# Weighting several returns over different time frames

I have a set of annualized returns over 4 time periods: 10yr, 5yr, 3yr and 1yr.

Is there a way to weight each return to have a "more representative" return?

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I know this question sounds odd, but I overheard that there is a typical approach to dealing with different time frames in returns. Maybe a exponential-based set of weights? – Victor Jun 20 '14 at 16:44
What's the goal of your analysis? – John Jun 20 '14 at 18:23
Just presentation, I won't use it for any calculation – Victor Jun 20 '14 at 19:19
Ignoring any GIPS-related issues, I think the weighted returns are too complicated for most presentations. That depends on your audience, of course, but people understand a table showing a few different periods of returns so I wouldn't try to fight that too much. If all else fails, do what your boss says. – John Jun 21 '14 at 2:17

You can weight the returns and use them in calculations as shown below.

From this site:-

The weighted mean is

and the weighted standard deviation is

So, making up some annualised returns for time spans, 1 yr, 3 yrs, 5 yrs & 10 yrs:

r = {0.01, 0.02, 0.03, 0.04}


and some weights based on decaying relevance with, say τ = 2

a = {e^(-1/τ), e^(-3/τ), e^(-5/τ), e^(-10/τ)}


and fixing their total to be 1

w = a/Σa


{0.660361, 0.242933, 0.0893701, 0.00733595}

the weighted mean return is 0.0144 and weighted s.d. is 0.00972

Note, by using weights that sum to 1 the formulae are simplified, Σw and (Σw)^2 = 1.

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