Allocation, Selection and Interaction Effects

Brinson attribution splits a portfolio's outperformance into pieces, but it's worth slowing down on exactly what each piece measures, because the three terms answer three different questions and get misread constantly. Allocation asks: did you put money in the right sectors? Selection asks: within a sector, did you pick the right names? Interaction asks: what happens when both answers are "yes" or both are "no" at once?

Three questions, one number each

For a single sector $i$, define the portfolio weight $w_p$, benchmark weight $w_b$, portfolio return $r_p$, and benchmark return $r_b$ within that sector. The three effects are:

$$\text{Allocation}_i = (w_{p,i} - w_{b,i}) \times r_{b,i}$$ $$\text{Selection}_i = w_{b,i} \times (r_{p,i} - r_{b,i})$$ $$\text{Interaction}_i = (w_{p,i} - w_{b,i}) \times (r_{p,i} - r_{b,i})$$

In words: allocation is your overweight or underweight in a sector, multiplied by how that sector itself did in the benchmark — it doesn't care what you actually owned within the sector, only whether you leaned toward or away from a sector that did well. Selection is the benchmark's own weight in that sector, multiplied by how much your picks beat the sector's benchmark return — it doesn't care whether you were overweight, only whether your stocks were better than average. Interaction catches the leftover: it's positive when you were both overweight and good at picking within that same sector, and negative when a good overweight call was paired with bad stock picking (or vice versa).

Worked example

A fund holds 40% in tech versus a benchmark weight of 25%. Tech returned 15% in the benchmark; the fund's tech holdings returned 20%.

1. Allocation. $(0.40 - 0.25) \times 0.15 = 0.15 \times 0.15 = 0.0225$, i.e. 2.25 percentage points, earned purely by overweighting a sector that beat the average.
2. Selection. $0.25 \times (0.20 - 0.15) = 0.25 \times 0.05 = 0.0125$, i.e. 1.25 percentage points, earned by picking better-than-average tech names, scaled by how much weight the benchmark itself gives tech.
3. Interaction. $(0.40 - 0.25) \times (0.20 - 0.15) = 0.15 \times 0.05 = 0.0075$, i.e. 0.75 percentage points — the bonus from having more money riding on the stocks that were also better picks.
4. Total sector contribution. $2.25\% + 1.25\% + 0.75\% = 4.25\%$, matching the direct calculation $0.40 \times 0.20 - 0.25 \times 0.15 = 0.08 - 0.0375 = 0.0425$.

Sum each of the three effects across every sector and you get total allocation, total selection, and total interaction for the whole portfolio, which together reconstruct the full active return.

What this means in practice

Most performance teams fold interaction into selection before presenting results to a client, because a standalone "interaction effect" line is hard to explain and often small. That's the Brinson-Fachler convention, versus the original Brinson-Hood-Beebower model, which reports all three separately. Either is defensible; what matters is being consistent within a firm, since the two conventions attribute the same total return differently across the allocation and selection lines.