What is the time-weighted return (TWR)?

The time-weighted return (TWR) measures how well the investments in a portfolio performed, independently of when money was added or taken out. It is the return a single euro would have earned if it had been invested at the start and left alone for the whole period.

That makes it the standard for comparing managers, funds and indices: a fund manager does not decide when investors send money, so the fund's performance should not depend on it. For a private portfolio it answers the question how good were the holdings, leaving the timing of the savings aside.

How it is calculated

The period is split into sub-periods at every cash flow — every deposit and every withdrawal. Within a sub-period no money comes in or goes out, so its return is a plain percentage change:

r_i = \frac{V_i^{\text{end}}}{V_i^{\text{start}}} - 1

where $V_i^{\text{start}}$ is the value right after the previous cash flow and $V_i^{\text{end}}$ the value right before the next one. The sub-period returns are then linked by multiplication — chained — into the return over the whole period:

\text{TWR} = \prod_{i=1}^{n} (1 + r_i) - 1

Multiplying rather than adding is what makes the result independent of the amount invested: every sub-period counts the same, whether the portfolio held €1,000 or €100,000 during it.

A simple illustration

A portfolio starts the year at €10,000. By the end of June it has grown to €11,000, and €10,000 of new savings is added, bringing it to €21,000. By December it has fallen to €19,950.

r_1 = \frac{11{,}000}{10{,}000} - 1 = 10\% \qquad r_2 = \frac{19{,}950}{21{,}000} - 1 = -5\%

\text{TWR} = (1 + 0.10) \times (1 - 0.05) - 1 = 4.5\%

The investments returned 4.5% over the year. Yet the investor put in €20,000 and ends up with €19,950 — fifty euros less. Both statements are true: the holdings did well in the first half, while little money was invested, and badly in the second, when most of it was. The TWR ignores that timing on purpose; the money-weighted return captures it.

Daily valuation and TTWROR

Portfolio trackers value the portfolio every day and chain the daily returns, so that no cash flow ever falls inside a sub-period. This is sometimes called the true time-weighted rate of return (TTWROR). For periods longer than a year the result is often shown annualized:

\text{TWR}_{\text{annual}} = (1 + \text{TWR})^{365/d} - 1

where $d$ is the number of days in the period.

When to use it

Comparing with an index, an ETF or a fund. They all report time-weighted returns, so only the TWR is a like-for-like comparison.
Judging the choice of investments separately from the timing of deposits and withdrawals.
Not for the question "what did the money earn?" A savings plan that bought heavily before a fall can show a positive TWR and still have lost money. For that question, the money-weighted return is the right measure.

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