INVESTMENTS The Capital Asset Pricing Model Chapter 5 INVESTMENTS 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 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 fiif n i iif n i ip rrwrrwrwr 111 ) ~ ( ~ )1( ~ ∑ = ?+= n i fiifp rrwrr 1 )( fi i p rr w r ?= ? ? INVESTMENTS Contribution of an asset to portfolio risk ? Recall that the variance of portfolio return is the sum of all entries of the following table 11 rw 22 rw ... nn rw 22 rw 11 rw nn rw ... 2 1 2 1 σw 1221 σww ... nn ww 11 σ 1221 σww 2 2 2 2 σw ... nn ww 12 σ ... ... ... ... nn ww 11 σ nn ww 12 σ ... 22 nn w σ 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 ∑ ≠ + ij ijjiii www σσ 2 22 INVESTMENTS The marginal contribution of asset i to 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 ] ~ , ~ cov[222)2( 222 2 ∑∑ ≠≠ =+=+ ? ? = ? ? ij piijjii ij ijjiii ii p rrwwwww ww σσσσ σ p ip p pi i p pi p rr ww σ σ σ σ σ σ == ? ? = ? ? ] ~ , ~ cov[ 2 1 2 INVESTMENTS 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 pip fi ip ip i rr w wr riskinalm returninalm RRR σσσ // / arg arg ? = ?? ?? == p fp p pip fi i rr RRR rr RRR σσσ ? == ? = / INVESTMENTS Alternative CALs M E(r) CAL (Global minimum variance) CAL (A) CAL (P) P A F PP&F A&F M A G P M 　 INVESTMENTS 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 fp pip fi rrrr σσσ ? = ? / )()( 2 fpipfp p ip fi rrrrrr ?=?=? β σ σ 2 / pipip σσβ = ip β fp rr ? INVESTMENTS Main points of modern portfolio theory 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 ? 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 im MCAPMCAP 1 ∑ = = n j j i i MCAP MCAP w 1 INVESTMENTS Derivation of CAPM ? Assumption: ? There are two dates: today and tomorrow; ? Individual investors are price takers ? Investments are limited to traded financial assets ? No taxes, and transaction costs ? There is a riskless asset: paying interest rate ? in zero net supply. ? Information is costless and available to all investors ? All investor have homogeneous expectations.. ? All investors hold efficient frontier portfolios. ? Demand of assets equals supply in equilibrium. f r INVESTMENTS implications ? 1 every investor put their money into two parts: (a) the riskless asset; (b) a single portfolio of risky assets—the tangency portfolio. ? 2. All investors hold risky assets in the same proportion: according to the tangency portfolio. ? 3 the tangency portfolio is market portfolio. INVESTMENTS A numerical illustration of CAPM ? CAPM requires that in equilibrium total asset holdings of all investors must equal the total supply of assets. ? There are only three risky assets, A, B, C. suppose the tangency portfolio is ? Wtangent=(WA,WB,Wc)=(0.25,0.5,0.25) ? There are only three investors in the economy, 1,2,3 with total wealth of 500, 1000, 1500 billion dollars, respectively. Their asset holdings are: 75015007500total 450900450-3003 2004002002002 1002001001001 CBArisklessinvestor INVESTMENTS CAPM ? The market portfolio is an efficient frontier portfolio. So we have ? Where )] ~ [(] ~ [ fmimfi rrErrE ?=? β 2 / mimim σσβ = INVESTMENTS Capital Market Line E(r) E(r M ) r f M CML 　 m 　 INVESTMENTS M = Market portfolio r f = Risk free rate E(r M ) - r f = Market risk premium E(r M ) - r f = Market price of risk = Slope of the CAPM M 　 Slope and Market Risk Premium INVESTMENTS ? The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio ? Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio Expected Return and Risk on Individual Securities INVESTMENTS Security Market Line E(r) E(r M ) r f SML M ? ? = 1.0 INVESTMENTS β = [COV(r i ,r m )] / σ m 2 Slope SML = E(r m ) - r f = market risk premium SML = r f + β[E(r m ) - r f ] Beta m = [Cov (r i ,r m )] / σ m 2 = σ m 2 / σ m 2 = 1 SML Relationships INVESTMENTS Example ? Consider two assets, A and B, together with the market: ? Which one of these two assets is more valuable.(1. Same expected payoff, 2. Positive and negative correlated with the market) 21PB 12PA 20100Pm 0.50.5Probability INVESTMENTS E(r m ) - r f = .08 r f = .03 β x = 1.25 E(r x ) = .03 + 1.25(.08) = .13 or 13% β y = .6 e(r y ) = .03 + .6(.08) = .078 or 7.8% Sample Calculations for SML INVESTMENTS Graph of Sample Calculations E(r) R x =13% SML m ? ? 1.0 R m =11% R y =7.8% 3% x ? 1.25 y ? .6 .08 INVESTMENTS Disequilibrium Example E(r) 15% SML ? 1.0 R m =11% r f =3% 1.25 INVESTMENTS ? Suppose a security with a β of 1.25 is offering expected return of 15% ? According to SML, it should be 13% ? Underpriced: offering too high of a rate of return for its level of risk Disequilibrium Example INVESTMENTS Understanding beta ? 1 an asset’s market beta is the measure of its risk. The higher an asset’s market beta, the higher its risk premium. ? 2 other things being equal, investors like assets that pay off when aggregate economic conditions are bad, thus assets having payoffs negatively correlated with those of market portfolio are desirable and earn a negative risk premium. INVESTMENTS continued ? 3. We can decompose return and risk as follows: ? With and . Then ifmimfi errrr ~ ) ~ ( ~ +?=? β Systematic component Unsystematic components 0] ~ [ = i eE 0] ~ , ~ cov[ = mi re ] ~ var[] ~ var[] ~ var[ 2 imimi err +=β Total risk systematic risk Unsystematic risk INVESTMENTS example ? Consider an asset having ? annual volatility of 40% ? market beta of 1.2 ? Suppose that the annual volatility of the market is 22%. What percentage of the total volatility of the asset is attributable to unsystematic risk? INVESTMENTS continued ? Unsystematic risk/total risk=0.3005/0.4=75% ? Empirically, roughly 70-80% of the total volatility of a typical stock is attributable to unsystematic risk. 3005.0 090304.0)var( ) ~ var(22.02.14.0 222 = = +×= e e e σ INVESTMENTS Example:Two securities with the same total risk ? Suppose that ? What is the total variance of each return? ? ? However %20= m σ 0.180.5software2 0.101.5steel1 Residual variance Market betaBusinessStock 19.01.02.05.1 222 1 22 1 2 1 =+×=+= emm σσβσ 19.018.02.05.0 222 2 22 2 2 2 =+×=+= emm σσβσ %5 19.0 2.05.0 %,47 19.0 2.05.1 22 2 2 22 2 1 = × == × = RR INVESTMENTS Beta of portfolio ? For a portfolio ,its market beta is ? explanation: consider two risky assets A and B with market betas , what is the beta of a portfolio of A and B? ? ),...,,( 21 np wwwW = v ∑ = = n i imipm w 1 ββ BA ββ , BA m mBmA m BA p ww RRwRRwRwwR ββ σσ β )1( ],cov[)1(],cov[])1(cov[ 22 ?+= ?+ = ?+ = INVESTMENTS Empirical implementation of CAPM ? Actual implementation of CAPM involves the following steps: ? 1. Construct a proxy for the market portfolio ? ---value-weighted market index, S&P 500 ect. ? 2. Estimate the current risk-free rate ? ---treasuries securities. ? 3. Estimate the premium on the market portfolio ? --- ? 4. Estimate the market betas of the individual assets ? Then for asset ? f r fmm rR ?=π im β i ) fmimfi rRrR ?+=（β INVESTMENTS Example: how would you obtain the required returns on IBM and AST ? Use the S&P500 as a proxy for market portfolio; ? Regress historic returns of IBM and AST on the returns on the return of S&P500, suppose the beta estimate is ? Use the historic excess returns on the S&P 500 to estimate average market premium ? obtain current riskless rate. Suppose it is ? Applying CAPM, we have ? ? 63.1,73.0 == ASTIBM ββ %6.8 500& =? fPs rR %4= f r %28.10086.073.0%4)( =×+=?+= fmIBMfIBM rRrR β INVESTMENTS Empirical Evaluation of the CAPM ? Does CAPM work well at pricing assets? We need to test it. ? What CAPM says? ? market portfolio is an efficient frontier portfolio; ? the formula ? Testing CAPM is to test the formula using market portfolio! ? Difficult with the test: ? (1) measuring the market portfolio; ? (2) measuring returns; ? (3) measuring beta ? ))(()( fmifi rRErRE ?=? β INVESTMENTS CAPM & Liquidity ? Liquidity ? Illiquidity Premium ? Research supports a premium for illiquidity - Amihud and Mendelson INVESTMENTS CAPM with a Liquidity Premium [ ] )()()( ifiifi cfrrErrE +?=? β f (c i ) = liquidity premium for security i f (c i ) increases at a decreasing rate INVESTMENTS Illiquidity and Average Returns Average monthly return(%) Bid-ask spread (%)