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Money-weighted vs time-weighted return

Money-weighted return is the IRR of dated cash flows: $10,000 out and five times $3,000 in earns 15.24 percent. Time-weighted return, on a stretch with no cash flows, is the CAGR: $10,000 to $18,000 over 6 years is 10.29 percent. They are different clocks, on different sheets.

 Money-weighted returnTime-weighted return
What it isThe internal rate of return of the dated series.The compound chain of holding-period returns, cash flows taken out.
Teaching sheet$10,000 out, five receipts of $3,000. The rate is 15.24 percent.$10,000 to $18,000 over 6 years, nothing paid in or taken out. The rate is 10.29 percent.
A second reading on that sheetAt 10 percent the same series has an NPV of $1,372.36. The 15.24 percent and that surplus are one fact.The same two balances over 3 years are 21.64 percent a year. The window is part of the number.
What a cash flow doesIt changes the weights. A deposit just before a good year raises the rate.It is stripped out. A deposit is a new unit, not growth of the old one.
When they agreeWhen nothing is paid in or taken out. Then money-weighted return is also the CAGR.On that same no-cash-flow stretch. Then time-weighted return is the CAGR.
When you would pick itAsking what the cash that was actually at work earned, given the dates you added and took out.Asking what one unit of money did, which is the usual fund-report clock.
On this page

Two clocks, two teaching sheets

Money-weighted return is IRR. Spend $10,000, collect $3,000 a year for five years, and the rate is 15.24 percent, 15.238237 percent before rounding. At that unrounded rate the NPV is 0. Discount the same series at 10 percent and the NPV is $1,372.36. Those are one fact: the cash that was at work beat 10 percent.

Time-weighted return, on a stretch with no deposits or withdrawals, is the CAGR. $10,000 to $18,000 over 6 years is 10.29 percent a year. The same two balances over 3 years are 21.64 percent a year. Up 60 percent then down 37.5 percent leaves $10,000 where it started: a time-weighted return of 0 percent.

The 15.24 percent and the 10.29 percent are not two readings of one holding. They are two different series. Mixing them in a table is how one person looks like two.

How money-weighted return works is the IRR page, with the IRR calculator under the answer. How time-weighted return works is the CAGR page. A cash flow is what splits them.

Same fund, two people, two money-weighted rates

Two people can hold the same fund and print different money-weighted returns, because they did not have the same cash in on the same dates. Time-weighted return is the clock that makes those two people comparable: it answers what one unit did, not what each client earned.

On a stretch with no cash flows the clocks agree, because there are no dates to disagree about. That is the CAGR sheet, not the IRR sheet.

This is educational material, not financial advice.

Worked examples

Money-weighted: \$10,000 out, five times \$3,000 in

Spend $10,000 now, collect $3,000 at the end of each of five years. What is the money-weighted return?

  1. Write the series: $10,000 out at time zero, then five payments of $3,000 in.
  2. Search for the rate that sets NPV to zero. It settles at r=0.1524r = 0.1524.
  3. At the unrounded rate the NPV is 0.

The money-weighted return is 15.24 percent a year, or 15.238237 percent before rounding.

The same series at 10 percent

What is that series worth at a 10 percent cost of capital?

  1. Discount the five $3,000 receipts at 10 percent. They add to $11,372.36.
  2. Take off the $10,000 outlay.
  3. NPV is $1,372.36.

NPV is $1,372.36 at 10 percent. The 15.24 percent money-weighted return and that surplus are one fact.

Time-weighted: \$10,000 to \$18,000 over 6 years

A holding was worth $10,000 at the start and $18,000 six years later, with nothing paid in and nothing taken out. What is the time-weighted return?

  1. Growth multiple: 18000/10000=1.818000 / 10000 = 1.8.
  2. Sixth root: 1.81/6=1.102923571.8^{1/6} = 1.10292357.
  3. Subtract 1: 10.29 percent a year.

The time-weighted return is 10.29 percent a year. With no cash flows, that is also the CAGR.

The same two balances over 3 years

Same $10,000 to $18,000, over 3 years. What is the time-weighted return?

  1. The multiple is still 1.8.
  2. Cube root: 1.81/3=1.216441.8^{1/3} = 1.21644.
  3. Subtract 1: 21.64 percent a year.

The time-weighted return is 21.64 percent a year. Same balances, shorter window, faster rate.

Common questions

Can I subtract 10.29 from 15.24 and call the gap a finding?

No. Those rates are on different series. The 15.24 percent has five receipts of $3,000 after an outlay. The 10.29 percent has two balances and no cash flows. Put them in one table only if you say so.

Which one is the return I earned?

Money-weighted, if you want the rate on the cash that was actually at work given your dates. Time-weighted, if you want what one unit of money did, which is how a fund is usually reported.

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.