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4
The Math of the Time-Weighted Return (TWR)

### Calculation:

1. Break up the period into sub-periods that are separated by the points in time where investments or redemptions occur.
2. For each sub-period, calculate the return. That is trivial because during the sub-period, there are no investments or redemptions. The begin value is the total asset value just after the investment or redemption at the beginning of the period. The end value is taken just prior to the investment or redemption that defines the end of the period.
3. Compound the returns obtained in Step 2.

It is perhaps noteworthy that the lengths of the periods, that is, the timing of the investments and redemptions, do not enter the calculation at all.

To emphasize again, this is the return upon which investors may base their claims. In other words, if the time-weighted return over some period of time equals x percent, then an investor who puts in P dollars at the beginning of that period may demand to be paid P × (100 + x) / 100 dollars at the end. This is further emphasized by the fact that in the mutual fund industry, reported returns are often accompanied by a chart that shows the growth of an initial investment of ten thousand dollars. That chart looks exactly like a chart of the cumulative return based on the mutual fund's reported period returns.

## An Example

Suppose that we're looking at a mutual fund that holds only one asset, the stock whose price chart from Jan 1, 2000 through Dec 31, 2005 is shown below. Now let us assume that the fund's holdings start out at 10,000 shares on Jan 1, 2000. Moreover, there was a 2,000 shares investment at the end of 2001 and a 4,000 share redemption at the end of 2002.

Date Number of Shares Owned Dollar Value
Jan 1, 2000 10,000 10,000 × 100 = 1,000,000
Dec 31, 2001 10,000 10,000 × 80 = 800,000
Jan 1, 2002 12,000 12,000 × 80 = 960,000
Dec 31, 2002 12,000 12,000 × 90 = 1,080,000
Jan 1, 2003 8,000 8,000 × 90 = 720,000
Dec 31, 2005 8,000 8,000 × 110 = 880,000