Transcript Empirical Applications of Capital Market Models
Empirical Applications of Capital Market Models Lecture XXVII
Capital Asset Pricing Models A basic question that must be addressed in the application of both CAPM and APT models is whether a risk-free asset exists.
In the basic Sharpe-Lintner CAPM model
i
im R f
im
m
i
m m
R f
R f
im
R f
Z i Z m
E R i
m
R R f f
Z i
i i Z m
i
Constructing a dataset of 43 stocks from the Center for Research into Security Prices (CRSP) dataset, using the return on the Standard and Poors 500 portfolio and using the 3 month treasury bill as the market portfolio
Sharpe-Lintner Results
i
i
11499 0.00651 (0.01459) 1.03884 (0.31641) 12140 -0.00267 (0.00522) 1.39450
(0.11317) 15553 0.00757
(0.00324) 0.30513
(0.07028) 18542 0.00223
(0.00494) 0.99853
(0.10708) 19166 0.00092
(0.00539) 1.08365
(0.11686) 19749 -0.00465
(0.01091) -0.00807
(0.23661)
Black’s Model An alternative to the model presented by Sharpe and Lintner is the zero-beta model suggested by Black
im
where
R
0
m
is the return on the zero-beta portfolio, or the minimum variance portfolio that is uncorrelated with the market portfolio
However, the model can be estimated assuming that the zero-beta return is unobserved as:
i
im
im
m
Which yields the empirical model
R it
i i R mt it
Black’s Results
i
i
11499 0.00636
(0.01485) 1.03803
(0.31591) 12140 -0.00422
15553 0.01033
(0.00531) (0.00330) 1.39097
(0.11303) 0.30890
(0.07028) 18542 0.00236
(0.00502) 0.98751
(0.10691) 19166 0.00056
(0.00548) 1.08609
(0.11667) 19749 -0.00057
(0.01110) -0.00944
(0.23618)
11499 12140 15553 18542 19166 19749 Comparison of Betas Sharpe-Lintner 1.03884
1.39450
0.30513
0.99853
1.08365
-0.00807
Black 1.03803
1.39097
0.30890
0.98751
1.08609
-0.00944
Tests for CAPM Efficiency ˆ Sharpe-Lintner Model ˆ
m
ˆ 1
T t T
1
R t
ˆ 1
T t T
1
Z t
ˆ
Z mt t T
1
Z mt
ˆ
m
2 ˆ
m
t T
1
Z t
ˆ
Z mt
Z t
ˆ
Z mt
ˆ
m
1
T t T
1
Z mt
ˆ ˆ
N
1 ,
T N
1 ,
T
1 ˆ
m
2 2
m
1 2
m
H
0 :
H
1 : 0 0
J
0
T
1 ˆ
m
2 2
m
1 ˆ 1 ˆ
Cross-Sectional Regression Fama, E. and J. MacBeth “Risk, Return, and Equilibrium: Empirical Tests.” 71(1973): 607 –36.
Using either set of betas, the question then becomes whether the expected returns are consistent with their betas.
i
0 1
i
0 1 Cross-Sectional Results Parameter Estimate 0.006062
(0.000592) 0.906548
(0.092863)
Two Tests Sharpe-Lintner test of the constant 0 0.006062
0.000592
10.23986
Betas explain the variations in expected returns 1 0.092863
1.006343
A little reformulation:
i
0 1
i
2
D i
where
D i
is a dummy variable which is equal to one if the stock is an agribusiness stock and zero otherwise
0 1 2 Risk Premium for Agribusinesses Parameter Estimate 0.006083 (0.000605) 0.913067 (0.097456) -0.000350 (0.001374)
Arbitrage Pricing Model As we discussed in previous lectures, the returns in the arbitrage pricing model are assumed to be determined by a linear factor model:
R t R t
M N
*1
f t
t
M k
*1
t b
M
R t
is a vector of
N
asset returns
f t
is a vector of
k
common factors
b
is a
N*k
matrix of factor loadings The arbitrage pricing equilibrium implies that the expected return on the vector of assets is a linear function of the factor loadings 0
b
Two ways to define the common factors: Endogenously based on returns Exogenously based on macroeconomic variables
Given the linear factor model above, the variance matrix for the returns on the vector of assets becomes:
where is a diagonal matrix.
Under this specification, we can estimate the vector of factor loadings by maximizing
L
ln 1
Stock Number 11499 12140 15553 18542 19166 19749 11499 Factor Loadings Factor 1 0.00576
0.03512
0.02569
0.01408
0.01922
-0.01091
0.00576
Factor 2 -0.05389
-0.06152
-0.00286
-0.05615
-0.05780
-0.00215
-0.05389
Parameter 0 1 2 Estimated Risk Premia Estimate 0.01123
(0.00200) 0.00397
(0.05674) -0.07589
(0.03665)
Augmenting the model to test for disequilibria in the equity market for Agribusinesses
i
0
b
1 1
i
b
2 2
i
3
D i
Parameter 0 1 2 3 Estimated Risk Premia Estimate 0.01061
(0.00198) -0.00940
(0.05568) -0.08497
(0.03601) 0.00413
(0.00229)