### What is the Sharpe Ratio

The Sharpe ratio was developed by Nobel laureate William F. Sharpe, and is used to help investors understand the return of an investment compared to its risk. The ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk. Subtracting the risk-free rate from the mean return allows an investor to better isolate the profits associated with risk-taking activities. One intuition of this calculation is that a portfolio engaging in “zero risk” investments, such as the purchase of U.S. Treasury bills (for which the expected return is the risk-free rate), has a Sharpe ratio of exactly zero. Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.

#### Sharpe Ratio

### BREAKING DOWN Sharpe Ratio

The Sharpe ratio has become the most widely used method for calculating risk-adjusted return; however, it can be distorted when applied to portfolios or assets that experience large positive and negative movements. A portfolio like this has a return distribution with a high degree of kurtosis ('fat tails') and/or negative skewness.

The Sharpe ratio also tends to fail when analyzing portfolios with significant non-linear risks, such as options or warrants. Alternative risk-adjusted return methodologies have emerged over the years, including the Sortino Ratio, Return Over Maximum Drawdown (RoMaD), and the Treynor Ratio to deal with these problems.

### Formula and Elements of the Sharpe Ratio

Modern Portfolio Theory states that adding assets to a diversified portfolio that have low correlations can decrease portfolio risk without sacrificing return. Adding diversification should increase the Sharpe ratio compared to similar portfolios with a lower level of diversification. For this to be true, investors must also accept the assumption that risk is equal to volatility which is not unreasonable but may be too narrow to be applied to all investments.

The Sharpe ratio is calculated as follows:

R_{p} = Return of portfolio

R_{f} = Risk-Free rate

σ_{p} = Standard deviation of portfolio’s excess return

The Sharpe ratio can be used to evaluate a portfolio’s past performance (ex-post) where actual returns are used in the formula. Alternatively, an investor could use expected portfolio performance and the expected risk-free rate to calculate an estimated Sharpe ratio (ex-ante).

### Applications of the Sharpe Ratio

The Sharpe ratio is often used to compare the change in overall risk-return characteristics when a new asset or asset class is added to a portfolio. For example, an investor is considering adding a hedge fund allocation to her existing portfolio that is currently split between stocks and bonds and has returned 15% over the last year. The current risk-free rate is 3.5%, and the volatility of the portfolio’s returns was 12% which makes a Sharpe ratio of 95.8%.

The investor believes that adding the hedge fund to the portfolio will lower the expected return to 11% for the coming year, but she also expects the portfolio’s volatility to drop to 7%. She assumes that the risk-free rate will remain the same over the coming year. Using the same formula, with the estimated future numbers, she finds the portfolio has an expected Sharpe ratio of 107%.

She has shown that although the hedge fund investment is lowering the absolute return of the portfolio, it has improved its performance on a risk-adjusted basis. If the addition of the new investment lowered the Sharpe ratio, it should not be added to the portfolio. This example assumes that the Sharpe ratio based on past performance can be fairly compared to expected future performance.

The Sharpe ratio can also help explain whether a portfolio's excess returns are due to smart investment decisions or a result of too much risk. Although one portfolio or fund can enjoy higher returns than its peers, it is only a good investment if those higher returns do not come with an excess of additional risk. The greater a portfolio's Sharpe ratio, the better its risk-adjusted performance. If the analysis results in a negative Sharpe ratio, it either means the risk-free rate is greater than the portfolio’s return, or the portfolio's return is expected to be negative. In either case, a negative Sharpe ratio does not convey any useful meaning.

### Criticisms and Alternatives

The Sharpe ratio uses the standard deviation of returns in the denominator as its proxy of total portfolio risk which assumes that returns are normally distributed. A normal distribution of data is like rolling a pair of dice. We know that over many rolls, the most common result from the dice will be 7 and the least common results will be 2 and 12. However, returns in the financial markets are skewed away from the average because of a large number of surprising drops or spikes in prices. Additionally, the standard deviation assumes that price movements in either direction are equally risky.

The Sharpe ratio can be manipulated by portfolio managers seeking to boost their apparent risk-adjusted returns history. This can be done by lengthening the measurement interval. This will result in a lower estimate of volatility. For example, the annualized standard deviation of daily returns is generally higher than that of weekly returns which is, in turn, higher than that of monthly returns. Choosing a period for the analysis with the best potential Sharpe ratio, rather than a neutral lookback period, is another way to cherry-pick the data that will distort the risk-adjusted returns.

It is also possible for an investor to take on too much risk in multiple portfolios and only promote the variation that was lucky enough to have the best returns. Even if the risk of the portfolio was very large, if the investor was lucky enough to receive some windfall profits, the Sharpe ratio in the “surviving” portfolio may still be very high.

A variation of the Sharpe ratio is the Sortino ratio which removes the effects of upward price movements on standard deviation to focus on the distribution of returns that are below the target or required return. The Sortino ratio also replaces the risk-free rate with the required return in the numerator of the formula.

R_{p} = Return of portfolio

R_{r} = Required return

TDD = Distribution of returns below the target or required return

Another variation of the Sharpe ratio is the Treynor ratio that uses a portfolio’s beta or correlation the portfolio has with the rest of the market. The goal of the Treynor ratio is to determine whether an investor is being compensated for taking additional risk above the inherent risk of the market.

R_{p} = Return of portfolio

R_{f} = Risk-Free rate

β_{p} = Portfolio’s beta

### Sharpe Ratio Summary

The Sharpe ratio adjusts a portfolio’s past performance—or expected future performance—for the excess risk that was taken by the investor. A high Sharpe ratio is good when compared to similar portfolios or funds with lower returns. The Sharpe ratio has several weaknesses including an assumption that investment returns are normally distributed.