Strategy Performance Attribution Vs Market Beta

When to Use

Use this skill when a strategy has outperformed and you need to know why before crediting it to skill or allocating more capital to it. It regresses realized strategy excess returns on the market excess return and, optionally, the Fama-French size (SMB) and value (HML) spreads, and reports the intercept — Jensen's alpha — with the standard error, $t$-statistic and $p$-value needed to judge whether it is distinguishable from zero. A levered long-only book in a bull market and a genuine alpha source produce the same headline return; only the decomposition separates them.

It also returns an exact additive attribution: risk-free contribution + alpha + $\sum_i \beta_i \bar{f_i}$ reconstructs the realized annual return, with unexplained_residual_pct as the closure check.

When NOT to Use

• As a forecast, or as evidence a strategy will keep working. Alpha here is an in-sample intercept over the window you supplied. For decay diagnosis use strategy-performance-decay-detection-vs-market-wide-decay.
• When the strategy's exposures are not stable over the window. One regression estimates one constant beta. A strategy that deliberately times its beta, or that changed mandate mid-window, will show an average that describes neither regime. Split the window and run it twice.
• To screen many strategies and report the significant ones. A 5% test applied to 20 candidates yields roughly one "significant" alpha by chance. Correct for it — factor-research-multiple-testing-correction.
• On non-equity strategies with equity factors. SMB and HML are US equity spreads. Regressing an FX carry or commodity strategy on them produces loadings that are noise. Choose factors that plausibly span the strategy's returns, or run CAPM against the right benchmark — benchmark-selection-for-strategy-evaluation.
• To rank strategies by their contribution to portfolio risk. That is a different decomposition — risk-adjusted-performance-attribution-per-strategy.
• As a risk control. This is a reporting engine. It does not size, throttle, or halt anything.

Prerequisites

• Strategy simple periodic returns (strategy_returns) as decimal fractions — a 0.53% day is 0.0053, not 0.53 — net of fees, financing and borrow.
• Benchmark returns (market_returns) on the same index and in the same units.
• Optional Fama-French factors (smb_returns, hml_returns), same index and units.
• Annual risk-free rate in percent (risk_free_rate_annual_pct, default 2.0), de-annualized geometrically to a periodic simple rate.
• A unique index, and a chronologically sorted one if you intend to use standard_errors="hac".
• At least $k+2$ aligned observations ($k$ = regressors including the intercept); the engine raises below that, and flags any sample shorter than one year via insufficient_history_warning.
• numpy, pandas, scipy (all already repo dependencies).

Workflow

1. Fix the unit and excess-return conventions before anything else:

   • Decision point — is market_returns a total return or already an excess return? Ken French's data library ships a column literally named Mkt-RF, which is already excess of the risk-free rate. Passing it with the default market_returns_are_excess=False subtracts the risk-free rate a second time and biases alpha by $\beta \times R_f$. Set the flag.
   • Decision point — units. French's files are in percent (a daily row reads 20260630, 0.73, 0.10, -0.62, 0.01); this engine expects decimals. Divide by 100. A percent/decimal mix scales every beta by 100 and is not detectable from the output alone.
   • SMB and HML are zero-investment long-short spreads and are used raw — never subtract the risk-free rate from them.

2. Align and validate:

   • Series are joined on the index and rows with any missing value are dropped. observations and dropped_observations record what actually entered the regression; check them, because a factor with a short history silently shortens the whole sample.
   • Duplicate index labels, infinities, non-numeric values, collinear factors, and samples below $k+2$ raise rather than producing a confident-looking number.

3. Fit the regression: $$R_{s,t} - R_{f,t} = \alpha + \beta_M (R_{M,t} - R_{f,t}) + \beta_S \mathrm{SMB}_t + \beta_H \mathrm{HML}_t + \varepsilon_t$$

   • Decision point — which standard errors? The default "ols" assumes residuals are iid and homoskedastic. If the strategy holds illiquid or stale-marked assets, uses overlapping holding periods, or is a momentum/trend book, its residuals are serially correlated, OLS standard errors are biased downward, and the alpha $t$-statistic is inflated. Use standard_errors="hac" (Newey-West, Bartlett kernel, default lag $\lfloor 4(n/100)^{2/9}\rfloor$) and report whichever you used. The coefficients are identical either way; only the inference changes.

4. Judge significance against the right threshold:

   • The engine compares $|t|$ against the exact two-sided Student-$t$ critical value with $n-k$ degrees of freedom, reported as t_critical_value, and returns alpha_p_value.
   • Decision point — 1.96 is the large-sample limit, not the threshold. At 252 daily observations the exact value is 1.9695, so the distinction rarely changes a verdict; at 60 monthly observations it is 2.0017 and 1.96 overstates significance. Read t_critical_value, not folklore.
   • Decision point — alpha_t_statistic of None does not mean zero. It means the residual variance is zero (a perfect in-sample fit, i.e. synthetic or degenerate data), so no sampling distribution exists. Read undefined_metrics for the reason.

5. Decompose and reconcile:

   • Each FactorAttributionBreakdown reports the loading, the annualized mean of the regressor as it entered the regression, and their product.
   • Verify the identity before quoting any number: risk_free_contribution_pct + annualized_jensens_alpha_pct + Σ return_contribution_pct must equal total_realized_annual_return_pct, and unexplained_residual_pct must be ~0. A non-zero residual is a bug, not a finding.

6. Act on the split, not the headline: an insignificant alpha with $\beta_M = 1.4$ is a levered index fund and should be priced as beta. A significant alpha with $\beta_S = 0.6$ is a small-cap tilt plus residual skill; the tilt is cheaply replicable and only the residual justifies an active fee.

Full procedure: see references/workflows.md. Standards reference: see references/standards.md. Printable pre-flight checklist: see assets/checklist.md.

Common Pitfalls

• Double-subtracting the risk-free rate from the market leg: French's Mkt-RF is already an excess return. Passing it as a total return deducts $R_f$ twice, shifting alpha by $\beta \times R_f$ — about $-2%$ a year at $\beta=1$ and a 2% rate, enough to turn a marginal alpha negative.
• Mixing percent and decimal units: French's factor files are in percent; strategy returns are usually decimals. Regressing 0.0053 on 0.53 returns a beta 100× too small and an alpha that absorbs the difference. Nothing in the output flags it — only a sanity check on the magnitudes will.
• Subtracting the risk-free rate from SMB and HML: they are zero-investment long-short spreads, already self-financing. Subtracting $R_f$ from them biases their loadings and corrupts the intercept.
• Computing factor statistics over a different sample than the regression: if SMB has a shorter history than the strategy, alignment shortens the whole regression. A factor mean taken over the full unaligned series and multiplied by a beta estimated on the short one is not an attribution of anything. Check dropped_observations and reconcile against unexplained_residual_pct.
• Reading t = 1.97 as proof of skill: it is a 5% test on one strategy. Run it on twenty and you expect one such result from noise even if every alpha is truly zero.
• Trusting OLS $t$-statistics on serially correlated residuals: a trend-following or illiquid book violates the independence Jensen's derivation assumes, and OLS standard errors understate the true uncertainty. The alpha does not change; your confidence in it should.
• Reading a high $R^2$ as "no alpha": $R^2$ measures the share of variance explained by the factors and says nothing about the mean the intercept captures. A strategy can be 95% explained by the market and still earn a large, significant alpha. The two questions are independent.
• Treating an insignificant alpha as an estimate of zero: it is an estimate that is imprecise. On a 60-day sample, alpha standard errors are wide enough that a genuinely excellent strategy will fail the test. insufficient_history_warning marks that case.
• Interpreting alpha_percentage_of_total_return on a losing strategy: it is None for a non-positive total return, because dividing a positive alpha by a negative total produces a negative share that reads as the opposite of the truth.
• Assuming one beta describes the whole window: a strategy that halves its exposure in a drawdown shows an average beta that matches neither half, and the residual variance it creates depresses the alpha $t$-statistic.

Verification

• Closed-form check (no library needed). With risk_free_rate_annual_pct=0.0, market_returns=[0.01, 0.02, 0.03, 0.04] and strategy_returns=[0.02, 0.03, 0.06, 0.07]: $S_{xy}=0.0009$, $S_{xx}=0.0005$, so $\beta=1.8$ and $\alpha=0.045-1.8(0.025)=0$ exactly. Residuals are $[0.002,-0.006,0.006,-0.002]$, $SS_{res}=8\times10^{-5}$, $SS_{tot}=0.0017$, so $R^2=0.9529$ and adjusted $R^2=0.9294$. With $\sigma^2=4\times10^{-5}$: $\mathrm{SE}(\alpha)=\sqrt{6\times10^{-5}}=0.0077460$ and $\mathrm{SE}(\beta)=\sqrt{0.08}=0.2828427$, giving $t_\beta=6.3640$. The 5% threshold at 2 df is 4.302653, not 1.96 — confirm t_critical_value reports it.
• Cross-validation. For a single-factor fit, market_beta, annualized_jensens_alpha_pct, r_squared, alpha_standard_error and the market p_value must match scipy.stats.linregress on the same excess series.
• Decomposition identity. For any input, unexplained_residual_pct must be zero to at least 8 decimal places.
• Excess-return flag. Running the same data with and without market_returns_are_excess=True must leave every beta unchanged and shift alpha by exactly $\beta_M \times R_{f,\text{period}} \times$ periods_per_year $\times 100$.
• Zero-investment factors. Changing risk_free_rate_annual_pct from 0% to 8% must leave the SMB and HML loadings bit-identical.
• Newey-West. With hac_lags=0 the estimator must reduce to the textbook HC1 sandwich $(X'X)^{-1}\left(\sum_t u_t^2 x_t x_t'\right)(X'X)^{-1}\cdot n/(n-k)$; the default lag at $n=504$ must be $\lfloor 4(5.04)^{2/9}\rfloor = 5$; and on an AR(1) residual with $\rho=0.6$ the HAC alpha standard error must exceed the OLS one.
• Negative checks — each must raise: an empty series, fewer than $k+2$ aligned rows, non-overlapping indices, duplicate index labels, an infinity, a non-numeric value, a factor collinear with the market, standard_errors="robust", hac_lags without standard_errors="hac", periods_per_year of 0 or 2.5, significance_level of 0 or 1, and risk_free_rate_annual_pct=-120.
• Undefined, not zero. A noiseless $\beta=1.5$ series must return alpha_t_statistic is None with a populated undefined_metrics, not $t=0$.
• Run python -m unittest discover -s skills/strategy-performance-attribution-vs-market-beta/scripts and confirm a 100% pass rate.

Related Skills

• risk-adjusted-performance-attribution-per-strategy
• benchmark-relative-performance-attribution
• benchmark-selection-for-strategy-evaluation
• strategy-performance-decay-detection-vs-market-wide-decay
• factor-research-multiple-testing-correction
• strategy-capacity-estimation-before-scaling-capital
• strategy-lifecycle-retirement-criteria