Return Frequency, Autocorrelation, and Sharpe Ratio Comparisons

Summary

The question asks whether beta should be calculated from cumulative or monthly returns, but the answers mainly discuss how sampling frequency and autocorrelation affect Sharpe ratios. They state that when returns are independent and identically distributed, Sharpe ratios should be consistent across daily, weekly, and monthly samples, with finer-frequency data providing more observations. When returns are autocorrelated, aggregation can change measured volatility and therefore the Sharpe ratio.

Negative autocorrelation, associated here with mean reversion, may make longer-horizon samples less volatile and raise the measured monthly Sharpe ratio relative to the daily one. Positive autocorrelation can have the opposite effect. The suggested practice is to compare results across frequencies and examine autocorrelation; spectral analysis may help identify seasonality. These points do not directly settle which return frequency is appropriate for estimating beta, so that question remains unanswered in the source. The discussion is a qualitative guide, not a universal rule, and refers readers to formal Sharpe-ratio analysis for specific autocorrelation structures.

Key ideas

• Under independent, identically distributed returns, Sharpe ratios are expected to be comparable across sampling frequencies.
• Autocorrelation can make measured volatility and Sharpe ratios vary with the return horizon.
• Negative autocorrelation may lower long-horizon volatility and raise the corresponding Sharpe ratio.
• Comparing frequencies and examining autocorrelation can reveal important return dynamics.
• The answers do not directly resolve the question about beta calculated from cumulative versus monthly returns.

Tags

# Beta: Cumulative vs. Simple Returns


# Beta: Cumulative vs. Simple Returns












How would calculating Beta using cumulative returns differ from Beta calculated with monthly returns? Is one more appropriate to use than another?

https://en.wikipedia.org/wiki/Beta_(finance)

## Answer by Gogo78 (score 1)

https://quant.stackexchange.com/a/47271

It's more appropriate the monthly returns than the cumulative one from experience. But all depends on why and for what you gonna use it ...

## Answer by lehalle (score 1)

https://quant.stackexchange.com/a/47348

If the returns of your time series are i.i.d. the Sharpe Ratios (SR) of the daily, weekly or month samples will be the same. Since the more data the better, you should take the daily one.

Nevertheless, if your time series is auto-correlated, the different sub-sampling will give you different SR, simply because

- if your time series is negatively auto-correlated (it mean reverts), then long term sub-samples will most probably have lower volatility, ie the monthly SR will be higher than the daily one;

- if it is positively auto-correlated (it is super-diffusive, it is quite rare), then long term sub-samples will most probably have larger volatility: the monthly SR will be smaller.

In general, you need to look at the SR at different frequencies and to have a look at your auto-correlation, just to know it. If you want to play a little bit with your data you can even apply a Fast Fourier Transform of it, to notice some seasonality effects (that would affect to SR too).

Have a look at The Statistics of Sharpe Ratios, by Andrew W. Lo for a formal approach of this (with closed form formula for some specific shapes of auto-correlations).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.