry standard.

A Basic Sharpe Ratio Example, Portfolio A

Assume portfolio A had or is expected to have a 10% rate of return with a standard deviation of 0.10. In the United States, US treasury bills are often used as the benchmark for risk free interest rates. During the 20th century, the treasury bills averaged a return of about 0.9%. In that case, R would 0.10, Rf would be 0.009, and s would be 0.10. The equation would be set up to read (0.10 – 0.009)/0.10, which calculates to 0.91. In other words, the Sharpe ratio for portfolio A would be 0.91. However, this number in and of itself is worthless without something to compare it to.

The Effect of Volatility, Portfolio B

If portfolio B shows more variability than portfolio A, but has the same return of return, it will have a greater standard deviation, but the same R. Assuming the standard deviation for portfolio B is 0.15, the equation would read (0.10 – 0.009) / 0.15. The Sharpe ratio for portfolio would be 0.61, so portfolio B would have a lower Sharpe ratio relative to portfolio A. This is not a surprising result, considering the fact that both investments offered the same return, but B had a greater risk. Obviously, the one which has less risk but offers the same return would be the preferred option.

Greater Return, Portfolio C

This next Sharpe ratio example should be even more intuitive. If two portfolios have the same risk (in mathematical terms, the same standard deviation), but one provides a greater return, the investment with the greater return is the better choice. This is one of the most basic principles of investing, and the Sharpe ratio can show it mathematically. If portfolio C has the same variability as portfolio B, but gives a return of 20%, its Sharpe ratio would be (0.20 – 0.009)/0.15 = 1.91. Not surprisingly, portfolio C’s ratio of 1.27 is significantly better than portfolio B’s 0.61.

Two Completely Different Options

It gets a bit more complicating when portfolio A is compared to portfolio C. Portfolio A has a lower rate of return, but it is also very low risk. Portfolio C offers higher returns for higher risk. Simply glancing at the details of the two portfolios is not enough to determine which one is a better investment. This is where the Sharpe ratio comes into play. Portfolio A has a ratio of 0.91 and portfolio C has a ratio o f 1.27, indicating that the risk of portfolio C is well worth the returns as compared to A.

When comparing two portfolios with the same risk or return, it is easy to see which one is a better choice. However, when looking at two options with completely different details it can be hard to determine which one provides the better return for risk. By plugging the numbers into the simple equation known as the Sharpe ratio, the return versus risk factor can easily be determined.