Time-Weighted vs Money-Weighted Returns

A fund manager runs a strategy that genuinely returns 10% one year and 10% the next. An investor who put money in right before a bad month and added more right before a good one will report an entirely different personal return than the manager's stated 10%/10%. Both numbers are correct. They are simply answering different questions: how good was the manager, versus how did this specific pool of money actually do, and reporting one when someone is asking the other is a common source of genuine confusion, and occasionally of genuine deception.

Two ways to grade a driving instructor

Imagine a driving instructor whose actual lesson quality is identical every week, but one student only takes lessons on weeks the instructor happens to be well-rested, and another takes lessons at random. Grading the instructor's skill should ignore when each student showed up; it should measure the quality of instruction itself, holding scheduling constant. Grading each individual student's progress, though, has to account for exactly when they showed up, because a student who only got lessons on the good weeks genuinely progressed faster in real terms. Time-weighted return measures the instructor; money-weighted return measures the student's actual experience.

The two calculations

Time-weighted return (TWR) breaks the period into sub-periods around each cash flow (deposit or withdrawal), computes the simple return in each sub-period, and geometrically links them:

$\text{TWR} = \left[\prod_{i=1}^{n}(1 + r_i)\right] - 1.$

In words: compute how the portfolio grew in each stretch of time between cash flows, ignoring how large those cash flows were, then chain the stretches together. Because the size of any deposit or withdrawal never enters the calculation, TWR isolates the manager's skill from the timing of investor cash flows they don't control.

Money-weighted return (MWR), essentially an internal rate of return (IRR), finds the single discount rate $r$ that makes the present value of every cash flow (including the starting and ending portfolio value) equal to zero:

$\sum_{t} \frac{CF_t}{(1+r)^t} = 0.$

In words: find the growth rate that, applied consistently to every dollar for exactly as long as it was actually invested, reconciles all the deposits, withdrawals, and the final value. Because MWR weights each period by how much money was actually at stake, large cash flows right before a strong or weak stretch dominate the answer.

Worked example

A $100 portfolio returns +20% in month 1, ending at $120. The investor then adds $100, bringing the balance to $220. In month 2 the portfolio falls 10%, ending at $198.

TWR: sub-period returns are +20% and −10%. $\text{TWR} = (1.20)(0.90) - 1 = 1.08 - 1 = 8\%$.

MWR: solve $-100 = \dfrac{-100}{(1+r)} + \dfrac{198}{(1+r)^2}$ for the two-month IRR (cash flows: −100 at $t=0$, −100 at $t=1$, +198 at $t=2$). Trying $r = -2\%$ per period: $-100/(0.98) - 100/(0.98)^2 + 198/(0.98)^2$... solving numerically gives $r \approx -1.3\%$ per month, roughly −2.6% over the two months. The investor's money-weighted experience is sharply negative, even though the manager's time-weighted skill was a healthy +8%, because the investor's second, larger dollar arrived right before the losing month.

What this means in practice

• GIPS-compliant performance reporting requires TWR specifically because it prevents a manager's reported return from being flattered or punished by clients' own contribution timing.
• MWR is the right number for an individual account or a private equity fund where the manager, not the client, controls the timing of capital calls and distributions, in which case a poor MWR genuinely reflects manager-driven timing skill (or lack of it).
• The gap between TWR and MWR is itself informative. A large negative gap, like the example above, tells you cash flows were badly timed relative to performance, information a single blended number would hide.

Practice

1. If deposits and withdrawals always happened at the very start of the measurement period (never mid-period), would TWR and MWR ever differ? Why or why not.
2. A private equity fund reports a strong IRR (MWR) but a much weaker public-market-equivalent benchmark comparison. What role did capital call timing likely play?
3. Compute the TWR and MWR for a $100 start, −10% in month 1 (to $90), a $200 deposit (to $290), then +15% in month 2. Which number would a manager prefer to advertise, and why?