3. Single Index Model - Single Factor Model (Single Index...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

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.

Ask a homework question - tutors are online