B.4 CAPM-1.pptx

# B.4 CAPM-1.pptx - Venti CAPM 1 Outline CAPM • The...

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Venti CAPM 1 Outline: CAPM The single-index model for: A single stock Portfolios of stocks Risk for the diversified investor: Beta Calculating Beta The Capital Asset Pricing Model Implications Estimation Uses Is it any good?

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Venti CAPM 2 Markowitz portfolio optimization model: Collect data on variances and covariances of all risky securities Identify efficient portfolios of risky assets Add risk-free asset and identify tangency portfolio Use risk tolerance to find optimal complete portfolio Markowitz portfolio optimization model is ideal in principle, but is computationally expensive For an N security portfolio: E(r)’s = N i 2 ’s = N ij ’s = (N 2 – N)/2 Total: (N 2 +3N)/2 With 100 Assets, need 10,300/2 = 5150 pieces of information to plot all 100 assets on an E(r) / graph. Markowitz Model
Venti CAPM 3 A more practical alternative is a single-index (or single factor) model Much easier to calculate portfolio variances Assumes that one common factor is responsible for the covariances between stocks This turns out to be systematic (market) risk All other variability is non-systematic (firm-specific) The single-index model is practical and simplifies security analysis Alternative to Markowitz Model

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Venti CAPM 4 Decompose the return on the ith security into two components: a common (systematic) factor that affects all firms a firm-specific factor r i = m+ e i The common factor may represent macroeconomic surprises, unanticipated business cycle movements, etc. More realistically, some securities may be more sensitive to m than others: r i = β i m+ e i Single-Index Model
Venti CAPM 5 r i = β i m+ e i Make two minor modifications: 1. Replace m with r m : the return on the (fully diversified) market portfolio which contains only systematic risk. 2. Let R denote the excess return : the rate of return in excess of the risk-free rate. R i = r i -r f R m = r m -r f Then, R i = β i R m + e i Single-Index Model

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6 The variance of the excess return on each security is the sum of two components: Var(R i )= Var( β i R m + e i ) = σ i 2 = β i 2 σ m 2 + σ ei 2 β i 2 σ m 2 : variance attributable to the common (systematic) factor σ ei 2 : variance attributable to the firm-specific (unsystematic) factor. The firm-specific components are uncorrelated across securities. Single-Index Model
7 The covariance between security i and security j is: Cov(R i ,R j ) = Cov( β i R m +e i , β j R m +e j ) = β i β j σ m 2 or σ ij = β i β j σ m 2 Single-Index Model

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8 If we know the β i for each security and R m and σ m , then we know everything we need to calculate r i , σ i 2 and σ ij (and thus calculate the efficient frontier). With 100 securities, we need to know only 102 parameters (the β i for each security and R m and σ m ) to estimate the 100 variance terms, 4,950 covariance terms, and 100 expected returns need to implement the Markowitz model.
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