Capital Asset Pricing Model

1
Lecture 8

• Capital Asset Pricing Model
• and
• Single-Factor Models

2
Outline

• Beta as a measure of risk.
• Original CAPM.
• Efficient set mathematics.
• Zero-Beta CAPM.
• Testing the CAPM.
• Single-factor models.
• Estimating beta.

3
Beta

• Consider adding security i to portfolio P to form
  portfolio C.
• ErC wiEri (1-wi)ErP
• sC2 wi2si22wi(1-wi)siP (1-wP)2sP2
• Under what conditions would sC2 be less than sP2?

4
Beta

• The value of wi that minimizes sC2 is

• wi gt 0 if and only if siP lt sP2 or

5
CAPM With Risk-FreeBorrowing and Lending
6
Security Market Line

• E(ri) rf E(rM) rfbi
• The linear relationship between expected return
  and beta follows directly from the efficiency of
  the market portfolio.
• The only testable implication is that the market
  portfolio is efficient.

7
Efficient Set Mathematics

• If portfolio weights are allowed to be negative,
  then the following relationships are mathematical
  tautologies.
• Any portfolio constructed by combining efficient
  portfolios is itself on the efficient frontier.

8
Efficient Set Mathematics

1. Every portfolio on the efficient frontier (except
   the minimum variance portfolio) has a companion
   portfolio on the bottom half of the minimum
   variance frontier with which it is uncorrelated.

9
Efficient Set Mathematics
Value of biP
Expected Return
bgt1
b1
P
0ltblt1
Z(P)
ErZ(P)
b0
blt0
Standard Deviation
10
Efficient Set Mathematics

1. The expected return on any asset can be expressed
   as an exact linear function of the expected
   return on any two minimum-variance frontier
   portfolios.

11
Efficient Set Mathematics

• Consider portfolios P and Z(P), which have zero
  covariance.

12
The Zero-Beta CAPM

• What if
• (1) the borrowing rate is greater than the
  lending rate,
• (2) borrowing is restricted, or
• (3) no risk-free asset exists?

13
CAPM With Different Borrowing and Lending Rates
B
Expected Return
M
L
rfB
Z(M)
EZ(M)
rfL
Standard Deviation
14
Security Market Line

• The security market line is obtained using the
  third mathematical relationship.

15
CAPM With No Borrowing
Expected Return
M
L
Z(M)
EZ(M)
rfL
Standard Deviation
16
CAPM With No Risk-Free Asset
Expected Return
M
Z(M)
EZ(M)
Standard Deviation
17
Testing The CAPM

• The CAPM implies that E(rit) rf
  biE(rM) - Rf
• Excess security returns should
• increase linearly with the securitys systematic
  risk and
• be independent of its nonsystematic risk.

18
Testing The CAPM

• Early tests were based on running cross section
  regressions. rP - rf a bbP eP
• Results a was greater than 0 and b was less than
  the average excess return on the market.
• This could be consistent with the zero-beta CAPM,
  but not the original CAPM.

19
Testing The CAPM

• The regression coefficients can be biased because
  of estimation errors in estimating security
  betas.
• Researchers use portfolios to reduce the bias
  associated with errors in estimating the betas.

20
Rolls Critique

• If the market proxy is ex post mean variance
  efficient, the equation will fit exactly no
  matter how the returns were actually generated.
• If the proxy is not ex post mean variance
  efficient, any estimated relationship is possible
  even if the CAPM is true.

21
Factor Models

• Factor models attempt to capture the economic
  forces affecting security returns.
• They are statistical models that describe how
  security returns are generated.

22
Single-Factor Models

• Assume that all relevant economic factors can be
  measured by one macroeconomic indicator.
• Then stock returns depend upon
• (1) the common macro factor and
• (2) firm specific events that are uncorrelated
  with the macro factor.

23
Single-Factor Models

• The return on security i is ri E(ri)
  biF ei.
• E(ri) is the expected return.
• F is the unanticipated component of the factor.
• The coefficient bi measures the sensitivity of ri
  to the macro factor.

24
Single-Factor Models

• ri E(ri) biF ei.
• ei is the impact of unanticipated firm specific
  events.
• ei is uncorrelated with E(ri), the macro factor,
  and unanticipated firm specific events of other
  firms.
• E(ei) 0 and E(F) 0.

25
Single-Factor Models

• The market model and the single-index model are
  used to estimate betas and covariances.
• Both models use a market index as a proxy for the
  macroeconomic factor.
• The unanticipated component in these two models
  is F rM - E(rM).

26
The Market Model

• Models the returns for security i and the market
  index M, ri and rM , respectively.
• ri E(ri) biF
  ei.
• ai bi E(rM) birM E(rM) ei
• ai bi rM ei

27
The Single-Index Model

• Models the excess returns Ri ri rf and RM
  rM rf .
• Ri E(Ri) bi F ei.
• ai bi E(RM) bi RM E(RM) ei
• ai bi RM ei

28
CAPM Interpretation of ai

• The CAPM implies that E(Ri) biE(RM).
• In the index model ai E(Ri) biE(RM) 0.
• In the market model ai E(ri) biE(rM)
  rf biE(rM) rf - biE(rM) (1 bi)rf

29
Estimating Covariances

• ei is also assumed to be uncorrelated with ej.
• Consequently, the covariance between the returns
  on security i and security j is
• Cov(Ri, Rj) bi bj sM2

30
Estimating a and b Using the Single-Index Model

• The model can be estimated using the ordinary
  least squares regression
• Rit ai biRMt eit
• ai is an estimate of Jensens alpha.
• bi is the estimate of the CAPM bi .
• eit is the residual in period t.

31
Estimates of Beta

• R square measures the proportion of variation in
  Ri explained by RM.
• The precision of the estimate is measured by the
  standard error of b.
• The standard error of b is smaller
• (1) the larger n,
• (2) the larger the var(RM), and
• (3) the smaller the var(e).

32
The Distribution of b and the 95 Confidence
Interval for Beta
33
Hypothesis Testing

• t-Stat is b divided by the standard error of b.
• P-value is the probability that b 0.
• Test the hypothesis that b g using the
  t-statistic

34
Estimating a And b Using The Market Model

• The model can be estimated using the ordinary
  least squares regression
• rit ai birMt eit
• ai equals Jensens alpha plus rf (1bi).
• bi is a slightly biased estimate of CAPM bi .
• eit is the residual in period t.

35
Comparison Of The Two Models

• Estimates of beta are very close.
• Use the index model to estimate Jensens alpha.
• The intercept of the index model is an estimate
  of a.
• The intercept of the market model is an estimate
  of a (1 b)rf

36
The Stability Of Beta

• A securitys beta can change if there is a change
  in the firms operations or financial condition.
• Estimate moving betas using the Excel function
  SLOPE(range of Y, range of X).

37
Adjusted Betas

• Beta estimates have a tendency to regress toward
  one.
• Many analysts adjust estimated betas to obtain
  better forecasts of future betas.
• The standard adjustment pulls all beta estimates
  toward 1.0 using the formula
• adjusted bi 0.333 0.667bi .

38
Non-synchronous Trading

• When using daily or weekly returns, run a
  regression with lagged and leading market
  returns.
• Rit ai b1Rmt-1 b2Rmt b3Rmt1
• The estimate of beta is Betai b1 b2 b3.