Money-weighted vs time-weighted rate of return

What is the money-weighted rate of return?

It is the discount rate that makes the present value of everything the investor put in equal to the present value of everything taken out, including the ending value. In other words, it is the internal rate of return (IRR) of the investor’s cash flows.

The money-weighted rate of return (also called the dollar-weighted rate of return ) is an internal rate of return applied to a portfolio. It uses only the investor’s external cash flows : money paid in or taken out by the investor, which the portfolio manager does not decide on investment grounds.

From the investor’s side, the first investment and every later contribution are negative cash flows; distributions paid out and the ending value are positive. From the fund’s side the signs flip. Either way the rate is the same. The rate solves net present value = 0:

$$0 = \sum_{t=0}^{T} \frac{CF_t}{(1 + r_{MW})^t}$$

Solve it with the cash flow worksheet or a spreadsheet’s IRR function.

Exam tip The money-weighted rate of return has no formula you can rearrange. On the exam you either solve a short case on the calculator, as above, or reason about which rate is higher. The interactive below trains the reasoning.

More money was invested in year 2 than in year 1, so year 2’s weaker return counts for more.

What is the time-weighted rate of return?

It measures how the assets performed, ignoring how much money was in them at each point. Split the period at every external cash flow, find the return of each piece and chain them.

Memorize

1. Split at every external cash flow.
2. Compute the holding period return of each sub-period.
3. Chain them: $(1 + r_1)(1 + r_2)\cdots - 1$.
4. If the total is longer than a year, take the geometric mean to annualize.

The result is the time-weighted rate of return .

$$r_{TW} = (1 + r_1)(1 + r_2)\cdots(1 + r_T) - 1$$

Over several years, compute each year’s time-weighted return and annualize with the geometric mean:

$$r_{TW} = \left[(1 + r_1)(1 + r_2)\cdots(1 + r_T)\right]^{1/T} - 1$$

The money-weighted rate (13.30%) is below the time-weighted rate (16.19%) because the investor added money just before the weaker year.

How do you handle flows within a year?

Work in sub-periods. For the money-weighted rate, net the flows on each date, solve for the sub-period internal rate of return, then annualize by compounding. For the time-weighted rate, chain the sub-period returns.

Money-weighted rate, step by step

Use it

1. List the size, timing and direction of every external flow, and net the flows that fall on the same date.
2. Solve for the internal rate of return per sub-period (a quarter, a month).
3. Annualize if the question asks for a yearly rate.

Time-weighted rate, step by step

Use it

1. Split the year at every external flow.
2. Find the holding period return of each sub-period from the values just before and after the flow.
3. Chain them with the time-weighted formula above.

How is a full year built from daily values?

Use it

Daily valuation is the standard: the portfolio is valued every day, so a year’s time-weighted rate links each day’s return, $(1 + r_1)(1 + r_2)\cdots(1 + r_{365}) - 1$. Using 252 trading days instead of 365 calendar days is also common, and the day count changes the result. The link is exact when flows happen only at the end of a day; otherwise it is an approximation.

For several years, compute each year’s time-weighted rate of return and annualize with the geometric mean, as in the formula above.

Which one should you use?

Use the time-weighted rate to judge a manager who does not control the flows. Use the money-weighted rate to show an investor’s own result, or to judge a manager who does control the timing of flows.

	Money-weighted	Time-weighted
What it measures	The investor’s actual result, including the timing of deposits	The growth of one unit invested from the start: the manager’s own decisions
Affected by size and timing of flows	Yes	No
Calculation	Internal rate of return of the net flows	Sub-period returns chained (geometrically linked)
Needs a valuation at each flow	No	Yes
Weakness	Can misrepresent performance when flows are frequent or large	Needs frequent valuations
Use for	Investors judging their own result; managers who control the flows, such as private equity	Managers of public-market portfolios who do not control the flows

Exam tip Timing of cash flows affects the money-weighted rate of return. If an answer says it affects the time-weighted rate, it is the trap.

What do the Global Investment Performance Standards require?

Memorize

The Global Investment Performance Standards (GIPS) are a voluntary standard for how investment firms present performance. They require time-weighted returns unless two conditions both hold: the manager controls the external cash flows, and the portfolio or fund is closed-end, fixed-life, has investors committed to provide capital on demand, or holds a significant share of illiquid investments. Then a money-weighted rate may be used.

Does the manager control external cash flows?the first test

• No Memorize time-weighted rate of return
• Yes Memorize and the fund has one of four features
  • Closed-end money-weighted allowed
  • Fixed life money-weighted allowed
  • Committed capital called on demand money-weighted allowed
  • Significant illiquid holdings money-weighted allowed

In practice, managers of listed shares and bonds use time-weighted returns. Private equity managers, who call committed capital when they find deals, may use money-weighted returns. The standards do not tell investors which measure to use, and for judging their own result investors usually prefer the money-weighted rate.

Who controls what:

• The manager controls reinvesting dividends and gains, whether to pay distributions out, and how and when new money is invested.
• Investors control when and how much they contribute or withdraw.

Go deeper Dividends in a money-weighted rate of return

Dividends and interest are part of the investor’s cash flows, so they enter the internal rate of return calculation. A dividend of USD 3 paid at t = 1 simply adds 3 to the flow at t = 1 (so −100 becomes −97) and the rate is solved again with the same calculator keys. No other step changes. The principle is simple, and it is the same for every pattern of flows. List every dollar that crosses the boundary between the investor and the portfolio, with its date, and solve for the rate that sets net present value to zero.

Go deeper Why the time-weighted rate needs so many valuations

To chain sub-period returns, you must know the portfolio’s value just before every external cash flow. Daily pricing makes that easy for a mutual fund, but it is a heavy requirement for assets that are valued rarely, such as property. Approximations exist, for example the Modified Dietz method, which weights each flow by the fraction of the period it was invested and avoids the interim valuations. It is mentioned here only so you recognize the name. It is not examinable in this module.

Go deeper When there is more than one IRR

Because the money-weighted rate is an internal rate of return, it inherits the quirks of that tool. When the cash flows change sign more than once, for example money in, money out, then money in again, the equation can have more than one solution, or none. That is unusual for a typical portfolio, but it appears in project finance and private investments. Capital allocation in Corporate Finance returns to this, together with the net present value rule that avoids the problem. For the exam, the practical advice is short: if a question gives a conventional pattern of one outflow followed by inflows, trust the single answer, and if the signs flip back and forth, expect the question to be about the caveat itself rather than the number.