Capm
ECON 405: Quantitative Finance
CAPM and APT
In this document, I use the package ”gmm”. You can get it the usual way through R or though the development website RForge for a more recent version. For the latter, you can install it by typing the following in R: > install.packages("gmm", repos="http://R-Forge.R-project.org") The data I use come with the package and can be extracted as follows: > > > > library(gmm) data(Finance) R > > > >
Rm F) 0.70956 0.70956 0.70956 0.70956
They use a particular test for multivariate linear models. If we look at the p-values, it says that we don’t reject the hypothesis that all αi are zero. We can therefore reestimate the model without the intercept: > res2 res2
Call: lm(formula = Z ~ Zm - 1) Coefficients: WMK UIS Zm 0.4770 1.3438 ZOOM Zm 0.7240
ORB 1.0524
MAT 0.7084
ABAX 0.7218
T 0.8037
EMR 0.9395
JCS 0.4137
VOXX 1.3517
We can then look at the systematic and non systematic risk of each asset: > > > > + + sigm > > > > > > > a > >
b > > > > > D Chisq) 1 2 10 8.2292 0.6065
2
Zero-beta CAPM (Black)
The zero-beta CAPM is based on the properties of the portfolio frontier. One of them tells us that for each efficient portfolio rp of risky assets, there exists a portfolio on the lower part of the portfolio frontier, rzp , which is uncorrelated with it. Its β defined as Cov(rp , rzp )/V ar(rp ) is therefore 0. That’s why the model is called the zero-beta CAPM. Let γ = E(rzp ), then the theory says that E(Rt − γ) = βE(Rpt − γ) We can estimate the model asset by asset using nonlinear least square (NLS). The formula must be in the form r = γ(1 − β) + βRm . Let g and b be γ and β, we can compare the estimates of γ as follows: > > + + + + + + model > > > > >
a > >
b > + + sigm > > > f1 > + +
SST > > > >
Rall
CAPM and APT
In this document, I use the package ”gmm”. You can get it the usual way through R or though the development website RForge for a more recent version. For the latter, you can install it by typing the following in R: > install.packages("gmm", repos="http://R-Forge.R-project.org") The data I use come with the package and can be extracted as follows: > > > > library(gmm) data(Finance) R > > > >
Rm F) 0.70956 0.70956 0.70956 0.70956
They use a particular test for multivariate linear models. If we look at the p-values, it says that we don’t reject the hypothesis that all αi are zero. We can therefore reestimate the model without the intercept: > res2 res2
Call: lm(formula = Z ~ Zm - 1) Coefficients: WMK UIS Zm 0.4770 1.3438 ZOOM Zm 0.7240
ORB 1.0524
MAT 0.7084
ABAX 0.7218
T 0.8037
EMR 0.9395
JCS 0.4137
VOXX 1.3517
We can then look at the systematic and non systematic risk of each asset: > > > > + + sigm > > > > > > > a > >
b > > > > > D Chisq) 1 2 10 8.2292 0.6065
2
Zero-beta CAPM (Black)
The zero-beta CAPM is based on the properties of the portfolio frontier. One of them tells us that for each efficient portfolio rp of risky assets, there exists a portfolio on the lower part of the portfolio frontier, rzp , which is uncorrelated with it. Its β defined as Cov(rp , rzp )/V ar(rp ) is therefore 0. That’s why the model is called the zero-beta CAPM. Let γ = E(rzp ), then the theory says that E(Rt − γ) = βE(Rpt − γ) We can estimate the model asset by asset using nonlinear least square (NLS). The formula must be in the form r = γ(1 − β) + βRm . Let g and b be γ and β, we can compare the estimates of γ as follows: > > + + + + + + model > > > > >
a > >
b > + + sigm > > > f1 > + +
SST > > > >
Rall