Energy Underweight in a Global Equity Portfolio: Allocation Sign, Interaction and Arithmetic versus Geometric Excess
An active global equity manager runs a $700 million portfolio against the MSCI World index. In the first quarter the portfolio returned 6.80% and the benchmark 6.00%. The manager is underweight energy - portfolio 4.0%, benchmark 9.0%; energy returned 14.0% - and overweight technology: portfolio 30.0%, benchmark 22.0%; technology returned 9.0%. The portfolio's energy holdings matched the sector return, and its technology holdings returned 10.5%. The benchmark total return was 6.00%. The client's system reports allocation −0.50%, selection +1.15% and interaction +0.15%, summing to +0.80%. A second system that folds interaction into selection reports selection +1.30%. The manager's factsheet says 'energy added 0.70% of relative return through underweighting' and shows only the arithmetic excess of 0.80%, although the geometric excess return is 0.75%. The consultant wants to know which figures are defensible and why underweighting the best sector is presented as a gain.
1. Compute the Brinson–Fachler allocation and selection effects for energy and explain how a −0.70% contribution difference became a claimed +0.70% gain.
Read the worked answer
Using Brinson–Fachler, with B = 6.00% benchmark total return:
- Allocation (energy) = (wp − wb)(Rb,sector − B) = (0.04 − 0.09)(14.0 − 6.0) = (−0.05)(8.0) = −0.40%
- Selection (energy) = wb(Rp,sector − Rb,sector) = 0.09 × (14.0 − 14.0) = 0.00%
- Interaction (energy) = (wp − wb)(Rp,sector − Rb,sector) = (−0.05)(0.0) = 0.00%
- Total energy effect = −0.40%
The manager's 0.70% is the contribution difference, not an allocation effect: 0.04 × 14.0 − 0.09 × 14.0 = 0.56 − 1.26 = −0.70%. That is the amount by which the underweight reduced the portfolio's absolute return relative to holding benchmark weights in energy. It is negative, and it is not the same concept as allocation, which measures the decision against the benchmark's average return.
The economic logic is straightforward: energy returned 14%, well above the 6% benchmark average. Holding 5 percentage points less than the benchmark in a sector that beat the average by 8 points costs 0.40% of relative return. Avoiding a sector only helps if that sector underperforms the benchmark average; being underweight the quarter's best sector hurts. The factsheet has the sign wrong, or is reporting the contribution difference as though it were an allocation gain, and either way it should be withdrawn.
2. Reconcile the arithmetic and geometric excess returns and explain what the interaction term represents and how the two systems can both be internally consistent.
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Arithmetic excess = Rp − Rb = 6.80 − 6.00 = +0.80%. This is the number that the Brinson effects sum to: allocation + selection + interaction = 0.80%.
Geometric excess = (1 + Rp)/(1 + Rb) − 1 = 1.0680/1.0600 − 1 = +0.7547%, roughly +0.75%. This measures the relative growth of the portfolio over the benchmark and is the more meaningful number for a client asking 'how much more money did I make relative to the index?' The two differ because arithmetic effects add linearly while returns compound; the gap widens as returns and effects grow.
Interaction exists because allocation and selection decisions are made together. It is (wp − wb)(Rp,sector − Rb,sector): the part of relative return attributable to holding a different weight and a different return within the sector. Brinson–Fachler isolates it; Brinson–Hood–Beebower uses a different allocation formula ((wp − wb) × Rb,sector) and produces a larger interaction term; many systems simply fold interaction into selection. All three are arithmetic identities, so both of the client's systems can sum correctly to 0.80% while attributing the same 15 bp differently.
The practical rules: never mix effects from two systems in one report, disclose which model and which excess-return definition is used, and if the manager wants to claim value added in a sector, show allocation, selection and interaction separately rather than a single contribution number.
3. What should the manager present to the consultant, and what should be corrected before the next client report?
Read the worked answer
Present one coherent attribution: Brinson–Fachler (or BHB), arithmetic excess of 0.80% reconciled line by line to allocation, selection and interaction, with the geometric excess of 0.75% shown alongside and labelled as a different measure. State the benchmark, the sector classification, the currency treatment and the linking method for the year-to-date figure.
Corrections required:
- Remove the claim that energy added 0.70%. Replace it with the true energy effect: allocation −0.40%, selection 0.00%, interaction 0.00%, total −0.40%. If the factsheet meant to say the underweight cost 0.40% relative to the benchmark, say that.
- Make the sign convention explicit. A manager underweight a sector that beats the benchmark average has a negative allocation effect; a manager overweight a sector that underperforms also has a negative allocation effect. Sign errors here are the most common attribution failure in client reporting.
- Do not present only the arithmetic excess while describing it as 'relative return' if the client's own system reports geometric. Label the measure.
- Reconcile the second system's +1.30% selection to the first system's +1.15% selection plus +0.15% interaction; if the systems use different allocation formulas, the effects are not comparable across reports.
- Check whether the reported benchmark return is total return (with income) and whether the portfolio return is gross or net of fees; mixing gross portfolio with net benchmark, or vice versa, invalidates the excess return.
A short reconciliation table showing both systems and the identity each satisfies is the cleanest way to answer the consultant without appearing to have two sets of books.