Capital Asset Pricing Model
Econ 424/Amath 462
Eric Zivot
Summer 2013
Updated: August 15, 2013

© Eric Zivot 2006

Relationship between E[R] and -0.2

-00.1

0.0

0.1

0.2

0.3

Monthly cc returns

-0.4

-0.3

sp500
sbux
msft
nordboeing

Q1

Q2 Q3
1999

Q4

Q1

Q2 Q3
2000

Q4

Q1

Q2 Q3
2001

Q4

© Eric Zivot 2006

Q1

Q2 Q3
2002

Q4

Q1

Q2 Q3
2003

Q4

Tangency Portfolio (computed in R) using rf = 0.01/12
Portfolio expected return:Portfolio standard deviation:
Portfolio Sharpep Ratio::Portfolio weights:
sbux
msft
nord boeing
1.3079 -0.5003 0.1604 0.0319

0.023903
0.17149
0.13452

Beta values computed from SI regression Ri = i + i*Rtan + ei
sbux
msft
nordboeing
0 73618 -0.038882
0.73618
0 038882 00.1026910269 00.037027
037027
Average returns on four assets
sbux
msft
nordboeing
0.017817 -0.000063638 0.0032024 0.0016875© Eric Zivot 2006

0.0100.00050.0

average reeturn

0.015
0Linear relationship between average return and beta0.00.20.4
beta wrt tangency
gy portfolio© Eric Zivot 2006

0.6

Regression of average returns on beta̂ i   0   1 ̂ i,tan   it
Coefficients:
Value Std. Errort value
Pr(>|t|)(Intercept) 1.0000e-003 0.0000e+000 7.4306e+014 0.0000e+000
beta.vec 2.3000e-002 0.0000e+000 7.6649e+015 0.0000e+000
Residual standard error: 1.86e-018 on 2 degrees of freedom
Multiple R-Squared: 1Note: Estimated coefficients are(Intercept) beta.vec0.00083333 0.023069
Risk-free rate and risk premium on tangency portfolio> 0.01/12[1] 0.00083333
0 00083333> tan.port$er - 0.01/12[1] 0.023069© Eric Zivot 2006

TESTING THE CAPM

© Eric Zivot 2006

Monthly Returns: Jan 1978 – Dec 1987
> colnames(berndt.dat)[1][] "CITCRP" "CONED" "CONTIL" "DATGEN" "DEC""DELTA"[7] "GENMIL" "GERBER" "IBM""MARKET" "MOBIL" "PANAM"[13] "PSNH""TANDY" "TEXACO" "WEYER" "RKFREE“# create excess returns by subtracting off risk free rate
> returns.mat = as.matrix(coredata(berndt.dat))> excessReturns.mat = returns.mat +returns.mat[,"RKFREE"]> excessReturns.df = as.data.frame(excessReturns.mat)© Eric Zivot 2006

Coefficients:
Value Std. Error t value Pr(>|t|)(Intercept)(p ) 0.001 0.009
0.065
0.948
MARKET 0.447 0.119
3.746
0.000
Residual standard error: 0.0703 on 58 degrees of freedom
Multiple R-Squared: 0.195

-0.2

-0.1

0.00.1

* = 0-00.3

Excess returns oon CITCRP

0.220.3

CAPM regression for CITCRP-0.2-0.1
Excess returns on MARKET© Eric Zivot 2006

0.00.1

Coefficients:
Value Std. Error t value Pr(>|t|)(Intercept) 0.000 0.007-0.0360.971
MARKET 0.339 0.089
3.818
0.000
Residual standard error: 0.0524 on 58 degrees of freedom
Multiple R-Squared: 0.201-0.1

0.0α = 0-0.2

Excess returrns on IBM0.1

CAPM regression for IBM-0.2-0.1
Excess returns on MARKET© Eric Zivot 2006

0.00.1Test CAPM: H0: i = 0ˆ it S E ( i )Decision: Reject H0 at 5% level if |t| > 2> tstats
CITCRP CONED CONTIL DATGENDEC
DELTA GENMILGERBER
0.06502 1.2141 -0.6703 -1.043 0.030922 0.62137 0.5411 -0.068848
IBMMOBIL
PANAM
PSNH TANDYTEXACO WEYER-0.036301 0.085956 -0.89396 -0.27455 1.997 -0.40445 -0.52

Conclusion: Do not rejectj
CAPM for any
y of the assets!

© Eric Zivot 2006

Relationship Between Average
RReturnandd> print(rbind(mu.hat,betas), digits=3)
CITCRP CONEDCONTILDATGENDEC DELTA GENMIL GERBER
mu.hat 0.0056 0.0098 -0.00319 -0.00372 0.00818 0.012 0.00535 0.00453
betas 0.4466 0.1405 0.38855 1.00562 0.70677 0.392 0.09874 0.46316
IBMMOBIL
PANAM
PSNH TANDY TEXACO
WEYER
mu.hat 0.00355 0.00838 -0.0053 0.000831 0.0427 0.00371 0.00325
betas 0.33901 0.67978 0.7466 0.218017 1.0308 0.64326 0.70789

Estimated Security Market Line (SML)ˆi   0   1ˆi  errori
CAPM   0  0,  1  ˆ M  rf  0.011© Eric Zivot 2006

0.02
2

TRUE and Estimated SML

Estimated SML
TRUE SML

Coefficients:
Value Std. Error t value Pr(>|t|)(Intercept) 0.000 0.006
0.068
0.947
b t
betas
00.011011 0.010
0 010
11.069069
00.304
304

0.0

0.01
0mu.ha
at

0.03

0.04

Jan 1978 – Dec 1982

0.2

0.4

0.6

0.8

betas

t  0  0  0 . 0 6 8 , t  1   M  rt 0 .0 1 1  0 .0 1 1
 00 .0 1 0© Eric Zivot 2006

1.0

TRUE and Estimated SML

0.0-0.01

Estimated SML
TRUE SML-0.02

mu.hat2

0.01

Jan 1983 – Dec 1987

00
0.002
0.2

04
0.406
0.6
betas2© Eric Zivot 2006

08
0.8

101.0

121.2