Sharpe ratio: return per unit of risk

Basics11 min read · 10 sources

The Sharpe ratio relates a portfolio's return to its risk: it shows how much return above the risk-free rate you get for each unit of volatility. This allows a fair comparison of portfolios with different levels of risk. It was developed by William F. Sharpe, who received the Nobel Memorial Prize in Economic Sciences in 1990 together with Harry Markowitz and Merton Miller.

The Sharpe ratio formula

The numerator is the excess return: what the portfolio earns over and above a safe investment. The denominator is the risk, measured as volatility. A Sharpe ratio of 0.5 therefore means: for every percentage point of volatility, there was half a percentage point of return above the money market.

Strictly speaking, you calculate with the excess returns per period: the money market rate of the same month is subtracted from each monthly return, and then the mean of these differences is divided by their standard deviation. This is how Sharpe (1994) defined the metric in his later version, also with a benchmark other than the money market. When interest rates fluctuate, this is more accurate than the formula above with annual figures.

The example shows the real benefit: with a little leverage (or a higher equity share), B could be brought to the risk level of A and would then have the higher return. This is exactly what the M² metric of Modigliani and Modigliani (1997) measures: the return of B if it had the same volatility as the benchmark.

Step by step from monthly returns

Here is how to calculate the Sharpe ratio from a series of monthly returns. Suppose a mixed portfolio had these returns over twelve months, and the money market returned a constant 0.25% a month:

The small difference arises because the annual return includes compounding, while the monthly formula uses the simple mean. With high volatility the gap grows: the geometric mean is then well below the arithmetic mean (see volatility drag). Above all, it is important to use the same method when comparing two portfolios.

What is a good Sharpe ratio?

The table is a rule of thumb, not a standard. Over short periods, the Sharpe ratio fluctuates strongly: a value of 1.5 from three good years on the stock market says little about the next three. Just how little is shown in the section on estimation error below. Over decades, a broad equity portfolio is usually at the lower end of the range, because its high return comes with high volatility; a mix with bonds can have a similar or higher Sharpe ratio with a lower return.

Only compare Sharpe ratios over the same period, with the same data frequency and the same risk-free rate. A fund measured over 2009–2021 saw almost nothing but rising markets and zero interest rates. That inflates the metric.

Where the metric comes from

Sharpe (1966) studied US mutual funds and looked for a measure that combines return and risk in a single number. He called it the reward-to-variability ratio, i.e. reward per unit of variability. The idea builds on the portfolio theory of Markowitz (1952) and on the capital market model that Sharpe (1964) had published two years earlier: if you can invest and borrow at the risk-free rate, all sensible portfolios lie on a straight line whose slope is exactly this metric.

It got its current name from other authors. In his 1994 paper, Sharpe extended the definition: instead of the risk-free rate, any benchmark can be subtracted. With an index as the benchmark, this gives the information ratio.

For advanced readers: how accurate is a Sharpe ratio?

A Sharpe ratio from historical data is an estimate with uncertainty. Lo (2002) derived its standard errors. If returns are independent and normally distributed, the following holds approximately:

On top of this comes autocorrelation: if good months tend to follow good months (for example with infrequently valued assets or trend strategies), the monthly volatility underestimates the annual risk, and multiplying by √12 overstates the Sharpe ratio. Lo gives a correction factor for this:

Example: suppose only the first autocorrelation is non-zero, at ρ₁ = 0.2. Then η = 12 / √16.4 = 2.96 instead of 3.46. A monthly Sharpe ratio of 0.15 thus becomes 0.44 instead of 0.52. The simple annualisation overstates by around 17%.

Many attempts, one best strategy

If you test a hundred variants of a strategy and show the best one, you will find a high Sharpe ratio by chance alone. Bailey and López de Prado (2014) correct for this with the deflated Sharpe ratio: it takes into account the number of trials, the length of the series and the skewness and kurtosis of the returns. Harvey et al. (2016) show the same problem for factor research: with hundreds of ideas tested, the usual t-statistic of 2 is not enough. More on this in the article backtest overfitting.

Limitations and common mistakes

• Wrong interest rate: with 0% instead of the actual money market rate, portfolios look better in periods of high interest rates. Portfolio Lab therefore deducts the time-varying euro rate.
• Wrong annualisation: from monthly data, Sharpe p.a. ≈ monthly Sharpe × √12. A monthly Sharpe ratio of 0.12 therefore corresponds to around 0.42 a year, not 1.44.
• Smoothed returns: infrequently valued assets (property funds, private equity) have artificially low volatility and therefore too high a Sharpe ratio.
• Option strategies and skewness: if you sell insurance, you earn steadily for a long time and lose rarely, but then a lot. The Sharpe ratio only sees the standard deviation, not the skewness; the drawdown and the Sortino ratio show more.
• Comparing negative values: if both portfolios are below the money market, the riskier one looks better, because the negative numerator is divided by a larger denominator. Negative Sharpe ratios are hardly suitable for ranking.
• Leverage does not change the Sharpe ratio: a portfolio with double leverage has (before borrowing costs) double the excess return and double the volatility, and therefore the same Sharpe ratio, but a completely different loss potential.

Related metrics

The Sortino ratio, as described by Sortino and Price (1994), divides only by downside volatility, the Treynor ratio by market risk beta, and the information ratio by the deviation from a benchmark index. The Calmar ratio, Omega and Ulcer Index are covered in the article more risk metrics. On the efficient frontier, the portfolio with the highest Sharpe ratio is the tangency portfolio, the target of the “Max. Sharpe ratio” optimisation.

For active funds, the fundamental law of Grinold (1989) links the information ratio to a manager's skill and the number of independent decisions. It explains why high metrics are so rare over long periods.

How Portfolio Lab calculates it

Portfolio Lab follows the definition with excess returns per period: the risk-free return of the same period is subtracted from each monthly or daily return. In euros, this is the time-varying money market rate, or alternatively a fixed rate converted to the period. The mean of these excess returns divided by their (sample) standard deviation is annualised with √12, √52 or √252, depending on the data frequency. The volatility displayed is that of the returns themselves; this is why the Sharpe ratio deviates slightly from (CAGR − rate) / volatility. Details are in the metrics methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.