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

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.”

## The Problem

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 short1).

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.

## The Simplest Case: No Investments or Redemptions

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 V2/V1. 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

## The Case of a Single Investment

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:

1) We actually prefer the term return on initial investment, or ROII for short. However, since the term time-weighted return is widely used, we will stay with it in this article.