Testing a CCAPM prefinal

B.Sc. Empirical research in economics using R
Seminar paper
Testing a consumption-based Capital
Asset Pricing Model with power utility
(SS 2012)
Catholic University of Eichstaett-Ingolstadt
Department of Business Administration
Written by:
Felix Gehrmann
Address
Phone:
+491717022540
E-Mail:
felix.gehrmann@ku.de
Matriculation number:
382728
Closing date:
July 9, 2012

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CONTENTS
I
Contents
1
Introduction
1
2
The Model in detail
1
2.1
Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2.2
Recent research . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
The Data
4
4
Empirical Results
7
4.1
Methodology
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4.2
Results and Interpretation . . . . . . . . . . . . . . . . . . . . . .
9
5
Conclusion
13

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LIST OF FIGURES
II
List of Figures
1
Return on the S&P 500 in excess of the risk-free asset from, net
consumption growth, net return on the risk-free asset and net re-
turn on the S&P 500 1962 to 2012 . . . . . . . . . . . . . . . . . .
5
2
Implied gamma from 1962 to 2011 . . . . . . . . . . . . . . . . . .
10
3
Box plot for the implied gamma from 1962 to 2011
. . . . . . . .
11
4
Implied beta from 1962 to 2008 . . . . . . . . . . . . . . . . . . .
14

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LIST OF SYMBOLS
III
List of Symbols

intertemporal preferences

coefficient of relative risk aversion
Ct
consumption in the period t
i
volatility of i
i;j
covariance of i and j
i;j
correlation of i and j
Ri;t
return of i in the period t
ri;t
natural logarithm of the capital letter
M
stochastic discount factor in the period t
X
cash flow in period t
W
financial wealth
w
weight of the investment
Y
labour income

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1
INTRODUCTION
1
1
Introduction
The financial markets are all about the perfect allocation of capital and if you
can do this better than everyone else then you can make a fortune and retire
very soon. But for doing that you need a (preferable legal) method. For this
purpose many scientists of the last half century have developed different models
to estimate where and how capital is allocated best. Following Wilmott (2011)
who says that the best model in finance is not the most complex one, but the best
model is the model best fitting the real world data,1 I am testing a very simple
consumption-based Capital Asset Pricing Model (CCAPM) in this paper.
I am testing the utility of the CCAPM with power utility in several steps: I
give you a short summary of how it is derived and a brief overview about the
current research in section 2. In section 4 I show you how you can use this
model to study the behaviour of agents regarding their relative risk aversion and
intertemporal preferences. Then I use this method to analyse which patterns
these factors have followed in the past according to the model, and argue weather
the results are reasonable or not and what this is implying about the explanatory
power of the model.
2
The Model in detail
This section gives you a summary of the model in section 2.1 and a brief overview
about the current research in section 2.2.
2.1
Derivation
To generate the implied values for relative risk aversion and intertemporal pref-
erences, we have first to set up the model which we want to test. Following that
we can solve the pricing equation for the implied variables which will be done in
section 4.
To keep the model as simple as possible I assume that asset prices are only
driven by two factors: On the one hand the future expected cash flow and a
stochastic discount factor (SDF). This is expressed in the following formula where
Mt+1 represents the SDF and Xi;t+1 the future cash flow. Pi;t represents the price
1This quote can also be found in Wilmott (2009)

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2
THE MODEL IN DETAIL
2
of an asset i in t:
Pi,t = Et[Mt+1Xi;t+1]
(1)
As Wheatley (1988) I assume that asset markets are perfect, the individuals are
homogenous and their number is constant through time, there is a single good
and a representative agent chooses an investment plan to maximize his expected
welfare. Further, to keep the model as simple as possible, I assume that our agent
has a myopic investment horizon, i.e. he only cares about today and the next
period. I do this because the resulting formula is the same as with an infinite
investment horizon, but it is simpler to derive the model with a limited horizon.
This yields in the maximization problem
max U (Ct) + tEt[U (Ct+1)]
(2)
Where represents the intertemporal preference and U () the utility function
of the agent. The agent has thus to decide which amount of his wealth he will
consume in t and which amount he should invest to consume the earnings in t + 1.
Therefore consumption (C) in t equals to
Ct = Wt + Yt - wPi;t
(3)
And in t + 1
Ct+1 = wPi;t+1 + Yt+1
(4)
Where Wt represents the financial wealth, Yt the labour income of the agent in
t.If we solve this problem, we get the following first order condition:
u (C
1 =
t+1)
tEt
Ri;t+1
(5)
u (Ct)
Where Ri;t+1 = Pi;t+1 and u represents the first derivative of the utility funcition
Pi;t
U . This is the so called Euler equation. I am assuming that the agent has a
constant relative risk aversion with the utility function
C1- - 1
U (C) =
(6)
1 -
where is representing the relative risk aversion of our agent. Using this function

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2
THE MODEL IN DETAIL
3
on equation 5 yields in
C
-
1 =
t+1
tEt
Ri;t+1
(7)
Ct
Taking the log of equation 7 yields in
C
-
0 = ln
t+1
t + ln Et
Ri;t+1
(8)
Ct
I assume that the consumption growth is log normal distributed. If we want
to solve this equation for the return we thus have to remember that if X is log
normally distributed
1
ln E[X] = E[ln X] +
V ar(X)
(9)
2
Thus we get
C
1
0 = ln
t+1
t + Et
- ln
+ ln Ri;t+1 + (2 + 22 - 2i;c)
(10)
C
i
c
t
2
Solving for the return gives us the final equation:
C
1
E
t+1
t[ri;t+1] = - ln + tEt
ln
- (2 + 22 - 2i;c)
(11)
C
i
c
t
2
where the small letter ri;t+1 represents the natural logarithm of the capital letter.
2 equals the variance of the consumption and 2 the variance of the return of
c
i
the asset i. i;c indicates the covariance of ri;t+1 and ln Ct+1 .
Ct
2.2
Recent research
Though Marquering (2006) and Campbell (2003) are arguing, that this model
cannot explain assets returns very good because one has to make unrealistic
assumptions about the risk aversion of the agent, it is proven to be better than
most other models, like the Fama French or three factor models, by Xiaohong
and Ludvigson (2009). Kim et al. (2012) find that the CCAPM has theoretical
preference to the portfolio based CAPM, which has been introduced by Sharpe
(1964). Further I am disagreeing with Marquering and Campbell because their
statements are based on Mehra and Prescott (1985) whose assumption about
realistic risk aversion levels is based either on unproven theory or on completely

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3
THE DATA
4
different models, which makes the usefulness of statement for the discussion on
the model arguable. This point is also supported by Kandel and Stambaugh
(1991). Hassan and Biljon (2010) prove that even a very simple model has a
very good explanatory power for the risk-free rate in the south African market.
Darrat et al. (2011) find further evidence for the CCAPM since the model is able
to explain different equity returns between countries.
3
The Data
We need three key inputs to make our calculations: The return of the risky
asset, consumption expenditures and the return of the risk free asset. I choose
the S&P 500 index as a proxy for the return of the risky asset. I choose the
S&P 500 index because, as a capitalization weighted index, it represents the
average investment into an US listed stock by the market participants. The US
personal consumption expenditures, in nominal dollars and seasonally adjusted,
are assumed to represent the consumption of the agent. I use total consumption
expenditures and not only the expenditure on non-durable goods. The data is
seasonally adjusted. Following Damodaran (2008) I choose the return on mid
term US government bonds as the risk-free rate. I am using non inflation indexed
bonds to stay consistent with the consumption expenditure data which is also not
adjusted for inflation. Further one can assume that there has been no realistic
default risk on these bonds in the past 50 years. The maturity is five years since
one has to consider so called reinvestment risk and it is assumed that the average
investment horizon is five years. I did observe the data for a time frame from the
31.3.1962 to the 31.3.2012. All data is sourced by Bloomberg L.P.
Figure 1 shows the annually excess return in nominal dollars on the S&P 500
index in US dollars in excess of holding a US Government bond with a maturity
of 5 years in plot a, the annually net US consumption growth in nominal dollars
in plot b, the annualized net return on coupon bearing US Government bonds
with a maturity of 5 years in plot c and the annual net return on the S&P 500
index in plot d.
Table 1 reports mean, median, standard deviation, minimum and the max-
imum of the used data as well as the covariance and correlation of the natural
logarithm of the annually gross US consumption growth in nominal dollars and
the natural logarithm of the annually gross return on the S&P 500 index. All
values except for the correlation are in percentages.

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3
THE DATA
5
0.4
0.10
0.2
a
b
0.0
0.05
-0.2
0.00
-0.4
1970
1980
1990
2000
2010
1970
1980
1990
2000
2010
0.14
0.4
0.2
0.10
c
d
0.0
0.06
-0.2
0.02
-0.4
1970
1980
1990
2000
2010
1970
1980
1990
2000
2010
Figure 1: Return on the S&P 500 in excess of the risk-free asset from, net con-
sumption growth, net return on the risk-free asset and net return on the S&P
500 1962 to 2012
Figure 1 shows the pattern of the used data from 1962 to 2012. Plot a represents the pattern
of the annual return on the S&P 500 in excess of the risk-free asset. b shows the pattern of
consumption growth. c is the path of the annualized return on US Government bonds with a
maturity of five years. d shows the annual return on the S&P 500.

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3
THE DATA
6
in
S&P
the
500
ends
US
the
for
wth
S&P
and
on
gro
earing
the
b
on
statistics
return
on
Q2/1962
(%)
7
coup
ollars
ually
d
consumption
44.03
12.56
14.39
46.5
from
on
Max
ann
summary
US
starts
the
the
nominal
ts
return
gross
index.
(%)
for
in
500
-41.33
-2.920
1.039
-39.68
sample
ually
Min
presen
ualized
o
return
ann
S&P
The
statistics
t
w
ann
e
net
w
the
th
(%)
net
Ro
of
on

17.318
2.7974
2.9977
17.307
the
ually
ariables.
v
summary
ears.
ann
y
for
return
the
orts
5
the
of
of
data
logarithm
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rep
for
3.291
6.986
6.222
9.091
y
gross
um
w
the
statistics
Median(%)
0.2719416
im
ro
ws
natural
ually
ax
)
maturit
first
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95
m
a
statistics
the
ann
(%
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of
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The
ro
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)
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.272
i
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with
Summary
1
7.113
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and
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ernmen
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ears.
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the
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ual
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us
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nominal
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um
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ro
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