Describe how performance attribution is done and apply it to split results into asset allocation, sector and security selection effects and their combined total.
The idea
Performance attribution explains why a portfolio's return differs from its benchmark's by stepping through the investment decisions. Asset allocation compares the portfolio's class weights with the benchmark's. Allocation = Σ (wP − wB) × rB. Here wP and wB are the portfolio and benchmark weights of a class, and rB is the benchmark's return in that class. Within a class, sector selection is the sector weight difference times the benchmark sector return, scaled by the portfolio's class weight. The three effects add exactly to the total excess return, portfolio return minus benchmark return. Security selection is measured within each sector, using portfolio weights. Security = wP × (rP − rB). A positive allocation effect from underweighting a falling class and a negative selection effect can coexist. The usual mistakes are using benchmark weights instead of portfolio weights in the selection term, or valuing the weight difference at portfolio returns instead of benchmark returns.
Formulas
Asset allocation effect
Allocation = Σ (wP − wB) × rB
- wP
- the portfolio's weight in the class
- wB
- the benchmark's weight in the class
- rB
- the benchmark's return in that class
Value the weight differences at benchmark returns. Underweights count too: a negative weight difference on a falling class adds value.
Security selection effect
Security = wP × (rP − rB)
- wP
- the portfolio's weight in the class or sector
- rP
- the portfolio's return there
- rB
- the benchmark's return there
Weighted by the portfolio's weight, so with the allocation effect it adds exactly to the total excess return. Using the benchmark weight is the trap.
Sector selection effect within an asset class
Sector = wP,class × Σ (wP,s − wB,s) × rB,s
- wP,class
- the portfolio's weight in the asset class
- wP,s
- the portfolio's weight in sector s within the class
- wB,s
- the benchmark's weight in sector s within the class
- rB,s
- the benchmark's return in sector s
Weights inside the sum are within the class. Scale by the portfolio's class weight, not the benchmark's.
A practice question
Performance attribution is used to:
- estimate the portfolio's effective asset mix from its returns alone.Why not: This is style analysis, which infers the asset mix from returns.
- explain the gap between portfolio and benchmark return through decisions such as allocation and selection. - correct
- test whether the portfolio's alpha is significantly different from zero.Why not: This is a t-test on a performance measure, not a decomposition of return.
- adjust the portfolio's return to match the market's risk level.Why not: This is the M2 approach, which matches the portfolio's risk to the market's.
Attribution decomposes excess return into allocation, sector and security selection effects that add up to the total.
A harder one
A portfolio holds 70% equities, 25% bonds and 5% cash; its benchmark holds 60%, 30% and 10%. Benchmark class returns are 8%, 3% and 1%. The portfolio's class returns are 10%, 2.4% and 1%. The total asset allocation effect (weight difference x benchmark return) is closest to:
- 0.83%Why not: Values the weight differences at the portfolio's class returns.
- 1.25%Why not: This is the total security selection effect, wP x (rP - rB).
- 0.60% - correct
- 0.80%Why not: Counts only the equity over-weight and omits the underweights in bonds and cash.
Allocation = (0.70 - 0.60) x 8 + (0.25 - 0.30) x 3 + (0.05 - 0.10) x 1 = 0.80 - 0.15 - 0.05 = 0.60%.
The hardest tier
A portfolio holds 65% equities and 35% bonds; the benchmark holds 60% and 40%. Benchmark returns are 12% for technology and 4% for energy (equal weights within the benchmark's equity) and 3% for bonds. Within the portfolio's equity, technology is 60% and energy 40%. Using sector effect = equity weight in the portfolio x sum of (portfolio sector weight - benchmark sector weight) x benchmark sector return, where the weights are within equity, the sector selection effect is closest to:
- 0.80%Why not: Omits the 65% equity weight of the portfolio.
- 0.52% - correct
- 0.26%Why not: This is the within-sector security effect, not the sector effect.
- 0.48%Why not: Scales by the benchmark equity weight (60%) instead of the portfolio's 65%.
Within-equity weight differences are +10% in technology and -10% in energy: 0.10 x 12 - 0.10 x 4 = 0.80. Scaled by the 65% equity weight, the sector effect = 0.52%.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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