Who controls the cash flow?

Learning goals

• Calculate time-weighted and money-weighted results under explicit cash-flow timing.
• Explain how index weighting and divisor changes affect a benchmark.

Draft · approximately 60 minutes, excluding further revision.

Break the record at external cash flows

Time-weighted return (TWR) links subperiod returns calculated between external deposits or withdrawals. Value the portfolio immediately before a flow and distinguish a flow from investment income retained inside the portfolio. Link the subperiod gross returns by multiplication. This reduces the effect of investor-controlled deposit timing when evaluating a manager.

Money-weighted return (MWR) is an internal rate of return that equates the present value of the investor’s signed contributions, withdrawals and ending value to zero. It describes the investor’s capital experience. Multiple sign changes may create multiple roots; some cash-flow patterns have no economically useful IRR. Identify the time unit and do not report a solver result without checking NPV.

A benchmark is a portfolio design

A price-weighted index gives more influence to higher-priced shares, regardless of company size. A capitalization-weighted index uses market capitalization, often free-float adjusted; an equal-weighted index requires periodic rebalancing as prices diverge. Their returns can differ without any calculation error. A total-return index reinvests distributions, whereas a price index excludes that income.

A stock split changes share price and share count but creates no wealth. A price-index divisor must be adjusted so a split alone does not change the index level. A capitalization index is unchanged by a pure split because price times shares is unchanged. Benchmark choice must match currency, investable universe, income treatment and strategy.

Cash-flow timing changes the investor experience

A portfolio starts at 100 and rises to 120 before a contribution of 100. The next period return is −10%, leaving 198. Time-weighted return links subperiod factors: 1.20 × 0.90−1=8%. It isolates portfolio performance from the external contribution. Do not count the contribution as investment income or divide the final gain by only the initial 100.

Money-weighted return solves the dated cash-flow equation. Here the two equal-length periods give −100−100/(1+r)+198/(1+r)^2=0, yielding r≈−0.6682% per period. The investor placed more money before the loss, explaining the difference from the positive linked return. Irregular dates require actual dated discounting; changing the sign convention midway or annualizing a two-period total as a one-period rate makes the comparison invalid.

Formula and assumptions

TWR = ∏(1 + R_subperiod) − 1; Σ CFₜ/(1 + IRR)^t = 0; index = Σ prices/divisor

Same manager returns, different investor experience

1. Start with 100. It rises to 110; the investor then adds 100, so the second period starts at 210. It falls 10%, ending at 189.
2. TWR = 1.10 × 0.90 − 1 = −1%. The two-period geometric rate is √0.99 − 1 ≈ −0.5013% per period.
3. For MWR solve −100 − 100/(1+r) + 189/(1+r)² = 0. With x = 1+r, 100x² + 100x − 189 = 0; positive x ≈ 0.962874, so r ≈ −3.7126% per period. More capital experienced the losing period.

Calculation lab

Errors to catch

• Subtracting deposits from ending value without recognizing their dates.
• Comparing a total-return fund with a price-only benchmark.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org