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Lecture 6: CAPM & APT. The following topics are covered: CAPM CAPM extensions Critiques APT. CAPM: Assumptions. Investors are risk-averse individuals who maximize the expected utility of their wealth

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lecture 6 capm apt
Lecture 6: CAPM & APT
  • The following topics are covered:
    • CAPM
    • CAPM extensions
    • Critiques
    • APT

L6: CAPM & APT

capm assumptions
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 – page 148)
  • 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.

L6: CAPM & APT

derivation of capm
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
    • (6.1)
  • 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:
    • (6.2)
    • (6.3)
  • 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.

L6: CAPM & APT

derivation of capm1
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:

L6: CAPM & APT

slide5

Extensions of CAPM

  • No riskless assets
  • Forming a portfolio with a% in the market portfolio and (1-a)% in the minimum-variance zero-beta portfolio.
  • The mean and standard deviation of the portfolio are:
  • The partial derivatives where a=1 are:
    • ;
    • ;
  • Taking the ratio of these partials and evaluating where a=1:
  • Further, this line must pass through the point and the intercept is . The equation of the line must be:

L6: CAPM & APT

extensions of capm
Extensions of CAPM
  • The existence of nonmarketable assets
    • E.g., human capital; page 162
  • The model in continuous time
    • Inter-temporal CAPM
  • The existence of heterogeneous expectations and taxes

L6: CAPM & APT

empirical tests of capm
Empirical tests of CAPM
  • Test form -- equation 6.36
    • the intercept should not be significantly different from zero
    • There should be one factor explaining return
    • The relationship should be linear in beta
    • Coefficient on beta is risk premium
  • Test results – page 167
  • Summary of the literature.

L6: CAPM & APT

roll 1977 s critiques
Roll (1977)’s Critiques
  • Roll’s (1977) critiques (page 174)
  • The efficacy of CAPM tests is conditional on the efficiency of the market portfolio.
  • As long as the test involves an efficient index, we are fine.
  • The index turns out to be ex post efficient, if every asset is falling on the security market line.

L6: CAPM & APT

arbitrage pricing theory
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

L6: CAPM & APT

deriving apt
Deriving APT
  • Return of the arbitrage portfolio:
  • To obtain a riskless arbitrage portfolio, one needs to eliminate both diversifiable and nondiversifiable risks. I.e.,

L6: CAPM & APT

deriving apt1
Deriving APT

As:

How does E(Ri) look like? -- a linear combination of the sensitivities

L6: CAPM & APT

slide12
APT
  • There exists a set of k+1 coefficients, such that,
    • (6.57)
  • If there is a riskless asset with a riskless rate of return Rf, then b0k =0 and Rf =
    • (6.58)
  • In equilibrium, all assets must fall on the arbitrage pricing line.

L6: CAPM & APT

apt vs capm
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.

L6: CAPM & APT

example
Example
  • Page 182
  • Empirical tests
    • Gehr (1975)
    • Reinganum (1981)
    • Conner and Korajczyk (1993)

L6: CAPM & APT

ff 3 factor model
FF 3-factor Model
  • http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_factors.html

L6: CAPM & APT