Performance Attribution

Performance Attribution

Core Concepts

Brinson-Fachler Attribution (Single Period)

The classic equity attribution model decomposes active return (portfolio return minus benchmark return) into three effects:

• Allocation effect: Value added by over/underweighting sectors relative to the benchmark
  • A_i = (w_p,i - w_b,i) × (R_b,i - R_b)
  • Rewards overweighting sectors that outperform the total benchmark
• Selection effect: Value added by picking better securities within each sector
  • S_i = w_b,i × (R_p,i - R_b,i)
  • Rewards outperforming the sector benchmark regardless of weight
• Interaction effect: Combined effect of both overweighting and outperforming (or vice versa)
  • I_i = (w_p,i - w_b,i) × (R_p,i - R_b,i)
  • Captures the joint benefit of overweighting a sector AND selecting better securities in it
• Total active return: R_p - R_b = Σ A_i + Σ S_i + Σ I_i

Where: w_p,i = portfolio weight in sector i, w_b,i = benchmark weight in sector i, R_p,i = portfolio return in sector i, R_b,i = benchmark return in sector i, R_b = total benchmark return.

Multi-Period Attribution

Single-period attribution does not compound across periods. Geometric linking methods are required:

• Carino method: Applies a smoothing factor to make arithmetic effects compound to the correct geometric total
• Menchero method: Uses a logarithmic approach for smoother decomposition
• GRAP (Geometric Return Attribution Program): Converts arithmetic effects to geometric equivalents
• Key principle: the sum of linked attribution effects must equal the total geometric active return over the full period

Factor-Based Attribution

Decomposes returns into exposures to systematic risk factors:

• Model: R_p = Σ β_k × F_k + α
  • β_k = portfolio's exposure (loading) to factor k
  • F_k = return of factor k during the period
  • α = residual return unexplained by factors (true alpha)
• Common factors: Market (MKT), Size (SMB), Value (HML), Momentum (UMD), Quality (QMJ), Low Volatility (BAB)
• Factor contribution: β_k × F_k for each factor
• Active factor contribution: (β_p,k - β_b,k) × F_k
• The model chosen (Fama-French 3, Carhart 4, Fama-French 5, Barra, Axioma) affects results

Fixed-Income Attribution

Decomposes bond portfolio returns into component sources:

• Yield return (income): Coupon income accrued during the period (yield × time)
• Roll return: Price appreciation as bonds "roll down" the yield curve toward maturity
• Curve change return: Impact of parallel and non-parallel yield curve shifts
  • Duration effect: -D × Δy (parallel shift)
  • Curve reshaping: key rate duration contributions
• Spread change return: Impact of credit spread changes: -spread_duration × Δspread
• Credit/default return: Losses from defaults or credit events
• Residual: Unexplained return (convexity effects, model error)

Currency Attribution

For international portfolios, returns decompose into:

• Local return: Return of the asset in its local currency
• Currency return: Gain/loss from exchange rate movements
• Cross-product: Interaction between local return and currency return
• Total return (base currency): R_base ≈ R_local + R_currency + R_local × R_currency
• Hedged return: Local return + hedge cost (forward premium/discount)
• Attribution of active currency decisions: actual currency exposure vs benchmark currency exposure

Holdings-Based vs Returns-Based Attribution

• Holdings-based: Uses actual portfolio positions; more accurate but requires detailed holdings data at each evaluation point
• Returns-based (style analysis): Regresses portfolio returns against a set of style indices (e.g., Sharpe style analysis); less precise but requires only return series
• Transaction-based: Most accurate; accounts for intra-period trading by using actual transaction records

Key Formulas

Worked Examples

Example 1: Brinson-Fachler equity attribution

Given: Two-sector portfolio (Tech and Healthcare). Portfolio: 35% Tech (returned 15%), 65% Healthcare (returned 8%). Benchmark: 25% Tech (returned 12%), 75% Healthcare (returned 6%). Total benchmark return: 0.25×12% + 0.75×6% = 7.5%. Calculate: Allocation, selection, and interaction effects for each sector, and total active return. Solution:

1. Total portfolio return: 0.35×15% + 0.65×8% = 5.25% + 5.20% = 10.45%.
2. Total active return: 10.45% - 7.50% = 2.95%.
3. Tech allocation effect: (0.35 - 0.25) × (12% - 7.5%) = 0.10 × 4.5% = +0.45% (overweight a sector that beat the benchmark).
4. Tech selection effect: 0.25 × (15% - 12%) = 0.25 × 3% = +0.75% (stock picks in Tech beat Tech benchmark).
5. Tech interaction effect: (0.35 - 0.25) × (15% - 12%) = 0.10 × 3% = +0.30% (overweight AND outperformed).
6. Healthcare allocation effect: (0.65 - 0.75) × (6% - 7.5%) = -0.10 × -1.5% = +0.15% (underweight a sector that lagged the benchmark).
7. Healthcare selection effect: 0.75 × (8% - 6%) = 0.75 × 2% = +1.50% (stock picks in Healthcare beat Healthcare benchmark).
8. Healthcare interaction effect: (0.65 - 0.75) × (8% - 6%) = -0.10 × 2% = -0.20% (underweight but outperformed — interaction is negative).
9. Totals: Allocation = 0.45 + 0.15 = 0.60%. Selection = 0.75 + 1.50 = 2.25%. Interaction = 0.30 + (-0.20) = 0.10%. Sum = 0.60 + 2.25 + 0.10 = 2.95% ✓.