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# Sharpe Ratio Formula and Definition With Examples

## What Is the Sharpe Ratio?

The Sharpe ratio compares the return of an investment with its risk. It's a mathematical expression of the insight that excess returns over a period of time may signify more volatility and risk, rather than investing skill.

Economist William F. Sharpe proposed the Sharpe ratio in 1966 as an outgrowth of his work on the capital asset pricing model (CAPM), calling it the reward-to-variability ratio. Sharpe won the Nobel Prize in economics for his work on CAPM in 1990.

The Sharpe ratio's numerator is the difference over time between realized, or expected, returns and a benchmark such as the risk-free rate of return or the performance of a particular investment category. Its denominator is the standard deviation of returns over the same period of time, a measure of volatility and risk.

### Key Takeaways

• The Sharpe ratio divides a portfolio's excess returns by a measure of its volatility to assess risk-adjusted performance
• Excess returns are those above an industry benchmark or the risk-free rate of return
• The calculation may be based on historical returns or forecasts
• A higher Sharpe ratio is better when comparing similar portfolios.
• The Sharpe ratio has inherent weaknesses and may be overstated for some investment strategies.
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## Formula and Calculation of Sharpe Ratio

In its simplest form,

\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}

Standard deviation is derived from the variability of returns for a series of time intervals adding up to the total performance sample under consideration.

The numerator's total return differential versus a benchmark (Rp - Rf) is calculated as the average of the return differentials in each of the incremental time periods making up the total. For example, the numerator of a 10-year Sharpe ratio might be the average of 120 monthly return differentials for a fund versus an industry benchmark.

The Sharpe ratio's denominator in that example will be those monthly returns' standard deviation, calculated as follows:

1. Take the return variance from the average return in each of the incremental periods, square it, and sum the squares from all of the incremental periods.
2. Divide the sum by the number of incremental time periods.
3. Take a square root of the quotient.

## What the Sharpe Ratio Can Tell You

The Sharpe ratio is one of the most widely used methods for measuring risk-adjusted relative returns. It compares a fund's historical or projected returns relative to an investment benchmark with the historical or expected variability of such returns.

The risk-free rate was initially used in the formula to denote an investor's hypothetical minimal borrowing costs. More generally, it represents the risk premium of an investment versus a safe asset such as a Treasury bill or bond.

When benchmarked against the returns of an industry sector or investing strategy, the Sharpe ratio provides a measure of risk-adjusted performance not attributable to such affiliations.

The ratio is useful in determining to what degree excess historical returns were accompanied by excess volatility. While excess returns are measured in comparison with an investing benchmark, the standard deviation formula gauges volatility based on the variance of returns from their mean.

The ratio's utility relies on the assumption that the historical record of relative risk-adjusted returns has at least some predictive value.

Generally, the higher the Sharpe ratio, the more attractive the risk-adjusted return.

The Sharpe ratio can be used to evaluate a portfolio’s risk-adjusted performance. Alternatively, an investor could use a fund's return objective to estimate its projected Sharpe ratio ex-ante.

The Sharpe ratio can help explain whether a portfolio's excess returns are attributable to smart investment decisions or simply luck and risk.

For example, low-quality, highly speculative stocks can outperform blue chip shares for considerable periods of time, as during the Dot-Com Bubble or, more recently, the meme stocks frenzy. If a YouTuber happens to beat Warren Buffett in the market for a while as a result, the Sharpe ratio will provide a quick reality check by adjusting each manager's performance for their portfolio's volatility.

The greater a portfolio's Sharpe ratio, the better its risk-adjusted performance. A negative Sharpe ratio means the risk-free or benchmark rate is greater than the portfolio’s historical or projected return, or else the portfolio's return is expected to be negative.

## Sharpe Ratio Pitfalls

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 return measurement intervals, which results in a lower estimate of volatility. For example, the standard deviation (volatility) of annual returns is generally lower than that of monthly returns, which are in turn less volatile than daily returns. Financial analysts typically consider the volatility of monthly returns when using the Sharpe ratio.

Calculating the Sharpe ratio for the most favorable stretch of performance rather than an objectively chosen look-back period is another way to cherry-pick the data that will distort the risk-adjusted returns.

The Sharpe ratio also has some inherent limitations. The standard deviation calculation in the ratio's denominator, which serves as its proxy for portfolio risk, calculates volatility based on a normal distribution and is most useful in evaluating symmetrical probability distribution curves. In contrast, financial markets subject to herding behavior can go to extremes much more often than a normal distribution would suggest is possible. As a result, the standard deviation used to calculate the Sharpe ratio may understate tail risk.

Market returns are also subject to serial correlation. The simplest example is that returns in adjacent time intervals may be correlated because they were influenced by the same market trend. But mean reversion also depends on serial correlation, just like market momentum. The upshot is that serial correlation tends to lower volatility, and as a result investment strategies dependent on serial correlation factors may exhibit misleadingly high Sharpe ratios as a result.

One way to visualize these criticisms is to consider the investment strategy of picking up nickels in front of a steamroller that moves slowly and predictably nearly all the time, except for the few rare occasions when it suddenly and fatally accelerates. Because such unfortunate events are extremely uncommon, those picking up nickels would, most of the time, deliver positive returns with minimal volatility, earning high Sharpe ratios as a result. And if a fund picking up the proverbial nickels in front of a steamroller got flattened on one of those extremely rare and unfortunate occasions, its long-term Sharpe might still look good: just one bad month, after all. Unfortunately, that would bring little comfort to the fund's investors.

## Sharpe Alternatives: the Sortino and the Treynor

The standard deviation in the Sharpe ratio's formula assumes that price movements in either direction are equally risky. In fact, the risk of an abnormally low return is very different from the possibility of an abnormally high one for most investors and analysts.

A variation of the Sharpe called the Sortino ratio ignores the above-average returns to focus solely on downside deviation as a better proxy for the risk of a fund of a portfolio.

The standard deviation in the denominator of a Sortino ratio measures the variance of negative returns or those below a chosen benchmark relative to the average of such returns.

Another variation of the Sharpe is the Treynor ratio, which divides excess return over a risk-free rate or benchmark by the beta of a security, fund, or portfolio as a measure of its systematic risk exposure. Beta measures the degree to which the volatility of a stock or fund correlates to that of the market as a whole. The goal of the Treynor ratio is to determine whether an investor is being compensated for extra risk above that posed by the market.

## Example of How to Use Sharpe Ratio

The Sharpe ratio is sometimes used in assessing how adding an investment might affect the risk-adjusted returns of the portfolio.

For example, an investor is considering adding a hedge fund allocation to a portfolio that has returned 18% over the last year. The current risk-free rate is 3%, and the annualized standard deviation of the portfolio’s monthly returns was 12%, which gives it a one-year Sharpe ratio of 1.25, or (18 - 3) / 12.

The investor believes that adding the hedge fund to the portfolio will lower the expected return to 15% for the coming year, but also expects the portfolio’s volatility to drop to 8% as a result. The risk-free rate is expected to remain the same over the coming year.

Using the same formula with the estimated future numbers, the investor finds the portfolio would have a projected Sharpe ratio of 1.5, or (15% - 3%) divided by 8%.

In this case, while the hedge fund investment is expected to reduce the absolute return of the portfolio, based on its projected lower volatility it would improve the portfolio's performance on a risk-adjusted basis. If the new investment lowered the Sharpe ratio it would be assumed to be detrimental to risk-adjusted returns, based on forecasts. This example assumes that the Sharpe ratio based on the portfolio's historical performance can be fairly compared to that using the investor's return and volatility assumptions.

## What is a Good Sharpe Ratio?

Sharpe ratios above 1 are generally considered “good," offering excess returns relative to volatility. However, investors often compare the Sharpe ratio of a portfolio or fund with those of its peers or market sector. So a portfolio with a Sharpe ratio of 1 might be found lacking if most rivals have ratios above 1.2, for example. A good Sharpe ratio in one context might be just a so-so one, or worse, in another.

## 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.

Article Sources
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1. Stanford University. "The Sharpe Ratio."

2. The Nobel Prize. "William F. Sharpe."

3. Fortune. "How the Finance Gurus Get Risk All Wrong."

4. MIT Sloan School of Management. "The Statistics of Sharpe Ratios."

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