Quiz # 3 Single-Index and Multifactor Models Consider the following two regression lines for stocks A and B in the following figure. rA - rf rB - rf rM - rf rM - rf Figure 1: Regressions of Stock A and B vs. Market which stock has higher firm-specific risk? Which stock has greater systematic (market) risk Which stock has higher R2 which stock has higher alpha? which stock has higher correlation with the market? Problem-statement: Discuss the questions, no analytical solution required. 1 Solution: The two figures depict the stocks security characteristic lines (SCL) Stock A has a higher firm-specific risk because the deviations from the SCL are larger for A than for B. Deviations are measured by the vertical distance of each observation from the SCL. Beta is the slope of the SCL, which is the measure of systematic risk. Stock B’s SCL is steeper, hence stock B’s systematic risk is greater. The R2 (or squared correlation coefficient) of the SCL is the ratio of the explained variance of the stock’s return to total variance, and the total variance is the sum of the explained variance plus the unexplained variance (the stock’s residual variance). β i2σ M2 R = 2 2 β i σ M + σ ε2 2 i Since stock B’s explained variance is higher (its explained variance is greater since its beta is higher, and its residual variance β B2σ M2 , which is β B2σ ε2 is smaller, its R2 is higher B than stock A’s. Alpha is the intercept of the SCL with the expected return axis. Stock A has a small positive alpha whereas stock B has a negative alpha, hence stock a’s Alpha is larger. The correlation coefficient is simply the square root of R2, sto stock’s B correlation with the market is higher. ρi ,M = R 2 = β i2σ M2 β i2σ M2 + σ ε2 i Try to solve the following questions 4,5 and 6 analytically. 2