Look again at the BHB allocation line for Financials: the portfolio was 4 points under-weight a sector that returned +6.0%, and BHB scores that as −0.24. Penalised for under-weighting a sector that went up. Now notice that the benchmark as a whole returned 5.88%. Financials returned 6.0%. The manager under-weighted a sector that performed almost exactly in line with the index: a decision that should have been worth approximately nothing, and BHB has charged her 24 basis points for it.
What problem does Brinson-Fachler fix?
The defect is that BHB measures a weighting bet against the segment's own return, when the economically meaningful question is against the return you would have got by putting the money in the benchmark instead. Brinson and Fachler's earlier 1985 formulation fixes it by subtracting the total benchmark return:
Aᵢ = (wPᵢ − wBᵢ) × (rBᵢ − RB)
Now the sign has a clean meaning: over-weighting an out-performing segment or under-weighting an under-performing one is positive; anything else is negative. On the same data, Financials drops from −0.24 to −0.0048: correctly, almost nothing, because Financials did almost nothing relative to the index. Energy rises from +0.48 to +0.83, correctly larger, because being 6 points under-weight a sector that lost 8% against an index that gained 5.88% was a 13.88-point relative call, not an 8-point one.
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Do Brinson-Fachler and BHB give the same total?
Here is the elegant part, and a fact worth carrying: the two conventions produce identical allocation totals. The difference between them is Σ (wPᵢ − wBᵢ) × RB = RB × Σ (wPᵢ − wBᵢ), and since both weight vectors sum to one, that sum is zero. Both give +1.40. The choice of convention changes the story you tell about individual segments while leaving the total untouched, which is precisely why it matters. Totals get reported; stories get people hired and fired.
Two riders. If the portfolio holds cash and the benchmark does not, the weights no longer sum alike and the identity above fails; you must either treat cash as a segment with its own benchmark return or handle the residual explicitly. And Brinson-Fachler's advantage disappears when there is no meaningful "total benchmark return" to subtract, as in some absolute-return mandates. Default to Brinson-Fachler; know why.