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
INVESTMENTS I vIClLlELI aSSer aind rrorrler porrolIos a Definition: the marginal return-to-risk ratio (RRR)of risky asset i in a portfolio p is RRR-marginal return or, /Ow E marginal risk do. law a Claim: for any frontier portfblio'p, the return-to-risk ratio of all risky assets must be the same RRR=i -RRR =P ip p a Just because Sharpe ratio of the frontier portfolio can not improved
INVESTMENTS Individual asset and frontier portfolios Individual asset and frontier portfolios Definition: the marginal return-to-risk ratio (RRR) of risky asset i in a portfolio p is: Claim: for any frontier portfolio p, the return-to-risk ratio of all risky assets must be the same: Just because Sharpe ratio of the frontier portfolio can not improved ip p i f p i p i i r r w r w m inal risk m inal return RRR σ / σ /σ / arg arg − = ∂ ∂ ∂ ∂ = = p p f p ip p i f i r r RRR r r RRR σ σ σ − = = − = /
INVESTMENTS Alternatiye cal e(r) CAL (P)CAL(A) dddddddddddd_d d』』』dd CAL (Global w minimum variance r G PP&f m a&f
INVESTMENTS Alternative Alternative CALs M E(r) CAL (Global minimum variance) CAL (P) CAL (A) P A F P P&F A&F M A G P M
INVESTMENTS n important formula ■ Rewriting a We have the following important results Where is the beta of asset i with respect to a frontier portfolio p a We can interpret the above relation as follows Given any frontier portfolio p(except the risk-free asset) Bip gives a measure of asset i's systematic risk r gives the premium per unit of systematic risk he risk premium on asset i equals the amount of its systematic risk times the premium per unit of the risk
INVESTMENTS An important formula An important formula Rewriting We have the following important results Where is the beta of asset i with respect to a frontier portfolio p. We can interpret the above relation as follows: Given any frontier portfolio p (except the risk-free asset) gives a measure of asset i’s systematic risk. gives the premium per unit of systematic risk. the risk premium on asset i equals the amount of its systematic risk times the premium per unit of the risk. p p f ip p i f r r r r σ σ σ − = − / ( ) ( ) 2 p f ip p f p ip i f r − r = r − r = β r − r σ σ 2 / βip = σ ip σ p βip p f r − r
INVESTMENTS Main points of modern portfolio theory a 1. Investors hold frontier portfolios a 2 Investors are concerned only with portfolio risk a In this chapter, we study how investors asset demand determines the relation between risk and return of assets in a market equilibrium---a model to price risky assets a The task for us now is to identify a frontier portfolio, which would give us a pricing model
INVESTMENTS Main points of modern portfolio theory Main points of modern portfolio theory and CAPM and CAPM 1. Investors hold frontier portfolios 2 Investors are concerned only with portfolio risk. In this chapter, we study how investors’ asset demand determines the relation between risk and return of assets in a market equilibrium---A model to price risky assets. The task for us now is to identify a frontier portfolio, which would give us a pricing model
INVESTMENTS The market portfolio Definition: the market portfolio is the portfolio of all risky assets traded in the market Definition: the market value(capitalization)of an asset is its total market price, i. e, the price one has to pay to buy all of it. Debt, A-share, B-share of a company) MCAPi(price per share)i*(# of shares outstanding )i a The total market capitalization of all risky assets is MCAP=∑MCAP a The market portfolio is the portfolio with weights in each risky asset I being MCAP MCAP
INVESTMENTS The market portfolio The market portfolio Definition: the market portfolio is the portfolio of all risky assets traded in the market. Definition: the market value (capitalization) of an asset is its total market price, i.e., the price one has to pay to buy all of it.(Debt, A-share,B-share of a company) MCAPi=(price per share)i*(# of shares outstanding)i The total market capitalization of all risky assets is The market portfolio is the portfolio with weights in each risky asset i being ∑= = n i MCAPm MCAPi 1 ∑= = n j j i i MCAP MCAP w 1