INVESTMENTS Chapter 5 The CapitakAsset Pricing mre
INVESTMENTS The Capital Asset The Capital Asset Pricing Model Pricing Model Chapter 5 Chapter 5
INVESTMENTS So far we have learned 1. Investor hold portfolios to reduce risk Non-systematic risks of individual assets does not matter only" systematic risks matter 1 2. Investors hold only frontier portfolio a The natural questions to ask next are a 1. How does an individual asset contribute to the risk of portfolios, especially the frontier portfolios 2. Can we be more specific about what"systematic risk IS a3. How is an assets systematic risk related to its expected return
INVESTMENTS Individual assets and frontier portfolio Individual assets and frontier portfolio So far we have learned: 1. Investor hold portfolios to reduce risk. “Non-systematic risks” of individual assets does not matter. only “systematic risks” matter. 2. Investors hold only frontier portfolios. The natural questions to ask next are: 1. How does an individual asset contribute to the risk of portfolios, especially the frontier portfolios? 2. Can we be more specific about what “systematic risk” is? 3. How is an asset’s systematic risk related to its expected return?
INVESTMENTS Contribution of ain asset to a portfolio a We assume the existence of a risk-free asset The return on a portfolio is ∑)+∑=r+∑(G2-r) a The expected portfolio return is r+∑(-r) 1= a The marginal contribution of risky asset i to the expected portfolio return is its risk premium a
INVESTMENTS Contribution of an asset to a portfolio Contribution of an asset to a portfolio We assume the existence of a risk-free asset, The return on a portfolio is The expected portfolio return is The marginal contribution of risky asset i to the expected portfolio return is its risk premium: ∑ ∑ ∑ = = = = − + = + − n i f i i f n i f i i n i p i r w r w r r w r r 1 1 1 ) ~( ~ (1 ) ~ ∑ = = + − n i p f i i f r r w r r 1 ( ) i f i p r r w r = − ∂ ∂
INVESTMENTS Contribution of an asset to portfolio ris. Recall that the variance of portfolio return is the sum of all entries of the following table 112 WIw W,w,0 W2Wno1 W W,w0 Ww.0 ●●● n The sum of the entries of the i-th-row and the i-th column is the total contribution of asset i to the portfolio variance +2∑g
INVESTMENTS Contribution of an asset to portfolio ris Contribution of an asset to portfolio risk Recall that the variance of portfolio return is the sum of all entries of the following table 1 1 w r 2 2 w r ... n n w r 2 2 w r 1 1 w r n n w r ... 2 1 2 w1σ w1w2σ12 ... w1wnσ1n w1w2σ12 2 2 2 w2σ ... w2wnσ1n ... ... ... ... w1wnσ1n w2wnσ1n ... 2 2 wnσ n The sum of the entries of the i-th-row and the i-th column is the total contribution of asset i to the portfolio variance ∑ ≠ + j i wi σ i wiwjσ ij 2 2 2
INVESTMENTS The marginal contribution of asset i to orroli yerian 0 portfolIo The marginal contribution of asset i to portfolio variance is its covariance with the portfolio Ow. a1 noa2+2∑nyg) )=2+2∑won=2cov The marginal contribution of asset i to portfolio StD is don 1 don cov, r 20.0
INVESTMENTS The marginal contribution of asset i to The marginal contribution of asset i to portfolio variance and to portfolio portfolio variance and to portfolio StD The marginal contribution of asset i to portfolio variance is its covariance with the portfolio The marginal contribution of asset i to portfolio StD is ] ~, ~ ( 2 ) 2 2 2cov[ 2 2 2 2 ∑ ∑ ≠ ≠ + = + = ∂ ∂ = ∂ ∂ j i i i j ij i p j i i i i j ij i i p w w w w w r r w w σ σ σ σ σ p ip p i p i p i p p r r w w σ σ σ σ σ σ = = ∂ ∂ = ∂ ∂ ] ~, ~ cov[ 2 1 2