Time-weighted and money-weighted return: what measures what?

In depth8 min read · 3 sources

The time-weighted return (TWR) measures how well an investment itself has performed, regardless of when money flowed in or out. The money-weighted return (MWR, usually calculated as the internal rate of return), by contrast, measures what you actually earned with your deposits and withdrawals. Both figures are correct, but they answer different questions and can be far apart.

Two questions, two returns

If you are assessing a fund, you want to know what the fund manager has made of one euro. The manager cannot influence when investors pay money in, so this must not distort the assessment. That is what the time-weighted return is for: it splits the period into sub-periods at every cash flow, calculates the return for each sub-period and chains them together.

If, on the other hand, you are assessing your own portfolio, you want to know how much wealth your payments have produced. What matters here is that a large amount shortly before a crash weighs more heavily than a small amount shortly before an upswing. This question is answered by the money-weighted return: the interest rate at which all payments together produce exactly the final wealth.

Worked example: a gain in the fund, a loss in the portfolio

The internal rate of return can only be found iteratively: you look for the interest rate at which −€10,000 today, −€10,000 in one year and +€19,800 in two years together have a present value of zero. Spreadsheets do this with the IRR function, or XIRR for irregular dates (IKV and XINTZINSFUSS in German versions of Excel).

The example shows the basic rule: if the timing of the payments is right, the money-weighted return is above the time-weighted return; if it is wrong, it is below. With a savings plan with equal contributions, the gap is usually small. With large one-off payments and with withdrawals in retirement, it can be considerable. More on this in the article sequence risk.

With a savings plan: what the two returns show

With a savings plan, money is added regularly. The time-weighted return then describes the ETF, and the money-weighted return the result of your plan. A simple example with three annual contributions shows how the two come about:

With even contributions over many years, such effects largely cancel each other out. But if you pause contributions, increase them after price rises or sell after crashes, you change the money-weighted return noticeably, for better or for worse. The article cost-average effect explains why regular saving nevertheless has no built-in return advantage.

When valuations are missing: Modified Dietz

The exact time-weighted return requires the portfolio value on every day with payments. If these values are missing, it is approximated with the Modified Dietz method, which is widely used in performance measurement (Bacon (2008) describes it together with the other methods). Each cash flow is weighted by the fraction of the period during which it was invested.

Which return for what?

Fund companies, index providers and factsheets therefore report time-weighted returns. Many banks' portfolio overviews, by contrast, show a money-weighted return or simply the gain in euros. If you compare the two, you are comparing apples with oranges.

What the research shows: investors earn less than their funds

Dichev (2007) calculated the money-weighted returns of all investors together for stock markets, i.e. taking into account when money flowed into the market and when it was withdrawn. The result: actual investor returns are systematically below buy-and-hold returns in almost all major international markets. On average, investors tended to buy after price rises and sell after losses. Dichev also concludes from this that the historical equity risk premium as investors experienced it was lower than often assumed (see equity risk premium).

Something similar shows up with frequent trading. Barber and Odean (2000) studied 66,465 households that held stocks with a large US discount broker from 1991 to 1996. The households that traded most earned 11.4% a year, while the market returned 17.9%; the average household achieved 16.4% and turned over 75% of its portfolio each year. The authors attribute this to overconfidence.

The lesson here is not a call never to pay in again. It only shows that a fund's return is an upper limit on what an investor achieves by buying and holding. Timing mistakes and the costs of trading come at the expense of the money-weighted return.

For advanced readers: properties and pitfalls

• Ambiguous IRR: if the payments change sign several times (deposit, withdrawal, deposit again), the equation can have several solutions. For pure savings plans or pure withdrawal plans, the IRR is unique.
• Annualisation: you should not scale up the time-weighted return of a sub-period of less than a year to a full year. Three good months with 6% do not reliably add up to 26% a year. The IRR is a rate per year by definition.
• Costs: both returns should be calculated after all costs. Order fees for savings plan contributions lower a portfolio's time-weighted return, even though the fund itself does not bear them.
• Scaling: the time-weighted return does not depend on the size of the wealth, but the money-weighted return does: a year with a lot of wealth counts for more. With withdrawals in retirement, the first years therefore weigh most heavily.
• Comparison with an index: to assess your own timing, you can “virtually” invest your actual payments in the index and compare the IRR of both variants. The difference is the effect of your own decisions.

Formally, the time-weighted return is the geometric mean of the sub-period returns, and the money-weighted return is a return weighted by the capital invested at each point. If returns are the same in all periods, the two coincide. The more returns fluctuate and the larger and more irregular the payments, the more they can differ.

How Portfolio Lab calculates it

The Backtest reports returns and all metrics on a time-weighted basis, with all costs but without the effect of inflows and outflows. This keeps portfolios with and without a savings plan comparable. What the payments achieve is shown by the money amounts: starting capital, net inflows and outflows and final value, in real terms and after taxes if required. The backtest does not calculate an internal rate of return; it can be determined in a spreadsheet from the final value and the payment schedule. Details are in the backtest methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.