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How money-weighted return works

Money-weighted return is the IRR of dated cash flows. Spend $10,000, collect $3,000 a year for five years, and the money-weighted return is 15.24 percent. Deposits and withdrawals change the rate because they change how much money was actually at work.

Internal rate of return

15.24%

$10,000.00 out now, then $3,000.00 back at the end of each of 5 years.

Net present value at 10.00%
$1,372.36
Margin over the hurdle
5.24 points
Cash back in total
$15,000.00

Accept. 15.24% beats the 10.00% the money costs you, so the project adds $1,372.36 of value.

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What the money costs you a year, the same period the cash flows and the IRR are in.

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In short

  • Money-weighted return is the internal rate of return of the dated series. On $10,000 out and five receipts of $3,000 it is 15.24 percent, or 15.238237 percent before rounding.
  • At that unrounded rate the NPV of the series is 0. That is the definition, not a second test.
  • A smaller series, $5,000 out and $9,000 back after four years, earns 15.83 percent. A higher rate is not more money. The outlay is half.
  • Deposits and withdrawals change the weights. A rate that ignores the dates and sizes of those flows is a time-weighted return, which is a different object.
  • How NPV and IRR work is the project version of the same root. This page is the personal-return reading.

The rate that sets NPV to zero

Money-weighted return is the internal rate of return of every cash flow that actually moved, dated when it moved.

0=∑t=0nCFt(1+r)t0 = \sum_{t=0}^{n} \frac{CF_t}{(1 + r)^t}

rr is the money-weighted return. Cash out is negative. Cash in is positive. Time zero is not discounted.

Spend $10,000 today, collect $3,000 at the end of each of the next five years. The rate that drives that series' NPV to zero is 15.24 percent a year, 15.238237 percent before rounding. At the unrounded rate the NPV is 0 by construction.

The IRR calculator on this page searches for that root. The IRR explorer holds the outlay still and lets you drag the yearly receipt.

How NPV and IRR work is the same equation asked for a project. A cash flow needs a date, or it is not in the sum.

The weights are the money that was actually there

Call it money-weighted because a period when more cash sat in the holding counts for more. A deposit just before a good year raises the rate. A deposit just before a bad year lowers it. A withdrawal does the opposite.

That is the point of the measure for a person who was adding and taking out. The rate answers: what compound return did the cash that was actually at work earn, given those dates?

It is also why two people in the same fund can print different money-weighted returns. They did not have the same cash in on the same dates.

A higher rate is not more money

A different shape: $5,000 out today, nothing for three years, $9,000 back at the end of year four. The IRR is 15.83 percent, 15.829219 percent before rounding.

That is a better rate than 15.24 percent, on half the outlay. Rank the two by money-weighted return and the small series wins. Rank them by NPV at 10 percent and the $10,000 series is worth $1,372.36 in today's money. A rate is silent on size.

NPV against IRR is that split. Money-weighted return inherits it, because it is IRR under another name.

Time-weighted is the other clock

Time-weighted return strips the deposits and withdrawals out and compounds the growth of one unit of money. On a stretch with no cash flows in or out, the two clocks agree: both are the CAGR of the two balances.

On a stretch with cash flows, they split. Money-weighted cares when you added. Time-weighted does not. How time-weighted return works is the CAGR reading. Money-weighted against time-weighted is the pair, on two different teaching sheets.

A fund reporting a time-weighted return is answering a different question from a household reporting a money-weighted one. Mixing them in a table is how one holding looks like two.

Searching, not a closed form

A five-year series is a degree-5 polynomial in the rate. There is no general radical formula. Every money-weighted return you have seen was found by searching: guess, price the series, adjust.

The one shape with a closed form is a single amount out and a single amount back, which is the third sheet: (9000/5000)1/4−1(9000/5000)^{1/4} - 1. That is also a CAGR. How the CAGR formula works is that special case.

What this page is not doing

It is not a time-weighted return, not a ranking of funds, and not a forecast. The three sheets are $10,000 out and five times $3,000 (15.24 percent), that same series discounted at 10 percent (NPV $1,372.36), and $5,000 out for $9,000 in four years (15.83 percent). This is educational material, not financial advice.

Worked examples

\$10,000 out, five times \$3,000 in

You spend $10,000 now, and it brings in $3,000 at the end of every year for five years. What is the money-weighted return?

  1. Write the series with signs: $10,000 out at time zero, then five payments of $3,000 in.
  2. The money-weighted return is the rate rr at which the discounted payments add back to the outlay: 10000=∑t=153000(1+r)t10000 = \sum_{t=1}^{5} \frac{3000}{(1+r)^t}.
  3. There is no way to rearrange that for rr, so search for it. At 10 percent the right side is worth more than $10,000, so 10 percent is too low. At 20 percent it is worth less, so 20 percent is too high.
  4. Keep halving the gap. It settles at r=0.1524r = 0.1524.
  5. Check: discount all five payments at that rate, take off the $10,000, and the NPV is 0.

The money-weighted return is 15.24 percent a year, or 15.238237 percent before rounding. At the unrounded rate the NPV is 0 by definition.

The same series at a 10 percent cost of capital

Your money costs 10 percent a year. What is that same $10,000 series worth at 10 percent, and how does that square with 15.24 percent?

  1. Discount each $3,000 payment back to today at 10 percent.
  2. The five discounted payments add to $11,372.36 of present value.
  3. Take off the $10,000 paid at time zero.
  4. The NPV is $1,372.36.

At 10 percent the series is worth $1,372.36 in today's money. The 15.24 percent money-weighted return and that positive NPV are one fact: the cash that was at work beat 10 percent.

\$5,000 now for \$9,000 in four years

You put $5,000 in today, nothing happens for three years, and $9,000 comes back at the end of year four. What is the money-weighted return?

  1. With one payment out and one payment back: 5000(1+r)4=90005000(1+r)^4 = 9000.
  2. Divide: (1+r)4=1.8(1+r)^4 = 1.8.
  3. Fourth root: 1.81/4=1.15831.8^{1/4} = 1.1583.
  4. Subtract 1: r=0.1583r = 0.1583 to four places, and at the unrounded rate the NPV is 0.

The money-weighted return is 15.83 percent a year, 15.829219 percent before rounding. That is a better rate than 15.24 percent on half the outlay. A higher rate is not the same thing as more money.

Common questions

Why do two people in the same fund print different money-weighted returns?

Because they did not have the same cash in on the same dates. A deposit just before a good year raises the rate. A deposit just before a bad year lowers it. Time-weighted return strips those dates out.

Is 15.24 percent a holding-period yield you can spend?

It is the rate that sets this series' NPV to zero. Reading it as a compound return across the whole life assumes interim receipts were put back to work at 15.24 percent. That reading is a claim you add, not something sitting in the arithmetic.

When does money-weighted equal time-weighted?

When nothing is paid in or taken out between the start and the finish. Then both clocks are the CAGR of the two balances.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.