Attribution effects are additive within a period and multiplicative across periods, and that mismatch is the single most common source of tables that do not tie.
Why do simple sums of attribution effects fail?
Take the first half's +2.35 active return and a second half of −0.95. The naive sum is +1.40. But returns compound: the portfolio's two-period return is 1.0823 × 0.9871 − 1 = 6.834% and the benchmark's is 1.0588 × 0.9966 − 1 = 5.520%, so the true compounded active return is +1.314%. The naive sum overstates it by 8.6 basis points, and the residual grows with the number of periods and the size of the returns. Over twelve monthly periods in a volatile year it is routinely tens of basis points: comfortably large enough to flip a "did she add value?" verdict, and invisible unless you check.
The fix that became standard is Cariño's (1999) logarithmic smoothing. Define a period coefficient and an overall coefficient:
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kₜ = [ln(1 + RPₜ) − ln(1 + RBₜ)] / (RPₜ − RBₜ) and K = [ln(1 + RP) − ln(1 + RB)] / (RP − RB)
then scale every single-period effect by kₜ / K before summing. The scaled effects reconcile exactly to the compounded active return. On our two half-years, k₁/K = 0.9918 and k₂/K = 1.0705: the first half's effects are shrunk about 0.8%, the second half's inflated about 7%, and the smoothed totals (allocation +0.918, selection +0.451, interaction −0.055) sum to +1.3138, the compounded active return, to fourteen decimal places. Menchero's (2000) optimised variant achieves the same reconciliation while minimising the distortion of individual periods. You will rarely code either from scratch; you must know they exist, because when a vendor's multi-period table ties out perfectly you should be able to say which linking algorithm made it tie, and when it does not tie you should be able to say why.
What are the four attribution models?
The Brinson frame is one of four attribution models, and using the wrong one is a category error:
- Holdings-based (Brinson): needs portfolio weights and returns by segment. Precise, and the standard for long-only equity mandates against a segmented benchmark. Its blind spot is intra-period trading: it assumes buy-and-hold within the period, so a manager who trades heavily inside the window will show effects that misattribute what actually happened.
- Transaction-based: reconstructs the portfolio from every trade. Exact, expensive, and the only honest choice for high-turnover books.
- Returns-based (style) analysis: Sharpe's constrained regression of fund returns on a set of style-index returns, requiring nothing but the return series. Cheap, external, ideal for screening funds you cannot see inside, and dependent on regression assumptions that QM1.02 teaches you to test. It tells you what a manager behaved like, not what she held.
- Factor-based: decomposes active return into exposures to priced factors (value, size, momentum, quality, duration, credit) plus a residual. The right frame when the benchmark is a factor portfolio, and the natural bridge from here to AA1.02's risk budgeting.
Fixed income deserves a warning of its own: a Brinson decomposition on a bond portfolio is close to meaningless, because the return drivers are duration, curve positioning, spread and currency, not "sector weights." Fixed-income attribution decomposes into a yield/carry effect, a parallel-shift (duration) effect, a curve-reshaping effect and a spread/selection effect. FI1.02 owns the duration and key-rate machinery those effects are built from; do not try to force a bond book through the frame taught here.