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5
The Math of the Time-Weighted Return (TWR)

According to our calculation method, we must divide the entire period (Jan 1, 2000 through Dec 31, 2005) into three sub-periods:

Begin Date End Date Begin Value End Value
Jan 1, 2000 Dec 31, 2001 1,000,000 800,000
Jan 1, 2002 Dec 31, 2002 960,000 1,080,000
Jan 1, 2003 Dec 31, 2004 720,000 880,000

The TWR for the fund is obtained by compounding the returns for the three periods:

In this case, there is of course an easy way to double-check that result: holding shares in this fund is tantamount to owning shares of the fund's one stock, and that stock has gained 10% over the time period in question.

In the real world, a mutual fund company would report an annualized return rather than the full five year return in a case like this, but that is a separate issue.

## Individual Investor's Accounts: TWR vs. FREQ

We started our derivation of the methodology for calculating the time-weighted return by looking at the financial obligations of mutual funds: if a fund reports a return of x percent for some period, then an investor who puts in P dollars at the beginning of the period may demand to be paid P × (100 + x) / 100 dollars at the end. Here's another way of expressing the same thing: suppose we're looking at a mutual fund over a period of time during which there were investments and redemptions. Now consider an imaginary fund that

• existed for the entire duration of the time period in question,
• had no cash flows (investments or redemptions) during that period,
• and had, at each point in time, the exact same asset allocation as the real mutual fund.

Then the real fund's TWR is the return on a single initial investment made in the imaginary fund at the beginning of the time period.

Now let us forget about mutual funds for a moment and consider an individual investor's financial account, like a 401k. Let us assume, as could be the case in the real world, that there are deposits and withdrawals. Technically and mathematically, this is no different from a mutual fund: it is an active portfolio which is subject to cash flows. Assuming that we know the amounts of the cash flows as well as the account values on the dates of the cash flows, we can calculate the account's time-weighted return. Using the insight that we have gained so far, we can now describe the meaning of this return in the context of individual investor's accounts.

Suppose you deposited \$5,000 into your 401k at some point in time in the past. Then you calculate the TWR from that point in time to the present. That return tells you what became of those \$5,000. It is a measure of the performance of that one particular deposit. It is not a measure of the performance of the account as a whole with all the other deposits and withdrawals that occurred in between. Anything could have happened there, depending on the timing and the amount of the other deposits and withdrawals. In other words, the time-weighted return tells you next to nothing about how much money you made or lost with that account. It strictly describes the performance of that one deposit.