Beta is often estimated by linear regression. A model often used is called the m
ID: 2795678 • Letter: B
Question
Beta is often estimated by linear regression. A model often used is called the market model, which is:
In this regression, Rt is the return on the stock and Rft is the risk-free rate for the same period. RMt is the return on a stock market index such as the S&P 500 index. i is the regression intercept, and i is the slope (and the stock's estimated beta). t represents the residuals for the regression. What do you think is the motivation for this particular regression? The intercept, i, is often called Jensen's alpha. What does it measure? If an asset has a positive Jensen's alpha, where would it plot with respect to the SML? What is the financial interpretation of the residuals in the regression?
Explanation / Answer
Jensen's alpha measures the returns of the stock/ portfolio, which are in excess of the expected returns, calculated using models such as the Capital Asset Pricing Model (CAPM), for a given level of risk (Beta). The amount by which a stock/ portfolio beats the market rate of return is termed as alpha. This excess usually represents a portfolio manager's ability to generate higher returns than the market and is not attributable to market conditions.
The simplest formula to calculate Jensen's Alpha is :
Alpha = Ra - [ Rf + x ( Rm - Rf ) ]
Where,
Ra : Actual return on the stock/ portfolio
Rf : Risk- free rate
Rm : Return on the market index
: Beta (Volatility of the stock/ portfolio against the market index)
E.g. Using CAPM, if the return on a portfolio of 5 stocks, given a set of market conditions, is 10% but the actual return earned is 13%, then the extra 3% (13%-10%) is the alpha return earned on the portfolio.
The Security Market Line (SML) shows the expected return of a portfolio, given its beta. Therefore, a portfolio with a zero alpha would plot on the SML, while a portfolio with a positive or negative alpha would lie above or below the SML, respectively.
The residuals or Standard Error of Estimate (SEE) indicates the accuracy of the regression model. The lower the SEE, the more accurately the model is able to predict the actual return.
Note: The SEE doesn't explain the extent to which the independent variable explains variations in the dependent variable.
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