A GreaterThanZero White Paper

Explains the meaning and purpose of the time-weighted return (TWR), and derives from
that the methodology for calculating it. The TWR is the return that is relevant for
mutual funds and similar investment vehicles, where multiple independent investors
contribute money. It is *not* very important for the individual investor who wishes
to gauge the performance of her accounts in the presence of deposits and withdrawals. For
that purpose, the FREQ (fixed rate equivalent) is the appropriate measure. For more on the
relationship between the FREQ and the TWR, see
the last
section of this paper, or the
article “Measuring the
Performance of Accounts with Deposits and Withdrawals”.

Once the intent and purpose of the time-weighted return are fully understood, it is in fact possible to give a more elegant and less formal argument for the correctness of the calculation method that we describe here. Therefore, this article should be viewed as an exercise in applying mathematical methods to issues of financial performance, and also as a challenge to simplify the proof that we give here.

No more than a high school math education is required for this article. However, the reader should be familiar with the material covered in the article “The Basics of Rate of Return, Compounding, and Annualization.”

Suppose you are the manager of a mutual fund, and you need to report the return of
your fund. That implies a financial obligation: every investor in the fund can look at the
return for the period of time that his or her money was in the fund, and then he or
she can demand to be paid accordingly. For example, suppose someone invests $10,000 in your
fund right when the market closes on Dec 31. On the evening of Jan 31 of the new year, the
fund publishes a return of 1% for that month. Now that investor may demand to be
paid $10,100. The sum of all investors' claims at any point in time must equal the current
total value of the fund's assets. That is the constraint that determines the way the fund's
return must be calculated. The result is called
the time-weighted return,
or TWR for short^{1)}.

In the real world, the TWR is almost always given in
annualized form,
in which case it
is more appropriate to call it the
time-weighted *rate of* return.
But annualization is not relevant to what we're discussing here.

Suppose we want to calculate the time-weighted return of a mutual fund for a particular reporting period, and nobody invested or redeemed anything during that period. Then the calculation is simple. All we need to know is the value of the fund's assets at the beginning and at the end of the period, say,

That means the total value of the fund's assets grew by a factor
of V_{2}/V_{1}.
It is clear that each investor's share in the "pot" has grown by exactly that
factor. Remember that the return on an investment is always equal to the factor by
which the investment has grown minus 1. Hence, the fund's TWR is

Things get more interesting if there were investments or redemptions during the period. Here, we will assume that a single investor put money into the fund at some point in time during the reporting period. We're going to need the following data: