How is Jensen's alpha calculated, and what does it actually tell you about manager skill?

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AcadiFi · Accepted Answer

**Jensen's alpha** measures the average return earned by a portfolio above (or below) the return predicted by the Capital Asset Pricing Model, given the portfolio's systematic risk: **Alpha = R_p - [R_f + Beta_p x (R_m - R_f)]** Or equivalently, it is the **intercept** from the time-series regression: **(R_p,t - R_f,t) = alpha + Beta x (R_m,t - R_f,t) + epsilon_t** **Interpretation** - **Alpha > 0:** Manager outperformed the CAPM benchmark on a risk-adjusted basis - **Alpha = 0:** Performance is exactly what the model predicts given the level of beta risk - **Alpha < 0:** Manager underperformed on a risk-adjusted basis **Worked Example** Cavendish Equity Fund over 60 months: - Average monthly return: 0.95% - Risk-free rate (monthly): 0.35% - Market return (monthly): 0.78% - Beta (from regression): 1.12 **CAPM expected return:** 0.35% + 1.12 x (0.78% - 0.35%) = 0.35% + 0.482% = 0.832% **Alpha = 0.95% - 0.832% = 0.118% per month = ~1.42% annualized** Cavendish outperformed by about 1.42% per year after adjusting for market risk. **Statistical Significance** A positive alpha does not necessarily indicate skill — it could be noise. You must check the **t-statistic**: **t = alpha / SE(alpha)** From Cavendish's regression: alpha = 0.118%, SE(alpha) = 0.065% **t = 0.118 / 0.065 = 1.82** At the 5% significance level (two-tailed), the critical value is approximately 2.0. Since 1.82 < 2.0, this alpha is **not statistically significant** — we cannot confidently reject the hypothesis that true alpha is zero. ```mermaid flowchart TD A["Run CAPM regression: 60 months"] --> B["Alpha = 1.42% annualized"] B --> C["t-statistic = 1.82"] C --> D{"t > 2.0?"} D -->|No| E["Not statistically significant"] D -->|Yes| F["Evidence of genuine skill"] E --> G["Could be luck or model misspecification"] ``` **Limitations of Jensen's Alpha** | Issue | Explanation | |---|---| | **Single-factor model** | CAPM only adjusts for market beta. Exposure to size, value, momentum, or other factors is attributed to alpha when it may just be factor loading | | **Beta instability** | If the manager times the market (varying beta), the regression misestimates both beta and alpha | | **Benchmark choice** | Alpha is relative to the chosen market proxy — different benchmarks yield different alphas | | **Survivorship bias** | Funds with negative alpha are closed; only survivors are analyzed | | **Short samples** | 36-60 months often insufficient for statistical significance | **Multi-Factor Alternative** Using a Fama-French 3-factor model: **(R_p - R_f) = alpha + B1(R_m - R_f) + B2(SMB) + B3(HML) + epsilon** Cavendish's 3-factor alpha might drop to 0.65% annualized — much of the CAPM alpha was explained by a small-cap tilt (positive SMB loading). **Exam Tip:** CFA questions may give regression output and ask you to (1) compute alpha, (2) assess statistical significance, and (3) explain why multi-factor models might reduce alpha. For more on attribution, explore our CFA question bank.

How is Jensen's alpha calculated, and what does it actually tell you about manager skill?

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