The Sharpe Ratio

The Sharpe ratio, proposed by William Sharpe in (Sharpe, 1966) and (Sharpe, 1975), measures a financial strategy’s performance by quantifying its excess reward to its variability. Formally, let $R_s$ be a random variable denoting the return of a given strategy, and let $r_f$ denote the risk-free rate of return, or the return one might get through an approximately risk-free investment such as a high-quality bond. The differential return $R_d$, sometimes called the excess return or residual return, captures the performance of the active strategy relative to the risk-free rate:

$R_d \triangleq R_s - r_f. \tag{1}$

Let $\mu_d = \mathbb{E}[R_d]$ be the expectation of the differential return, and let $\sigma_d = \sqrt{\mathbb{V}[R_d]}$ denote the standard deviation of the differential return. Then the ex-ante Sharpe ratio $S$ is

$S \triangleq \frac{\mu_d}{\sigma_d}. \tag{2}$

If we have historic data, we can compute an ex-post Sharpe ratio using the standard estimators for $\mu_d$ and $\sigma_d$:

$\begin{aligned} \hat{\mu}_d &= \frac{1}{T} \sum_{t=1}^T r_d(t), \\ \hat{\sigma}_d &= \frac{1}{T-1} \sum_{t=1}^T \left( r_d(t) - \hat{\mu}_d \right)^2. \end{aligned} \tag{3}$

Above, $r_d(t)$ denotes the realized (non-random) differential return at time period $t$. The ex-ante or historic Sharpe ratio is

$\hat{S} \triangleq \frac{\hat{\mu}_d}{\hat{\sigma}_d}. \tag{4}$

I will not distinguish between ex-ante and ex-post Sharpe ratios except when the distinction is important.

Information ratio

Often, we are interested in the differential return of a strategy relative to a benchmark strategy with return $R_b$. This can be quantified using the information ratio (IR), which is the Sharpe ratio generalized by replacing the risk-free rate with the return of a benchmark strategy:

$\text{IR} = \frac{\mathbb{E}[R_s - R_b]}{\sqrt{\mathbb{V}[R_s - R_b]}}. \tag{5}$

Since the benchmark is often a market index (a passive investment), IR is often described as the ratio of active return to active risk. The denominator is sometimes referred to as the tracking error.

In my experience, Equation $5$ is a common definition of IR. For example, this is how both (Grinold & Kahn, 2000) and Wikipedia define it. However, in (Sharpe, 1998), William Sharpe argues that the Sharpe ratio is this more general formulation, i.e. that it is the information ratio. This is not how the Sharpe ratio was discussed in (Sharpe, 1966). There, the Sharpe ratio was defined with respect to a risk-free asset. This makes sense, because William Sharpe derived the Sharpe ratio from the capital asset pricing model, particularly from the slope of the linear efficient frontier. He admits this later in (Sharpe, 1998), writing

Originally, the benchmark for the Sharpe Ratio was taken to be a riskless security. In such a case the differential return is equal to the excess return of the fund over a one-period riskless rate of interest.

Thus, I am adhering to Sharpe’s original paper for the definition of the Sharpe ratio and to (Grinold & Kahn, 2000) for the definition of information ratio.

I mention these issues not to litigate the history of the Sharpe ratio, but to underscore that the Sharpe ratio is often conflated for the information ratio, and this may happen because different sources use different definitions.

Time aggregation

Since the magnitude of the Sharpe ratio depends on the magnitude of the returns, the Sharpe ratio is time-dependent. We cannot naively compare the Sharpe ratio of a strategy that has traded for one hour to a strategy that has traded for one month. However, time aggregation can be easily-approximated, although not necessarily well-approximated, using the square-root-of-time rule.

Formally, let $S_1$ denote the Sharpe ratio for one time period of interest (e.g. one day), and let $S_T$ denote the Sharpe ratio after $T$ such periods (e.g. $T$ days). Then the square-root-of-time rule suggests

$\sqrt{T} S_1 \approx S_T. \tag{6}$

This approximation is only as good as the underlying assumptions are true. If the returns of a strategy are correlated, for example, then the i.i.d. assumption of the square-root-of-time rule no longer holds.

A convention is to report the annualized Sharpe ratio. Imagine that we have two strategies, one resulting in daily returns and second resulting in hourly returns. We could compare them by scaling just one Sharpe ratio into the time scale of the other. However, it is useful to always work at the same time scale. Thus, we might convert the first strategy’s Sharpe ratio to an annualized Sharpe by multiplying it by $\sqrt{252}$, assuming $252$ trading days in a year, and then convert the second strategy’s Sharpe ratio to an annualized Sharpe by multiplying it by $\sqrt{24 \cdot 252}$. This would allow for an “apples to apples” comparison of annualized Sharpe ratios.

Relationship to $t$-statistics

Recall that a $t$-statistic is the ratio of the deviation between an estimated parameter value and its hypothesized true value to its standard error. If we think of the parameter of interest as the differential mean $\mu_d$ and if we hypothesize that the true value is zero, the $t$-statistic for $\mu_d$ is

$t_{\mu} = \frac{\hat{\mu}_d}{\text{s.e.}(\hat{\mu}_d)}, \tag{7}$

where the standard error is

$\text{s.e.}(\hat{\mu}_d) = \frac{\sigma_d}{\sqrt{T}} \tag{8}$

for a true standard deviation $\sigma_d$. Since $\sigma_d$ is often unknown, we can replace it with the sample standard deviation, allowing us to express the $t$-statistic as

$t_{\mu} = \sqrt{T} \left( \frac{\hat{\mu}_d}{\hat{\sigma}_d} \right). \tag{9}$

Thus, we can see that the ex-post Sharpe ratio (Equation $4$) is equivalent to the $t$-statistic for $\mu_d$ up to scaling constant. In particular, the annualized ex-post Sharpe ratio is equivalent to the $t$-statistic when the number of data points is equal to the number of trading days, i.e. when using daily returns.