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%

Factor Attribution

Alpha-Beta Decomposition

R_p = α + β × R_m + ε

α (alpha): excess return, manager skill
β (beta): market exposure, systematic risk
ε (epsilon): residual, idiosyncratic risk

Regression method: OLS regression, with at least 60 data points

Multi-Factor Attribution (Fama-French Extension)

R_p - R_f = α + β_mkt × (R_m - R_f) + β_smb × SMB + β_hml × HML + β_mom × MOM + ε

| Factor | Meaning | China A-share Proxy |
|------|------|--------|
| MKT | Market | CSI 300 return |
| SMB | Small-cap premium | CSI 500 - CSI 300 |
| HML | Value premium | high-PB group - low-PB group |
| MOM | Momentum | top past-12M winners - bottom group |

Factor Exposure Analysis Template

### Factor Exposure Analysis

| Factor | Beta | t-stat | Significance | Interpretation |
|------|------|---------|--------|------|
| Market (MKT) | 0.85 | 12.3 | *** | Below 1, defensive profile |
| Small-cap (SMB) | 0.25 | 3.2 | ** | Small-cap tilt |
| Value (HML) | -0.15 | -1.8 | * | Growth tilt |
| Momentum (MOM) | 0.30 | 4.1 | *** | Significant momentum exposure |
| **Alpha** | **0.8% / month** | **2.5** | ** | **Significant alpha** |

R² = 0.72 → factors explain 72% of return variation
Alpha = 0.8% / month = 10% / year, significant

Market-Timing Evaluation

Treynor-Mazuy Model

R_p - R_f = α + β × (R_m - R_f) + γ × (R_m - R_f)² + ε

γ > 0 and significant → timing ability exists (adds risk in bull markets, cuts risk in bear markets)
γ ≤ 0 → no timing ability

Henriksson-Merton Model

R_p - R_f = α + β × (R_m - R_f) + γ × max(R_m - R_f, 0) + ε

γ > 0 → portfolio beta is higher in bull markets (successful timing)

Practical Timing Metrics

Benchmark Comparison Framework

Benchmark Selection

Risk-Adjusted Performance Metrics

Rolling Analysis

Use rolling windows (such as 12 months) to analyze:
- Rolling Sharpe: strategy stability
- Rolling alpha: whether alpha persists
- Rolling beta: whether market exposure is stable
- Rolling information ratio: persistence of benchmark outperformance

Suggested windows: 252 days for daily data, 12-36 months for monthly data

Analysis Framework

Step 1: Aggregate Analysis

1. Cumulative return vs benchmark
2. Excess-return decomposition (annual / monthly)
3. Summary risk metrics (volatility / max drawdown / Sharpe)

Step 2: Attribution Decomposition

1. Brinson attribution (if sector information is available)
2. Factor attribution (alpha / beta / factor exposure)
3. Timing attribution (TM / HM models)

Step 3: Style Analysis

1. Large cap vs small cap exposure
2. Growth vs value exposure
3. Style drift detection (rolling style analysis)

Step 4: Conclusions and Recommendations

1. Main sources of excess return
2. Whether risk exposure is reasonable
3. Suggested improvement directions

Output Format

## Performance Attribution Report

### Performance Overview
| Metric | Strategy | Benchmark | Excess |
|------|------|------|------|
| Cumulative return | +85.2% | +32.1% | +53.1% |
| Annualized return | 12.5% | 5.8% | +6.7% |
| Annualized volatility | 18.2% | 20.5% | - |
| Sharpe | 0.69 | 0.28 | - |
| Information Ratio | 0.82 | - | - |

### Attribution Breakdown
| Source | Contribution (annualized) | Share |
|------|-----------|------|
| Sector allocation | +2.1% | 31% |
| Stock selection | +3.8% | 57% |
| Timing | +0.8% | 12% |

### Factor Exposure
[factor exposure table]

### Conclusion
Excess return mainly comes from stock selection (57% contribution), followed by sector allocation.
Alpha is significant (`t=2.5`), indicating real stock-picking ability.
Watch the risk of excessive small-cap exposure (`SMB beta=0.25`).

Notes

1. Attribution ≠ prediction: attribution explains the past; it does not guarantee persistence in the future
2. Benchmark selection affects attribution: switch the benchmark and alpha may disappear, so benchmark choice must be appropriate
3. Data frequency: daily attribution is noisy, monthly attribution is more stable but has fewer samples; recommended workflow is daily computation with monthly reporting
4. Survivorship bias: delisted stocks may be excluded in backtests, creating false alpha
5. Multiple-testing problem: if you test 100 strategies, about 5 may appear significant by chance (p=0.05); use multiple-comparison correction
6. Factor data requirement: factor attribution requires factor return data, which can be obtained from tushare or self-constructed
7. Attribution in backtest reports: metrics.csv already provides basic metrics after a backtest; this skill adds deeper attribution analysis