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