Using Strategy Returns to Calculate an Ex Post Sharpe Ratio
Summary
The document distinguishes the return of an underlying asset from the return series generated by a trading strategy. An asset’s daily price changes describe its buy-and-hold performance, while a strategy’s wealth changes describe the performance an investor actually experiences when following that algorithm. For evaluating a strategy, the Sharpe ratio should use the strategy’s realized return series; the asset’s ratio is useful as a benchmark for comparison.
The response clarifies that a Sharpe ratio divides mean excess return by standard deviation, not variance. For an ex post estimate, calculate the sample mean and sample standard deviation of the realized returns rather than using expected returns. The annualization convention shown assumes daily observations and no risk-free rate. The document does not discuss serial correlation, changing exposure, transaction costs, or uncertainty in the estimated ratio, so comparisons should account for those factors separately.
Key ideas
- Use returns on strategy wealth to measure the realized Sharpe ratio of a trading algorithm.
- Use the underlying asset’s return series to measure its own performance or as a buy-and-hold benchmark.
- The Sharpe ratio uses standard deviation as its denominator.
- An ex post Sharpe ratio uses the sample mean and standard deviation of observed returns.
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# Simple Sharpe Ratio Question Related to Trading Strategy
# Simple Sharpe Ratio Question Related to Trading Strategy
Given a price vector $(p_1,p_2,...,p_n)$ for some stock, then the corresponding return at $k$th day is described by $$ R_k = \frac{p_{k+1} - p_k}{p_k} $$
On the other hand, let $W_k$ be wealth at day $k$ induced by some trading algorithm. Then I have another ''return" (under some trading algorithm) given by $$ r_k = \frac{W_{k+1} - W_k}{W_k} $$
Assuming no risk-free rate. I want to use annualized Sharpe ratio to characterize my trading performance, then I got huge confusion for the following two possibilities: [The Sharpe Ratio formula below are fixed. ]
$$SR_1 = \sqrt{252} \frac{E[R_k] }{Std(R_k)}$$
$$SR_2 = \sqrt{252} \frac{E[r_k] }{Std(r_k)}$$
which one is the correct annualized Sharpe ratio? Any suggestion is appreciated.
## Answer by Forgottenscience (score 4, accepted)
https://quant.stackexchange.com/a/32376
First, you do not divide by the variance, but the standard deviation when calculating Sharpe ratios. Secondly, none of them are wrong, but $SR_1$ is the expected Sharpe ratio of the asset you are trading, and the second is the expected Sharpe ratio of your strategy. As a trader you care about the latter, but the first can be interesting to see if you actually beat just buying and holding the instrument in question.
What this means is that, notationally, you don't input the ex ante (expected) return in your Sharpe ratio calculation if you want to evaluate your results ex post (after they happened.) So you calculate your return series $r = \{r_1, r_2, r_3, \ldots, r_t\}$ and calculate the sample mean, $\hat{\mu}_{r}$ and similarly with the standard deviation $\hat{\sigma}_r$. This gives you the realized or ex post Sharpe ratio.
To make it explicit per Alex C's suggestion, calculating the ex post Sharpe ratios for both the strategy and the asset can be used to do what I describe in the first paragraph.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.