# Sharpe Ratio

Sharpe ratio is a measure of excess return earned by investment per unit of total risk. It is calculated by dividing excess return (which equals return minus risk free rate) by standard deviation of the investment returns.

Investment management requires a trade-off between risk and return. Investments that have high risk must be compensated by high return. For example, comparing return on a high growth technology stock with return on a mature utility stock is meaningless unless we also factor-in the difference in risk level. Sharpe ratio standardizes investment returns so that they are comparable across investment portfolios, companies, investment classes, industries, etc.

## Formula

Sharpe ratio is calculated by dividing the difference between portfolio return and risk-free rate by the standard deviation of the portfolio. Investment return is the actual realized return or expected return on an investment or portfolio over a period.

Risk free rate is the rate earned on risk-free assets. Yield on government treasury bills is normally used a proxy for risk-free rate.

Standard deviation is a statistic which measures the total risk i.e. volatility of an investment portfolio.

All the three inputs in the formula must be for the same time period. For example, if the excess return is for one year, the standard deviation must also be for a year. Please note that monthly standard deviation can't be annualized by simply multiplying it with 12.

The excess return is sometimes calculated with reference to another benchmark instead of the risk-free rate.

## Interpretation

Risk-free investments such a Treasury bills have zero Sharpe ratio because their investment return equals risk free rate and the standard deviation of their returns is zero.

Investors target a return which is in line with their risk tolerance. Risk averse investors are willing to take on more risk only if there is excess return. Sharpe ratio measures just that i.e. the amount of excess return per unit of risk.

A higher Sharpe ratio is better.

## Example

Alphamania is an asset management company which has only two funds: Alpha Driller (AD), an oil and gas focused fund, and Alphologics (AL), a technology focused fund.

For the year ended 31 June 20X7, Alpha Driller earned annual return of 6.49% and Alphologics earned 10.86%.

Secure Pensions, Inc. (SP) has placed 5% of their total assets equally in both the funds. Considering the annual returns, SP’s trustee suggested that the whole 5% assets should be placed in Alphologics. Let’s see if the suggestion carries any merit. On the face of it, a decision based only on the realized or expected return is a poor decision because it ignores the associated risks. Investment decisions should be made by considering both the risk tolerance and return per unit of risk.

Here is the monthly returns for both the funds from July 20X6 to June 20X7:

Alpha Driller Alphalogics
0.20% 2.00%
0.50% 1.00%
-0.30% 3.00%
0.60% -8.00%
1.00% 1.00%
2.00% 2.50%
0.25% 0.50%
0.10% 3.00%
-0.20% 5.00%
-0.30% -3.00%
0.00% 2.00%
2.50% 2.00%

Let’s assume that the risk-free rate for the period is 1.5%.

Using MS Excel STDEV function, we find out that Alpha Driller and Alphologics have standard deviation of 0.90% and 3.39% respectively.

As we already have values for the investment return and risk-free rate, we can work out Sharpe ratios for both investments:

 Sharpe Ratio (Alpha Driller) = 4.9% − 1.5% = 5.54 0.90%
 Sharpe Ratio (Alphalogics) = 10.86% − 1.5% = 2.76 3.39%

This shows that even though Alphologics earned a return higher than Alpha Driller, it doesn’t mean that it is a better investment. In fact, when the risk is accounted for, Alpha Driller appears to be a better fund as evident from its higher Sharpe Ratio.

## Disadvantages of the Sharpe Ratio

Even though Sharpe ratio is useful, some of its assumptions are problematic.

First, it uses standard deviation which is a measure of total risk of a portfolio or investment. Since unsystematic risk can be diversified, beta coefficient, which is a measure of the systematic risk is a better indicator of risk in the context of a diversified portfolio. Treynor’s ratio, which is a variant of the Sharpe’s ratio, attempts to address this weakness of the Sharpe ratio by using beta coefficient in the denominator instead of standard deviation.

Second, it assumes that investment returns are normally distributed, which is not the case for many investment classes such as derivatives, etc. Sortino ratio attempts to address this second weakness.