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Distinguish a portfolio's time-weighted return from its dollar-weighted return and explain when each is the right one to use.

FRM® Part II · Risk Management and Investment Management · Portfolio Performance Evaluation · FRM2.IM.7.a

The idea

The time-weighted return (TWR) chains the returns earned in each sub-period between external cash flows, with the portfolio valued just before each flow. For two sub-periods with returns r1 and r2, the chained return is: TWR = (1 + r1) × (1 + r2) − 1. Because every sub-period is weighted equally, the size and timing of contributions and withdrawals have no effect. So TWR measures what the manager did with the assets and is the right basis for comparing managers. The dollar-weighted return (DWR) is the internal rate of return on the portfolio's cash flows, so it also reflects when money came in or out. Use DWR to judge the investor's own experience, for example a limited partner in a private fund who decides contribution timing. The usual mistake is judging a manager by DWR when clients, not the manager, chose the timing of the flows.

Formulas

Time-weighted rate of return

time-weighted return = (1 + r₁) × (1 + r₂) − 1

r₁, r₂
the holding period return of each sub-period, cut at every external cash flow

Value the portfolio just before each flow and compound, never add, the sub-period returns. Annualise a multi-year figure geometrically, not with the arithmetic mean.

Dollar-weighted return (IRR of the investor's flows)

0 = Σ CFₜ ÷ (1 + DWR)ᵗ

CFₜ
the net cash flow at time t: contributions negative, withdrawals and ending value positive
DWR
the dollar-weighted return, solved for
t
the time of each flow, in periods

Solve for DWR. It reflects flow timing, so it is below the time-weighted return when most money was invested in the weaker period.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement best describes a portfolio's time-weighted return?

  1. It is the internal rate of return of all contributions, withdrawals and the ending value, which reflects flow timing.Why not: Describes the dollar-weighted return, which is the IRR of the cash flows.
  2. It links sub-period returns between external flows, so flow size and timing do not affect it. - correct
  3. It is the simple arithmetic average of the sub-period returns, taken without regard to any cash flows.Why not: Averages arithmetically instead of compounding (geometric linking) the sub-period returns.
  4. It is the ending portfolio value divided by the beginning value, minus one.Why not: Ignores interim cash flows altogether, so contributions would be counted as investment gains.

The time-weighted return values the portfolio before each external flow and compounds the sub-period returns, giving each sub-period equal weight. The internal rate of return of the flows is the dollar-weighted return.

A harder one

Two investors hold the same fund over two years. Investor X invests at the start. Investor Y invests the same amount at the start and adds a very large sum after a strong first year, just before a weak second year. Which statement is correct?

  1. Y's time-weighted return is lower than X's because more money was exposed to the weak year.Why not: Confuses time-weighted with dollar-weighted; the time-weighted return does not depend on how much money is invested.
  2. Both investors have the same dollar-weighted return because they hold the same fund.Why not: Ignores that the dollar-weighted return depends on the investor's own cash flows.
  3. The fund's time-weighted return is the same for both; Y's dollar-weighted return is lower. - correct
  4. Y's dollar-weighted return is higher than X's because Y invested more capital overall.Why not: Size alone does not raise the return; more money was exposed to the poor year, which pulls the IRR down.

The time-weighted return is a property of the fund and the same for both. Y put more money at work in the poor second year, so Y's internal rate of return is below X's.

The hardest tier

An investor puts USD 100 into a fund. At the end of year 1 the fund is worth USD 120 and the investor adds USD 80. At the end of year 2 the fund is worth USD 210. The annual dollar-weighted return is closest to:

  1. 12.2%Why not: This is the annualised time-weighted return, sqrt(1.20 x 1.05) - 1, not the investor's IRR.
  2. 16.7%Why not: Total gain USD 30 over total contributions USD 180, not annualised and ignoring timing.
  3. 8.0%Why not: Annualises the 16.7% total gain over two years, which ignores that USD 80 was invested for only one year.
  4. 10.3% - correct

Solve -100 - 80/(1+r) + 210/(1+r)² = 0. With x = 1/(1+r): 210x² - 80x - 100 = 0, x = 0.9063, so r = 10.3%. The time-weighted return is 12.2%; the IRR is lower because most of the money was invested during the weaker second year.

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