performance-attribution

Performance Attribution Analysis

Overview

Decompose portfolio excess returns into explainable sources: sector allocation, stock selection, factor exposure, timing contribution, and more. This helps explain why a strategy made or lost money, rather than only how much it made or lost.

Brinson Attribution Model

Single-Period Brinson-Fachler Model

Total excess return = portfolio return - benchmark return

Decomposed into three parts:
1. Allocation effect: sector-weight deviation × sector benchmark return deviation
2. Selection effect: stock selection within a sector × sector benchmark weight
3. Interaction effect: weight deviation × stock-selection deviation

Mathematical formulas:

Let w_p,i = portfolio weight of sector i
    w_b,i = benchmark weight of sector i
    r_p,i = portfolio return of sector i
    r_b,i = benchmark return of sector i
    R_b   = total benchmark return

Allocation_i = (w_p,i - w_b,i) × (r_b,i - R_b)
Selection_i  = w_b,i × (r_p,i - r_b,i)
Interaction_i = (w_p,i - w_b,i) × (r_p,i - r_b,i)

Total excess = Σ(Allocation_i) + Σ(Selection_i) + Σ(Interaction_i)

Example Brinson Attribution

### Brinson Sector Attribution

| Sector | Portfolio Weight | Benchmark Weight | Portfolio Return | Benchmark Return | Allocation | Selection | Interaction |
|------|---------|---------|---------|---------|---------|---------|---------|
| Food & Beverage | 20% | 10% | 15% | 8% | +0.3% | +0.7% | +0.7% |
| Electronics | 15% | 12% | 5% | 10% | -0.1% | -0.6% | +0.2% |
| Banks | 5% | 20% | 3% | 2% | +0.3% | +0.2% | -0.2% |
| Others | 60% | 58% | 8% | 7% | +0.0% | +0.6% | +0.0% |
| **Total** | 100% | 100% | 9.5% | 6.2% | **+0.5%** | **+0.9%** | **+0.7%** |

Excess return 3.3% = allocation effect 0.5% + selection effect 0.9% + interaction effect 0.7% + residual 0.2%

Multi-Period Attribution (Linked Brinson)

Directly summing single-period attribution creates residuals (compounding effect). Common approaches:

Method 1: arithmetic linking (simple sum of each period's attribution)
  - Advantage: simple
  - Disadvantage: residual remains

Method 2: Carino logarithmic linking
  - Advantage: no residual
  - Disadvantage: more complex

Practical recommendation: arithmetic linking is enough for monthly attribution; residuals are usually <0.1%