Sharpe Ratio

The Sharpe ratio is a measure of risk-adjusted performance: it shows how much return an investment earned above a risk-free alternative for each unit of total risk it took. Total risk is measured by the standard deviation of returns. A higher Sharpe ratio means the investor was paid more for each unit of ups and downs they had to sit through. It is the best-known measure of risk-adjusted return, which lets investments with very different levels of volatility be compared on a like-for-like basis.

The measure was introduced by William F. Sharpe in a 1966 Journal of Business paper on mutual fund performance, where he called it the "reward-to-variability ratio". Sharpe revisited and generalised it in a Fall 1994 article in the Journal of Portfolio Management titled "The Sharpe Ratio", by which time the name had stuck. It is one of the most widely quoted statistics in fund analysis and is often shown next to alpha, beta and R-squared.

Imagine two routes to work. One is a motorway that usually saves ten minutes but sometimes jams badly. The other is a slower back road that almost never surprises you. Asking only which route is faster misses the point; a fair comparison asks how much time each saves for the stress and uncertainty it brings.

The Sharpe ratio asks that question about investments. It takes the extra return earned over a safe option, such as government bills, and divides it by how bumpy the ride was. Two funds with the same return are not equal if one got there with far bigger swings. The smoother fund earns the higher Sharpe ratio, because it delivered the same reward with less discomfort.

The ex-post (historical) Sharpe ratio is calculated as:

• $S$ is the Sharpe ratio.
• $R_p$ is the average return of the portfolio over the period.
• $R_f$ is the risk-free rate, usually the return on short-term Treasury bills.
• $\sigma_p$ is the standard deviation of the portfolio's returns, or, in Sharpe's 1994 version, of its returns in excess of the risk-free rate or another benchmark.

Suppose Fund A returned 9% a year on average with a standard deviation of 10%, and Treasury bills yielded 4%. Its Sharpe ratio is (9% − 4%) ÷ 10% = 0.5. Fund B returned 12% with a standard deviation of 20%. Its ratio is (12% − 4%) ÷ 20% = 0.4. Fund B made more money, but Fund A gave more return per unit of risk.

Because returns are often measured monthly, a monthly Sharpe ratio is usually annualised by multiplying it by the square root of 12. A monthly ratio of 0.15 becomes about 0.15 × 3.46 ≈ 0.52 a year. This shortcut assumes returns are independent from month to month.

The ratio is popular because it fits one number to a whole portfolio and allows comparisons across very different strategies:

• Fund comparison. Fact sheets, fund databases and consultants rank funds in the same category by Sharpe ratio, typically over three, five and ten years.
• Portfolio construction. In modern portfolio theory, the portfolio of risky assets with the highest Sharpe ratio is the "tangency portfolio"; combining it with cash or borrowing gives the best available trade-off between risk and return. This links the ratio directly to the risk–return trade-off.
• Strategy testing. Quantitative and algorithmic trading teams use it to compare back-tested strategies, and hedge funds often quote it in marketing material.
• Leverage decisions. Leverage scales both excess return and risk, so it does not change the Sharpe ratio in theory. A strategy with a high ratio but low return can, in principle, be levered up to a target return.

As a rough market convention, a long-run Sharpe ratio for a broad equity index has historically been well under 1, so sustained figures above 1 for a diversified fund are notable and figures above 2 or 3 deserve scrutiny.

• Treynor ratio. Divides excess return by beta instead of standard deviation, measuring reward per unit of market risk. It suits a fund that is one part of a diversified whole.
• Sortino ratio. Uses only downside deviation, so upside swings are not penalised.
• Information ratio. Divides the return above a benchmark by tracking error. Sharpe's 1994 paper treats this as a form of the Sharpe ratio with the benchmark in place of the risk-free asset.

• Standard deviation treats gains and losses alike. A fund with occasional large gains is "penalised" in the same way as one with large losses.
• It assumes roughly normal returns. Strategies that sell options or carry hidden credit risk can show a high Sharpe ratio for years and then suffer a rare, severe loss of the kind described as a black swan event.
• Smoothed prices inflate it. Assets valued by appraisal rather than daily trading, such as private equity or property, show artificially low volatility and so high ratios. Andrew Lo's 2002 Financial Analysts Journal paper on the statistics of Sharpe ratios showed how serial correlation distorts annualised figures.
• Results depend on the period. A ratio measured during a long bull market may say little about how a strategy behaves in a downturn.
• Negative values are hard to rank. When excess returns are negative, a higher standard deviation can perversely make the ratio look better.

• William F. Sharpe: The Sharpe Ratio (Stanford University)
• William F. Sharpe – Facts (Nobel Prize)