Separate compound returns from arithmetic active returns

Start from a consistent wealth convention

The report initializes model and benchmark wealth at one. For each ordered period it multiplies wealth by one plus that period’s decimal return. Cumulative return is final wealth minus one. This calculation assumes the supplied returns already represent the intended period performance; it cannot infer fees, distributions, cash-flow adjustments or trade execution from a single number.

CFA Institute’s GIPS handbook describes geometric linking in performance measurement. The mathematical reference is useful, but this teaching report does not implement the full standards or claim compliance. It does not calculate returns from holdings, adjust subscriptions and redemptions, or validate whether an input labeled net actually includes relevant costs.

Work through an up-and-down path

Take two invented model returns of plus ten percent and minus ten percent. Wealth moves from one to 1.1, then to 0.99. Cumulative return is minus one percent even though the arithmetic mean return is zero. Equal positive and negative percentage changes operate on different wealth bases and do not cancel geometrically.

Now give the benchmark plus five percent and minus five percent. Its final wealth is 0.9975, a cumulative return of minus 0.25 percent. Model cumulative return minus benchmark cumulative return is minus 0.75 percentage points. The report labels that difference separately instead of treating it as a linked series of active returns.

Calculate the active series independently

For the same example, period active returns are model minus benchmark: plus five percentage points and minus five percentage points. Mean active return is zero. Its sum is also zero, yet the cumulative return difference remains minus 0.75 percentage points. Those results coexist because multiplication and period-by-period subtraction answer different questions.

Compounding the active returns themselves would give minus 0.25 percent, a third number with a different interpretation. The engine does not report that as relative wealth or alpha. If a study needs a model-to-benchmark wealth ratio, it must specify that formula explicitly; it should not relabel whichever existing number looks most favorable.

Keep investment interpretation narrower than arithmetic

A positive cumulative difference does not show that risk exposures, leverage, turnover or implementation conditions were comparable. It also does not establish statistical significance or persistence. A model can outperform one selected benchmark while another benchmark is more appropriate to its intended mandate. The reason for benchmark selection belongs in the review record.

The fixture schema contains no position, transaction or factor series. Consequently, it cannot estimate attribution, execution costs or regression alpha. Its report is still useful for catching arithmetic confusion and inconsistent labels, provided readers can see the supplied return window, benchmark identity assumptions and exact definitions beside the results.

Verify scale and exports

Test the decimal convention with a one-percent input of 0.01. Entering one instead would represent a one-hundred-percent return, which lies inside the broad numerical guard but answers a different question. Type checks alone cannot detect every unit mistake; an acceptance review needs concrete expected values and clear input labels.

Retain full precision for computation and round only for presentation. Export model cumulative return, benchmark cumulative return, their difference and mean active return as separately named fields. These original synthetic calculations demonstrate a transparent report contract. They neither describe realized investment performance nor certify that a model’s future results will resemble its review window.

Keep in the decision record

• Link model and benchmark returns geometrically and separately.
• Mean active return is not the cumulative return difference.
• The report does not calculate alpha or cost-adjusted performance.

Source notes

1. CFA Institute — GIPS Standards Handbook for Firms, November 2020
2. NIST/SEMATECH e-Handbook — Measures of Scale
3. W3C — Data on the Web Best Practices, Recommendation, 31 January 2017

Sources inform the stated methods and governance context. All example monitoring observations are synthetic. No regulatory approval or production monitoring certification is implied.

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