Choosing Return Windows for Sharpe Ratio Analysis
Summary
The document considers how to chart Sharpe ratios across daily, weekly, and monthly periods using a long history of one-minute price bars. It distinguishes estimates calculated over expanding histories from estimates calculated within each period, then discusses what makes comparisons across time scales meaningful.
The answer frames the Sharpe ratio as a measure of return relative to uncertainty and relates an annualized Sharpe ratio to a t-statistic under Gaussian assumptions, with the observation length affecting statistical reliability. It suggests resampling returns and estimating how often returns exceed a chosen threshold as another way to assess performance. If returns are independent and identically distributed at the smallest interval, different aggregation scales should yield similar estimates; dependence can change that, and mean reversion can raise measured Sharpe at longer horizons by reducing measured volatility. The discussion gives no worked example and refers readers to further statistical treatment, so it does not settle a universally best window or plotting method.
Key ideas
- A Sharpe ratio compares average excess return with return variability.
- Under Gaussian assumptions, an annualized Sharpe ratio scales with the square root of the observation length when related to a t-statistic.
- Resampling returns can help estimate how often performance exceeds a chosen threshold.
- Independent, identically distributed returns support comparable Sharpe estimates across aggregation intervals.
- Dependence and mean reversion can make Sharpe estimates vary with the return horizon.
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# Sharpe Ratio Graphed Over Time
# Sharpe Ratio Graphed Over Time
I looked and could not find a suitable answer to my question already, so:
What is the best way to calculate the Sharpe Ratio over time, given I have about a decade's worth of 1-minute candlesticks?
I want to chart the Sharpe Ratio per: Day, Week, Month over that 10 year period.
Sharpe Ratio formula: $$ \frac{\bar{r}_p-\bar{r}_f}{\sigma_p} $$
I have two possibilities, but maybe they are incorrect:
- Begin the data from the beginning of the population and end it at the open of the time slice in question (or at the end, but be consistent).
- Begin the data from the open of the time slice, and end it on the close of the time slice.
Are either of these "correct"? Does charting the Sharpe over time have any meaningful value and is it generally done? I haven't found any other examples of doing this.
Thanks a bunch!
## Answer by lehalle (score 3)
https://quant.stackexchange.com/a/59307
First of all, Sharpe Ratio (SR) is meant to assess the uncertainty surrounding the expected returns of your PnL. In short: you divide by the standard deviation of the returns because you trust less a time series of PnL with a large standard deviation than with a small one.
Nevertheless it is in fact not the best indicator; the best one is the t-test, that reflects the probability that your PnL is positive, under Gaussian assumptions. There is a simple relationship between the SR and the t-test: when your returns are annualized and known over $y$ years, one can write:
$$\mbox{t-test}=\mbox{SR}\cdot \sqrt{y}.$$
It is natural since the longer you observed a SR, the more reliable your expected returns.
It seems that you have in mind to compute different estimates of your SR over different time windows. I would say that it is in the spirit of bootstrapping the SR. My view is that is is better to bootstrap the returns and to directly compute the probability that the returns are larger than a given threshold. You can do it as soon as you compute the returns $r$ over $N$ different time windows $\omega_1,\ldots,\omega_N$. Hence an estimator of the probability that the expected returns are larger than an arbitrary threshold $\theta$ is the number of $r(\omega_n) > \theta$ divided by $N$.
Then you ask the question of the time scale at which it is "better" to compute the returns and obtain a reliable SR. If the returns at the smallest time scale are i.i.d., then all time scale will give the same estimate, and hence it is better to take the smallest time scale (to use as much points as possible). If they are not i.i.d. it is far more complicated. See The Statistics of Sharpe Ratios, by Andrew Lo. It is for instance obvious that is the returns are mean reverting, the largest the time scale, the lower the standard deviation and as a consequence the largest the SR.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.