CFA Level I · CFA Level I Exam

Benchmarking Returns: formula sheet

Full chapter guide

Key formulas

Active return: Active return = Portfolio return − Benchmark return; Positive means the portfolio beat the benchmark. Use the same period and same return basis for both.
Absolute target return: Target = fixed % or (inflation + spread); Does not depend on market performance. Used by absolute return or liability-driven mandates.
Custom blended benchmark: Benchmark return = Σ (weight i × index return i); Weights must sum to 100%. Use the strategic asset allocation weights.
SAMURAI checklist: Specified in advance, Appropriate, Measurable, Unambiguous, Reflective of current investment opinions, Accountable, Investable; A valid benchmark should meet all seven. Exam questions usually show one failing.
Active return: Active return = Portfolio return − Benchmark return; Only meaningful if the benchmark is valid. A poor benchmark makes active return misleading.
Investable test: Could an investor hold the benchmark's securities in their weights at reasonable cost?; If not, passive replication is not a real alternative.
Price-weighted index value: Index = Σ prices ÷ divisor; Initial divisor is often the number of stocks. Adjust the divisor after splits so the index value does not change.
Price weight of a security: w(i) = P(i) ÷ Σ P; Highest-priced stock has the greatest influence.
Equal-weighted index return: R = (R1 + R2 + … + RN) ÷ N; Simple average of returns, assuming weights reset to 1/N at the start of the period.
Market-cap weight: w(i) = (P(i) × Shares(i)) ÷ Σ (P × Shares); Use free-float shares for a float-adjusted index.
Float-adjusted market cap: Price × shares outstanding × free-float factor; Free float factor is the share of shares available to the public.
Index return from index levels: Return = (Index end ÷ Index start) − 1; Price return only unless the index is a total return index.
Weighted return: R(index) = Σ w(i) × R(i); Works for any scheme, with the weights at the start of the period.
Divisor after a split (price-weighted): New divisor = Σ adjusted prices ÷ old index value; Choose the new divisor so the index is unchanged right after the split.
Benchmark return (blend): R_B = Σ (w_i × R_i); Use benchmark weights w_i, which sum to 1. R_i is the return of each benchmark component.
Portfolio return (blend): R_P = Σ (w_i × R_i); Use the portfolio's actual weights and the portfolio's own segment returns.
Active return: Active return = R_P − R_B; Same period, same basis (both gross or both net). Result is in percentage points.
Tracking difference: Tracking difference = R_fund − R_index; A single-period or cumulative gap, not a standard deviation.
Tracking error: Tracking error = standard deviation of (R_P,t − R_B,t) over t; Measures variability of active returns. It is not the average active return.
Active return with geometric linking: Cumulative active return = [Π(1 + R_P,t) − 1] − [Π(1 + R_B,t) − 1]; Compound each series separately over the periods, then subtract. Do not add yearly active returns.
Active return: Active return = Portfolio return − Benchmark return; Only meaningful if the benchmark matches the manager's style and universe.
Qualities of a valid benchmark: Unambiguous, investable, measurable, appropriate, reflective of current investment opinions, specified in advance, owned (the manager is aware of the benchmark and accepts accountability for it); Use these seven qualities to test any proposed benchmark.
Custom benchmark return: R(custom) = w1 × R(index 1) + w2 × R(index 2) + …, with Σw = 1; Weights should reflect the manager's intended exposures.
Survivorship bias effect (qualitative): Not a formula: survivorship bias tends to overstate returns and understate risk.; Failed funds or securities drop out of the data, leaving better-performing survivors. It is a tendency, not a calculated number.

Quick revision

A benchmark is a reference portfolio used to judge performance against a mandate.
Active return = portfolio return − benchmark return, over the same period.
Benchmark return = Σ (weight × component return).
A valid benchmark is specified in advance, so it is not chosen after results are known.
Check the seven qualities: unambiguous, investable, measurable, appropriate, reflective of current investment opinions, specified in advance, and accountable (the manager accepts the benchmark as the yardstick for their performance).
Market-cap weighting gives larger companies larger weights and adjusts naturally as prices move.
Price weighting depends on share price, so a high-priced stock has more influence regardless of company size.
Equal weighting gives each security the same starting weight and needs rebalancing to stay equal.
A benchmark must fit the mandate: a small-cap manager should not be judged against a large-cap index.
Peer-group comparisons can suffer from survivorship bias and different mandates.
Compare returns on a like-for-like basis: same currency, same period, same treatment of income.
Positive active return means outperformance, but it says nothing alone about risk taken.

Common mistakes

Treating any index as an appropriate benchmark for any portfolio. Fix: Check that the index matches the asset class, style, region and risk of the mandate.
Calling a fixed target such as inflation plus 3% a relative benchmark. Fix: If the target does not move with a market or group, it is absolute.
Confusing Unambiguous with Measurable. Fix: Unambiguous is about knowing exactly what the holdings and weights are. Measurable is about being able to calculate its return regularly.
Treating Investable as the same as Appropriate. Fix: Appropriate is about fit with the mandate. Investable is about whether you can actually buy and hold it, for example because of liquidity or access.
Averaging returns for a price-weighted or cap-weighted index. Fix: Use the weights the method implies. A simple average belongs only to equal weighting.
Using ending values to set weights. Fix: Weight with starting values. Return over a period is earned on the beginning weights.
Using the portfolio's actual weights to compute the benchmark return Fix: Benchmark return always uses benchmark weights. Portfolio return uses portfolio weights.
Confusing tracking error with active return Fix: Active return is a difference in returns. Tracking error is the standard deviation of those differences over time.
Confusing style drift with benchmark mismatch Fix: Check timing. Mismatch is present from the start; drift develops as the manager changes holdings.
Saying survivorship bias understates returns Fix: Removing failed funds leaves winners, so average returns are overstated and risk understated.

Exam tips

Look for the words 'regardless of market' or 'fixed target'. They signal an absolute benchmark.
Questions often ask which type fits a mandate. Match the mandate first, then pick the type.
Know the main weakness of each type: peer groups are not investable and can have survivorship bias.
In blended benchmark questions, compute carefully and check the sign of active return before choosing.
Questions are standalone with three options, so find the clue phrase first, then eliminate the two qualities that clearly hold.
Several qualities can seem to fit. Choose the one the stem's key detail points to most directly.
Expect scenarios about timing (Specified in advance), liquidity (Investable) and manager agreement (Accountable).
Learn each quality as an idea in plain words, because wording can differ from the acronym.