Linking Attribution Across Periods

A performance team runs Brinson attribution every month and gets clean, correct numbers: allocation, selection, and interaction sum exactly to that month's active return. The client, however, wants to know what drove the year's outperformance, not each month's in isolation. Stitching twelve correct monthly answers into one correct annual answer turns out to be its own problem, because the fund's annual active return is a compounded quantity, not a sum, while the attribution effects were computed as sums within each month.

Why you can't just add the months

Suppose allocation contributed +0.4% in January and +0.6% in February. Adding gives +1.0%. But the portfolio's actual return compounded across both months, and so did the benchmark's, so the true two-month active return is $(1+R_p)/(1+R_b) - 1$ using compounded $R_p$ and $R_b$, not the sum of the two months' active returns. The gap between the naive sum and the true compounded figure is called the linking residual, and it grows with volatility and with the number of periods being chained.

Worked example

Active returns of +2%, -1%, and +3% over three months.

1. Naive sum. $2\% - 1\% + 3\% = 4\%$.
2. True compounded active return, assuming these are the portfolio-vs-benchmark ratios each month: $(1.02 \times 0.99 \times 1.03) - 1$. Compute step by step: $1.02 \times 0.99 = 1.0098$; $1.0098 \times 1.03 = 1.0401$. So the true figure is $4.01\%$.
3. The residual. $4.01\% - 4\% = 0.01$ points here — tiny with three small numbers, but the same arithmetic with twelve volatile months routinely produces residuals of tens of basis points, enough to visibly disagree with a client's own return calculation.

A linking algorithm's job is to take that residual and distribute it back across the individual allocation, selection, and interaction lines so each month's linked effects sum exactly to the linked total, rather than leaving an unexplained plug on the report.

What this means in practice

Every institutional attribution report that spans more than one period is running a linking method behind the scenes, most commonly the Cariño algorithm, and the choice of method is disclosed in GIPS-compliant reporting because different methods distribute the same residual differently across periods and sectors. A performance analyst who doesn't know which linking convention their system uses can't reliably explain why last year's per-sector attribution doesn't sum to the number on the client's statement.