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Unformatted text preview: Single Factor Model (Single Index Model)
• initial model focus is on an individual asset, and actual or realized returns • two sources of risk 1. Systematic risk arising from a macroeconomic factor (M) that impacts (to varying degrees) all assets. 2. Nonsystematic (firm-specific) risk arising from (unexpected) events that impact the specific firm. • focus is on excess returns: Ri = ri - rRF • let a market index represent the macro factor Ri = α i + ßiRM + ei ⇒ ri = rRF + αi + ßI(rM - rRF)+ ei • αi = stock's excess return if RM = 0, i.e. the "market is neutral" • ßiRM = part of return due to movements in the overall market • ßi = sensitivity of the asset to the overall market • ei = part of return (and unexpected events) specific to this firm • σ2M > 0 and σ2e > 0 ⇒ i.e. the market and firm aspects are sources of risk • Cov(ei, ej) = 0 and Cov(ei, M) = 0 • Var(αi) = 0, Cov(αi, ei) = 0 and Cov(αi, M) = 0 2 Using the Single Index Model • need far less estimated inputs here than in Markowitz and CAPM Ri = α i + ßiRM + ei • αi = stock's excess return if RM = 0, i.e. the "market is neutral" • RM = part of return due to movements in the overall market • ßi = sensitivity of the asset to the overall market • ei = part of return (and unexpected events) specific to this firm • ß2i σ2M = variance attributable to movements in the market • ß2i σ2M = systematic variance (risk) • σ2e = variance attributable to firm-specific uncertainty • σ2e = nonsystematic variance (risk) ⇒ ⇒ total risk = systematic risk + firm-specific risk total risk = ß2i σ2M + σ 2e • Cov(ri, rj) = ßi ßj σ2M ⇒ • Cov(ri, rM ) = ßiσ2M covariance = product of betas x market index risk • Correlation(ri, rj) = ρ iM x ρ jM ⇒ ⇒ ρij = product of correlations with the market index risk all variances and covariances are determined by the betas and the market risk Portfolios • αP = ∑wiαi • ßP = ∑wi ßi • eP = ∑wi2ei ⇒ eP = 0 in a well-diversified portfolio ⇒ all the previous results hold for a portfolio • ß2Pσ2M = systematic variance (risk) • σ2eP = ∑wi2σ2ei = nonsystematic variance (portfolio-specific risk) • total variance = systematic variance + nonsystematic variance ⇒ σ2eP = 0 in a well-diversified portfolio Calculations 1. For a single asset, either total risk or σe is given. Calculate the systematic risk. Total risk = systematic risk + nonsystematic risk. 2. For a portfolio, calculate the systematic risk and the nonsystematic risk. • Total risk = systematic risk + unsystematic risk. ⇒ can also calculate the total variance using the standard linear function for the assets in the portfolio. 4 Example: Suppose the following has been estimated. • RA = 1.0% + .9RM + eA • RB = -2.0% + 1.1RM + eB • σ2(eA ) = 8% • σ2(eB ) = 10% • σ2M = 20% 1. Calculate the standard deviation of each stock and the covariance and correlation between the two stocks. 2. Suppose we construct a portfolio X that has 40% in A and 60% in B. Find the portfolio alpha, beta, error term, total variance, systematic variance, portfolio-specific variance and the covariance of the portfolio with the market. 3. Suppose we construct a portfolio P that has 50% in X, 40% in the market index and 10% in the risk-free asset. Find the portfolio alpha, beta, error term, total variance, systematic variance and portfolio-specific variance. ...
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This note was uploaded on 09/27/2009 for the course UGBA 133 taught by Professor Distad during the Summer '08 term at University of California, Berkeley.
- Summer '08