ASSET VALUATION MODELS - CAPM & APT

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CAPM: Assumptions • Investors are risk-averse individuals who maximize the expected utility of their wealth • Investors are price takers and they have homogeneous expectations about asset returns that have a joint normal distribution (thus market portfolio is efficient) • There exists a risk-free asset such that investors may borrow or lend unlimited amount at a risk-free rate. • The quantities of assets are fixed. Also all assets are marketable and perfectly divisible. • Asset markets are frictionless. Information is costless and simultaneously available to all investors. • There are no market imperfections such as taxes, regulations, or restriction on...

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CHAPTER FIVE: ASSET VALUATION MODELS - CAPM & APT 06/08/2011 1
CAPM: Assumptions • Investors are risk-averse individuals who maximize the expected utility of their wealth • Investors are price takers and they have homogeneous expectations about asset returns that have a joint normal distribution (thus market portfolio is efficient) • There exists a risk-free asset such that investors may borrow or lend unlimited amount at a risk-free rate. • The quantities of assets are fixed. Also all assets are marketable and perfectly divisible. • Asset markets are frictionless. Information is costless and simultaneously available to all investors. • There are no market imperfections such as taxes, regulations, or restriction on short selling. 06/08/2011 2
Derivation of CAPM • If market portfolio exists, the prices of all assets must adjust until all are held by investors. There is no excess demand. • The equilibrium proportion of each asset in the market portfolio is market value of the individual asset – wi  market value of all assets • A portfolio consists of a% invested in risky asset I and (1-a)% in the market portfolio will have the following mean and standard deviation: – ~ ~ ~ E ( R p )  aE ( Ri )  (1  a) E ( Rm ) – ~  ( R p )  [a 2 i2  (1  a) 2  m  2a(1  a) im ]1 / 2 2 • A portfolio consists of a% invested in risky asset I and (1-a)% in the market portfolio will have the following mean and standard deviation: • Find expected value and standard deviation of R p with respect to the percentage of the portfolio as follows. ~ E ( R p ) ~ ~  E ( Ri )  E ( Rm ) a 06/08/2011 3
Derivation of CAPM ~  ( R p ) 1 22 [a  i  (1  a) 2  m  2a(1  a) im ]1/ 2  [2a i2  2 m  2a m  2 im  4a im ]  2 2 2 a 2 • Evaluating the two equations where a=0: ~ E ( R p ) ~ ~  E ( Ri )  E ( Rm ) a 0 a ~  ( E p )   m2 1 2 1 / 2 ( m ) (2 m  2 im )  im  2 a 0 m a ~ ~ ~ 2 E ( R p ) / a E ( Ri )  E ( Rm ) • The slope of the risk-return trade-off:  ~ a 0 ( im   m ) /  m  ( R p ) / a 2 • Recall~that the slope of the market line is: E ( Rm )  R f ; m • Equating the above two slopes: ~ ~ ~ E ( Rm )  R f E ( Ri )  E ( Rm )  m ( im   m ) /  m 2  ~ ~ E ( Ri )  R f  [ E ( Rm )  R f ] im m 2 06/08/2011 4
Extensions of CAPM 1. No riskless assets 2. Forming a portfolio with a% in the market portfolio and (1-a)% in the minimum-variance zero-beta portfolio. 3. The mean and standard deviation of the portfolio are: – E ( R p )  aE ( Rm )  (1  a) E ( Rz ) ~  ( R p )  [a 2 m  (1  a) 2  z2  2a(1  a)rzm z m ]1 / 2 – 2 4. The partial derivatives where a=1 are: E ( R p ) – ;  E ( Rm )  E ( R z ) a  ( R p ) – ; 1  [a 2 m  (1  a) 2  z2 ]1 / 2 [2a m  2 z2  2a z2 ] 2 2 a 2 5. Taking the ratio of these partials and evaluating where a=1: E ( R p ) / a E ( Rm )  E ( R z ) –   ( R p ) / a m Further, this line must pass through the point E(R ),  ( Rm ) 6. and the intercept m is E ( Rz ) . The equation of the line must be: E ( Rm )  E ( Rz ) – ] p E ( R p )  E ( Rz )  [ m 06/08/2011 5
Arbitrage Pricing Theory • Assuming that the rate of return on any security is a linear function of k factors: Ri  E ( Ri )  bi1F1  ...  bik Fk   i Where Ri and E(Ri) are the random and expected rates on the ith asset Bik = the sensitivity of the ith asset’s return to the kth factor Fk=the mean zero kth factor common to the returns of all assets εi=a random zero mean noise term for the ith asset • We create arbitrage portfolios using the above assets. • n w 0 • No wealth -- arbitrage portfolio i i 1 • Having no risk and earning no return on average 06/08/2011 6
Deriving APT • Return of the arbitrage portfolio: n R  w R p i i i 1   wi E ( Ri )   wi bi1 F1 ...   wi bik Fk   wi i i i i i • To obtain a riskless arbitrage portfolio, one needs to eliminate both diversifiable and nondiversifiable risks. I.e., 1 wi  , n  ,  wi bik  0 for all factors n i 06/08/2011 7
Deriving APT R p   wi E ( Ri ) i  w E(R )  0 i i i wb  0 for each k As: i ik i How does E(Ri) look like? -- a linear combination of the sensitivities 06/08/2011 8
APT • There exists a set of k+1 coefficients, such that, – ~ E ( Ri )  0  1bi1  ...  k bik • If there is a riskless asset with a riskless rate of return Rf, then b0k =0 and Rf = 0 – E( Ri )  R f  1bi1  ...  k bik • In equilibrium, all assets must fall on the arbitrage pricing line. 06/08/2011 9
APT vs. CAPM • APT makes no assumption about empirical distribution of asset returns • No assumption of individual’s utility function • More than 1 factor • It is for any subset of securities • No special role for the market portfolio in APT. • Can be easily extended to a multiperiod framework. 06/08/2011 10