ASSET VALUATION MODELS - CAPM & APT doc

CHAPTER FIVE: ASSET VALUATION MODELS - CAPM & APT 06/08/2011 1 2 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 3 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 – • 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: – – • 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 with respect to the percentage of the portfolio as follows. ) ~ () ~ ( ) ~ ( mi p RERE a RE    assetsallofvaluemarket assetindividualtheofvaluemarket w i  ) ~ ()1() ~ () ~ ( mip REaRaERE  2/12222 ])1(2)1([) ~ ( immip aaaaR   p R 06/08/2011 4 Derivation of CAPM • Evaluating the two equations where a=0: • The slope of the risk-return trade-off: • Recall that the slope of the market line is: ; • Equating the above two slopes: ]42222[])1(2)1([ 2 1 ) ~ ( 2222/12222 imimmmiimmi p aaaaaaa a R       ) ~ () ~ ( ) ~ ( 0 mia p RERE a RE     m mim immma p a E     2 22/12 0 )22()( 2 1 ) ~ (       mmim mi a p p RERE aR aRE   /)( ) ~ () ~ ( /) ~ ( /) ~ ( 2 0       m fm RRE  ) ~ ( mmim mi m fm RERE RRE   /)( ) ~ () ~ ( ) ~ ( 2     2 ]) ~ ([) ~ ( m im fmfi RRERRE    06/08/2011 5 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: – – 4. The partial derivatives where a=1 are: – ; – ; 5. Taking the ratio of these partials and evaluating where a=1: – 6. Further, this line must pass through the point and the intercept is . The equation of the line must be: – )()1()()( zmp REaRaERE  2/12222 ])1(2)1([) ~ ( mzzmzmp raaaaR   m zm p p RERE aR aRE  )()( /)( /)(      )(),( mm RRE  )( z RE p m zm zp RERE RERE   ] )()( [)()(   )()( )( zm p RERE a RE    ]222[])1([ 2 1 )( 2222/12222 zzmzm p aaaa a R       06/08/2011 6 Arbitrage Pricing Theory • Assuming that the rate of return on any security is a linear function of k factors: 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. • • No wealth arbitrage portfolio • Having no risk and earning no return on average ikikiii FbFbRER   )( 11 0 1    n i i w 06/08/2011 Deriving APT • Return of the arbitrage portfolio: • To obtain a riskless arbitrage portfolio, one needs to eliminate both diversifiable and nondiversifiable risks. I.e., 7      i ii i kiki i ii i ii n i iip wFbwFbwREw RwR  )( 11 1   i ikii factorsallforbwn n w 0,, 1 06/08/2011 Deriving APT 8   i iip REwR )( 0)(   i ii REw How does E(Ri) look like? a linear combination of the sensitivities keachforbw i iki 0  As: 06/08/2011 9 APT • There exists a set of k+1 coefficients, such that, – • If there is a riskless asset with a riskless rate of return R f , then b 0k =0 and R f = – • In equilibrium, all assets must fall on the arbitrage pricing line. 0  ikkii bbRE   ) ~ ( 110 ikkifi bbRRE   )( 11 06/08/2011 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. 10 06/08/2011 . CHAPTER FIVE: ASSET VALUATION MODELS - CAPM & APT 06/08/2011 1 2 CAPM: Assumptions • Investors are risk-averse individuals who maximize the expected. mia p RERE a RE     m mim immma p a E     2 22/12 0 )22()( 2 1 ) ~ (       mmim mi a p p RERE aR aRE   /)( ) ~ () ~ ( /) ~ ( /) ~ ( 2 0       m fm RRE  ) ~ ( mmim mi m fm RERE RRE   /)( ) ~ () ~ ( ) ~ ( 2     2 ]) ~ ([) ~ ( m im fmfi RRERRE    06/08/2011 5 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. invested in risky asset I and (1-a)% in the market portfolio will have the following mean and standard deviation: – – • A portfolio consists of a% invested in risky asset I and (1-a)% in the market