Lecture 8 Capital Asset Pricing Model and Single-Factor Models.
-
Upload
marvin-quinn -
Category
Documents
-
view
218 -
download
2
Transcript of Lecture 8 Capital Asset Pricing Model and Single-Factor Models.
![Page 1: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/1.jpg)
Lecture 8
Capital Asset Pricing Modeland
Single-Factor Models
![Page 2: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/2.jpg)
Outline
Beta as a measure of risk. Original CAPM. Efficient set mathematics. Zero-Beta CAPM. Testing the CAPM. Single-factor models. Estimating beta.
![Page 3: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/3.jpg)
Beta
Consider adding security i to portfolio P to form portfolio C.
E[rC] = wiE[ri] + (1-wi)E[rP]
C2 = wi
2i2+2wi(1-wi)iP +(1-wP)
2P2
Under what conditions would C2 be
less than P2?
![Page 4: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/4.jpg)
Beta
The value of wi that minimizes C2 is
22
2
2 PiPi
iPPiw
wi > 0 if and only if iP < P2 or
012 .P
iPiP
![Page 5: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/5.jpg)
CAPM With Risk-FreeBorrowing and Lending
![Page 6: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/6.jpg)
Security Market Line
E(ri) = rf + [E(rM) – rf]i
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.
![Page 7: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/7.jpg)
Efficient Set Mathematics
If portfolio weights are allowed to be negative, then the following relationships are mathematical tautologies.
1. Any portfolio constructed by combining efficient portfolios is itself on the efficient frontier.
![Page 8: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/8.jpg)
Efficient Set Mathematics
2. 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.
![Page 9: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/9.jpg)
Efficient Set Mathematics
Expected Return
StandardDeviation
P
Z(P)E[rZ(P)]
Value of iP
![Page 10: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/10.jpg)
Efficient Set Mathematics
3. 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.
PQP
PQiPQPQi rErErErE 2
![Page 11: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/11.jpg)
Efficient Set Mathematics
Consider portfolios P and Z(P), which have zero covariance.
iPPZPPZ
P
iPPZPPZi
rrErE
rrErErE
2
![Page 12: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/12.jpg)
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?
![Page 13: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/13.jpg)
CAPM With Different Borrowing and Lending Rates
Expected Return
rfL
E[Z(M)]
rfB
M
B
L
Z(M)
StandardDeviation
![Page 14: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/14.jpg)
Security Market Line
The security market line is obtained using the third mathematical relationship.
iMZMMZ
M
iMMZMMZi
rrErE
rrErErE
2
![Page 15: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/15.jpg)
CAPM With No Borrowing
Expected Return
rfL
E[Z(M)]
ML
Z(M)
StandardDeviation
![Page 16: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/16.jpg)
CAPM With No Risk-Free Asset
Expected Return
E[Z(M)]
M
Z(M)
StandardDeviation
![Page 17: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/17.jpg)
Testing The CAPM
The CAPM implies that
E(rit) = rf + i[E(rM) - Rf]
Excess security returns shouldincrease linearly with the security’s systematic risk andbe independent of its nonsystematic risk.
![Page 18: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/18.jpg)
Testing The CAPM Early tests were based on running
cross section regressions.
rP - rf = a + bP + 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.
![Page 19: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/19.jpg)
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.
![Page 20: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/20.jpg)
Roll’s 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.
![Page 21: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/21.jpg)
Factor Models Factor models attempt to capture the
economic forces affecting security returns.
They are statistical models that describe how security returns are generated.
![Page 22: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/22.jpg)
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.
![Page 23: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/23.jpg)
Single-Factor Models The return on security i is
ri = E(ri) + iF + ei.
E(ri) is the expected return. F is the unanticipated component
of the factor. The coefficient i measures the
sensitivity of ri to the macro factor.
![Page 24: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/24.jpg)
Single-Factor Models
ri = E(ri) + iF + 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.
![Page 25: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/25.jpg)
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).
![Page 26: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/26.jpg)
The Market Model
Models the returns for security i and the market index M, ri and rM , respectively.
ri = E(ri) + iF + ei.
= i + i ErM) + i[rM – E(rM)] + ei
= i + i rM + ei
![Page 27: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/27.jpg)
The Single-Index Model
Models the excess returns Ri = ri – rf and RM = rM – rf .
Ri = E(Ri) + i F + ei.
= i + i ERM) + i [RM – E(RM)] + ei
= i + i RM + ei
![Page 28: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/28.jpg)
CAPM Interpretation of i
The CAPM implies that E(Ri) = iE(RM).
In the index model i = E(Ri) – iE(RM) = 0.
In the market model
i = E(ri) – iE(rM)
= rf + i[E(rM) – rf] - iE(rM)
= (1 – i)rf
![Page 29: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/29.jpg)
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) = i j M2
![Page 30: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/30.jpg)
Estimating and 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 Jensen’s alpha.
bi is the estimate of the CAPM i .
eit is the residual in period t.
![Page 31: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/31.jpg)
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).
![Page 32: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/32.jpg)
The Distribution of b and the 95% Confidence Interval for Beta
![Page 33: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/33.jpg)
Hypothesis Testing t-Stat is b divided by the standard
error of b. P-value is the probability that Test the hypothesis that = using
the t-statistic
706322990
017851
2
t~..
..t
nt~bErrorStd
bt
![Page 34: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/34.jpg)
Estimating And Using The Market Model
The model can be estimated using the ordinary least squares regression
rit = ai + birMt + eit
ai equals Jensen’s alpha plus rf (1–i. bi is a slightly biased estimate of
CAPM i .
eit is the residual in period t.
![Page 35: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/35.jpg)
Comparison Of The Two Models
Estimates of beta are very close. Use the index model to estimate
Jensen’s alpha. The intercept of the index model is an
estimate of . The intercept of the market model is an
estimate of + (1 – )rf
![Page 36: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/36.jpg)
The Stability Of Beta A security’s beta can change if
there is a change in the firm’s operations or financial condition.
Estimate moving betas using the Excel function
=SLOPE(range of Y, range of X).
![Page 37: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/37.jpg)
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 .
![Page 38: Lecture 8 Capital Asset Pricing Model and Single-Factor Models.](https://reader034.fdocuments.in/reader034/viewer/2022051618/56649d225503460f949f7e38/html5/thumbnails/38.jpg)
Non-synchronous Trading
When using daily or weekly returns, run a regression with lagged and leading market returns.
Rit = ai + b1Rmt-1 + b2Rmt + b3Rmt+1
The estimate of beta is
Betai = b1 + b2 + b3.