Return Contribution for Annual Returns

Question

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 1

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