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
Do not retype these formulas into throwaway Python. They are implemented and
tested in src/quantlib/attribution.py; import them.
Single-Period Brinson-Fachler Model
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 active return = Σ(Allocation_i) + Σ(Selection_i) + Σ(Interaction_i)
**The decomposition itself has no residual term.** The three effects sum to
`R_p - R_b` identically, for any sector returns whatsoever, provided the
portfolio and benchmark weights carry the same total. `brinson_fachler` enforces
the weight-sum precondition and raises rather than returning a decomposition
that does not tie out.
A residual is therefore never a property of the algebra — but it is a real and
expected property of a *reported* attribution, because the inputs are a
snapshot. Intra-period trading, cash drag, corporate actions and FX translation
all move the actual portfolio return away from the one these weights and sector
returns imply. So:
- residual inside the decomposition, given the inputs → **impossible**; if you
see one, the arithmetic or the weight convention is wrong;
- residual between the decomposition and the reported fund return → **normal**;
quantify it and attribute it to its source rather than absorbing it silently
into selection. This is what the `/attrib` reconciliation gate asks for.
```python
from src.quantlib.attribution import brinson_fachler
result = brinson_fachler(
portfolio_weights={"Tech": 0.40, "Financials": 0.10, "Energy": 0.30, "Health": 0.20},
benchmark_weights={"Tech": 0.25, "Financials": 0.30, "Energy": 0.25, "Health": 0.20},
portfolio_returns={"Tech": 0.12, "Financials": 0.04, "Energy": -0.02, "Health": 0.07},
benchmark_returns={"Tech": 0.10, "Financials": 0.05, "Energy": -0.01, "Health": 0.06},
)
result.portfolio_return # 0.0600
result.benchmark_return # 0.0495
result.active_return # 0.0105
result.allocation # 0.0045
result.selection # 0.0015
result.interaction # 0.0045
# 0.0045 + 0.0015 + 0.0045 == 0.0105 exactly (residual ~3e-18, machine epsilon)
for effect in result.sectors:
print(effect.sector, effect.allocation, effect.selection, effect.interaction, effect.total)
A sector return may be omitted only where the matching weight is zero. A benchmark sector you did not own therefore shows zero selection and zero interaction, and the whole effect lands in allocation — you cannot demonstrate stock-picking skill in something you never held.
Example Brinson Attribution
Rendered from the call above, so every figure below is reproducible:
### Brinson Sector Attribution
| Sector | Portfolio Weight | Benchmark Weight | Portfolio Return | Benchmark Return | Allocation | Selection | Interaction |
|------|---------|---------|---------|---------|---------|---------|---------|
| Tech | 40% | 25% | 12% | 10% | +0.7575% | +0.50% | +0.30% |
| Financials | 10% | 30% | 4% | 5% | -0.0100% | -0.30% | +0.20% |
| Energy | 30% | 25% | -2% | -1% | -0.2975% | -0.25% | -0.05% |
| Health | 20% | 20% | 7% | 6% | +0.0000% | +0.20% | +0.00% |
| **Total** | 100% | 100% | 6.00% | 4.95% | **+0.45%** | **+0.15%** | **+0.45%** |
Active return 1.05% = allocation 0.45% + selection 0.15% + interaction 0.45%. No residual.
Multi-Period Attribution (Linked Brinson)
Single-period effects add, but returns compound, so simply summing each period's
effects does not reproduce the compounded active return. Take the four-sector
period above and two more like it (the exact three are the _three_periods
fixture in tests/quantlib/test_attribution.py, so you can run them): summing the
three active returns gives 2.8500%, while the compounded active return is 3.0318%
— an 18.2bp error that grows with the horizon and the return level.
Use Carino logarithmic linking, implemented as carino_link. It is
residual-free, and its per-period scaling factor depends only on that period's
total portfolio and benchmark return — never on the effects being linked — so
linking is deterministic and cannot be steered by how sectors were bucketed.
(Menchero linking is also residual-free but distributes a correction term derived
from the effects themselves; Carino needs less machinery for the same guarantee.)
k = (ln(1 + R_P) - ln(1 + R_B)) / (R_P - R_B) # over the whole horizon
k_t = (ln(1 + R_p,t) - ln(1 + R_b,t)) / (R_p,t - R_b,t) # for period t
linked effect = Σ_t (k_t / k) × effect_{i,t}
from src.quantlib.attribution import brinson_fachler, carino_link
periods = [brinson_fachler(**month) for month in monthly_inputs]
linked = carino_link(periods)
linked.active_return # compounded, not summed
linked.allocation, linked.selection, linked.interaction
linked.scaling_factors # one k_t / k per period, exposed so a report can be audited
for sector in linked.sectors:
print(sector.sector, sector.total)
# allocation + selection + interaction == linked.active_return exactly
Arithmetic linking is acceptable only when you explicitly report the residual.
Since carino_link costs one function call and leaves none, prefer it.
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
| Metric | Calculation | Meaning |
|---|---|---|
| Bull capture ratio | portfolio return in bull markets / benchmark return | >100% = outperforming |
| Bear capture ratio | portfolio return in bear markets / benchmark return | <100% = better downside defense |
| Timing hit rate | proportion of months where market direction was called correctly | >55% = shows skill |
| Correlation between position changes and market | corr(position_change, future_return) |
>0 = timing is correct |
Benchmark Comparison Framework
Benchmark Selection
| Strategy Type | Recommended Benchmark | China A-share Code |
|---|---|---|
| China A-share large cap | CSI 300 | 000300.SH |
| China A-share small cap | CSI 500 / CSI 1000 | 000905.SH |
| China A-share broad market | CSI All Share | 000985.SH |
| Hong Kong equities | Hang Seng Index | HSI |
| US equities | S&P 500 | SPX |
| Crypto | BTC | BTC-USDT |
| Multi-asset | 60/40 portfolio | self-constructed |
Risk-Adjusted Performance Metrics
| Metric | Formula | Excellent | Good | Average |
|---|---|---|---|---|
| Sharpe | (R_p - R_f) / σ_p |
>1.5 | 1.0-1.5 | 0.5-1.0 |
| Sortino | (R_p - R_f) / σ_down |
>2.0 | 1.5-2.0 | 1.0-1.5 |
| Calmar | R_p / MaxDD |
>1.0 | 0.5-1.0 | 0.2-0.5 |
| Information Ratio | (R_p - R_b) / TE |
>1.0 | 0.5-1.0 | 0.2-0.5 |
| Treynor | (R_p - R_f) / β |
used comparatively |
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
- Attribution ≠ prediction: attribution explains the past; it does not guarantee persistence in the future
- Benchmark selection affects attribution: switch the benchmark and alpha may disappear, so benchmark choice must be appropriate
- Data frequency: daily attribution is noisy, monthly attribution is more stable but has fewer samples; recommended workflow is daily computation with monthly reporting
- Survivorship bias: delisted stocks may be excluded in backtests, creating false alpha
- Multiple-testing problem: if you test 100 strategies, about 5 may appear significant by chance (
p=0.05); use multiple-comparison correction - Factor data requirement: factor attribution requires factor return data, which can be obtained from
tushareor self-constructed - Attribution in backtest reports:
metrics.csvalready provides basic metrics after a backtest; this skill adds deeper attribution analysis - Brinson is implemented, not improvised:
src/quantlib/attribution.pyholds the tested single-period and Carino-linked decomposition. Import it. Hand-written attribution code that reports a single-period residual is a bug in that code, not a property of the model