Weighting Annualized Returns Across Unequal Time Horizons

Summary

The document proposes combining annualized returns measured over several horizons with a weighted mean and weighted standard deviation. It illustrates assigning each horizon a weight that decays exponentially with its length, controlled by a chosen decay parameter, then normalizing the weights so they sum to one. This gives more influence to shorter, more recent periods and less to longer horizons.

The example uses hypothetical returns for four horizons and reports the resulting normalized weights, weighted mean, and dispersion. These figures demonstrate the calculation rather than establish that this weighting is statistically or economically optimal. The choice of decay rate is subjective and determines how quickly older horizons lose influence. The note does not address dependence between overlapping return windows, nor whether annualized returns from different spans are directly comparable for a particular investment decision.

Key ideas

• A weighted mean can combine annualized returns measured over different horizons.
• Exponential decay weights can give shorter horizons greater influence.
• Normalize the raw horizon weights so their sum is one.
• The selected decay parameter controls how quickly longer horizons lose influence.
• The example demonstrates arithmetic but does not justify a universally representative weighting scheme.

Tags

# Weighting several returns over different time frames


# 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?

## Answer by Chris Degnen (score 1)

https://quant.stackexchange.com/a/12742

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

From this site:-

http://disc.sci.gsfc.nasa.gov/giovanni/additional/users-manual/G3_operation_time_series_stats.shtml

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.