One portfolio report ends with more money. Another traded less. Which did better? G93 separated deposits from investment returns. A question remains: better than what, and which decisions account for the difference?
Comparing finish times from two different running routes would be unhelpful without understanding the routes. Establish a common reference, measure how the portfolio deviated, and then break down the difference. A remaining unexplained number does not become alpha simply because it needs a label.
Choose the measuring stick first
You will write a comparable benchmark specification, recalculate active return, tracking error, and the information ratio from matching net returns, and reconcile a single-period allocation, selection, interaction, and fee table. Multiperiod attribution linking and regression models remain further study, not prerequisites for this exercise.
Performance measurement describes account changes; attribution investigates sources; appraisal considers process quality. CFA Institute distinguishes those jobs and requires attention to reconciliation, benchmark fit, and inference limits.[1] A mismatched comparison can invalidate the interpretation even when its arithmetic is correct.
Here, designate C3-50's quarterly rule as the teaching benchmark and the five-percentage-point threshold rule as the candidate. This is an explicit comparison convention using already known synthetic results, not an empirical preregistration made before unknown outcomes. Both use the same stock, bond, and cash total returns, initial $10,000, $1,000 contribution before M7 returns, currency, execution windows, and five-basis-point fee policy.
The same fee rate need not produce the same fee amount: different trading should retain its different cost. Pretax and after-tax results, price indexes and distribution-inclusive returns, or a deposit-inflated balance and another account's return rate are not automatically comparable. A benchmark useful for comparing rebalancing rules may be inappropriate for an industry stock or hedged strategy.
Subtract matching monthly net returns
Using T22's convention, divide each month's equity after execution and fees by starting equity after any contribution, then subtract one. M7 includes the $1,000 already received in its denominator. Do not remove it again outside the calculated return. G93 supplies the cash-flow method; we use its reconciled results.
Active return here means candidate net return minus benchmark net return for the same period. It is not return above a risk-free rate or regression alpha. Keep monthly differences in explicit percentage-point units before calculating their variability.
| Month | Quarterly net return | Threshold net return | Threshold minus quarterly |
|---|---|---|---|
| M1 | 3.300000% | 3.300000% | 0.000000 pp |
| M2 | −5.465634% | −5.465634% | 0.000000 pp |
| M3 | 1.932586% | 1.932609% | 0.000023 pp |
| M4 | 3.600001% | 3.598435% | −0.001566 pp |
| M5 | −2.745174% | −2.743955% | 0.001220 pp |
| M6 | 1.505937% | 1.506010% | 0.000073 pp |
| M7 | 0.682924% | 0.683760% | 0.000836 pp |
| M8 | −7.667407% | −8.400326% | −0.732919 pp |
| M9 | 2.368285% | 2.620198% | 0.251914 pp |
| M10 | 1.950087% | 1.907313% | −0.042775 pp |
| M11 | −1.212412% | −1.180917% | 0.031496 pp |
| M12 | 3.155499% | 3.082237% | −0.073261 pp |
Displayed rounding can prevent the printed columns from subtracting exactly. Calculate from the full-precision original ledger. At M7 the threshold account had used new cash to replenish risky assets while the quarterly account awaited M9. Both stocks and bonds then fell in M8, producing a larger negative active difference. That explains the given path; it does not mean the decline was predictable beforehand.
Tracking error is not an audit error
Tracking error, or TE, is the sample standard deviation of these twelve monthly active returns. It measures variability in departures from the benchmark, not accounting mistakes or only months of underperformance. Positive and negative deviations both enter.
The monthly mean active return is approximately negative 0.0470799 percentage points. Subtract that mean from each observation, square the deviations, sum them, divide by eleven, and take the square root. Monthly TE is about 0.2300644 percentage points. Eleven is the sample denominator, n minus one; a population calculation dividing by twelve is a different convention.
Using decimal returns, the mean is approximately negative 0.000470799 and TE 0.002300644. Converting both consistently leaves their ratio unchanged. Changing only the numerator's units would introduce a hundredfold error. Record the units alongside intermediate values rather than relying on the final cell's format.
The information ratio, or IR, in this chapter is mean active return divided by matching-frequency TE. It is approximately negative 0.204638 monthly. Kidd describes this active-return/active-risk form, its benchmark and period dependence, and another form based on model residuals.[2] Identify which form a report uses instead of assuming the name specifies everything.
A negative IR says that the sample's mean active return was negative. It does not say every month underperformed, or fully describe correlation, drawdown, and tail losses. Low TE means close tracking, not a safe account. A portfolio can follow a falling benchmark closely while losing substantial money.
A familiar approximate annualization multiplies monthly TE by the square root of twelve and the monthly mean by twelve. Annualized IR then equals monthly IR times the square root of twelve. Under that convention, TE is about 0.7969664 percentage points and IR negative 0.708886. Matching frequency, adequate stability, and negligible serial dependence matter. Scaling twelve synthetic observations cannot create twelve years of evidence.
Constant active returns have zero standard deviation, leaving IR undefined even with a positive mean. One observation is insufficient for a sample standard deviation. Missing months cannot be filled with zeros to create twelve observations. MathWorks likewise illustrates undefined IR when relative-return variation is zero.[3]
Full-period differences need compounding
G93 gives full-period TWR of about 0.6523872% for the quarterly rule and 0.0186399% for the threshold rule. The difference is approximately negative 0.6337473 percentage points. Summing the monthly active differences instead gives approximately negative 0.5649585 points. There is no contradiction: subtracting separately compounded returns is not the same as adding monthly differences.
The threshold endpoint is about $69.5983 below the quarterly endpoint. Dividing that dollar gap by the initial $10,000 does not replace the TWR difference. The later contribution participated for only part of the record. Preserve the existing flow boundaries rather than choosing a convenient new denominator.
IR uses an arithmetic mean and same-frequency standard deviation. TWR describes a compounded account path. The dollar difference retains the wealth effect of the amounts actually present. Keep all three names attached to their questions; do not substitute the cumulative TWR gap into IR's arithmetic-mean numerator.
Use a separate single-period card for attribution
Now introduce C6-51, an independent synthetic card in dollars. Both portfolios begin at $10,000, with no external flows or trading during the period. Starting weights and category total returns are supplied. This is not a C3-50 month and will not be inserted into its twelve-period ledger.
| Category | Candidate starting weight | Benchmark starting weight | Candidate category gross return | Benchmark category gross return |
|---|---|---|---|---|
| Stocks | 70% | 60% | 10% | 8% |
| Bonds | 20% | 30% | 1% | 2% |
| Cash | 10% | 10% | 0% | 0% |
Candidate gross return is 70% times 10% plus 20% times 1%, or 7.2%. The benchmark earns 60% times 8% plus 30% times 2%, or 5.4%. Gross active return is 1.8 percentage points. The case supplies category returns; it does not pretend to possess actual stock-selection evidence explaining their difference.
Use the specified Brinson–Hood–Beebower-style three-part decomposition.[4] For each category, allocation is the weight difference times the benchmark category return. Selection is the benchmark weight times the category-return difference. Interaction is the weight difference times the return difference. Other attribution conventions exist; first establish which model the report applies.
| Category | Allocation | Selection | Interaction |
|---|---|---|---|
| Stocks | 0.8 pp | 1.2 pp | 0.2 pp |
| Bonds | −0.2 pp | −0.3 pp | 0.1 pp |
| Cash | 0 | 0 | 0 |
| Total | 0.6 pp | 0.9 pp | 0.3 pp |
For stocks, the extra ten percentage points of weight times the benchmark's 8% stock return give 0.8 points of allocation. At the benchmark's 60% weight, the extra two points of category return give 1.2 points of selection. The extra ten points of actual weight also participate in that two-point return difference, adding 0.2 points of interaction.
Interaction is a directly calculated cross-product, not an unexplained leftover. If a report combines selection and interaction into one displayed selection effect, do not add interaction again. Here 0.6 plus 0.9 plus 0.3 reconciles to 1.8 gross active points. That identity establishes consistency with the chosen decomposition, not causation or skill.
Give fees and an unexplained residual separate places
The card additionally specifies candidate fees of $15 and benchmark fees of $5, deducted from cash at period-end. Category gross returns did not include them. Relative to $10,000 starting capital, these are 0.15 and 0.05 percentage points; the candidate's additional fee drag is negative 0.10 points.
Reported ending account values are $10,703 for the candidate and $10,535 for the benchmark. The benchmark agrees with a 5.4% gross gain less $5. The candidate's 7.2% gross gain less $15 would imply $10,705. Its recorded endpoint is another $2 lower. Keep that amount as an unclassified residual of negative 0.02 percentage points, not an invented fee or confirmed human mistake.
Net active return is 1.68 percentage points. Reconcile it as 0.6 allocation, plus 0.9 selection, plus 0.3 interaction, minus 0.10 additional fees, minus the 0.02 unclassified residual. The total matches the account records, but the residual still needs evidence. Investigate valuation, timing, fees, and data before claiming the difference is fully explained.
Departures from an idealized result are not all avoidable poor execution. Changing the benchmark, categories, intraperiod trading, or cash-flow treatment can change the decomposition. Multiperiod attribution additionally needs a linking method for compounding; this single-period 1.68-point explanation cannot be pasted onto C3's full-period result.
Change a weight and preserve the evidence boundary
Explain why low TE does not ensure safety. Then identify the problem in calling interaction a residual, combining it into selection, and also adding it separately in the total. Write your answers before proceeding.
Reference answers · reveal
Answer guide: Small benchmark-relative variation leaves the benchmark's own risk intact. Interaction is an explicit product under this model, unlike an unknown residual. Combining it into selection and adding it again duplicates a contribution. Fix the definitions before reconciling rather than adjusting labels to fit a desired total.
For transfer, change only the C6-51 candidate's starting weights to 60% stocks, 30% bonds, and 10% cash. Keep category returns and benchmark unchanged. Recalculate gross active return and the three components. Can the old $10,703 endpoint provide this version's net active return?
Answer guide: Candidate gross return becomes 6.3%, exceeding the benchmark's 5.4% by 0.9 points. All weight differences are zero, so allocation and interaction are zero; selection remains 0.9 points. The old endpoint belongs to the 70/20/10 path and cannot be reused. New fees, actual account records, and any residual need new inputs. Changing one assumption does not authorize retaining every earlier outcome.
Save the T08 return-source card
Start from the T22-reconciled account. Record benchmark identity and version, why it matches the question, currency, dates, flows, fees, distributions, complete same-frequency returns and active sequence, sample size, mean, standard-deviation denominator, IR definition, and annualization assumptions. Retain attempts involving different benchmarks or deleted months.
For attribution, separately state the model, starting weights, category scope and returns, included or excluded fees, interaction presentation, residual, and unresolved material. Check active return from the original sequence and reconcile components, fees, and residual. Keep the single-period card distinct from the multiperiod ledger. An unexplained component is not proven alpha.
Compare first, explain second, judge within the evidence
Fix a benchmark appropriate to the task and build a matching cash-flow and net-return comparison. Active return, TE, and IR describe a particular sample relative to that reference; another reference or interval can change the conclusion.
Then decompose differences under a stated model, with separate places for interaction, costs, and unknown residuals. Reconciliation is the beginning of explanation, not proof of skill or cause. Leave unsupported claims unresolved rather than giving them a more confident name.
References
- CFA Institute: Portfolio Performance Evaluation, 2026 public overview. Measurement, attribution, appraisal, reconciliation, and benchmarks; not the paid reading.
- Deborah Kidd: The Sharpe Ratio and the Information Ratio, 2011. Active-return/active-risk form and benchmark dependence.
- MathWorks: Using the Information Ratio, R2026b. Matching-frequency differences, their standard deviation, and an undefined zero-denominator result. No MATLAB run is claimed.
- Carl Bacon: Performance Attribution—History and Progress, CFA Institute Research Foundation, 2019, printed pages 13–15. BHB decomposition and alternative treatment of interaction.
Sources inspected October 30, 2026. C3-50 and the separate C6-51 card are synthetic. No real-strategy alpha or professional skill is validated. AI-assisted draft; independent AI full-text and bilingual review is complete. Human financial review and target-reader testing have not been performed.