Explaining active return by allocation and selection
Series XXI-B case study · set 2, case 3 of 6 · 5 questions · 2 marks each
Question 1 of 5Case practice
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A portfolio has two sectors for one period. Portfolio weights are 60% equities and 40% bonds. Benchmark weights are 50% each. The portfolio’s equity sleeve returns 12% and bond sleeve 6%. Benchmark equity and bond returns are 10% and 5%. All weights are opening weights, no cash flows or costs occur, and the weights sum to one. Use a specified three-part attribution: allocation = sum[(portfolio weight − benchmark weight) × benchmark sector return]. Selection = sum[benchmark weight × (portfolio sector return − benchmark sector return)]. Interaction = sum[(weight difference) × (sector-return difference)]. Report all contribution results in percentage points, not percentage changes in wealth.
All amounts and contract terms in this case are hypothetical.
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The case and its answers
Every question of this case, with its answer and why each other option is wrong.
A portfolio has two sectors for one period. Portfolio weights are 60% equities and 40% bonds. Benchmark weights are 50% each. The portfolio’s equity sleeve returns 12% and bond sleeve 6%. Benchmark equity and bond returns are 10% and 5%. All weights are opening weights, no cash flows or costs occur, and the weights sum to one. Use a specified three-part attribution: allocation = sum[(portfolio weight − benchmark weight) × benchmark sector return]. Selection = sum[benchmark weight × (portfolio sector return − benchmark sector return)]. Interaction = sum[(weight difference) × (sector-return difference)]. Report all contribution results in percentage points, not percentage changes in wealth.
All amounts and contract terms in this case are hypothetical.
Q1What is the portfolio’s total return?
Option A. 7.50%
Why it is wrong: This is the benchmark return.
Option B. 9.60% — Correct answer
Option C. 18.00%
Why it is wrong: This adds sleeve returns without weights.
Option D. 9.00%
Why it is wrong: This gives the two sleeves equal weights.
Answer: B. 0.60 × 12% + 0.40 × 6% = 9.60%. Remember: Portfolio return uses portfolio weights and portfolio sleeve returns.
Hint, shown before you answer in practice mode: Weight each portfolio sleeve’s return by its portfolio weight.
XXIB-S2-C03-Q1easynumerical
Q2What is the active return over the benchmark?
Option A. 1.50 percentage points
Why it is wrong: This is only the selection contribution.
Option B. 2.10 percentage points — Correct answer
Option C. 0.50 percentage points
Why it is wrong: This is only the allocation contribution.
Option D. 2.80 percentage points
Why it is wrong: This confuses relative percentage outperformance with a percentage-point gap.
Answer: B. Benchmark return = 0.50×10% + 0.50×5% = 7.50%. Active return = 9.60% − 7.50% = 2.10 percentage points. Remember: Define whether outperformance is a difference or a relative ratio.
Hint, shown before you answer in practice mode: First calculate the benchmark’s equally weighted return.
XXIB-S2-C03-Q2mediumnumerical
Q3What allocation contribution follows the formula provided?
Option A. 0.10 percentage points
Why it is wrong: This is the interaction contribution.
Option B. 2.10 percentage points
Why it is wrong: This is total active return.
Option C. 1.50 percentage points
Why it is wrong: This is the selection contribution.
Option D. 0.50 percentage points — Correct answer
Answer: D. (0.60−0.50)×10% + (0.40−0.50)×5% = 1.00 − 0.50 = 0.50 percentage points. Remember: An overweight can help when its benchmark sleeve outperforms the underweighted sleeve.
Hint, shown before you answer in practice mode: Apply the weight differences to benchmark sector returns.
XXIB-S2-C03-Q3mediumnumerical
Q4What selection contribution follows the specified formula?
Option A. 0.50 percentage points
Why it is wrong: This is allocation.
Option B. 1.50 percentage points — Correct answer
Option C. 2.00 percentage points
Why it is wrong: This considers equity selection alone without its benchmark weight.
Option D. 1.60 percentage points
Why it is wrong: This uses portfolio weights and combines selection with interaction.
Answer: B. 0.50×(12−10) + 0.50×(6−5) = 1.00 + 0.50 = 1.50 percentage points. Remember: Use the exact attribution convention supplied.
Hint, shown before you answer in practice mode: Weight each sleeve’s excess return by its benchmark weight.
XXIB-S2-C03-Q4mediumnumerical
Q5What interaction contribution completes the reconciliation?
Option A. −0.10 percentage points
Why it is wrong: This reverses the summed interaction sign.
Option B. 0.00 percentage points
Why it is wrong: Nonzero weight and selection differences produce interaction here.
Option C. 0.60 percentage points
Why it is wrong: This does not reconcile the specified components.
Option D. 0.10 percentage points — Correct answer
Answer: D. 0.10×2 + (−0.10)×1 = 0.10 percentage points. Allocation 0.50 + selection 1.50 + interaction 0.10 = active return 2.10. Remember: Check that attribution components reconcile to active return.
Hint, shown before you answer in practice mode: Multiply each weight difference by its sleeve-return difference.
XXIB-S2-C03-Q5hardnumerical
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