Sharpe Ratio

In one sentence: The Sharpe ratio says how much extra return you earned for each unit of ups and downs, called risk, that you accepted to earn it.

$S = \frac{R_p - R_f}{\sigma_p}$

Read it out loud: “S equals R p minus R f, all divided by sigma p. S is the Sharpe ratio. R p is the return of the investment, R f is the risk-free return, and sigma p is the volatility, the size of its ups and downs.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Percent and decimal: A percent is a part out of 100, so 8% means 8 parts in 100. In a formula you write it as a decimal, 8% becomes 0.08, 2% becomes 0.02 and 12% becomes 0.12. Both the return and the volatility are rates per year, so the percent units cancel when you divide, and the Sharpe ratio is a plain number with no unit.
Return: The return is how much an investment grew, as a fraction of the money put in, over a period such as one year. A return of 8% a year means that 100 dollars invested at the start is worth 108 dollars after one year, before any fees or taxes.
Risk-free rate: The risk-free rate is the return you could earn with almost no chance of losing money, usually the yield on a very safe government debt. It is the baseline. An investment that carries real risk should pay more than this baseline, or there is no reason to take the risk.
Standard deviation and volatility: The standard deviation is a number that measures how far results usually spread away from their average. A small standard deviation means results stay close to the average, and a large one means they swing widely. When it is applied to investment returns, it is called volatility.
Subscripts like $R_p$ and $\sigma_p$: A small letter written below another letter, like the p in $R_p$ , is a subscript. It is a label, not a power. The p stands for portfolio, the word for a mix of holdings, and f in $R_f$ stands for free, as in risk-free. The Greek letter sigma, written $\sigma$ , is the usual name for volatility.

What each symbol means

Symbol	Meaning
$S$	The Sharpe ratio, a single number with no unit. It is a ratio of two rates that both use the same unit, per year, so the units cancel. A higher number means more extra return for each unit of risk.
$R_p$	The return of the investment over one year, written as a decimal. A value of 0.08 means 8% a year. It is the amount the investment actually earned.
$R_f$	The risk-free rate over the same year, written as a decimal. A value of 0.02 means 2% a year, the return of a very safe asset.
$\sigma_p$	The volatility of the investment’s yearly returns, a standard deviation written as a decimal per year. A value of 0.12 means the yearly returns typically land about 12% above or below their average.

The formula piece by piece

$R_p - R_f$: The extra return: how much the investment earned above the safe rate. With $R_p = 0.08$ and $R_f = 0.02$ , the extra return is 0.06, or 6% a year. This is the reward for taking risk.
$\sigma_p$: The risk: how much the yearly returns swing. Dividing by it turns the reward into reward per unit of risk, so investments with different swings can be compared fairly.
$\frac{R_p - R_f}{\sigma_p}$: The extra return divided by the risk, which is the Sharpe ratio itself. A bigger number means more reward for each unit of swing.
$S = \frac{R_p - R_f}{\sigma_p}$: The whole formula in one sentence: the Sharpe ratio equals the extra return above the risk-free rate, divided by the volatility.

What the formula is doing, and why

The Sharpe ratio answers one question: was the extra return worth the risk? An investment that earns 8% a year sounds good, but if its results swing wildly, a calm investment that earns 6% might be the better deal for the risk you carry. The formula turns both facts into one score so that investments can be compared. It is named after the economist William F. Sharpe.

Why subtract the risk-free rate? You could earn the risk-free rate with almost no worry, so that part of the return is not a reward for taking risk. Subtracting it leaves only the extra return that comes from accepting the ups and downs. In the example, 8% minus 2% leaves 6%, which is the payment for taking the risk.

Why divide by volatility? Volatility measures how much the returns jump around. Dividing the extra return by it asks how much reward you received for each unit of swing. If two investments have the same extra return, but one swings twice as much, its score is half as large. That is why an investment with a lower total return can still come out ahead.

What a good number looks like. A higher Sharpe ratio is better when the comparison is fair. Two investments can only be compared when both are measured over the same period, usually years, and against the same risk-free rate. There is no single threshold that separates good from bad, because the result depends on the assets and the period you look at.

What the number does not tell you. The ratio is calculated from past returns, and it treats volatility as the measure of risk. It cannot predict future returns, and it does not capture every kind of danger, such as a sudden loss that a calm history did not show. It describes what happened, and it is not a recommendation.

An everyday picture

Think of a car’s fuel economy. A car that travels 15 kilometers on each litre of fuel is more efficient than one that travels 10, even if the second car is faster. The Sharpe ratio works the same way. It measures how much reward you get for each litre of risk you burn. A fund that earns 10% a year but swings wildly can be less efficient than a fund that earns 8% with calm swings, as a fast, thirsty car can cover fewer kilometers per litre than a slower, economical one.

Worked example: Comparing two funds by return per unit of risk

Two made-up investment funds are compared over one year, with a risk-free rate of 2% a year. Fund A returned 8% a year, and its yearly returns typically swing by 12% a year, so its volatility is 12%. Fund B returned 10% a year, and its volatility is 20%. In decimal form the inputs are $R_f = 0.02$ , Fund A has $R_p = 0.08$ and $\sigma_p = 0.12$ , and Fund B has $R_p = 0.10$ and $\sigma_p = 0.20$ .

Find the extra return of Fund A. The Sharpe ratio first subtracts the risk-free rate, the return you could earn with almost no risk, so that only the reward for taking risk is left: $R_p - R_f = 0.08 - 0.02 = 0.06$ , which is 6% a year.
Divide the extra return by the volatility. Dividing asks how many units of swing the extra 6% pays for, where one unit is the 12% swing of Fund A. The calculation is $S = \frac{0.06}{0.12} = 0.5$ , so Fund A has a Sharpe ratio of 0.5.
Repeat the same steps for Fund B. Fund B has an extra return of $0.10 - 0.02 = 0.08$ , which is 8% a year. Dividing by its 20% volatility gives $S = \frac{0.08}{0.20} = 0.4$ , so Fund B has a Sharpe ratio of 0.4.
Compare the two scores. Fund B earned more in total, yet its ratio is lower, because its swings are large compared with the extra return it earned. Fund A gave a bigger reward for each unit of risk, even though its total return was smaller.
Read the result carefully. The two numbers come from made-up, round inputs, and they describe past behavior only. They show how the ratio works, not which fund a reader should hold.

Answer: Fund A has a Sharpe ratio of 0.5 and Fund B has 0.4. Fund B’s larger return came with much larger swings, so in these illustrative numbers Fund A gave more extra return for each unit of risk.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Mixing percent and decimal in one formula. If the return is written as 8% (0.08) but the risk-free rate is typed as 2 instead of 0.02, the subtraction gives a wrong result, so convert every rate to a decimal first.
Forgetting to subtract the risk-free rate. Dividing the raw return by volatility also counts the part of the return that needed no risk at all, which overstates the reward for risk.
Comparing ratios measured over different periods or against different risk-free rates. The numbers are only comparable when the time span and the baseline are the same.
Using a volatility measured per month together with a return measured per year. Both the return and the volatility must cover the same period, usually one year.
Taking a high Sharpe ratio as a promise of future gains. The ratio is calculated from past returns, and the future can behave very differently.

Where you will meet it

Comparing mutual funds and other investment portfolios in fund reports and academic research.
Judging a trading strategy by the return it earned relative to the risk it took.
Ranking fund managers by how efficiently they turn risk into return.
Portfolio construction, where the mix of assets is chosen to get the best Sharpe ratio for a given level of risk.

Questions people ask

Is a higher Sharpe ratio always better?

Within a fair comparison, yes, because a higher number means more extra return for each unit of risk. The number is only as reliable as its inputs, though. A ratio based on a short history, or on prices that were smoothed, can look better than the investment really was.

What is a good Sharpe ratio?

There is no single answer. A value below 0 means the investment earned less than the risk-free rate. Values near 1 are often described as good over long periods, but that is a rule of thumb. The number is meaningful only when it is compared with others measured in the same way.

Why does the Sharpe ratio have no unit?

Both the return and the volatility are measured in percent per year. When you divide one by the other, the percent units cancel, leaving a plain number. That is why 0.5 has no name like dollars or years attached to it.

Does the Sharpe ratio tell me which investment to buy?

No. It describes how an investment’s past returns compare with its past swings. It does not weigh your goals, your time frame or your other money, so it is a measurement, not a recommendation.

Volatility (Standard Deviation of Returns): Volatility says how far an investment’s returns usually land above and below their average, so a bigger number means a bumpier ride.
Capital Asset Pricing Model (CAPM): CAPM says the return you should expect from an investment is a safe rate plus extra return for the market risk it carries, scaled by its beta.
Expected Return of a Portfolio: The expected return of a portfolio is the average of its parts’ expected returns, where each part counts in proportion to how much of the money it holds.