Why Portfolio Returns Differ from Products of Asset Returns
Summary
The document explains why a portfolio’s cumulative return generally differs from multiplying the cumulative returns of its component assets. For each period, the portfolio return is the sum of the weighted asset returns. Multiplying the assets’ gross returns instead introduces a cross-product term between their period returns, so the calculation is not equivalent to the portfolio’s return series.
The explanation also gives an investment interpretation: the product does not represent a feasible allocation held through time, because the capital exposed to each asset changes with prior returns, while subsequent-period returns arrive at different times. Thus portfolio performance should be compounded from the portfolio’s own periodic returns. The text references a spreadsheet but supplies no numerical example or fuller discussion of rebalancing assumptions; its algebraic point applies to the stated weighted-return setup.
Key ideas
- A portfolio’s period return is the sum of its weighted asset returns.
- Multiplying component gross returns adds a cross-product term absent from the portfolio return.
- Cumulative portfolio performance is found by compounding the portfolio’s own period returns.
- The product of separate asset returns does not generally describe a realizable portfolio allocation through time.
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# Cumulative portfolio returns vs. product of cumulative asset returns
# Cumulative portfolio returns vs. product of cumulative asset returns
I wasn't able to find something that addressed this specifically with the search terms I was using, though I am sure an answer exists here.
[Please reference the image below]
Columns B & C are weighted asset returns (i.e. raw asset return * weight). Column D represents a portfolio of the two assets and is the sum of the weighted returns for each period. Cells B3 & C3 are the cumulative returns of the weighted asset returns. Cell D3 is the cumulative return of the portfolio. Cell F3 is the product of the two cumulative asset returns.
QUESTION: Why are do the calculations in the yellow cells produce slightly different results?
I think this may be an example of where I've forgotten some fairly basic arithmetic concepts, but intuitive explanations would greatly be appreciated. Link to spreadsheet below. Thanks.
Link to spreadsheet
## Answer by Diego F Medina (score 1, accepted)
https://quant.stackexchange.com/a/35874
Mathematically you are asking: $$\prod_i(1+B_i+C_i) "=" \prod_i(1+B_i)\prod_i(1+C_i)\,,$$ which usually does not happen as: $$(1+B_i)(1+C_i)=1+B_i+C_i+B_iC_i\,,$$ so there is $B_iC_i$ factor that you are missing.
Intuitively the product of the returns does not correspond to any investment strategy. There is no way to fix the amount of money you invest so that after an investment period you get: $$(1+B_i)(1+C_i)\,,$$ as having the amount $(1+B_i)$ requires you to be at investment period $i+1$ when you cannot obtain a return of $C_i$ but rather $C_{i+1}$ or $B_{i+1}$. Therefore you should not expect it to be equal to the return of investing weighted amounts on your assets.
Please check @AlexC comment, where he gives you an example of why multiplying the individual returns is an incorrect way to compute the return of the composed portfolio.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.