Why Annualized Asset Return Contributions Do Not Add

Summary

The document asks whether a portfolio’s annualized return can be decomposed into annualized contributions from its individual assets. It defines each period’s portfolio return as the sum of asset returns weighted by their portfolio weights, then annualizes the compounded portfolio series. Annualizing each asset’s weighted return series separately generally does not produce components that sum to the portfolio’s annualized return.

The answer points to a portfolio performance attribution methodology and states the key distinction: segment contributions can add to total portfolio performance when expressed cumulatively, but the corresponding annualized figures no longer necessarily add. The reason is that compounding and annualization are nonlinear operations, while the period-by-period weighted returns combine linearly. The document gives no alternative allocation procedure or worked example, so it clarifies the mathematical limitation rather than prescribing a unique way to assign annualized contribution. Any contribution analysis should specify whether it reports cumulative attribution or annualized rates.

Key ideas

• Portfolio return in each period is the weighted sum of the assets’ period returns.
• Compounding and annualizing the portfolio return series is not equivalent to annualizing each asset contribution separately.
• Cumulative segment contributions can sum to cumulative portfolio performance.
• Annualized asset contributions generally do not sum to the annualized portfolio return.

Tags

# Return Contribution for Annual Returns


# Return Contribution for Annual Returns












I have a portfolio $p$ made up of a bunch of assets $i$ where the weights change slowly over time. The returns over each period $j$ composed of the weighted returns of the asset $r^j_p$

$$ r^j_p = \sum_i w^j_i r^j_i $$

The annualized return $\bar{r}_p$ given $n_y$ periods per year and $n$ total periods is:

$$ \bar{r}_p = \prod_j (1+r^j_p)^{(n_y/n)} -1 $$

Is there a financially reasonable way to understand return contribution on this portfolio. I.E. find a $\bar{r}_i$ such that

$$ \bar{r}_p = \sum_i \bar{r}_i $$

If I annualize the weighted return series of each asset the sum of those of course do not at to the portfolio annualized returns.

## Answer by anand k (score 3)

https://quant.stackexchange.com/a/42863

This reference paper from Morningstar explains why they don't add up

https://corporate.morningstar.com/US/documents/MethodologyDocuments/MethodologyPapers/TotalPortfolioPerformanceAttributionMethodology.pdf

"When contributions are expressed in cumulative terms, segment contributions sum to that of the total portfolio. However, once annualized, these numbers no longer add up."

Here segments can just be securities and not just groups like sectors. So mathematically they won't add up in annualized terms

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