Alpha (finance)

Alpha is a measure of an investment's performance relative to the return that would be expected from a specified benchmark, market model, or set of risk factors.

In a regression-based framework, alpha is commonly represented by the intercept of a return regression.

For a simple market model:

$R_i={\alpha }_i+{\beta }_i{R}_m+{ϵ}_i$

where:

• $R_i$ = return of asset or portfolio $i$;
• ${\alpha }_i$ = alpha or regression intercept;
• ${\beta }_i$ = sensitivity of the asset or portfolio to the market;
• $R_m$ = market return;
• ${ϵ}_i$ = residual return.

The estimated alpha is the intercept obtained from the regression:

${\hat{\alpha }}_i={\overline{R}}_i-{\hat{\beta }}_i{\overline{R}}_m$

for a regression with an intercept estimated by ordinary least squares.

Alpha relative to a risk-free rate

In asset pricing, returns are often expressed as excess returns over the risk-free rate.

A simplified market model is:

$R_i-R_f={\alpha }_i+{\beta }_i(R_m-R_f)+{ϵ}_i$

where:

• $R_f$ = risk-free rate;
• $R_i-R_f$ = asset's excess return;
• $R_m-R_f$ = market's excess return.

The corresponding alpha is:

${\alpha }_i=E[R_i-R_f]-{\beta }_{i}E[R_m-R_f]$

In this formulation, alpha represents the portion of the asset's expected excess return not explained by its estimated exposure to the market factor.

Jensen's alpha

Jensen's alpha is an application of alpha based on the Capital asset pricing model (CAPM).

The CAPM-implied expected return is:

$E[R_i]=R_f+{\beta }_i(E[R_m]-R_f)$

Jensen's alpha is:

${\alpha }_i=E[R_i]-[R_f+{\beta }_i(E[R_m]-R_f)]$

Using realized or sample average returns, an estimated form is:

${\hat{\alpha }}_i={\overline{R}}_i-[{\overline{R}}_f+{\hat{\beta }}_i({\overline{R}}_m-{\overline{R}}_f)]$

A positive alpha indicates that the observed or estimated return exceeded the return predicted by the specified CAPM relationship, while a negative alpha indicates underperformance relative to that prediction.

Alpha and beta

Alpha and beta represent different components of a regression-based return model.

$\text{Alpha}\to \text{Intercept}\to \text{Return not explained by included factors}$

$\text{Beta}\to \text{Slope}\to \text{Sensitivity to included factors}$

For the market model:

$R_i={\alpha }_i+{\beta }_i{R}_m+{ϵ}_i$

alpha is represented by the intercept, while beta is represented by the slope coefficient.

Alpha in multifactor models

Alpha can also be estimated using multiple risk factors.

A general multifactor model can be written as:

$R_i-R_f={\alpha }_i+\sum _{k=1}^K{\beta }_{ik}{F}_k+{ϵ}_i$

where:

• $F_k$ = return or value of factor $k$;
• ${\beta }_{ik}$ = exposure of asset $i$ to factor $k$;
• $K$ = number of included factors;
• ${\alpha }_i$ = regression intercept.

The estimated alpha is therefore:

${\hat{\alpha }}_i=\overline{(R_i-R_f)}-\sum _{k=1}^K{\hat{\beta }}_{ik}{\overline{F}}_k$

In this context, alpha represents the average return component not accounted for by the factors included in the model.

Portfolio alpha

Alpha can be calculated for an investment portfolio in the same manner as for an individual security.

A simplified portfolio model is:

$R_p-R_f={\alpha }_p+{\beta }_p(R_m-R_f)+{ϵ}_p$

where:

• $R_p$ = portfolio return;
• ${\alpha }_p$ = portfolio alpha;
• ${\beta }_p$ = portfolio market beta;
• $R_m$ = market return;
• $R_f$ = risk-free rate.

The estimated portfolio alpha is:

${\hat{\alpha }}_p=\overline{(R_p-R_f)}-{\hat{\beta }}_p\overline{(R_m-R_f)}$

A positive estimated alpha indicates that the portfolio produced a return above that predicted by the specified regression model, after accounting for its estimated market exposure.

Alpha versus active return

Alpha and active return are related but are not identical concepts.

Active return is generally defined as the difference between a portfolio's return and its benchmark return:

$R_{\text{active}}=R_p-R_b$

Alpha, by contrast, is generally measured relative to a specified asset-pricing or factor model.

A simplified distinction is:

$\text{Active Return}=R_p-R_b$

$\text{Alpha}=\text{Actual Return}-\text{Model-Implied Return}$

Therefore, a portfolio can have positive active return relative to a benchmark without necessarily having positive alpha relative to a multifactor model.

Alpha and information ratio

The Information ratio measures active return relative to the volatility of active return:

$\text{Information Ratio}=\frac{R_p-R_b}{\sigma (R_p-R_b)}$

Alpha is different because it measures performance relative to the expected return generated by a specified risk model.

Thus:

$\text{Information Ratio}\to \text{Active Return per Unit of Active Risk}$

$\text{Alpha}\to \text{Return Relative to a Specified Pricing Model}$

Alpha and Sharpe ratio

The Sharpe ratio measures excess return relative to total return volatility:

$\text{Sharpe Ratio}=\frac{R_p-R_f}{\sigma (R_p)}$

Alpha does not directly divide excess return by volatility.

The distinction can therefore be expressed as:

$\text{Sharpe Ratio}\to \frac{\text{Excess Return}}{\text{Total Risk}}$

$\text{Alpha}\to \text{Return Relative to Model-Implied Return}$

Statistical significance of alpha

An estimated alpha is subject to sampling uncertainty.

A common hypothesis test is:

$H_0:{\alpha }_i=0$

against:

$H_1:{\alpha }_i\ne 0$

A corresponding t-statistic can be written as:

$t_{\alpha }=\frac{{\hat{\alpha }}_i}{\mathrm{SE}({\hat{\alpha }}_i)}$

A statistically significant alpha provides evidence that the estimated intercept differs from zero under the assumptions of the specified regression model.

Statistical significance does not necessarily imply that the alpha is economically large or persistent.

Interpretation and limitations

Alpha is model-dependent. A reported alpha is meaningful only relative to the benchmark, factors, estimation period, and methodology used to calculate it.

A positive alpha does not necessarily demonstrate superior investment skill. It may result from:

• omitted risk factors;
• incorrect model specification;
• measurement error;
• benchmark selection;
• sampling variation;
• temporary market conditions;
• non-linear exposures;
• changing factor exposures.

Similarly, a zero alpha does not necessarily mean that an investment strategy has no value; it indicates that the strategy's return is not distinguishable from the return predicted by the specified model, subject to the statistical assumptions.

Alpha can also vary substantially across different models. An investment may have positive alpha under a single-factor market model but little or no alpha after additional factors are included.

Example

Suppose an asset has an average annual return of 12%, a risk-free rate of 4%, and an estimated beta of 1.1. If the market has an average annual return of 10%, the CAPM-implied return is:

$R_{\text{CAPM}}=4\mathrm{\%}+1.1(10\mathrm{\%}-4\mathrm{\%})$

$R_{\text{CAPM}}=10.6\mathrm{\%}$

The estimated Jensen's alpha is therefore:

$\alpha =12\mathrm{\%}-10.6\mathrm{\%}=1.4\mathrm{\%}$

The asset therefore generated an estimated alpha of 1.4 percentage points per year relative to the specified CAPM relationship over the period considered.

See also

• Jensen's alpha
• Beta (finance)
• Active return
• Information ratio
• Sharpe ratio
• Capital asset pricing model
• Factor model
• Factor investing
• Regression analysis
• Multiple linear regression
• Ordinary least squares
• Risk premium
• Risk-adjusted return