What is the money-weighted return (IRR / XIRR)?

The money-weighted return is the yearly rate of return that the money in a portfolio actually earned, taking into account when and how much was invested. Mathematically it is the internal rate of return (IRR) of the portfolio's cash flows: the single yearly rate at which every deposit, every withdrawal and the final value balance out exactly.

When the cash flows fall on irregular dates, as they do in any real portfolio, it is calculated with exact day counts. That version is called the XIRR, after the spreadsheet function that computes it.

The formula

Take every cash flow $C_k$ on its date $t_k$ : deposits are negative (money leaving the investor's pocket), withdrawals and the final value are positive. The money-weighted return is the rate $r$ that solves:

\sum_{k} \frac{C_k}{(1 + r)^{(t_k - t_0)/365}} = 0

where $t_0$ is the date of the first cash flow. There is no formula that gives $r$ directly: it is found numerically, by trying rates until the sum reaches zero.

A simple illustration

€10,000 is invested on 1 January and grows to €11,000 by the end of June. Another €10,000 is added on 1 July, and the portfolio ends the year at €19,950. The money-weighted return is the rate $r$ for which:

-10{,}000 - \frac{10{,}000}{(1 + r)^{0.5}} + \frac{19{,}950}{1 + r} = 0

Solving gives $r \approx -0.33\%$ . The money earned slightly less than nothing, although the investments themselves returned +4.5% time-weighted over the same year. The difference is timing: twice as much money was invested in the half-year that went badly as in the one that went well.

How to read it

It is personal. Two investors in the same fund get the same time-weighted return, but different money-weighted returns if they invested at different times.
Above or below the time-weighted return. A money-weighted return above the time-weighted return means money tended to arrive before good periods; below it, before bad ones.
It is always a yearly rate, so it can be compared directly with the interest on a savings account or a loan: it is the rate a savings account would have had to pay to produce the same result from the same deposits.

Limitations

Not for comparisons with indices or funds. It depends on the size and timing of the investor's own deposits, which an index or a fund does not have.
Short periods look extreme. A few weeks of gains, expressed as a yearly rate, can produce implausibly large figures.
Unusual cash flows. When large withdrawals are followed by new deposits, the equation can have more than one solution, and the figure should be read with care.

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