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Jensen's alpha

The estimated intercept in a regression of portfolio excess returns on a benchmark, conditional on the model, sample, costs and statistical significance.

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Who this is for — Readers interpreting a performance-model intercept without turning a positive alpha into automatic proof of skill.

Jensen's alpha is the estimated intercept in a regression that relates a portfolio's excess return to that of a market benchmark:

Rₚ,ₜ − R_f,ₜ = α + β(Rᵦ,ₜ − R_f,ₜ) + εₜ

In the original model, β is linear sensitivity to the market and α is the mean component not explained by that relationship. Alpha is conditional on benchmark, risk-free rate, sample and model. Change the factor set or universe and what remains in the intercept can change too.

Jensen's alpha: an intercept conditional on a model Alpha and beta depend on benchmark, factors, frequency and sample Jensen's alpha: an intercept conditional on a model Alpha and beta depend on benchmark, factors, frequency and sample rₚ − r_f = α + β(r_m − r_f) + ε αβε Alpha α Estimated intercept aftercontrolling for the chosenmarket factor. Beta β Estimated slope of excessreturn relative to the market. Residual ε Part not explained by the modelin one observation. Specification Changing benchmark or factorscan change alpha withoutchanging returns. Cyclepedia · educational diagram: state conventions, period and data
Beta and alpha are estimated in the same regression; residuals, standard error and model specification determine what can be inferred.

Beta and alpha

Under the linear-regression assumptions:

β = Cov(Rₚ−R_f, Rᵦ−R_f) / Var(Rᵦ−R_f)

Beta is the estimated slope; alpha is the intercept. A beta of one is neither good nor bad: it describes an average one-for-one relationship in the sample. A positive alpha does not identify how much of the outcome came from omitted exposures, leverage, cost, options or timing.

Jensen designed the measure to evaluate funds relative to the asset-pricing theory available at the time. The original paper explicitly considers standard errors, t-statistics and sample size. Reporting only the intercept removes a central part of the inference.

Specifying the regression

A reproducible result states benchmark and return variant; risk-free source, currency and duration; frequency, dates and window; simple or log returns; price or total return; gross or net basis; missing-data and outlier treatment; estimator and standard errors; treatment of autocorrelation and heteroskedasticity; observation count; and R².

The risk-free rate must match the return frequency and currency. Subtracting an annual rate directly from monthly returns gives an intercept on the wrong scale.

Benchmark and omitted factors

A domestic large-cap index may be unsuitable for a global, small-cap, bond or multi-asset portfolio. Alpha then absorbs part of the benchmark misfit. Adding factors can reduce, increase or reverse the intercept:

Rₚ−R_f = α + β₁F₁ + β₂F₂ + … + ε

This does not make every multifactor model true. Each factor adds definitions, data and model-selection risk. Alpha always means “relative to the specified model”.

For nonlinear payoffs, a linear regression may conceal convexity, optionality or time-varying exposure. Rolling regressions and interactions can expose instability, but they also introduce further choices.

Example

A hypothetical monthly regression estimates α = 0.15% and β = 0.8. When benchmark excess return is zero, the model estimates average portfolio excess return of 0.15% in the sample; each percentage point of benchmark excess return is associated linearly with 0.8 percentage points from the portfolio.

Compound annualisation gives (1+0.0015)^12−1 ≈ 1.81%, but this transformation does not make the estimate significant. Standard error, autocorrelation, stability and cost are still needed. Multiplication by 12 is another convention and yields a slightly different number.

Alpha, active return and attribution

Object Definition Main dependence
Active return portfolio return minus benchmark return benchmark variant
Jensen's alpha intercept of a CAPM regression model, beta and risk-free rate
Information ratio mean active return divided by tracking error benchmark and relative dispersion
Attribution accounting decomposition of return classification, weights and linking

A portfolio can have positive active return and near-zero alpha when outperformance is associated with beta or factors. Attribution can assign an outcome to sectors, selection and weights without proving causality or skill.

Significance, selection and costs

An estimated alpha has uncertainty. A t-statistic and confidence interval help assess compatibility with zero under the model, but do not by themselves correct for many funds or strategies tested, survivorship and backfill, selected windows, style drift, trading costs and fees, capacity, market impact or a model chosen after looking at the data.

Gross alpha may be consumed by cost. Net alpha may not scale to larger capital. A statistically significant result may be economically small or confined to one market regime.

Common error — Calling every return above an index alpha. Jensen's alpha is a model intercept; the simple difference is active return.

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