# 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.

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.

• "product of cumulative asset returns" (i.e. formula in F2) is not AFAIK a correct method of computing returns on a portfolio. Commented Aug 31, 2017 at 23:00
• For example: split your money 50/50 between Stock A and Stock B. In next period Stock A is unchanged and Stock B goes to zero. Then you lost $\frac{1}{2}$ you money while Formula F2 incorrectly says you lost 100%. Commented Sep 1, 2017 at 0:02
• Thanks. So what would be the correct procedure if I was trying to impute portfolio $values based solely on periodic contributions to portfolio returns? In other words, is there a way to calculate how many dollars are invested in each asset at each period using only the data in the spreadsheet? I am trying to see how poertfolio weights drift over time without rebalancing. Commented Sep 1, 2017 at 20:22 ## 1 Answer 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.