Geometric vs Arithmetic Attribution

Single-period Brinson attribution is clean: allocation plus selection plus interaction adds up exactly to active return. The trouble starts the moment you have more than one period. A fund's return over a year is not the sum of its twelve monthly returns, it's the compounded product of them, because each month's gain or loss is earned on top of the previous month's balance. Attribution effects inherit the same problem: if allocation added 0.3% in January and 0.2% in February, the annual allocation effect is not simply 0.5%.

Two honest ways to be wrong

Arithmetic attribution computes allocation, selection, and interaction each period the ordinary way, then sums them across periods: $\text{Total Allocation} = \sum_t \text{Allocation}_t$. This is simple and each period's number means exactly what it always meant. The catch is that summing ignores compounding — a portfolio that outperforms by 1% in a month where the market fell has actually helped the client more than the raw 1% suggests, because that 1% is protected from the next month's losses being applied to a smaller base.

Geometric attribution instead defines active return multiplicatively, $1 + R_{active} = (1 + R_p) / (1 + R_b)$, and looks for a way to decompose that ratio into allocation and selection pieces, then chains periods together by multiplication rather than addition: $(1 + R_{active,total}) = \prod_t (1 + R_{active,t})$. This correctly reproduces the portfolio's actual compounded excess return, but the individual allocation and selection terms no longer add to the total in a way that's easy to explain to a client, since products don't decompose as cleanly as sums.

Worked example

A portfolio's active return is +2% in month one and +3% in month two.

1. Naive arithmetic sum. $2\% + 3\% = 5\%$.
2. Actual compounded active return. The portfolio needs its own compounding, not just the sum: $(1.02 \times 1.03) - 1 = 1.0506 - 1 = 5.06\%$.
3. The gap. $5.06\% - 5\% = 0.06$ percentage points — small here because both numbers are small, but the gap grows quickly with larger returns or more periods, since compounding effects are multiplicative and errors accumulate.

A fund reporting five years of monthly attribution and simply summing 60 months of allocation effects can be off from the fund's actual five-year excess return by a full percentage point or more, purely from ignoring compounding.

What this means in practice

Most institutional performance systems use geometric attribution, or a smoothing method like the Cariño linking algorithm, specifically to make the sum of period effects reconcile exactly with the fund's actual compounded excess return — a number that shows up on every client statement and cannot be allowed to disagree with the attribution report sitting next to it. Arithmetic attribution survives mainly for single-period reporting, where there's no compounding problem to solve, and as a simpler mental model when teaching the concept for the first time.