Brinson Portfolio Attribution Lab

Institutional performance attribution engine. Decompose active equity excess return ($\alpha = R_P - R_B$) into Sector Allocation, Security Selection, and Interaction effects across Brinson-Fachler (1985) and Brinson-Hood-Beebower (1986) frameworks.

Active Alpha ($R_P - R_B$)
+3.14%
Port: 17.20% | Bench: 14.06%
Sector Allocation Effect
+1.42%
+142 bps
Stock Selection Effect
+1.55%
+155 bps
Interaction Effect
+0.17%
+17 bps
Market Strategy Presets:
Sector Holdings, Returns & Attribution Breakdown
Editable Values
Sector Port Wt ($w_P$) % Bench Wt ($w_B$) % $\Delta w$ % Port Ret ($R_P$) % Bench Ret ($R_B$) % $\Delta R$ % Allocation % Selection % Interaction % Total Contrib %
Tip: Adjust any sector weight or return above. Total allocation and selection effects dynamically update in real time.
Visual Attribution Analytics
Visual analysis: The waterfall decomposes total alpha into institutional building blocks. The quadrant scatter identifies overweight outperforming sectors (+/+) vs underweight laggards.
Institutional Attribution Sensitivity Matrices
Matrix 1: Tech Active Weight vs Tech Alpha on Total Contribution (bps)

Evaluates Information Technology sector total contribution (bps) across varying active tilts ($\Delta w$) and within-sector stock picking outperformance ($\Delta R$).

$\Delta R_{\text{Tech}}$ Active Weight Tilt ($\Delta w_{\text{Tech}}$)
-4.0% -2.0% 0.0% +2.0% +4.0%
Matrix 2: Benchmark Return ($R_B$) vs Active Weight on BF vs BHB Spread (bps)

Quantifies the exact allocation difference ($A_i^{\text{BF}} - A_i^{\text{BHB}} = -\Delta w_i \times R_B$). Illustrates why BHB produces distorted results during strong market bull runs.

Bench $R_B$ Active Weight Tilt ($\Delta w_i$)
-4.0% -2.0% 0.0% +2.0% +4.0%
Institutional Performance Attribution Theory (Brinson & Beyond)
1. The Brinson-Hood-Beebower (BHB 1986) Formulation

Gary P. Brinson, L. Randolph Hood, and Gilbert L. Beebower published their seminal 1986 Financial Analysts Journal paper "Determinants of Portfolio Performance", demonstrating that asset allocation accounted for ~93.6% of the variation in quarterly returns across 91 large US pension funds. The mathematical framework separates total active return:

$$R_P - R_B = \sum_{i=1}^N (w_{P,i} - w_{B,i}) R_{B,i} + \sum_{i=1}^N w_{B,i} (R_{P,i} - R_{B,i}) + \sum_{i=1}^N (w_{P,i} - w_{B,i}) (R_{P,i} - R_{B,i})$$

Where $A_i^{\text{BHB}} = (w_{P,i} - w_{B,i}) R_{B,i}$ is the Allocation Effect, $S_i = w_{B,i}(R_{P,i} - R_{B,i})$ is Selection, and $I_i = (w_{P,i} - w_{B,i})(R_{P,i} - R_{B,i})$ is Interaction.

2. The Brinson-Fachler (BF 1985) Superiority

Gary Brinson and Nimrod Fachler (1985) recognized that BHB suffered from the benchmark-neutral allocation paradox: a manager overweighting a sector with positive return is credited with positive allocation alpha even if the sector lagged the broader market benchmark!

$$A_i^{\text{BF}} = (w_{P,i} - w_{B,i}) \cdot (R_{B,i} - R_B)$$

Because weights sum to 100% ($\sum (w_{P,i} - w_{B,i}) = 0$), the total allocation effect sums identically whether subtracting $R_B$ or not, but sector-by-sector scores accurately reflect whether the allocation bet aided or harmed the portfolio relative to the market.

3. 3-Factor vs. 2-Factor Model in Board Reporting

The 3-factor model isolates the Interaction term ($I_i$). While mathematically pure, interaction represents an ambiguous joint decision: did the return come from being overweight, or from picking good stocks? In institutional reporting, managers frequently combine selection and interaction into Total Selection:

$$S_i^* = S_i + I_i = w_{P,i} \cdot (R_{P,i} - R_{B,i})$$

This weights the manager's stock picking skill by the actual portfolio weight allocated, providing a clean 2-factor attribution (Allocation + Total Selection = Active Return).

4. Multi-Period Geometric Linking (Carino 1999)

Single-period attribution is additive, but multi-period returns compound geometrically: $1 + R = \prod (1 + R_t)$. Simple sums of monthly attribution terms fail to match cumulative active alpha. David Carino (1999) introduced continuous logarithmic linking:

$$k_t = \frac{\ln(1 + R_{P,t}) - \ln(1 + R_{B,t})}{R_{P,t} - R_{B,t}} \times \frac{R_P - R_B}{\ln(1 + R_P) - \ln(1 + R_B)}$$

Multiplying each period's attribution terms by $k_t$ preserves exact mathematical additivity over any multi-year evaluation horizon.

Interactive Performance Attribution Mastery Quiz

1. Why do institutional consultants and CFA standards recommend Brinson-Fachler (BF) over Brinson-Hood-Beebower (BHB)?

2. Under the Brinson-Fachler model, how can an active manager generate positive Sector Allocation alpha?

3. In the 3-factor Brinson decomposition, how is the pure Security Selection effect defined?

4. What fundamental problem does Carino (1999) logarithmic linking solve in multi-period performance attribution?

Frequently Asked Questions: Brinson Performance Attribution

The critical difference lies in the definition of the Sector Allocation Effect. Brinson-Hood-Beebower (BHB 1986) calculates allocation effect as $(w_P - w_B) \times R_{B,i}$, which credits an overweight sector as long as its return is positive, regardless of whether it outperformed the broad benchmark. Brinson-Fachler (BF 1985) calculates allocation effect as $(w_P - w_B) \times (R_{B,i} - R_B)$, measuring sector return relative to total benchmark return $R_B$. In BF, an overweight position only yields positive allocation alpha if that sector beat the overall market benchmark.

In the standard 3-factor Brinson decomposition: Allocation Effect is $(w_{P,i} - w_{B,i}) \times (R_{B,i} - R_B)$ under BF or $(w_{P,i} - w_{B,i}) \times R_{B,i}$ under BHB; Selection Effect is $w_{B,i} \times (R_{P,i} - R_{B,i})$, measuring within-sector stock picking skill using benchmark weights; and Interaction Effect is $(w_{P,i} - w_{B,i}) \times (R_{P,i} - R_{B,i})$, capturing the joint impact of simultaneous active weight and active selection decisions.

The 3-factor model separates the Interaction term, but interaction is notoriously difficult to explain to institutional trustees and clients because it represents an ambiguous cross-product. In the 2-factor model, interaction is folded directly into the selection effect: Total Selection = $w_{P,i} \times (R_{P,i} - R_{B,i})$. This credits the manager's stock selection skill evaluated over the actual portfolio capital allocated to that sector.

The BHB allocation paradox occurs when a manager overweights a sector that generated a positive absolute return (e.g., +4%) in a market where the overall benchmark gained +15%. Under BHB, $(w_P - w_B) \times R_{B,i}$ produces a positive allocation score even though the decision dragged down overall relative performance. Brinson-Fachler eliminates this paradox by subtracting the total benchmark return: $(w_P - w_B) \times (4\% - 15\%) = \text{negative allocation alpha}$, correctly penalizing the manager.

Single-period attribution effects are strictly additive. However, over multiple periods, portfolio returns compound geometrically ($1 + R_P = \prod (1 + R_{P,t})$), causing simple sums of single-period attribution effects to deviate from cumulative excess return. Carino (1999) logarithmic linking solves this by multiplying each single-period attribution term by a continuous scaling factor $k_t = \frac{\ln(1 + R_{P,t}) - \ln(1 + R_{B,t})}{R_{P,t} - R_{B,t}} \times \frac{R_P - R_B}{\ln(1 + R_P) - \ln(1 + R_B)}$, ensuring exact multi-period additivity.
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