The portfolio returned 8.23%, the benchmark 5.88%, so the active return is +2.35 points. Where did they come from?
There are only two ways to differ from a benchmark: hold different amounts of the segments it holds, or hold different things inside those segments. Brinson, Hood and Beebower formalised that intuition in 1986 into a decomposition that is still the industry's default, and its power is that it is an identity: the pieces sum to the active return exactly, always, with no residual, when computed correctly. That reconciliation is your error-detector for the rest of your career.
Brinson attribution formula and example
Set up the notation. For each segment i (sector, region, asset class; the frame is general): wPᵢ and wBᵢ are portfolio and benchmark weights, rPᵢ and rBᵢ are portfolio and benchmark returns within that segment. Total returns are RP = Σ wPᵢ rPᵢ and RB = Σ wBᵢ rBᵢ. Then, for each segment:
- Allocation
Aᵢ = (wPᵢ − wBᵢ) × rBᵢ: the effect of the weighting decision alone, holding security selection at benchmark. You over-weighted a segment; you earned its benchmark return on the extra weight. - Selection
Sᵢ = wBᵢ × (rPᵢ − rBᵢ): the effect of picking better (or worse) securities inside the segment, at benchmark weight. - Interaction
Iᵢ = (wPᵢ − wBᵢ) × (rPᵢ − rBᵢ), the cross-term: the extra weight applied to the extra return. It is what you get for over-weighting a segment in which you also picked well.
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And the identity: Aᵢ + Sᵢ + Iᵢ = wPᵢ rPᵢ − wBᵢ rBᵢ, so Σ(Aᵢ + Sᵢ + Iᵢ) = RP − RB. Expand the three terms on any segment and watch the cross-products cancel: one line of algebra, worth doing once by hand, because afterwards you will never again accept a table that does not tie out.
On Worked Example 2's five-sector US portfolio the totals are allocation +1.40, selection +0.80, interaction +0.15, sum +2.35: the active return, to the last decimal. The reading is the whole point: this manager's year was two-thirds a weighting story. She was 8 points over-weight the best sector and 6 points under-weight the worst, and those two decisions alone produced +1.60 of the +1.40 net allocation effect; her stock-picking added a real but smaller +0.80. Told only "she beat the index by 2.35%," you would not know whether you were paying for a top-down macro call or a bottom-up research process: different products, with different capacity, repeatability and fees.
How should the interaction effect be treated?
Interaction is the term people mishandle, and there are three defensible treatments. Report it separately, as in the three-factor BHB form above. Fold it into selection with portfolio weights, S'ᵢ = wPᵢ(rPᵢ − rBᵢ), the two-factor form most commercial systems use: here selection becomes +0.95, exactly 0.80 + 0.15. The third option is to fold it into allocation. What is not defensible is dropping it, which breaks the identity, or presenting it as a distinct investment decision. Nobody sets out to generate interaction; it is the arithmetic consequence of two decisions pointing the same way: genuine skill when a manager concentrates weight where her edge is strongest, noise otherwise. Say which convention you used before you show anyone a number.