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The Math of the Time-Weighted Return (TWR)

If you want to measure the performance of your account as a whole, that is, the performance of all your money rather than just the one initial investment, you should be using the FREQ (fixed rate equivalent). The FREQ is the annual interest rate of the fixed rate account (think a somewhat idealized savings account) that replicates the performance of your real account. In other words, imagine that all deposits and withdrawals would have been made to an imaginary savings account with a fixed interest rate rather than your real account. Then you can ask, "Which annual interest rate for the savings account would have resulted in the same ending balance than your real account?" That rate is the FREQ. The FREQ is a very real measure. If your FREQ comes in as, say, 0.5%, then you would indeed have more money now had you put it all in a real savings account whose interest rate never fell below 0.75%.

Measuring performance the way the FREQ does, taking into account deposits and withdrawals, is also referred to as money-weighted performance measurement.

In summary, we see that for an individual investor's account, the time-weighted return is not a measure of the performance of the account, or, for that matter, of the investor's money. One could perhaps call it a "musing" on one's money: "I wonder what became of those particular $5,000 that I inherited from my great-aunt." We have seen behavioral psychologists advice both for and against such an attitude, where one considers different chunks of one's money separately, depending on how or when they were attained.

It should be easy for the reader to convince herself that in the absence of deposits and withdrawals during the measuring period, the TWR and the FREQ are the same. (That's assuming that the TWR is annualized. In the case of the TWR, annualization is really a separate issue; one may or may not annualize the TWR. The FREQ, on the other hand, is inherently an annual rate.)

It is perhaps worth pointing out once more that both the TWR and the FREQ are naturally defined in terms of certain related imaginary accounts. The TWR is the (annualized or unannualized) return of an account that has the same underlying portfolio, that is, the same asset allocation, as the real account, but no deposits and withdrawals. The FREQ, on the other hand, is the annual interest rate of the fixed rate account which, when subjected to the same deposits and withdrawals as the real account, has the same ending balance as the real account.

Finally, a word on the input that is needed for calculating the TWR and the FREQ, respectively. For the TWR, we need a beginning date and balance, an ending date and balance, the amounts of the deposits and withdrawals, and the account balances on the dates of the deposits and withdrawals. (The begin and end date are needed for annualization only.) For the FREQ, which is much more important for individual investors, the required information is different: what's needed is a beginning date and balance, an ending date and balance, and the dates and amounts of the deposits and withdrawals. For the FREQ, we do not, repeat not, need the account balances on the dates of the deposits and withdrawals. That is fortunate because banks and brokerages do not usually provide the account balances on the days of deposits and withdrawals on a customer's monthly or quarterly statement.