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# Sharpe Ratio

## 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. Volatility is a measure of the price fluctuations of an asset or portfolio.

### Key Takeaways

• 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.
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## Formula and Calculation of Sharpe Ratio

﻿ \begin{aligned} &\textit{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}\\ &\textbf{where:}\\ &R_{p}=\text{return of portfolio}\\ &R_{f} = \text{risk-free rate}\\ &\sigma_p = \text{standard deviation of the portfolio's excess return}\\ \end{aligned}﻿

The Sharpe ratio is calculated as follows:

1. Subtract the risk-free rate from the return of the portfolio. The risk-free rate could be a U.S. Treasury rate or yield, such as the one-year or two-year Treasury yield.
2. Divide the result by the standard deviation of the portfolio’s excess return. The standard deviation helps to show how much the portfolio's return deviates from the expected return. The standard deviation also sheds light on the portfolio's volatility.

## What the Sharpe Ratio Can Tell You

Subtracting the risk-free rate from the mean return allows an investor to better isolate the profits associated with risk-taking activities. The risk-free rate of return is the return of an investment with zero risks, meaning it's the return investors could expect for taking no risk. The yield for a U.S. Treasury bond, for example, could be used as the risk-free rate.

Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.

The Sharpe ratio is one of the most widely used methods for calculating risk-adjusted return. Modern Portfolio Theory (MPT) states that adding assets to a diversified portfolio that has 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 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).

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.

## Example of How to Use 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 their 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 the Sharpe ratio of 95.8%, or (15% - 3.5%) divided by 12%.

The investor believes that adding the hedge fund to the portfolio will lower the expected return to 11% for the coming year, but also expects the portfolio’s volatility to drop to 7%. They assume that the risk-free rate will remain the same over the coming year.

Using the same formula, with the estimated future numbers, the investor finds the portfolio has the expected Sharpe ratio of 107%, or (11% - 3.5%) divided by 7%.

Here, the investor 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 Difference Between Sharpe Ratio and Sortino Ratio

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, making the formula the return of the portfolio less the required return, divided by the 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 with the rest of the market. Beta is a measure of an investment's volatility and risk as compared to the overall 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. The Treynor ratio formula is the return of the portfolio, minus the risk-free rate, divided by the portfolio’s beta.

## Limitations of Using Sharpe Ratio

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 seven, and the least common results will be two and twelve.

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 look-back period, is another way to cherry-pick the data that will distort the risk-adjusted returns.

## What is a Good Sharpe Ratio?

Sharpe ratios above 1.0 are generally considered “good," as this would suggest that the portfolio is offering excess returns relative to its volatility. Having said that, investors will often compare the Sharpe ratio of a portfolio relative to its peers. Therefore, a portfolio with a Sharpe ratio of 1.0 might be considered inadequate if the competitors in its peer group have an average Sharpe ratio above 1.0.

## How is the Sharpe Ratio Calculated?

To calculate the Sharpe ratio, investors first subtract the risk-free rate from the portfolio’s rate of return, often using U.S. Treasury bond yields as a proxy for the risk-free rate of return. Then, they divide the result by the standard deviation of the portfolio’s excess return. Note that, in using the standard deviation, this formula implicitly assumes that the portfolio’s returns are normally distributed, which may not in fact be the case.

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