View source for Jensen's alpha

'''[[Jensen's alpha]]''' is a [[Risk-adjusted return|risk-adjusted return]] measure that evaluates the performance of an investment or portfolio relative to the return predicted by the [[Capital asset pricing model]] (CAPM). It measures the excess or shortfall of an investment's actual return relative to its CAPM-implied expected return, given its level of systematic risk.

==Calculation==

Jensen's alpha is commonly calculated as:

<math>
\alpha_p
=
R_p
-
\left[
R_f
+
\beta_p
\left(
R_m-R_f
\right)
\right]
</math>

where:

* <math>\alpha_p</math> = Jensen's alpha;
* <math>R_p</math> = portfolio return;
* <math>R_f</math> = [[Risk-free rate|risk-free rate]];
* <math>\beta_p</math> = portfolio [[Beta (finance)|beta]];
* <math>R_m</math> = market portfolio return.

The expression inside the brackets represents the return predicted by the CAPM.

==CAPM relationship==

The [[Capital asset pricing model]] estimates the expected return of a portfolio as:

<math>
E(R_p)
=
R_f
+
\beta_p
\left[
E(R_m)-R_f
\right]
</math>

Jensen's alpha compares the actual portfolio return with this expected return:

<math>
\alpha_p
=
R_p-E(R_p)
</math>

Substituting the CAPM expected return gives:

<math>
\alpha_p
=
R_p
-
\left[
R_f
+
\beta_p
\left(
R_m-R_f
\right)
\right]
</math>

==Interpretation==

A '''positive Jensen's alpha''' indicates that the portfolio generated a return above the return predicted by the CAPM for its level of systematic risk.

A '''negative Jensen's alpha''' indicates that the portfolio generated a return below the CAPM-implied return.

An alpha of zero indicates that the portfolio's return was equal to the return predicted by the CAPM.

For example, if:

* portfolio return = 14%;
* risk-free rate = 4%;
* market return = 10%;
* portfolio beta = 1.2;

then:

<math>
\alpha_p
=
14\%
-
\left[
4\%
+
1.2(10\%-4\%)
\right]
</math>

<math>
\alpha_p
=
14\%-11.2\%
=
2.8\%
</math>

The portfolio therefore generated a Jensen's alpha of 2.8 percentage points.

==Systematic risk==

Jensen's alpha evaluates performance after accounting for [[Systematic risk|systematic risk]] as measured by beta.

Beta is commonly calculated as:

<math>
\beta_p
=
\frac{\operatorname{Cov}(R_p,R_m)}
{\operatorname{Var}(R_m)}
</math>

where:

* <math>\operatorname{Cov}(R_p,R_m)</math> = covariance between portfolio and market returns;
* <math>\operatorname{Var}(R_m)</math> = variance of market returns.

Because Jensen's alpha is based on CAPM, it does not directly compensate for or measure [[Unsystematic risk|unsystematic risk]].

==Jensen's alpha and Sharpe ratio==

The [[Sharpe ratio]] measures excess return relative to total portfolio volatility:

<math>
\text{Sharpe Ratio}
=
\frac{R_p-R_f}
{\sigma(R_p)}
</math>

Jensen's alpha instead measures the difference between actual return and the return predicted by CAPM:

<math>
\alpha_p
=
R_p
-
\left[
R_f
+
\beta_p(R_m-R_f)
\right]
</math>

The key distinction is that the Sharpe ratio evaluates excess return relative to total volatility, while Jensen's alpha evaluates performance relative to the systematic risk predicted by CAPM.

==Jensen's alpha and Treynor ratio==

The [[Treynor ratio]] measures excess return per unit of systematic risk:

<math>
\text{Treynor Ratio}
=
\frac{R_p-R_f}
{\beta_p}
</math>

Jensen's alpha instead measures the absolute difference between actual return and CAPM-implied return:

<math>
\alpha_p
=
R_p
-
\left[
R_f
+
\beta_p(R_m-R_f)
\right]
</math>

Both measures use beta as a measure of systematic risk, but they provide different information about portfolio performance.

==Active management==

Jensen's alpha is commonly used to evaluate [[Active management|actively managed]] portfolios.

An active manager may seek to generate returns above those predicted by CAPM by selecting securities or adjusting portfolio exposures.

A positive Jensen's alpha may indicate that the portfolio generated excess performance relative to its CAPM-implied return, although the result depends on the validity of the CAPM assumptions and the selected measurement period.

==Regression approach==

Jensen's alpha can also be estimated using a regression of portfolio excess returns against market excess returns.

The CAPM regression can be represented as:

<math>
R_p-R_f
=
\alpha_p
+
\beta_p(R_m-R_f)
+
\epsilon_p
</math>

where:

* <math>\alpha_p</math> = Jensen's alpha;
* <math>\beta_p</math> = portfolio beta;
* <math>\epsilon_p</math> = residual return not explained by the model.

In this formulation, Jensen's alpha is the regression intercept.

==Measurement period==

Jensen's alpha depends on the period over which returns, beta, and the risk-free rate are measured.

A consistent calculation should use compatible:

* portfolio returns;
* market returns;
* risk-free rates;
* beta estimates;
* observation frequencies.

Differences in measurement periods or methodologies can therefore produce different estimates of Jensen's alpha.

==Limitations==

Jensen's alpha has several limitations.

These include:

* dependence on the assumptions of CAPM;
* sensitivity to the selected market benchmark;
* reliance on beta estimates;
* sensitivity to the measurement period;
* possible changes in beta over time;
* inability to account for all factors that may explain asset returns.

A positive alpha should therefore not automatically be interpreted as evidence of superior investment skill.

==Importance==

Jensen's alpha is an important measure of [[Portfolio management|portfolio performance]] because it evaluates an investment's return relative to the return predicted by the CAPM for its level of systematic risk. It is commonly used in investment analysis to assess the historical performance of actively managed portfolios and funds.

==See also==

* [[Capital asset pricing model]]
* [[Risk-adjusted return]]
* [[Sharpe ratio]]
* [[Treynor ratio]]
* [[Sortino ratio]]
* [[Information ratio]]
* [[Active return]]
* [[Beta (finance)]]
* [[Risk-free rate]]
* [[Risk premium]]
* [[Systematic risk]]
* [[Unsystematic risk]]
* [[Portfolio management]]
* [[Active management]]

[[Category:Investment]]
[[Category:Portfolio management]]
[[Category:Financial metrics]]
[[Category:Risk-adjusted return]]
[[Category:Performance measurement]]
[[Category:Financial risk]]
[[Category:Capital asset pricing model]]