Money-weighted returns and root certificates

What you will learn

Derive a contribution-timed money-weighted return equation, prove its unique positive gross-factor root and construct signed-residual brackets separating investor rates from linked horizon returns and chip gains.

Prerequisites: Linked returns and external cash flows · Approximation error and decision margins

Choose the cash-flow perspective and clock

A portfolio return and an owner experience answer different questions. Linked returns deliberately neutralize external flows. A money-weighted return instead asks which single constant rate would reproduce the owner’s dated contributions and final receipt. It is a rate obtained from a cash-flow equation, not an average of the actual period returns. Two owners holding the same investment through the same price path can have different money-weighted rates when they contribute at different times. That difference is meaningful only after stating the measurement boundary and who controls the flows.

Our fictional account starts with 100 chips at tick zero and has two equal-length investment periods. Their gross factors are 1.5 then 0.8, or the reverse. One nonnegative additional contribution C arrives at tick t, where t is zero, one or two. A tick-one contribution arrives after the first period return and before the second; a tick-two contribution arrives after all investment returns but immediately before liquidation. Contributions buy exposure at fair value without changing the supplied returns. All holdings are liquidated at tick two. There are no fees, taxes, borrowing, withdrawals before liquidation or price impact. Fractional chips are allowed. These are deterministic bookkeeping experiments, not forecasts or investment advice.

Derive a rate from dated capital rather than averaging returns

Use the investor perspective for signs: initial capital and contributions are payments, while liquidation value V is a receipt. Write q=1+r, requiring q>0. At tick two the initial 100 has been committed for two periods, so its constant-rate equivalent is 100q². The contribution has been committed for only 2−t periods, so its equivalent is Cq^(2−t). Equating accumulated payments to the terminal receipt gives 100q²+Cq^(2−t)=V. Equivalently, the investor’s net present value is −100−C/q^t+V/q²=0. Multiplying that equation by positive q² gives the same accumulated-value condition with the opposite residual sign.

Define F(r)=100(1+r)²+C(1+r)^(2−t)−V. A negative F says the candidate constant rate accumulates too little capital; increase the rate. A positive F says it accumulates too much; decrease the rate. This is the sign convention used in the workshop. A deposit is not a positive investor receipt merely because it increases the account balance. At t=0 combine simultaneous payments into 100+C. At t=2 the contribution and the matching portion of liquidation cancel in the present-value ledger. Neither endpoint case should manufacture performance from transferring the owner’s own chips.

Work a complete boundary-contribution example

Let C=100 arrive at tick one with +50% then −20% returns. The original 100 grows to 150. The contribution raises investable capital to 250; multiplying by 0.8 gives liquidation value V=200. Net chip gain is 200−100−100=0. The flow-neutral two-period return is 1.5×0.8−1=0.2, or 20%. That is compatible with zero owner gain because the later hundred participates only in the losing second interval. The two-period geometric equivalent of the linked return is sqrt(1.2)−1, about 9.544512% per period; this is still a different question from reproducing the dated owner payments.

The owner equation is 100q²+100q=200. Divide by 100 and factor q²+q−2=(q−1)(q+2)=0. The algebraic roots are q=1 and q=−2. Only q=1 satisfies q>0, so r=0% per period is the admissible money-weighted rate. Independently discount the investor ledger at zero: −100−100+200=0. A second arithmetic check uses trial rates: at −1%, F=100×0.99²+100×0.99−200=−2.99; at +1%, F=100×1.01²+100×1.01−200=3.01. Thus the rate lies between them. A narrower bracket [−0.5%,0.5%] has residuals −1.4975 and 1.5025 and meets the workshop’s one-percentage-point width requirement.

Prove uniqueness and build a numerical certificate

For this contribution-only family, F is strictly increasing for every q>0. Its derivative with respect to r is 200q+2Cq when t=0, 200q+C when t=1, and 200q when t=2. Every expression is positive. At q approaching zero, F approaches −V for t=0 or t=1, or C−V for t=2. Each is negative because positive initial capital survives the positive return factors. As q grows without bound the 100q² term diverges. Continuity gives existence of a root, and strict increase proves uniqueness. The rejected negative gross factor in the worked quadratic is outside this economic domain, not a second admissible answer.

A bracket is a proof object: choose L≤U with L>−100%, evaluate F(L)≤0≤F(U), and state its width. Monotonicity then certifies that the unique rate lies inside. Bisect a wide bracket, retain the half whose endpoint residuals have opposite signs, and repeat until the requested precision is reached. Residuals have chip units; rates and interval widths have percentage units. A small residual alone is not a rate-error guarantee without a slope bound. The activity therefore checks both signs and a maximum width of one percentage point. Exact equality at an endpoint is valid, including a zero-width interval for an exactly known root. Display rounding is only for readability; grading uses unrounded values with a small floating-point tolerance.

Move the contribution and challenge the interpretation

Keep C=100 but move it to tick zero. Now 200×1.5×0.8=240, so 200q²=240 and r=sqrt(1.2)−1, about 9.544512% per period. Move it instead to tick two: the original capital becomes 120 and the late deposit makes the terminal receipt 220. The equation 100q²+100=220 again gives q²=1.2. Both endpoint clocks share a rate, although their chip gains are 40 and 20. The middle clock produced zero rate and zero gain. All three share the same 20% linked two-period return. Neither raw chip gain nor linked return can be substituted for the cash-flow equation.

Now reverse the periods while keeping the contribution at tick one. Original capital falls to 80, the contribution creates 180, and 50% growth produces 270. The equation becomes q²+q−2.7=0. Its positive root is (−1+sqrt(11.8))/2, giving approximately 21.755640% per period. Verify a practical certificate with [21%,22%]: residuals are −2.59 and 0.84 chips. Linked return stays 20%, but gain becomes 70 because the additional capital now enters before the positive interval. Save both middle-clock certificates to demonstrate the mechanism. This is a controlled timing comparison, not evidence of an ability to predict favorable contribution dates.

Report the rate with its limitations

Name the rate’s horizon whenever reporting it. Here r is per equal investment period, and (1+r)²−1 is its two-period compounded equivalent. Only if each period is one year would r be an annual rate. Unequal dates require dated exponents rather than silently assigning equal periods. The cited professional handbook uses dated flows for actual reporting; this lesson’s finite clock isolates the reasoning and does not establish standards compliance. A terminal valuation also behaves as a synthetic liquidation receipt, so inaccurate valuations can distort the inferred rate even when the numerical solver is correct.

Our uniqueness proof depends on nonnegative contributions followed by a positive terminal receipt. It does not automatically extend to arbitrary interleaved withdrawals and payments, where multiple admissible rates or no useful rate may occur. Keep that restriction visible rather than trusting a solver’s first answer. Money-weighted performance describes the supplied capital experience, including flow timing. It does not by itself isolate a manager’s contribution, establish risk-adjusted skill, or forecast an attainable future rate. Retain the dated ledger, rate convention and a root certificate so another learner can reproduce exactly what was measured.

Hands-on: Construct a money-weighted certificate

Move dated contributions, bracket the unique constant owner rate with signed residuals and compare capital experiences.

Common mistakes

• Confusing account deposits with positive investor receipts.
• Applying two periods of growth to a late contribution.
• Accepting a negative gross factor merely because it solves a quadratic.
• Reporting the arithmetic mean 15% or linked 20% as the investor per-period rate.
• Treating a narrow same-sign residual interval as a root certificate.
• Annualizing without a calendar convention or extending the uniqueness proof to arbitrary withdrawals.

Keep these ideas

• Date every owner payment and terminal receipt before writing an equation.
• A constant-rate accumulated-value equation weights each payment by its remaining exposure.
• Nonnegative contributions give a strictly increasing residual and one admissible root.
• Certify a rate with endpoint signs and width; report horizon separately from linked return and chip gain.

Further reading

• GIPS Standards Handbook for Firms: IRR and Provision 2.A.29 ↗

  Read IRR discussion at HTML lines 1715–1728 and Provision 2.A.29 discussion at 1792–1833: dated cash-flow signs, terminal value, timing sensitivity, period conventions and possible multiple answers. Supports definitions only. Our equal-period contribution-only monotonic proof, residual brackets, chip examples and diagnostic questions are original. No standards-compliance or video-review claim. Checked 2026-10-25.