BHB Attribution Analyser

Period Data Input

Presets:

Sector	Wp%	Rp%	Wb%	Rb%

Wp: —

Wb: —

Wp = portfolio weight · Rp = portfolio return · Wb = benchmark weight · Rb = benchmark return
In formula workings, pp = percentage points (weight difference); effects are expressed as % of portfolio value.

—

Portfolio Return

—

Benchmark Return

—

Active Return

—

Allocation

—

Selection (BHB)

—

Interaction (BHB)

—

Selection (BF)

—

Allocation (BF)

Attribution Table

BHB BF

Sector	Wp%	Wb%	Rp%	Rb%	Alloc (BHB)	Sel (BHB)	Int (BHB)	Alloc (BF)	Sel (BF)	Int (BF)	Working

Step-by-Step Formula Workings

Enter data in the Analyse tab to see formula workings.

Chart A — Attribution Effects by Sector (BHB)

Allocation (BHB)

Selection (BHB)

Interaction (BHB)

Allocation (BF)

Chart B — Waterfall: Active Return Build-Up

Chart C — Sector: Weight Differential vs Return Differential

Multi-Period Attribution (Geometric Linking — Cariño Method)

How geometric linking works: Single-period arithmetic effects do not simply sum across periods. The Cariño method (1999) scales each period's effects by a linking coefficient k_t = (ln(1+rp_t) − ln(1+rb_t)) / (rp_t − rb_t) so that the effects chain multiplicatively and sum to the true multi-period active return.

Use the Period Data Input (Analyse tab) to enter multiple periods using the period tabs. Each period is linked here.

Add at least 2 periods in the Analyse tab.

Multi-Period: Cumulative Attribution Effects

Enter data in the Analyse tab to generate insights.

Theory & Model Reference

The Brinson-Hood-Beebower (BHB) model (1986) decomposes a portfolio's active return — the difference between portfolio return and benchmark return — into three sources: allocation, selection, and interaction. Together they account for 100% of active return with no residual.

BHB — Allocation Effect (per sector j)

A_j = (Wp_j − Wb_j) × Rb_j

Did the manager overweight sectors that outperformed (had high benchmark return)?

BHB — Selection Effect (per sector j)

S_j = Wb_j × (Rp_j − Rb_j)

Did the manager pick stocks in each sector that beat the sector benchmark?

BHB — Interaction Effect (per sector j)

I_j = (Wp_j − Wb_j) × (Rp_j − Rb_j)

Joint effect of overweighting a sector AND achieving above-benchmark returns within it. This is a mathematically necessary third term in the 3-factor decomposition — not a separate deliberate decision. Some practitioners assign it to selection (giving a 2-factor model).

BF — Allocation Effect (Brinson-Fachler, 1985)

A_j(BF) = (Wp_j − Wb_j) × (Rb_j − Rb_total)

Uses sector benchmark return relative to the overall benchmark. Positive only when the manager overweights a sector that beats the total benchmark — a stricter and more informative standard than BHB allocation. Key identity: total BHB allocation = total BF allocation when ΣWp = ΣWb = 100%. Differences only appear at the sector level.

BF — Selection & Interaction (3-factor practitioner form)

S_j(BF) = Wb_j × (Rp_j − Rb_j) [same as BHB] I_j(BF) = (Wp_j − Wb_j) × (Rp_j − Rb_j) [same as BHB]

This tool implements the widely used 3-factor BF variant (BF allocation + BHB selection + BHB interaction), which is standard in the CFA/CIPM curriculum. The original BF (1985) paper used only 2 effects, defining selection as Wp_j×(Rp_j−Rb_j) so that it absorbed the interaction term. Both forms are academically valid; the 3-factor form is more common in practice.

Identity: Active Return = Sum of All Effects

AR = Σ(A_j + S_j + I_j)

For BHB and BF alike. The three effects always sum exactly to the active return. No residual.

Multi-Period Linking (Cariño, 1999)

k_t = [ln(1+rp_t) − ln(1+rb_t)] / (rp_t − rb_t) Linked A = [Σ_t k_t × A_t] / K_total

Converts single-period arithmetic effects into geometrically linked multi-period effects that sum to the true compounded active return.

References:
Brinson, G.P., Hood, L.R. & Beebower, G.L. (1986). Determinants of Portfolio Performance. Financial Analysts Journal, Jul–Aug, 39–44.
Brinson, G.P. & Fachler, N. (1985). Measuring Non-US Equity Portfolio Performance. Journal of Portfolio Management, Spring, 73–76.
Cariño, D. (1999). Combining Attribution Effects Over Time. Journal of Performance Measurement, Summer, 5–14.