Sharpe Ratio Formula

Question

In calculating Sharpe Ratio, I have come across a variety of equations that are similar but differ in wordings. I am curious to know which one is used within best practice. Here are all the different equations I have found as well as their source:

1. Trading and Money Management by Bruce and Greene - 2014 pg. 171 $$\frac{\bar{r}_p-\bar{r}_f}{\sigma_p}.$$ "The Sharpe Ratio for a portfolio divides the portfolio's excess return (i.e. its return net of the risk free rate) by the portfolio's standard deviation"
2. Fundamentals of Investing by Smart, Gitman, Joehnk - 2013 pg. 509 \begin{align*} SR&=\frac{\text{Total Portfolio Return - Risk Free Rate}}{\text{Portfolio Standard Deviation Of Return}} \\ &=\frac{r_p-r_f}{s_p}. \end{align*} "Sharpe's measure of portfolio performance, developed by William F. Sharpe, compares the risk premium on a portfolio to the portfolio's standard deviation of return. The risk premium on a portfolio is the total portfolio return minus the risk-free rate."
3. Quantitative Finance link here $$SR(s) = \frac{x_s - r}{\sigma_s},$$ where for the time period under evaluation: $x_s$ represents the average return of the portfolio and $r$ represents average return of the risk-free rate.
4. Wikipedia link here for ex-ante Sharpe Ratio $$SR=\frac{E[R_a-R_b]}{\sigma_a}=\frac{E[R_a-R_b]}{\sqrt{var[R_a-R_b]}},$$ where $R_a$ is the asset return, $R_b$ is the risk-free return (such as a U.S. Treasury security). $E[R_a-R_b]$ is the expected value of the excess of the asset return over the benchmark return, and $\sigma_a$ is the standard deviation of the asset excess return.
5. Investopedia link here $$\frac{R_p-R_f}{\sigma_p},$$ where $R_p$ is the return of the portfolio, $R_f$ is the risk free rate, and $\sigma_p$ is the standard deviation of excess returns.

6. Journal of Portfolio Management enter link description here

   -Ex-Ante Sharpe: Let $\tilde{R}_f$ represent the return on fund $f$ in the forthcoming period and $\tilde{R}_b$ the return on a benchmark portfolio or security. In the equations, the tildes over the variables indicate that the exact values may not be known in advance. Define $\tilde{d}$, the differential return, as: $$\tilde{d}=\tilde{R}_f-\tilde{R}_b.$$

   Let $\bar{d}$ be the expected value of $\tilde{d}$ and $\sigma_d$ be the predicted standard deviation of $\tilde{d}$. The ex ante Sharpe Ratio ($S$) is : $$S=\frac{\bar{d}}{\sigma_d}.$$

   -Ex-post Sharpe Ratio: Let $R_{f,t}$ be the return on the fund in period $t$, $R_{b,t}$ the return on the benchmark portfolio or security in period $t$, and $D_t$ the differential return in period $t$: $$D_t=R_{f,t}-R_{b,t}$$ Let $\bar{D}$ be the average value of $D_t$ over the historic period from $t=1$ through $T$: $\bar{D}=\frac{1}{T}\sum\limits^T_{t=1}{D_t}$ and $\sigma_D$ be the classic standard deviation of $D_t$. Then, the ex-post Sharpe is $$\frac{\bar{D}}{\sigma_D}.$$

Clearly, there are some differences because of the lack of clarification. I believe best practice would be to use 6b (ex-post Sharpe) because it is the most clear.

Lastly, what is the difference between Post and Ante Sharpe? They both seem to be taking the average/expected excess return of the portfolio and risk-free rate divided by the standard deviation(excess return).

Answer 1

As you correctly pointed out, all these formulae are kind of related and try to capture the same notion. The Sharpe ratio is a measure which relates (excess) return and risk (measured by volatility) and hence, gives a metric to compare different assets (may be stocks, indices, portfolios, etc). Obviously, agents prefer a high Sharpe ratio.

The standard asset pricing definition would be \begin{align*} \frac{\mathbb{E}_t[R_{i,t+1}]-R_{f,t}}{\sqrt{\mathbb{V}\mathrm{ar}_t[R_{i,t+1}]}}, \end{align*} i.e. the expected future return of asset $i$ in the next period minus the risk-free rate (which is assumed to be predictable) divided by the conditional standard deviation of the future return. This number is then called (conditional) Sharpe ratio or market price of risk.

The problem with this ''theoretical'' concept is that you simply have no clue what it is in real life. We don't know the distribution of future returns, in particular we don't know the first two moments. This is why you may want to refer to this number as the ''ex-ante'' Sharpe ratio, i.e. a forward-looking Sharpe ratio.

When practitioners employ this number, they estimate the expectation by the simple sample mean and the standard deviation by the sample standard deviation of a time series of historical returns. The length of the time series and its frequency (daily, weekly etc) depends on your applications.

The above idea contrasts to the notion of the ''ex-post'' or ''realised'' Sharpe ratio. This number compares the outperformance of an asset in the past to its volatility and is hence computed as ratio of sample mean to sample standard deviation. Thus, the ''ex-post'' Sharpe ratio asks for the risk-adjusted historical return of an asset.

So, one number looks into the future whereas the other number measures an already observed outperformance. The latter, the ''ex-post'' Sharpe ratio is additionally employed in order to estimate (or forecast) the future Sharpe ratio.

In general, both numbers are relevant but serve different purposes. The conditional Sharpe ratio may be used to assess asset pricing models (see Hansen Jagannathan bound), whereas the ''ex-ante'' Sharpe ratio is a useful number upon which investors would like to condition their investment decisions. Finally, the ''ex-post'' Sharpe ratio can be used to compare historical returns of stocks, funds etc.

Finally, you can of course consider the Sharpe ratio as outperformance between two risky assets. Then, it is the expected/realised difference between their returns divided by the standard deviation of this outperformance. This corresponds to the wikipedia formula which considers two assets, $a$ and $b$. It is however more common to use the risk-free asset as benchmark.

Answer 2

Truly, the only difference in your definitions is between ex-post and ex-ante.

It’s worth noting that the Sharpe ratio is the slope of your SML. This plots expected return for a unit of volatility (risk). Ex-ante means a prediction. You predict what will happen. Your prediction for the risk free asset in this 2 variable universe MUST have zero volatility, because volatility is risk, and the asset is “risk-free”. Therefore, your volatility only comes from excess returns (your risky asset).

In an ex-ante universe, excess return vol = portfolio vol because there’s no “other” kind of volatility. The risk free rate is not a random variable. It is constant (c).

Var(X - c) = Var(X)

In an ex-post universe, you look back. This is mainly for benchmarking. In keeping with your ex-ante assumption, you say “all of my portfolio volatility is attributable to my risky asset.” So you can take the standard deviation of historical returns. In reality, there is no risk free asset. So your risk free proxy moves around and there is risk. I think that’s where you’re confused. But in practice that assumption is fine, because that risk should be equivalent across portfolios since there is no correlation between the risky (X1 and X2) and risk less asset(Y):

Var(X1 - Y) = Var(X1) + Var(Y) Var(X2 - Y) = Var(X2) + Var(Y)

Also, bear in mind that the time period you choose to measure ex-post vol is as much an assumption as anything else. Weekly, daily, monthly? That’s why ex-post is used for comparison. Consistently applied, these assumptions should be of little importance.