# Cumulative returns of rolling returns

So say I have a series of 252 period rolling simple returns. How can I work out real cumulative returns from that? Is it possible?

$$\ \frac{P_{252}}{P_0}-1 , \frac{P_{253}}{P_1}-1, \frac{P_{254}}{P_2}-1$$ and so on. Where P is price at time t

When you apply the usual formula cumulative return formula, unless I'm doing something very wrong, it explodes towards infinity.

$$\ (1 + r_0) * (1 + r_1) * (1 + r_2) ... - 1$$

If you convert to log returns and then sum, the cumulative return ends up being much, much higher than it is supposed to be as calculated from daily non-rolling returns.

• The returns for [0,252] and for [1,253] are largely overlapping, so of course if you use the compounding formula there is tremendous duplication and you get a big return. As far as I can see there is no way to recover the original daily returns from these rolling returns. – Alex C Mar 5 '18 at 0:23
• You can recover the cumulative return if only you would sample and compound the periodic return at regular intervals. Pretty simple, really. – David Addison Mar 5 '18 at 1:37
• To follow up on what David Addison said, if you have the returns [0,252], [252,504], [504,756], ... i.e. the annual returns, then you can compound those. – Alex C Mar 5 '18 at 20:00