Estimating returns per year by scaling them using the returns of an index

Dear geniuses,

I'm doing research on some return series covering different periods. My data consists of transactions and I can thus calculate the returns on individual transaction pairs.
As I want to compare the returns per period, but most transactions overlap periods I would like to split/estimate the returns per period.
I first did this assuming the returns are exponentially increasing over the whole transaction time, and then calculated the return per period by looking at the number of years of the transaction that fall in the period.

However, the assumption that the increase is the same in each year is rather big. Therefore, I would like to give it a more realistic division by looking at an index that is constructed using the same data. (It concerns housing prices over 100 years). So I want to divide the returns of each transaction to the periods it overlaps assuming the returns took the same path as the returns of the index.

But I have no clue how to do it in a proper way.

I've been playing around with it, but if I just calculate the fraction of the total return for each year of the index, and then multiply this by the total return of transaction pair to check, I don't get to the correct total return.

I hope you can help!

I had a spark of wisdom and solved the issue.

It was a lot simpler than expected.
RETURN_Transaction_t-t+i = RETURN_Transaction_t-tn ^ (RETURN_INDEX_t-t+i / RETURN_INDEX_t-tn)