# Alpha

A measure of excess return. Alpha indicates to what extent a manager is outperforming or underperforming passive investment strategies with similar macro exposures. For example, if a manager had a 20% beta to the S&P 500 and the S&P 500 was up 10% for the year, then we would expect the manager to have earned 2%, 2% = 10% × 20%. If the manager actually earned 9%, then the excess of 7%, 7% = 9% − 2%, is the manager’s alpha for the year.

More formally, alpha can be determined by regressing the manager’s returns against a benchmark. If we represent the manager’s return at time t by Rt, and the market return at time t by Rm,t, then our regression equation would be:

Rt = α + βRm,t + εt

where, α and β are constants, and ε is a mean zero error term. The traditional use of α and β by statisticians for the constants in regression equations is the reason that we use alpha and beta in finance today when measuring outperformance and market exposure, respectively. Just as with our calculation for beta, the calculation of alpha is often done using excess returns, that is we would subtract the risk free rate from both Rt and Rm,t before calculating the regression equation. When this is done, the alpha calculated is sometimes referred to as Jensen’s alpha.

While the preceding regression equation could be calculated using the manager’s realized returns, the regression calculation assumes that β is constant. For active managers with changing exposures a more accurate procedure is to measure the beta of the portfolio each day, and then calculate the alpha based on the realized returns of the manager and the market. This process is then repeated each day. While more computationally intensive, this method will produce far more consistent results when evaluating the performance of active managers. See Performance Attribution Analysis and Ex Ante vs Ex Post Performance Attribution Analysis for further discussion of these two alternatives.

A portfolio with no beta has no systematic risk and should earn the risk-free rate. Jensen's alpha takes into account the risk-free rate by replacing Rt and Rm,t in the equation above with the equivalent excess returns. Jensen's alpha is consistent with the capital asset pricing model (CAPM).

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