[font={defaultattr}][font={defaultattr}]we all know the standard way of annualisation of volatility for a series of returns is to multiply by SQRT (252).
What if I have a sequence of overlapping 5 day PL returns used as a series of returns and in addition another sequence of overlapping 5 day Moving Average PL returns as a series of returns. How do I then annualise the volatility in each of these cases? Is is multiply by SQRT (252/5) in the first case and multiply by SQRT (252 *5) in the second case? Doesnt necessarily move the needle in either case, could I be thinking about it wrong?[/font][/font]

[font={defaultattr}][font={defaultattr}]Thanks v much[/font][/font]

[font={defaultattr}][font={defaultattr}]J.[/font][/font]

I have quite a bit of experience with timeseries but not necessarily in finance, but I’ll respond anyway
You corrected for the “smoothing” of the moving averages in the second case, so probably that’s why you don’t see the difference. If you wanted to annualise the smoothed series itself, you’d not multiply by sqrt(5), since its returns are daily returns. You want to estimate the underlying risk, though, so you “reverse” the smoothing effect by multiplying by sqrt(5). This can overestimate your volatility, because you assume that the returns are independent- which is obviously not true if they are a moving average.

PS, you can possibly use the two series to estimate the autocorrelation, but it would be “mathwashing” I’m afraid

Yeah I think we are going to forget about the autocorrelation bit which I know is a big assumption under moving average. So just trying to get my simple mind around annualising using SQRT 252 X SQRT 5 in the moving average case. Any chance you can run this past me one more time please? I appreciate it..

I understand you’re asking why averaging over 5 days smooths/ reduces the volatility by sqrt(5). It should be intuitive if you think about it, but here are some hints I quickly scribbled (too tired to wrestle with latex here, sorry!)

If you don’t want to “fix” the smoothing effect, you just do the same as for daily returns to convert it to annual measure.

For problems of this kind I’m a big fan of doing controlled experiments. You don’t need to be fancier than the rand() function in Excel, but there are of course more powerful tools available. Generate your daily return series and aggregate it into weekly P&L and your MA and calculate their statistical properties. That should be doable in a few minutes, and will hopefully show you how the (known) daily return vol is transformed. You can create a very long series to see converged results, and also perhaps of interest, shorter ones to learn something about the sample variation of your estimators.

Actually just re read your notes, u mentioned that if you don’t want to “fix” the smoothing effect, you just do the same as for daily returns to convert it to annual measure. - So we want to work off the new smoothed series i.e. calculate the vol of the smoothed series, thats the whole point as the traders think we have artifical vol - so in this case we DONT bring in the SQRT (5) term - right?

I’d guess that there’s a standard way to calculate the Sharpe Ratio in this situation (which, from what I understand, someone complicated by introducing the sqrt 5 and confused you). If the standard procedure skips de-smoothing the volatility, that makes sense to me. You’re basing your calculation on available data without introducing additional assumptions about the market. The sqrt 5 adjustment can be incorrect in many ways: ignores correlations, volatility clustering, market structure - not to mention that averaging could sometimes lead to higher volatility (eg, during very rapid daily fluctuations). Still, my only experience with trader’s kitchen is cooking in one, so take what I’m writing with a pinch of salt Better follow expert advice from bearish above.

Using overlapping data introduces a bias in the standard deviation of variance estimates you might want to correct for (it understates the standard deviation). Cochrane adjustment corrects for this and there are other similar adjustments:

https://www.cambridge.org/core/journals ... 9EDAC20863