returns - How to calculate the annual contribution of a fund to a portfolio of funds?

let's assume I have a portfolio of two funds (call them F1 and F2), where, by convention, there is a monthly compounding of the returns.

On a monthly basis, the contribution of each fund will just be Weight_Fi*Return_Fi.

On an annual basis, though, since the assumption is that there is monthly compounding in the whole portfolio (in the sense that Return_portfolio_year = (1+Return_portfolio_jan)*(1+Return_portfolio_feb)...) I cannot think of any straightforward way to calculate the annual contribution of each fund.

Answer

These problems arise when you compute arithmetic return contributions: in a given month, you want the sum of the funds' contributions to equal the portfolio return. The sum of these single-month contributions over more than one month will never equal the total portfolio returns over more than one month.

One way to include the compounding effect is to 'pretend' that a segment's return contribution in one period is reinvested in the overall portfolio in succeeding periods.

Here is an example, with R code. There are just two periods: first F1 makes 10%, then F2 makes 10%. The weights of F1/F2 are kept constant at 50% each.

library("PMwR")
weights <- rbind(c( 0.5, 0.5),
                 c( 0.5, 0.5))

R <- rbind(c( 10,   0),
           c( 0 , -10))/100


rc(R, weights, segment = c("F1", "F2"), timestamp = 1:2)

The output will be

$period_contributions
  timestamp   F1   F2 total
1         1 0.05 0.00  0.05
2         2 0.00 0.05  0.05

$total_contributions

    F1     F2  total 
0.0525 0.0500 0.1025 

F1 makes a higher overall contribution because its single-period contribution occured earlier and hence is compounded by the overall portfolio return.

(The PMwR package, of which I am the author, is available from GitHub https://github.com/enricoschumann/PMwR .)