3

The Math of the Time-Weighted Return (TWR)
From the beginning of the period to the point in time where the redemption happened,
the value of the fund's assets grew from V_{1} to
V_{2}. Therefore, at the beginning of the period, the
value of the investment that was later redeemed was

It follows that at the beginnig of the period, the total value of the other investors' shares in the fund was

This means that these investors' claim has grown from

to V_{3} over the course of the entire period. Their claim
has thus grown by a factor of

The return that the fund must publish for that period is thus

Again, we see that this return is the result of a compounding. The first part,
V_{2}/V_{1},
describes the return of the fund from the beginning of the period to the point just prior to
the redemption. The second part,

describes the return of the fund from the point just after the redemption to the end of the period.

The following recipe for for calculating the time-weighted return over a period of time is the obvious generalization of Formulas (1) and (2) to situations where there is more than one investment or redemption. Giving a rigorous proof that this generalization is valid is actually a bit tedious. The proof is by induction on the number of cash flows (investments or redemptions). The idea is to partition a reporting period with more than one cash flow into two subperiods each of which has fewer cash flows. One then applies the induction hypothesis to each subperiod and combines the results via compounding. We leave the details as an exercise to the reader.

- The total asset value at the beginning of the period
- For each investment and redemption, the total asset value just before the investment or redemption, and the amount of the investment or redemption
- The total asset value at the end of the period