"When it comes to money, everybody is of the same religion." - Voltaire
Most people would agree that they want to make and have money, but very few people would agree to the level of risk they are willing to take on to make that money. Therefore, risk must be the first issue you address when you are looking at choosing your investments. (For more insight, see Determining Risk And The Risk Pyramid.)
The Sharpe ratio is designed to measure a unit of reward for each unit of risk taken. Let's take a look at this simple ratio in more detail.
Sharpe Ratio Dynamics
The Sharpe ratio, developed by Nobel Laureate William Sharpe, is designed to measure how many excess units of returns an investor can achieve over the risk-free rate for each unit of risk taken.
Let's take a look at the efficient frontier chart below to better illustrate the concept of risk, return and the Sharpe ratio.
|Figure 1: Efficient Frontier - if you plot all the investment choices that you have at your disposal - stocks, bonds and portfolios of stocks and bonds, etc. - on the chart above, the resulting chart will be bounded by an upward sloping curve known as the efficient frontier.|
Without taking on risk, you can achieve a level of return as indicated on the chart by the risk-free portfolio, the U.S. Treasury.
To achieve an additional X percent of return, you will need to take Z level of risk. Portfolio A represents your risk and return payoff. The Sharpe ratio of Portfolio A can simply be defined as X divided by Z. Portfolios B and C will deliver a higher level of returns should you choose to take additional risk beyond Z.
Unlike portfolio B and C, portfolios A' and A'' will deliver a higher level of returns for the same level of risk Z. Thus, A'' is preferable to A' and A' is preferable to A. The Sharpe ratio of A' is defined as X+Y divided by Z.
Therefore, the Sharpe ratio of A' is higher than that of A. Given the same level of risk Z, it can be concluded that any portfolio providing X plus additional returns should be considered superior. The additional achievable returns will be limited by the efficient frontier. Applying this same methodology, we can also presume that Portfolios B and C are superior if their Sharpe ratios are shown to be higher to that of A. (To learn more, check out Understanding The Sharpe Ratio and The Sharpe Ratio Can Oversimplify Risk.)
Breaking Down the Sharpe Ratio
A common mathematical definition of the Sharpe ratio for a portfolio is the excess returns of the portfolio over the risk-free rate divided by the portfolio's standard deviation.
Here is an illustration of the Sharpe ratio in the same efficient frontier chart:
It can be concluded that for a given level of risk (sp), Portfolio A can achieve a higher Sharpe ratio by following the blue arrow toward the efficient frontier or, for a given level of return (Rp), Portfolio A can also achieve higher Sharpe ratio by following the red arrow toward the efficient frontier.
Sharpe Ratio and Risk
The charts and the formula demonstrate that the Sharpe ratio penalizes the excess returns by adding of risk as defined by standard deviation. The standard deviation is also commonly referred to as the total risk. Mathematically, the square of standard deviation is the variance, Markowitz's definition of risk. (For further reading, see Understanding Volatility Measurements.)
So why did Sharpe choose the standard deviation to adjust excess returns for risk and why should we care? We know that Markowitz defined variance as something not to be desired by investors. Variance is defined as a measure of statistical dispersion or an indication of how far away it is from the expected value. The square root of variance, or standard deviation, has the same unit form as the data series being analyzed and is such more commonly used to measure risk.
The following example illustrates why investors should care about variance:
An investor has a choice of three portfolios, all with expected returns of 10% for the next 10 years. The average returns in the table below indicates the stated expectation. The returns achieved for the investment horizon is indicated by annualized returns, which takes compounding into account. As the data table and the chart clearly illustrates below, the standard deviation takes returns away from the expected return. If there is no risk, zero standard deviation, your returns will equal your expected returns.
|Expected Average Returns|
|Year||Portfolio A||Portfolio B||Portfolio C|
Risk and reward must be evaluated together when considering investment choices; this is focal point presented in modern portfolio theory. In a common definition of risk, the standard deviation or variance takes rewards away from the investor. As such, the risk must always be addressed along with the reward when you are looking to choose your investments. The Sharpe ratio can help you determine the investment choice that will deliver the highest returns while considering its risk.
To learn more, read Modern Portfolio Theory: An Overview.