The Security Market Line (SML) graphically depicts the market price of risk. The risk measure (x-axis) is beta (not σ in CML graph) The beta for the individual security is β i = cov( r i , r M ) σ M 2 = σ iM σ M 2 Regression line: ( r i − r F ) = α i + β i ( r M − r F ) + ε i The beta is the measure of the amount of systematic risk of a stock and can be estimated from the above regression. Estimated beta is the slope of the regression line (call security characteristic line SCL here). In the above graph, we use excess returns.
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory 3 From the above SML, the slope of the SML is the expected market risk premium (E(R M ) – R F ) because market beta equals 1. Asset i’s risk premium = (Price of risk) x (Amount of Risk) = [(r M – rf) / β M ] x β i . Since β m = 1, risk premium = [(r M – rf)] x β i Required return for a security = Compensation for its systematic risk (risk premium) plus the risk-free rate.
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory 4 The above is from running a regression. The dependent variance (Y) is Microsoft’s monthly return for the past five years and the independent variable (X) is S&P500’s monthly return. The estimated beta is about 0.89 in our estimation. If you go to Yahoo finance (US), the beta shown is 0.69. The difference is due to the chosen time period and the use of monthly returns or weekly returns. Anyway, this is only for illustration.:) From the results, it shows that 32.35% of the Microsoft’s return variability can be explained by S&P500’s return variability (proxy for the market variability). 67.65% of the return variability is due to the firm- specific reason or unsystematic risk. Remember the below from Ch6. Measuring Components of Risk σ i 2 = where; β i 2 σ m 2 + σ 2 (e i ) σ i 2 = total variance β i 2 σ m 2 = systematic variance σ 2 (e i ) = unsystematic variance σ MSFT 2 = β MSFT 2 σ M 2 + σ e i 2 = 0.889593 2 (0.001502181) + 0.05035 2 = 0.003723912 This is slightly different from 0.003685668 (variance of MSFT’s returns from the results above) but close. In our formula above, we assume the market return M and error terms are uncorrelated. The SML can also be used to illustrate a security’s alpha. Alpha is another common performance measure. (The other one is Sharpe ratio.) Using excess returns for the dependent and independent variables, alpha can be used immediately. Positive alpha indicates outperformance. That is, the actual or obtained risk premium is greater than the required risk premium. Negative alpha indicates underperformance. Using returns instead of excess returns, we need to do some calculations first. Estimated regression line and the SML: r i = α i + β i r M + ε i Estimated regression line E ( r i ) = r F + β i ( E ( r M ) − r F ) SML(CAPM) E ( r i ) = (1 − β i ) r F + β i E ( r M ) SML(re-group) Compare the intercepts of the estimated regression line and the SML.
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