# 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)

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 ...

• It would be used for the calculation of Beta in CAPM Aug 23, 2019 at 11:19
• well for sure it's more appropriate to use Simple returns. Aug 23, 2019 at 13:54
• Could you explain why that is? Aug 23, 2019 at 13:58

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).

• I think this is backwards (negative auto-correlation induces lower monthly volatility, and hence higher Sharpe ratio than daily, and positive auto-correlation induces higher monthly volatility, hence lower Sharpe ratio) but in any case, the question is about beta estimation, not Sharpe ratio estimation. Aug 28, 2019 at 11:56
• you are right, sorry, I edit my answer Aug 28, 2019 at 14:34