Portfolio Performance Attribution

Decomposing portfolio returns into allocation, selection, and factor contributions.

When to Activate

• User needs to explain why a portfolio outperformed or underperformed its benchmark
• Implementing Brinson-Fachler attribution for equity portfolios
• Building factor-based attribution using regression or risk models
• Performing fixed income attribution (duration, credit, curve)
• Linking single-period attribution across multiple periods

Core Concepts

What Is Attribution?

Attribution decomposes the active return (portfolio return minus benchmark return) into components that explain where and how value was added or lost. Two major frameworks:

1. Holdings-based (Brinson): uses portfolio and benchmark weights and returns by sector/region
2. Returns-based (factor regression): regresses portfolio returns on factor returns to identify exposures
3. Risk-model-based: uses a multi-factor risk model to decompose returns into factor and specific components

Brinson-Fachler Attribution

Decomposes active return into three effects for each sector s:

Allocation Effect: did the manager overweight sectors that outperformed?

AE_s = (w_p,s - w_b,s) * (R_b,s - R_b)

Selection Effect: did the manager pick better stocks within each sector?

SE_s = w_b,s * (R_p,s - R_b,s)

Interaction Effect: combined effect of overweighting sectors where stock picks were also good

IE_s = (w_p,s - w_b,s) * (R_p,s - R_b,s)

Total active return: R_p - R_b = ΣAE_s + ΣSE_s + ΣIE_s

Where:

• w_p,s, w_b,s: portfolio and benchmark weight in sector s
• R_p,s, R_b,s: portfolio and benchmark return in sector s
• R_b: total benchmark return

Factor-Based Attribution

Uses a multi-factor risk model (e.g., Barra, Axioma, Northfield):

R_p - R_f = Σ (β_p,k - β_b,k) * f_k + α_specific

Where β_p,k is the portfolio's exposure to factor k, and f_k is the factor return. Decomposes into:

• Factor timing: return from active factor bets
• Specific return: stock-level alpha (idiosyncratic return)

Methodology

Step 1: Brinson Attribution Implementation

import pandas as pd
import numpy as np

def brinson_fachler(portfolio_weights, benchmark_weights,
                     portfolio_returns, benchmark_returns):
    """
    Single-period Brinson-Fachler attribution by sector.

    All inputs are Series indexed by sector.
    """
    total_bench_return = (benchmark_weights * benchmark_returns).sum()

    allocation = (portfolio_weights - benchmark_weights) * (benchmark_returns - total_bench_return)
    selection = benchmark_weights * (portfolio_returns - benchmark_returns)
    interaction = (portfolio_weights - benchmark_weights) * (portfolio_returns - benchmark_returns)

    result = pd.DataFrame({
        'port_weight': portfolio_weights,
        'bench_weight': benchmark_weights,
        'port_return': portfolio_returns,
        'bench_return': benchmark_returns,
        'allocation': allocation,
        'selection': selection,
        'interaction': interaction,
        'total_active': allocation + selection + interaction
    })

    # Verify: sum of total_active should equal portfolio return - benchmark return
    port_return = (portfolio_weights * portfolio_returns).sum()
    bench_return = (benchmark_weights * benchmark_returns).sum()
    assert abs(result['total_active'].sum() - (port_return - bench_return)) < 1e-10

    return result

Step 2: Multi-Period Linking

Single-period attribution does not compound correctly across periods. Linking methods:

Carino Method (logarithmic smoothing):

def carino_linking(single_period_attributions, portfolio_returns, benchmark_returns):
    """
    Link single-period Brinson attribution across T periods using Carino's method.
    """
    T = len(portfolio_returns)

    # Cumulative returns
    cum_port = np.prod(1 + portfolio_returns) - 1
    cum_bench = np.prod(1 + benchmark_returns) - 1

    # Carino scaling factors
    k_t = []
    for t in range(T):
        rp = portfolio_returns[t]
        rb = benchmark_returns[t]
        if abs(rp - rb) < 1e-12:
            k_t.append(1.0 / (1 + cum_port))
        else:
            ln_ratio_t = np.log(1 + rp) - np.log(1 + rb)
            ln_ratio_total = np.log(1 + cum_port) - np.log(1 + cum_bench)
            k_t.append((ln_ratio_t / (rp - rb)) / (ln_ratio_total / (cum_port - cum_bench)))

    # Scale each period's attribution by k_t
    linked = {}
    for effect in ['allocation', 'selection', 'interaction']:
        linked[effect] = sum(
            k_t[t] * single_period_attributions[t][effect]
            for t in range(T)
        )

    return linked

GRAP Method (geometric): another popular linking approach used by major custodians.

Step 3: Factor-Based Attribution

def factor_attribution(portfolio_exposures, benchmark_exposures,
                       factor_returns, specific_returns):
    """
    Attribution using a multi-factor risk model.

    portfolio_exposures: DataFrame (dates x factors) of portfolio factor exposures
    benchmark_exposures: DataFrame (dates x factors) of benchmark factor exposures
    factor_returns: DataFrame (dates x factors) of realized factor returns
    specific_returns: Series of portfolio-level specific returns
    """
    active_exposures = portfolio_exposures - benchmark_exposures

    # Factor contribution = active exposure * factor return (each period)
    factor_contrib = active_exposures * factor_returns

    # Group by factor category
    style_factors = ['Value', 'Momentum', 'Quality', 'Size', 'Volatility']
    industry_factors = [f for f in factor_returns.columns if f.startswith('IND_')]

    attribution = {
        'style': factor_contrib[style_factors].sum(axis=1).sum(),
        'industry': factor_contrib[industry_factors].sum(axis=1).sum(),
        'specific': specific_returns.sum(),
        'total_active': factor_contrib.sum(axis=1).sum() + specific_returns.sum()
    }

    return attribution

Step 4: Fixed Income Attribution

Fixed income attribution requires different decomposition:

def fi_attribution(portfolio, benchmark):
    """
    Fixed income attribution components:
    1. Income return (coupon/yield carry)
    2. Treasury curve effect (parallel shift + twist + curvature)
    3. Spread change effect (credit spread widening/tightening)
    4. Selection effect (individual security alpha)
    """
    components = {}

    # Income effect: difference in portfolio vs benchmark yield
    components['income'] = portfolio['yield'] - benchmark['yield']

    # Duration effect: sensitivity to parallel rate move
    rate_change = benchmark['rate_change']
    components['duration'] = -(portfolio['duration'] - benchmark['duration']) * rate_change

    # Curve positioning: key rate duration attribution
    for tenor in ['2Y', '5Y', '10Y', '30Y']:
        krd_active = portfolio[f'krd_{tenor}'] - benchmark[f'krd_{tenor}']
        components[f'curve_{tenor}'] = -krd_active * benchmark[f'rate_change_{tenor}']

    # Spread effect
    spread_change = benchmark['spread_change']
    components['spread_duration'] = -(portfolio['spread_duration'] - benchmark['spread_duration']) * spread_change

    # Credit selection (residual)
    components['selection'] = portfolio['total_return'] - benchmark['total_return'] - sum(components.values())

    return components

Step 5: Attribution Reporting

def attribution_report(attribution_df, title="Performance Attribution"):
    """
    Standard attribution report format.
    """
    report = f"""
    {title}
    {'='*60}

    Portfolio Return:     {port_ret:>8.2%}
    Benchmark Return:     {bench_ret:>8.2%}
    Active Return:        {active_ret:>8.2%}

    --- Decomposition by Sector ---
    {'Sector':<20} {'Alloc':>8} {'Select':>8} {'Inter':>8} {'Total':>8}
    {'-'*60}
    """
    for sector, row in attribution_df.iterrows():
        report += f"    {sector:<20} {row['allocation']:>8.2%} {row['selection']:>8.2%} "
        report += f"{row['interaction']:>8.2%} {row['total_active']:>8.2%}\n"

    report += f"    {'-'*60}\n"
    report += f"    {'TOTAL':<20} {attribution_df['allocation'].sum():>8.2%} "
    report += f"{attribution_df['selection'].sum():>8.2%} "
    report += f"{attribution_df['interaction'].sum():>8.2%} "
    report += f"{attribution_df['total_active'].sum():>8.2%}\n"

    return report

Examples

Equity Attribution by Sector

# Sector weights and returns
sectors = ['Technology', 'Healthcare', 'Financials', 'Energy', 'Consumer']
port_w = pd.Series([0.35, 0.20, 0.15, 0.10, 0.20], index=sectors)
bench_w = pd.Series([0.28, 0.15, 0.18, 0.12, 0.27], index=sectors)
port_r = pd.Series([0.08, 0.05, 0.03, -0.02, 0.04], index=sectors)
bench_r = pd.Series([0.06, 0.04, 0.04, -0.01, 0.03], index=sectors)

result = brinson_fachler(port_w, bench_w, port_r, bench_r)
# Allocation to Tech (overweight in strong sector) = positive
# Selection in Tech (8% vs 6%) = positive

Interpreting Results

Key insights to surface:

• Was alpha from sector allocation (top-down) or stock selection (bottom-up)?
• Which sectors had the largest attribution? Is it consistent over time or a one-off?
• Is the interaction effect large? If so, the manager may be making correlated allocation and selection bets
• Does attribution match the stated investment process? (A bottom-up manager should show selection, not allocation)

Quality Gate

• Attribution sums exactly to total active return (no residual beyond rounding)
• Multi-period attribution uses a proper linking method (Carino, GRAP, or Menchero)
• Sector/region classification is consistent between portfolio and benchmark
• Transaction effects excluded from holdings-based attribution (or accounted for separately)
• Currency effects separated from local returns for international portfolios
• Factor-based attribution uses a risk model consistent with the portfolio's universe
• Interaction effect interpreted correctly — not lumped into allocation or selection arbitrarily
• Reported at multiple levels: total, sector, country, and individual security
• For fixed income: duration, curve, and spread effects isolated before reporting selection
• Attribution is stable and explainable — large unexplained residuals indicate model problems