Resume ................................................................................................................................................. 2
1. Introduction ...................................................................................................................................... 3
1.1 Problem Statement ..................................................................................................................... 4
1.2 Method and structure ................................................................................................................. 5
1.2.1 Considerations concerning validity ..................................................................................... 7
1.2.3 Delimitation and definitions................................................................................................ 7
1.2.4 Validity, sensitivity and reliability ...................................................................................... 8
Part 2
2. Finance theory ................................................................................................................................ 12
2.1 Introducing financial models ................................................................................................... 12
2.1.2 Resume .............................................................................................................................. 18
2.2 The CAPM ............................................................................................................................... 18
2.2.1 MV .................................................................................................................................... 19
2.2.2 Introducing the cross section and time series dimension of CAPM ................................. 19
Figure 1 Cross section and Time series dimension ............................................................ 20
2.3 Assumptions ............................................................................................................................. 22
2.3.1The Market Portfolio and Rolls Critique ........................................................................... 24
2.4 The APT model ........................................................................................................................ 26
2.4.1 The empirical APT model ................................................................................................. 27
2.4.2 The F&F three factor model.............................................................................................. 28
2.5 Long-term vs. short-term investors .......................................................................................... 30
2.6 Conditional CAPM and Unconditional CAPM ....................................................................... 32
2.7 Resume ..................................................................................................................................... 33
Part 3
3. Data series in Denmark .................................................................................................................. 35
3.1 Problems................................................................................................................................... 35
3.2 The Sample period ................................................................................................................... 36
3.3 Data .......................................................................................................................................... 37
3.3.1 Data description ................................................................................................................ 38
3.3.2 Data screening ................................................................................................................... 38
3.4 Resume ..................................................................................................................................... 41
Part 4
4. Econometric tests of the Danish capital market ............................................................................. 44
4.1 The portfolios ........................................................................................................................... 44
Table 1 Correlation Matrix for branch ............................................................................... 45
Table 2 Correlation Matrix for BE/ME.............................................................................. 45
Table 3 Descriptive statistic ............................................................................................... 45
4.2 The tests ................................................................................................................................... 46
Figure 2 CAPM and BE/ME portfolios ............................................................................. 47
Table 4 Beta & t-tests......................................................................................................... 47
Table 5 Estimating the CAPM ........................................................................................... 49
Figure 3 Estimating Branch portfolios ............................................................................... 50
4.2.1 So where did it go wrong. ................................................................................................. 51
Table 6 Project sum up....................................................................................................... 52
4.3 Resume ..................................................................................................................................... 53
Part 5
5. Conclusion, interpretation and perspectives .................................................................................. 56
5.1 Conclusion ............................................................................................................................... 56
5.2 Perspectives .............................................................................................................................. 57
Bibliography
Figures and Tables
Disk overview
Enclosures
Attached Disks
1
Resume
Based on the criticism and failure of the CAPM it is surprising how influential it is in companies,
projects and at universities. Due to the problems arising with the CAP model, asset pricing theory
recognized the theoretical possibility that factors are needed, in order to explain why some average
returns are higher than others (i.e. size vs. volatility).
The F&F model is built on the theory of APT models and is a further expansion of the CAPM. The
aim of this project is to illustrate, test and model the unconditional S&L CAPM on the Danish
capital market. A perspective of why more factors are needed in order to model the Danish market
will be presented. It is found that assumptions, model restrictions and sample biases distort the
picture and no conclusions are made on how to model the Danish capital market.
2
1. Introduction
Since the introduction of the Capital Asset Pricing Model (CAPM) in the mid sixties by William
Sharpe and John Lintner, it has been one of the most empirical tested and studied models in
finance1. If investors hold a mean-variance efficient portfolio then the market portfolio2 will be a
mean-variance efficient portfolio. The model is theoretical strong, however, based on several
stringent assumptions it is too unrealistic to hold in reality. The model has been expanded,
assumptions have been weakened and different views and criticisms of the model have been
discussed and tested over the years.
Through early empirical tests of the CAPM theoretical failings became apparent as, among others,
the result of many simplifying and stringent assumptions3 causing difficulties in implementing valid
tests of the model (Fama, 2004). Today the model has been further developed through tests, articles
and debates. The basic model4 is still the stunk of the capital asset theory by linking risk and return
to describe the behaviour of assets return on the capital market.
Over the years the evolvement of CAPM has proved some success in empirical work. General
results have compiled how characteristics such as size5 that may expect to give high average return
also turn out to have high betas. Important failures of the CAPM that became apparent after the mid
sixties are, among others, the so called small firm effect; the smallest firms earn an average return a
few percent too high given their betas6. Strategies that one might expect to result in high average
returns, such as very volatile stocks7, turned out not to have high average returns when the stocks
did not have high betas. Other general failures became apparent as the data material and time
horizon changed and as the tests and theoretical assumptions were further discussed and dealt with.
Markowitz (1959) laid the groundwork for the CAPM, by casting the investor’s portfolio selection problem in terms of
expected return and variance of return. He argued that the investor would optimally hold a mean variance efficient
portfolio – a portfolio with the highest expected return for a given level of variance. Sharpe and Lintner used
Markowitz´s work.
2
The portfolio of all invested wealth. See also Roll´s critique.
3
Assumptions: Enclosure 1
4
CAPM: E[Zi] = αi + βi*(Rm-Rf) + ei
5
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html
6
Empirical results which, compared to what is predicted by the CAPM in a time series test, in an OLS cross section
regression gives a CAPM-line which is “pushed” a few cm away from the predicted CAPM line in a mean return-Betas
diagram.
7
Intuitively risk and return goes along. High risk = high return. Lower risk = lower return.
1
3
These problems, failures and theoretical questions are the drive for the problem statement of this
project.
Based on the criticism and failure of the model it is surprising that it worked for so long and up till
now has had such an impact in real life calculations of i.e. a firms capital costs, in corporate project
valuation and in evaluating portfolio managers. Due to the problems arising with the CAP model,
asset pricing theory recognized the theoretical possibility that factors are needed (state variables or
sources of priced risk beyond movements in the market portfolio) in order to explain why some
average returns are higher than others (i.e. size vs. volatility).
Following, the ground for a new and more attractive model for empirical tests was laid. This led to
the development of one of the most popular multifactor models8 that now dominates empirical
research. The F&F model is built on the theory of APT models and is a further expansion of the
CAPM aiming to improve the statistical tests hereof and relax the stringent assumptions that until
now have dominated the popular asset pricing model.
Empirical tests of the F&F three-factor model became very popular. The model successfully
explains the average returns of size and book-market sorted portfolios with a three factor model
consisting of the market, a small minus big portfolio (SMB) and a high minus low portfolio (HML).
Though the models and the theory behind have improved, articles, debates and scientific tests still
develops what have been central for asset pricing theory for more than 40 years.
1.1 Problem Statement
The aim of this project is to present and describe different financial models and test the CAPM on
the Danish capital market. Anomalies that the CAPM cannot explain in the American market, but
which are explained in the widely accepted three factor model tested by Fama and French (Fama,
1996) will be introduced.
The final tests of the models are included in order to describe how well they depict reality. It will be
discussed how reasonable the assumptions behind the models appear, to describe the behaviour of
returns in the Danish capital market.
Zt = α + βZkt + et ; E[et] = 0 ; E[etet`] = ∑ ; ∑ = variance-covariance matrix of the disturbances . Where i denote the
asset i and t denotes the timeperiod. Zt is an (Nx1) vector of excess returns for N assets, or portfolio of assets. Z kt is the
(Kx1) vector of factor portfolio excess returns, and α and e t are (Nx1) vectors. Campbell (1997).
8
4
The project will be divided into a theory description of the models, a presentation of the Danish
return data, the tests and finally the interpretation and presentation of the empirical results of the
project.
The structure of the paper can be illustrated by the following:
Many financial models are present to describe the capital markets. First focus will be on an
introduction to some of the most widely used models, and thereafter a discussion will be undertaken
regarding the assumptions behind them:
1. The single index and multi index models will be introduced and following two open models –
the CAPM and APT model – will be illustrated.
1.2 Are the assumptions behind the above mentioned models reliable and valid in a real world
context?
The second part of the project will more specific relate to the Danish capital market and the tests of
the CAPM.
Through empirical analyses and discussions the author will seek to describe:
2. Are the available data found and the instruments used to analyse them, valid to test the
CAPM on the Danish capital market?
2.1 Can the CAPM describe and model the Danish capital market based on BE/ME portfolios
and Branch portfolios?
2.2 Are the test results reliable and valid, and what is there to learn about the Danish capital
market?
1.2 Method and structure
This project will be based on both empirical and theoretical references. This section will give
insight into the structure of the paper and the considerations concerning the methodical procedure,
with respect to data collection, use of theoretical and empirical material and reliability and validity
of the paper.
5
The project is divided into five main areas: the introduction, a theoretical discussion, data
presentation, empirical analyzes and finally a conclusion. Each section will end with a sum up of
the most important conclusions and results.
Part 2 the theoretical discussion
Over the years many attempts have been done to come up with a model that holds in theory as well
as in real life. To understand the relevance and difficulty of describing and modelling the return on
the Danish capital market, it is important to get an overview and a basic understanding of the
subject. This project will present basic finance theory, some of the models laying the ground hereof
and the complications with regard to the structure of the capital market. The first part of the project
is basically theoretical and lays the ground for the rest of the project.
3. Data presentation
The data presentation is the introduction to the tests of the project and is the ground of the project. It
is one of the most important cornerstones for the outcome. Data collection with regard to the Danish
capital market, however, is very time consuming. Due to the sensitivity to possible biases of the
statistic and empirical research methods, it could be devastating and troublesome for the project if
no focus is laid here. Since the Danish capital market is very rule changing and volatile, no
representative records are found on return data, and due to the minimum of time for the project, the
data collection and processing has not been to its fullest. As a consequence hereof these problems
will be discussed and treated in the best possible way.
The chosen time period dates back to 19899. This year was a turnaround for data and data collection
on the Danish capital market. Next to that the data are more reliable and valid compared to earlier
periods and is considered the best possible representative sample.
4. The Empirical analyses
The empirical analyses will be based on material found and presented in “data presentation” and the
theory described.
The models used are due to relevant statistical material and references. Excel will be used to
estimate the model though many other programs could have been preferred10.
9
Data from OMX is present on DataStream from august 1989
e.g. Eviews or SAS.
10
6
Despite many considerations and amount of published statistical material covered, it is important to
take account of the chosen models and the paper’s validity and significance. The empirical results
will be compared to international studies to compare the Danish capital market to international
market tests. The aim is to provide a better understanding of the structure of the Danish capital
market, and to compare different variables that might have an influence to describe the return on the
Danish capital market.
5. Evaluation and Conclusion
This part of the project contains a short sum up of the theoretical discussion, the results of the
empirical studies and the main conclusions of the paper. Also, this section will include a discussion
of future research and perspectives of the project.
1.2.1 Considerations concerning validity
Validity is a guaranty for correct results by using accepted and correct performed methods. The
study will aim to reassure that the validity of the paper and the evaluation and conclusions contain a
certain degree of quality.
An intensive research in finance literature was conducted to give a broad perspective on the
theoretical spectrum. It is assumed that all theoretical analyses and empirical studies that are used
and reflected to during the project, to some degree are affected by the respective writer’s subjective
stands and purpose, and a critical evaluation will be conducted throughout the paper, to increase the
validity of the project.
With respect to the empirical study of the paper, a thorough discussion will deal with biases and
statistical problems. In section 1.2.4 the three most important quality criteria will be presented;
validity, sensitivity and reliability (Zikmund, 2000).
1.2.3 Delimitation and definitions
The main emphasis will be on stocks and the pricing of the stock market. Bonds and derivatives will
be considered as an alternative to stocks.
7
Microstructure effects like bid-ask spreads, trade costs, liquidity etc will not be taken into account
during the project due to the enormous amount of material it would cover, which may have negative
consequences to the output of the project.
The project will be presented solely from the view of the investor. Due to relevance, to limit the
amount of material and not to change the direction of the project toward subjects as corporate
finance, any interests concerning the company will not be taken into account.
It is important to point out that this project reflects the Danish capital market. However, a reference
to foreign studies of international capital markets will be necessary for comparing and evaluating
the results of the tests.
It is important to keep in mind that the tests solely refer to the short term investor. To include the
long term investor will in worse case both increase the insecurity of the tests and increase the
amount of material, since other and more demanding features should be accounted for.
The CAPM and ATP models are “open” models. Since the introduction, variations and test methods
have been presented in various papers, and to cover it all would seem endless. Only few tests will
be explored in this paper. However, variations of the tests and models will be presented and
explained if found relevant for the project.
1.2.4 Validity, sensitivity and reliability
Three criteria must be fulfilled to reassure that the tests and the project are valid: validity,
sensitivity and reliability (Zikmund, 2000, pp.279). These three criteria will be presented in the
following.
Reliability is when the result of the conducted analysis is consistent. There are two ways to test for
reliability; a pre test or to split the material into two and check whether the two halves provide the
same results. The second method is best performed with two independent persons. However, to split
the material in two would easily cause statistical errors since the material would not be
representative. A pre test and comparisons to other equivalent tests however, would be relevant for
this project. A number of tests are performed to ensure consistency. Despite the statistical bias in
the data material it is concluded that the analyses are approximately consistent.
8
The collection and selection of data was done mainly by the author and with help from a student
worker at the Aarhus Business School. This may however also have consequences for the reliability
of the data. Since only one person is analysing the data, it is important to keep in mind that biases
come up due to weakness of the data collection.
Next to that, Parum (1998) mentions three independent attempts for estimating time series of stock
returns in Denmark11. However, a major issue that comes up is the lack of continuity in Danish
return data due to changes in regulations, tax etc.
Despite the issues mentioned above, it is concluded that the reliability criteria is partly fulfilled.
Validity is whether the models will test what actually is to be tested. Parum (1998) mentions some
important criteria that should be discussed when dealing with Danish return data.
1. What is the “best” data series for stock returns that can be estimated by comparing different
data sources, and how far should one go back in time with respect to data quality.
2. Are there any systematic and important failings in the historical total stock data due to nonrepresentative random samples, inappropriate estimates of the direct return of the market
portfolio, lacking adjustments for bankruptcy etc.
3. Where to put work effort to further improve the data quality of historical total return stock
data
a. Is it worth the trouble, when considering the huge insecurity on the estimates due to
investor-tax, the spread on the estimated mean values, the basic problems when
using before-tax return data, the switch from private to institutional saving and the
internationalisation of capital markets?
4. Can historical stock market data be used for estimating future expected risk premiums for
Danish companies?
The above mentioned become relevant when evaluating the material found on DataStream.
DataStream is a wide used database that contains a vast number of data related to among others the
Danish capital market.
Parum (1998) conclude that attempts in Denmark to create series of Danish return data, lack
documentation hereof as well as a discussion of the quality of the data. When looking at the
collected material and the results of the statistical tests, one can only agree with Parum(1998), and
11
Parum, august 1998, page 5 Christiansen & Lystbæk (1994), Lund og Engsted (1996) and Nielsen & Risager (1997)
9
for that reason it is important to keep in mind the lack of consistent data available from the Danish
capital market.
Next to that the models are relying on many stringent assumptions, which are not fulfilled.
However, this will be discussed in a later chapter.
For the continuation of this paper, it is concluded that the criteria are approximately fulfilled.
Sensitivity refers to the ability to capture the variation in the respective tests and whether any
variation in the material is perfectly captured. Since the material on the subject is of some size there
has not been the time for the author to go through all available material. Thereby the choice of
material and the estimation process to some extent become partly subjective based on certain sets of
articles and databases. It would be relevant to improve the discussion of e.g. tax problems,
dividends, changing regulations and changing levels of inflation. Parum (aug 1998) states that it can
hardly be recommended to use historical return data from stocks and bonds in Denmark to estimate
relevant future risk premiums for Danish stock registered corporations.
According to Ang (2005), the unconditional model, which will be tested here, does not fully capture
the variation in the model. This issue will be dealt with in a later chapter.
Due to the complexity of the data material, (Parum, aug. 1998), it is concluded that the sensitivity
criterion is approximately fulfilled.
10
Part 2
Theoretical discussion
To understand and construct models of how the world works, intuitively one must set assumptions
to simplify what are thought to have only a small effect. As the physicist builds models of a
movement, the economist builds models to describe the movement of stock prices. In part 2, the
simplest form of an equilibrium model (CAPM) will be presented based on the most stringent
assumptions. Finally a new theory of asset pricing will be introduced: APT. However, the final test
of a model is not how reasonable the assumptions appear, but rather how well it describes reality.
11
2. Finance theory
This part will focus on and present basic finance theory, including the theory behind the CAPM and
the APT. The aim is to give the reader a general overview of the many tests and articles that deals
with the two interesting models. The CAP model is characterized by empirical problems that may
reflect theoretical failings, the result of many assumptions or difficulties implementing valid tests of
the model. These problems and failings of the models will be discussed.
The CAPM and APT are some of the most simple and wide used models for estimating firms cost
of capital and evaluate the performance of managed portfolios. The CAPM is the oldest, simplest
and most prominent asset pricing theory in finance. The CAP model is built on several assumptions
which among other things cause empirical failings. This led to the development of the APT models,
and most well known the three factor model first introduced in articles by Fama and French. This
section will give a thorough introduction to finance and some of the models that influence the
theories and empiric of models of equilibrium in the capital markets.
2.1 Introducing financial models
The number of articles, theories and alternative material on how to describe the capital market and
calculate expected return is numerous. The focus of this project will be one of many central points
in traditional finance theory; the trade off between expected return and risk. The CAPM illustrates
how a difference in expected return on respective assets is reflected in differences in risk and the
risk premium (Rm – Rf).
There are basically two different approaches when estimating expected return. One is to estimate
future expected returns based on historical data and the other is to do a more theoretical approach.
Historical return data
When the estimation of the return of the market portfolio is based on historical return data, it is
assumed that investor’s expectations to the future will equal the past. The estimate is found by
calculating an average of the actual historical realised return of the market portfolio. Such a
procedure could be argued for if no other information were available. However, other information is
available like e.g. the rate of inflation and the risk free nominal rate which both varies considerably
over time. Calculating the expected markets return based on historical return data is, according to
Parum (1998), not to be recommended. This is due to two reasons. First the risk free asset varies
12
considerably over time which will give varying expected risk premiums. Secondly, the expected
nominal return of the market portfolio may depend on the rate of inflation.
A prevailing view is that it is more appropriate to take a starting point in the existing return of a risk
free asset and then add a risk premium from the historical return series as the historical average
realised abnormal return of the market portfolio relative to the risk free asset. Assuming that there
exist a normal stable risk premium for the market portfolio then the expected future risk premium
can be estimated by the average past risk premium. This assumption imply an expectation that
investors today claim the same reward for taking on risk as they did for 10,30 or 70 years ago 12.
The single index model
The single index model is a very used model that simplifies the estimation of the assets correlation
structure. Since it is a static description of single assets correlation with the market, the model is not
to identify with an economic financial model and it is not to compare with CAPM since no
equilibrium assumptions are made.
Dealing with many assets the number of correlations to estimate is huge and the single index model
becomes adequate. Assuming that one underlying factor, typically the market return drives all
returns, then for the single asset i
-
Rit = αi + βit*Rmt + eit 13 , cov(eit;ejt) = 0
(1)
And
αit = αi + eit14
σi^2 = βi^2σm^2 + σei^2
σij = βiβjσm^2
12
A basic hypothesis in traditional finance theory is the Efficient Market Hypothesis (EMH); that prices reflect all
relevant information and only changes when new information is in the market. This means that abnormal risk adjusted
returns are said to become unpredictable and have an expectation equal to zero. The concept of informationally efficient
markets can be defined in terms of the random walk hypothesis. Let rt = pt – p(t-1) = µt + εt and assume that E(εt|F(t1)) = 0, where F(t-1) is some information set used to forecast returns. This is Famas definition of capital market
efficiency (EMH). It is purely statistical description. Thus, under EMH one cannot expect a return that differs from the
historical average. However, an implication of this model is that technical trading strategies using F(t-1) will not work.
Classical tests here of include tests for serial correlation, event studies, and tests for the random walk model and for
correlation between asset returns and past asset related information, depending on what form for efficiency to test.
Engsted (2004)
13
E(ei,t;(Rm-rm)) = 0. For at portfolio of assets, rp = ∑(xi*ri = ∑[xi(αi+βi*Rm), where xi is the portfolio weights xi =
Zi/∑Zi. Note that there is a time (t) connected to return (Rit) but not to α.
14
eit is a stochastic variable, and it is assumed that E(ei) = 0
13
From the equation,
-
cov(eit,ejt)15 = σi,i = βi*βj*σm2
(2)
the covariance depends on market risk. This (2) is an important main assumption in the single index
Alfa and Beta in equation (1) is a constant and a risk factor respectively. Alfa is independent of the
market whereas Beta is dependent and related to the market. Beta expresses the assets systematic
risk, the risk that cannot be eliminated. From the two equations an important conclusion can be
made, that σe,i is unsystematic risk that can be diversified16.
Every single asset is related to the market alone, and not to any other assets. Every single portfolio
manager only needs to know the relation between his own branch and the market. The asset i risk in
the single index model can be defined as follow:
-
Define αp = ∑Xiβi for i=1...N , βp = ∑ Xiβi for i=1…N
(3)
 Rp = αp + βp*Rm
(4)
 σ^2p = β^2p*σ^2m + ∑X^2i*σ^ei for i=1…N 17
(5)
Again note that the individual risks (σei^2) can be diversified away, but that each assets contribution
to the total portfolio variances (βi^2σm^2) cannot be diversified away.18
From (5) we have
-
σ ^2p = β^2p*σ^2p + ∑X^2i*σ^ei for i=1…N, assume that Xi=1/N
(6)
 σp^2 = βp^2σm^2 + 1/N∑1/Nσei^2 for i=1…N
 σp^2 = βp^2σm^2 + 1/Nσei^2
 σp^2 → βp^2σm^2 for N → ∞
If the single index model holds, then even small portfolios will have the individual risks diversified
away19.σei is the assets unsystematic risk.
Beta (β) is an expression for the part that cannot be diversified away, also called systematic risk.
Beta only takes positive values. When Beta is higher than 1 the assets are more risky than the
market (“growth stocks”). Secondly, when beta is less than one the asset is less risky than the
15
Assume independency between the residuals. If dependency, then data is still unbiased however not as efficient and
not with the lowest bias. That is, autocorrelation exists and something is missing in the model which can explain the
dependent variable.
16
Engsted (2004) σi^2 = βi^2*σm^2 + σei^2, where σm^2 = E(Rm-rm)^2 , note that for portfolio varians σp^2 = βp^2*σm^2
+ 1/N[∑1/N*σe,i^2] , so the higher N, the lower is the weight of σe,i ^2.
17
Engsted (2004) and Campbell (1997) : Σ^2 p = ∑∑XiXjβiβjσ^2m + ∑Xiσei^2 = ((∑Xiβi)(∑Xjjj)σm^2) = βp^2σm^2 +
∑xi^2σei^2 and Rp=∑xiRi = ∑xi(αi+βiRm) = ∑xiαi + ∑xiβiRm = A + B = αp + βp
18
Engsted(2004)
19
Elton m.fl (2003) Table 7.2
14
market (“value stocks”) and finally for beta equal to one the asset has the same risk as the market20.
Beta (βi) express the single assets risk assuming that the variance of the market return (σm^2)
describes the general risk in society. However, βi cannot be given any fundamental economic
explanation21.
However, no equilibrium assumption is made in this model and it is a simple static description of
single assets correlation with the whole market, under the assumption that single assets
unsystematic risk is non-correlated. CAPM is not identical with the single index model22, still the
single index model is wide used in practice to forecast the future correlation structure between
single assets. This model assumes that there is only one underlying factor causing that stock prices
move, typically the market index. However, empirically and fundamentally other things besides the
market index influences the way that stock prices move. This led to the multi index model.
The multi index model
The multi index model assumes that other factors than the market index influence why stocks vary.
Returns of N assets is described by a common set of indices of L>1 factors
-
Index {I1,t ; I2,t .... IL,t}
An important assumption in the multi index model is that E(ei,ej) = 0; The only reason why stocks
vary together is because of a common co-movement with the set of indices that have been specified
in the model. The test aims to evaluate whether the specified model is significant and approximately
good. The theoretical frame for the model is
-
Ri = ai + bi,1*I1 + bi,2*I2 + … + b*iLI*L+ci23
(7)
Assume that the main purpose is to generate forecasts of the future returns, then (7) can be used
directly. However if the model is to be used to describe the correlation structure between assets for
a portfolio, then one has to make the indexes independent24 of each other. If the Index {I1,t ; I2,t ....
IL,t} is uncorrelated
-
Ri,t = ai +bi1I1,t + bi2I2,t + … + biLIL,t+ci,t
(8)
Engsted (2004). Βi = σi/(σm^2) = [∑ [(Rit-rit)(Rmt-rmt)]]/[∑(Rmt-rmt)^2] and corri,m = σi,m/(σi*σm) = (βi*σm^2)/(σi*σm) =
βi*(σm/σi) and pij = σij/(σi*σj) = (βi*βj*σm^2)/(σi*σj)
21
Elton (2003) pp 149-152
22
Elton (2003) ch. 13
23
σi^2 = βi1^2*σi1^2 + βi2^2*σi2^2 + … + σei^2 and σij = βiβjσi1^2 + βi2βj2*σi2^2 + … and Rp = ∑(xi*Ri) and E[ei(Ij-Ii)]
= 0 and cov(ei,ej)=0
24
Orthogonal. However this procedure will not be discussed in this project. But see Elton (2003)
20
15
And for asset i25,
-
Ri = ai +bi1I1 + bi2I2+ … + biLIL
-
σi^2 = bi1^2σI1^2 + bi2^2σI2^2 + … + biL^2σIL^2 + σci^2
-
σij = bi1bj1σI1^2 + bi2bj2σI2^2 + … + biLbjLσIL^2
(9)
The factors can be chosen from e.g. two models, the industrial index model and the fundamental
multi index model. The industrial index model is composed by two factors I1 and I2, representing
the Market Return (Rm) and the return in a specific branch26 respectively. The fundamental multi
index model is composed by macro economic variables and the market return.
The multi index model have historically shown better empirical results than the single index model
to describe the historical correlation structure when calculating the expected return, E(Ri).
However, the single index model forecasts the future correlation structure better than the multi
index model27.
Price formation models
Financial price formation theory describes the creation of equilibrium returns and risk premiums in
the market. Most well known and accepted models are the CAPM, APT (Fama and French) and the
C-CAPM.
If an economist is to value an investment project or private equity in a Danish company the
company’s capital cost will be estimated and used to discount the expected cash flows. Assume
further that a company has the same risk as the market portfolio. In equilibrium the relevant capital
cost to value the company’s expected cash flows is the market portfolios expected return, which is
the expected return investors would give up to invest in a company from the market portfolio
The CAP model is an intuitive tool to measure the relation between risk and return. It is widely used
in theoretical applications as in practice. The theory and model behind the CAPM makes it
intuitively easy to use in practice. When doing econometric tests, the goal is to reduce bias and
improve the theoretical and econometric results of the model. To build the model on returns instead
of observed prices advances the model theoretically as well as in practice.
25
Engsted (2004)
A firms return can be affected by the market and several industries.
27
Engsted (2004)
26
16
-
CAPM: Ri = RF + βi(Rm-RF)
(10)
Where βi = Cov(Ri,RM)/σM^2 => βi can be estimated by regression
The empirical way to write (10) is:
-
Ri = RF + βi(Rm-RF) + ei 28
 Ri – RF = αi + βi(Rm-RF) + ei
(11)
CAPM can be tested in two dimensions, a time series and a cross section dimension. In a time series
dimension Beta is estimated and the hypothesis that Alfa is zero is tested. However, in a crosssection dimension the return on Beta is regressed.
CAPM is a one-factor model where the expected return alone depends on Beta, that is, the
covariation with the market portfolio. Empirical evidence shows that there is a positive relation
between return and beta. However, CAPM is rejected statistically in several studies and cannot
among other things explain the observed above average return on small companies or companies
with high B/M values.
This led to the development of different models under the framework of APT, Arbitrage Pricing
Theory. APT is a more general pricing model that among others covers the famous Fama & French
three factor Model. Based on important assumptions the equation for the expected equilibrium
return of asset i according to APT can be explained by
-
E(Ri) = RF + λ1bi1 + .... + λ1bi1
where λ1,….λj are Risk-Premiums of the riskfactors I1…Ij. (12)
The APT model includes many of the same assumptions as the CAPM. It does not tell anything
about what the factors I1…Ij represents, how many factors to include or what the size of the RP29
should be.
Under the framework of APT, Fama and French developed the most well known multifactor model,
where the expected return of single assets is related to three factors valued by a market portfolio, a
“size” portfolio and a “value” portfolio.
The C-CAPM is to some extent the most satisfactorily price formation model in finance. It
explicitly models the relation that savings (e.g. investing in financial assets) is done with respect to
consumption in later periods, and the risk-averse investor will balance out the consumption profile
over time by using the financial markets.
28
29
ei is a stochastic eror that explains and captures expectations-errors.
Risk Premium Ri - RF
17
Asset i is priced from the assets correlation with investors consumption. The discount factor
become time varying and stochastic, and the time-varying risk-adjusted above average return has an
expected value of zero (EMH). Unfortunately lots of empirical tests state that C-CAPM has
empirical problems like e.g. the “equity premium puzzle”, the historical very high above average
return on stocks relative to short bonds.
2.1.2 Resume
According to the CAPM single assets and portfolios are priced from the covariation with the market
portfolio, which is to compare with the single-index model. However, in the single index model, the
βi is a static description of the assets sensitivity to variations in an arbitrary market index. CAPM´s
βi provides us with an economic equilibrium description of the assets pricing from the market
portfolio, which in theory incorporates all asset types – financial as non financial. As many other
financial theories the CAPM is based on simple assumptions that does not hold exact in a real life
context. It becomes an empiric question how well CAPM describes real financial markets.
The above chapter was meant as an introduction to finance and should provide the reader with
different angles to explain the capital market. What will be in focus throughout this paper is the
CAPM and following the APT model. The choice is based on pure interest and it become clear that
the CAPM in spite of lots of articles, improvements and research, still is one of the biggest
challenges in empirical finance. The model fit is a huge challenge. However, research has managed
to improve the empirical results and overcome most complications over the years.
2.2 The CAPM
The CAPM is one of the most popular models in finance. It lays the ground for financial price
formation models like C-CAPM, APT etc. The model has been tested and further developed since
its introduction in the mid sixties. Followed by APT, F&F introduced the famous three factor model
in the mid seventies.
18
2.2.1 MV
The basis for the CAPM is found in the MV analysis marking the start of asset-pricing theory30.
MV analysis is a classical static one period portfolio theory that lays the ground for the single and
multi index, CAPM and APT models.
The central point of MV analysis is that the lower the correlation between the assets the less
portfolio variance and the less risky is the portfolio. By diversification it is possible to reduce or
eliminate the unsystematic risk. However, the systematic risk cannot be diversified for which reason
focus will be toward the unsystematic risk. Note that from equation (5) and (6) the lower the
correlation between the assets, the lower the portfolio variance and the lower the risk.
Graphically efficient portfolios can be illustrated as the efficient frontier, that is, portfolios with a
minimum risk (variance) for a given expected return.
The two main results of the MV regard elimination of unsystematic risk by combining portfolios of
assets with low correlation, and how the optimal combination of the risky portfolio and the risk free
asset is found by splitting up the portfolio decision between the risky portfolio and the risk free
asset. Based on these conclusions, Markowitz31 laid the groundwork for the CAPM with his article
of investors’ portfolio selection problem in terms of expected return and variance of return back in
1959. He argued that “… investors would optimally hold a mean-variance efficient portfolio, that is,
a portfolio with the highest expected return for a given level of variance32.
2.2.2 Introducing the cross section and time series dimension of CAPM
CAPM is one of the most tested models in finance. The far most studies found are on US data. This
project will test the CAPM on the Danish Capital Market. The CAPM can be tested in a time-series
and a cross-section. See figure 1
30
Engsted (2004): Investors risk aversion depends on the utility function.
Assumptions:
- Investor has quadratic utility of wealth and/or arithmetic returns are normal distributed. That is, utility is
defined over the mean value and variance of the return.
- Investor wishes to maximize utility
- Investor prefer more than less and is risk averse
- Either R is normal distributed or a quadratic utility function
31
32
Campbell, Lo and Mackinley (1997), pp 181
Campbell, Lo and Mackinley (1997), pp 181
19
Figure 1 Cross section and Time series dimension
T\N
1
2
1
R
R
2
R
R
..
R
3
...
i
..
N
t
..
T
Note: Red color: Times series regression. This dimension is used to estimate Beta values by normal OLS
Green color: Cross section regression. This dimension cannot be used to estimate Beta since no variation can be
found.
As can be seen from Figure 1 the CAPM can be tested in two different dimensions.
The basic CAPM is given by
-
Ri = RF + βi(RM – RF) + ei
(11)
Let T be a sample of observations, t = 1, … T, on Rit, RFt and RMt. The number of assets (portfolios)
is N: i = 1, …., N. Then the time series regression for each of the N assets is given by
-
Rit - RFt = αi + βi(RMt – RFt) + eit
(12)
The time series test is a test for whether αi = 0, whereby (11) becomes equal to (12). This test can be
done by N t-tests, one for each regression. If αi = 0 then the model shows consistency.
However assume that αi ≠0 in (12). A possible explanation could be that investors don’t find the RF
risk free. Tests conducted by Black, Jensen and Scholes33 showed that the intercepts varied from
zero. If (12) is rewritten, then the Zero Beta CAPM to test is
-
Rzt = αi + RFT(1-βi) + βiRMt + eit
(12a)
Then if the zero Beta explains prices, then eliminating βiRMt and solving for
-
αi = (Rzt – RF)(1-βi)
Since Beta only takes positive values, see (10), where Rzt > RF means that αi < 0 for βi > 1 and αi >
0 for βi < 1which is consistent with Zero-beta CAPM34.
33
Elton (2003) pp 345
Consistency is found when RFT(1-β) = RZT(1-βi). So we have αi + RFt(1-βi) = RZT(1-βi).=> αi = RZT(1-βi).- RFt(1-βi) =
(RZt – RFt)
34
20
From the time series regressions we have the estimated β`s and the residual variances, σei^2, for
each asset i = 1,…,N. Now if we let βi be a regressor in the T cross-section regressions, one for each
t we have
-
Ri = γo + γ1βi + ηi35
(13)
According to CAPM, the only factor to describe the expected return is βi, and as a result other
variables than βi cannot be significant – e.g. the σei^236 from the time series regression in (12). Next
to that CAPM describes a linear relation between βi and Ri so that βi^2 has to be insignificant.
Finally this means that for the regression
-
Ri = γo + γ1βi + γ2βi^2 + γ3σei^2 + ηi 37
(14)
And as a result
-
γ0 = RF
-
γ1 = (RM-RF) > 0
-
γ 2 = γ3 = 0
The model can be tested by traditional t-tests or simultaneous F-tests.
Instead of estimating every single asset and test the hypothesis that αi = 0 with separate t-tests for all
i = 1,…N, all N equations can be estimated simultaneous and tested with one simultaneous test.
With this method possible correlation between the errors eit across assets will be accounted for
which may give a stronger test. This method will be used in the project.
In matrix form we have
-
Yt = α + β(RMt – RFt) + et
(15)
Where Yt , α, β and et is Nx1 vectors and the NxN is the covariansmatrix for et. Define the
following:
-
E(etet´) = ∑
-
E(RMFt) = ụMF
-
E(RMFt - ụMF)^2 = σMF^2
Then the following multivariate test-statistic can be used to test the hypothesis that α = 0. The Wald
test:
-
J0 = α´[Var(α)]^-1α = T(1+(ụMF^2/σMF^2))^-1α´∑^-1α
38
(16)
Where γo and γ1 are regression parameters, ηi is the error, γo = Rf , γ1 = (RM-RF) > 0 which can be tested by
traditional t-tests
36
Cov(Ri,σei) = 0 ; No relation between Ri and σei
37
See Elton (2003) pp 349
35
21
Under H0: α = 0 then J0 is asymptotic χ2. If CAPM holds exact then α = 0 and J0 = 0. The less the
value of J0 the better CAPM holds.
Assuming that the error et is normal shaped, then a F-test for H0 : α = 0 is suitable with a GRS test,
which is a very often used for the tests of the CAPM
-
J1 = ( (T-N-1)/N)(1+( ụMF^2/σMF^2))^-1α´´∑^-1α
(17)
These tests will be used for this project.
Resume
According to the CAPM the equilibrium return of asset “i” is formed by the risk free rate RF, plus
the risk premium of the asset equal to the market price of risk times the amount of risk. However,
tests of equation (2) have both a cross section dimension and a time series dimension.
The CAPM implies that the expected return of an asset must be linearly related to the covariance of
its return with the return of the market portfolio. But the CAPM also suffers from many stringent
assumptions, which makes the model unrealistic and a theoretical tool rather than a real life
application.
2.3 Assumptions
The CAPM holds provided that severe stringent assumptions are fulfilled. But for all of these
assumptions to hold in reality is not realistic. The following assumptions apply39,
1. No transaction costs; this is a reasonable assumption. It could add a great deal of complexity
if transaction costs were to be included. If transaction costs were present the return from the
asset would be a function of whether or not the investor owned it before the decision period.
Assumption is not fulfilled.
2. Assets are infinitely divisible; any investor can buy one krone of e.g. A.P Moeller stocks.
Assumption is not fulfilled.
3. Absence of personal income tax; the individual is indifferent to the form in which the return
on the investment is received (dividends or capital gains). If income tax and capital gains
taxes were of equal size, then the major results of the model would hold. Unfortunately this
is not the case in Denmark. Referring to Parum (august 1998) Denmark has changed tax
38
39
Elton (2003)
Elton (2003) pp 293
22
rules and frames over the years. This assumption is NOT fulfilled. However, this
assumption is assumed to be fulfilled to lower model-complexity.
In the Danish tax system we have company- as well as investor tax, and it is investors demand for
return after investor tax that drives the valuation process.
Before tax-CAPM and after tax-CAPM are two equilibrium theories that are not to compare, and
according to Parum (1998) it is complicated to evaluate whether one or the other is right when
dealing with empirical data. Before tax-CAPM assumes that investors marginal tax level is
identical. In this analyse it is assumed that investors are taxed identically of all financial assets.
4. An individual cannot affect the price of a stock by his buying or selling action; perfect
competition. No single investor can affect the prices by an individual action, but investors in
total determine prices by their actions. This is a common assumption when dealing with
economic and finance, all things being equal it simplifies the tests and the analyses.
Assumption fulfilled.
5. Investors are expected to make decisions solely in terms of expected values and standard
deviations of the returns on their portfolios; assumed fulfilled.
6. Unlimited short sales are allowed; this model can be derived under either of the descriptions
of short sales. So whether or not this assumption is fulfilled does not have any further
impact for the tests40.
7. Unlimited lending and borrowing at the risk free rate; The investor can lend or borrow any
amount desired at a rate of interest equal to the rate of interest for risk less securities. This
assumption is not fulfilled.
8. Homogeneity of expectations  a classical assumption that are partly fulfilled.
a. Investors are assumed to be concerned with the mean and variance of returns (or
prices over a one single period) and all investors are assumed to define the relevant
period in exactly the same manner
b. All investors are assumed to have identical expectations with respect to the necessary
inputs to the portfolio decision.
9. All assets are marketable; all assets, including human capital, can be sold and bought in the
market. This assumption is not fulfilled.
It should now be clear that many of the assumptions behind the CAPM are untenable and do not
40
Engsted (2004), Elton (2003)
23
hold in the real world. But how much are the empirical tests distorted by these assumptions and are
these assumptions a necessity to describe the actual performance of the capital market? It is
interesting to look into the econometric tests of the CAPM that might reveal a pattern of the
influence of these assumptions and what statistical and econometrical foundations that are lacking
for this model to make the best possible fit. However, these considerations will not be dealt with in
this paper.
Resume
This chapter went over the many assumptions that the CAPM is based on. The conclusions illustrate
how the model fails due to many unrealistic assumptions. Though the assumptions simplify the
model, the main effect clearly shows in the data, data processing and validity of the tests. One of the
most prominent critiques is Rolls critique, which have been a classic discussion of the CAPM’s
validity and model fit since the seventies.
2.3.1The Market Portfolio and Rolls Critique
An important point in the debate of the CAPM and the validity of the model is Roll’s critique. Rolls
critique claim that the CAPM may appear to be rejected in tests not because it is wrong but because
the proxies for the market return are not close enough to the true market portfolio available to
investors.
The ninth assumption, that all assets are marketable, has been the central point for Rolls Critique,
but causes a conflict since the market portfolio according to Rolls critique should incorporate all
assets41.
For simplicity of the data processes and to fulfil the ninth assumption it is a necessity to assume that
investor’s wealth only consist of stocks. This makes it possible to approximate the market portfolio
return by a broad stock market index, RM, as the expected return on OMX. This simplifies the
procedures and the ninth assumption is fulfilled since assets are marketable, however, the “market
proxy” is not perfect.
Roll come up with two relevant points. Firstly, tests of the CAPM are whether the market portfolio
is mean-variance efficient. The CAPM holds if the market proxy used is Mean Variance Efficient
(MVE). If the proxy is not MVE the relationship between E[R] and Beta will not hold. Secondly
Roll points out, that since the market portfolio is not identifiable it does not make sense to test the
41
One cannot sell eg. human capital and to incorporate real estate become too complex
24
CAPM. From the above analyses it was argued whether the market proxies used included all assets;
stocks as well as Real Estate and human capital. This would complicate the process and the
conclusion of Rolls statements is that the CAPM is useless since it is not testable. Instead Roll
advocates the use of APT. The usefulness according to Rolls critique may state though, that if the
CAPM tests do not hold then the so-called “market” (or market proxy) is not MVE.
Empirically what is understood with a market portfolio is a broad compounded portfolio of stocks.
However, the questions are whether it relates to the OMX, the Nordic stock exchanges, European
stock exchanges or a global stock market, is it a value weighted or equally weighted portfolio etc.
Most tests from the US are based on some national stock market portfolio like the NYSE and S&P
500.
Most presented American market portfolios are strong positively correlated as a consequence of the
American stock markets relatively influential size. A (national) American market portfolio may
therefore be strong positively correlated with an (international) world market portfolio.
The situation however, is different for a more modest stock market like the Danish, where the
choice of a market proxy will have other consequences for the test results. In the last couple of
years, the Danish stock market has gone through an internationalisation process with the EU, a
common Nordic market etc. According to Parum (1998) historical returns of stocks and bonds in
Denmark cannot be used to estimate future expected risk prices for Danish stock registered
companies42, since big parts of the Danish capital market in the past have been valued by Danish
investors. In the future the valuation will to a higher degree take place in an international space.
Nordea and Danske Bank as an example has departments internationally that trade stocks.
Using past data from the CSE43 to estimate expected risk premiums for a Nordic, European or
global capital market for that reason may seem some disorientated. In the future, segmentation of
the capital markets will still take place. If past return data are to be used, it seems relevant to use
data from a number of European countries as a consequence of internationalisation of the stock
markets.
Resume
The model assumptions are central for the theory but also very stylized and simplified.
42
43
Parum (aug 1998)
Copenhagen Stock Exchange
25
For simplicity it is assumed that investor only hold stocks, whereby the market portfolio can be
approximated by a broad market index like the OMX (DK) index or S&P500 (US). In theory
however, the market portfolio should contain all components of investor’s wealth and not only
stocks but also bonds, human capital, real estate etc. Richard Roll argued that one cannot observe
the true market portfolio exactly and that the CAPM therefore cannot be tested. Instead econometric
tests of the CAPM show how efficient the chosen proxy is for the market portfolio.
The CAPM is like any other economic theory a vulgarization of reality and cannot be expected to
hold exactly. What becomes interesting in light of Rolls critique is how well the model holds as an
approximation and whether the chosen proxy is satisfactory due to the internationalisation of the
capital markets.
2.4 The APT model
The CAPM is a one factor model, where the expected return alone depends on Beta and the
covariation with the market portfolio. However, several important characteristics regarding returns
variation over time and across assets, the CAPM cannot explain. This led to the development of a
multi factor model, APT. The most popular multi-factor model today is F&F three factor model,
where the expected return under the frames of APT is related to three factors, a market portfolio, a
size portfolio and a value portfolio.
The APT model is a more general asset pricing theory than the CAPM. However, CAPM is a case
of APT.
If the single index model holds and the index is the return of the market portfolio, then the APT
becomes identical with the cross section CAPM assuming that RF is constant44. If the multi index
model holds, the market return is not a factor, the N>1 factors (index) together describes the market
return and if each assets systematic risk can be described by the correlation with the market
portfolio (βi), then the N-factor APT is consistent with the CAPM45.
Like the CAPM and other economic models, the APT “suffers” from stringent assumptions, which
hardly holds in reality. Some few assumptions are added to those for the CAP model.
44
45
See (7)-(9) Rit = ai + bi*Rmt + eit => Ri = RF + λi * bi1 + eit, where λi = RM-RF, bi = βi = [cov(Ri,bi)/σ(bi)]*(RM-RF)
Engsted (2004)
26
1. Basically the same assumptions as in the CAPM; homogenous expectations, perfect competition,
no taxes or transaction costs, risk free rate, short sale allowed… etc.
2. The “law of one price” (LOP) is an important assumption, saying that assets with the same risk
have the same price. The “no arbitrage” argument assumes that it is not possible to generate an
abnormal return.
3. Return on an asset can be described by a linear multi index model (7), so the model has to be
linear in the parameters b1...bj.
4. Is not an assumption but rather a statement/notification. There are no assumptions about the
utility function defined over the mean value and variance, or about the return distribution. However,
the linearity assumption is not as restrictive as first assumed, since any of the indexes can be a
nonlinear function of a variable. Often however, it may be assumed that investor is risk adverse46.
2.4.1 The empirical APT model
A relation for the expected equilibrium return is written as
-
E (Ri) = RF + λ1bi1 + … + λjbij ,
(18)
where λ1 … λj are the risk premiums for each risk factor I1,..Ij. This equation is a general relation
that assumes LOP (no- arbitrage). APT says nothing about what the factors represent, how many
factors it takes or what the sign or size of the risk premiums are. To give the APT an empirical
content, three factors will be specified in the following principal component factor analyzes.
The principal component and factor analyzes deals with N assets and a covariance matrix Ω47. For
the principal analysis a limited number of L< N principal components (factors) are withdrawn by
the eigenvalues and eigenvectors that together describes the main part of the variation in the N
returns. The L principal components are then used as factors I1,…., Il 48.
In the factor analyses each of the N returns are regressed on the L factors from the Principal
Component analyses. The regression coefficients, factor loadings, then become estimates on
bi1,..,biL. Finally, the returns are regressed on the L factor loadings and estimates of the riskpremiums λ1,…., λL are found. Empirically it has been proved that 3-4 factors are fully to describe
the main part of the cross section variation in stock returns (see Elton, table 16, 2).
46
Elton, pp 365
Engsted (2004) Ω = (1 – λ)^2 – ρ12^2 = 0
48
| Ω - λI | = 0 ,
47
27
Certain problems come up with this model that makes it hard to work with. Since sign and size on
the factor loadings and the risk premiums are arbitrary, the factors are considered hard if not
impossible to interpret economically. However, one suggestion in literature is to interpret λ as how
big a part of the variation that can be explained by a certain factor. This interpretation will not be
dealt with further here, but will be left for the reader to explore on its own.
The macro economic factors can be found in two different ways.
One way is to specify the bi´s directly by observing company or sector specific variables, e.g. D/P,
P/E, beta, size, sector dummies etc49. Another way is to specify the basic macroeconomic factors
behind the systematic risk, like inflation, BNP growth, rate-spreads50. After the factors have been
specified the λ´s can be estimated by a regression analyses. The analysis seeks to cover the
economic interpretation of the factors and how they are priced.
Another approach is to specify the factors as portfolios made from company or sector specific
characteristics. The approach is the same but the factors are estimated as portfolio returns. A
prominent example of this approach is the Fama and French model, where three factors are
specified as follows:
1. The difference between the market return and the risk free return (Rm – Rf)
2. The difference between the return on small companies and the return on big companies, the
size factor (SMB)
3. The difference between the return on stocks with high book-to-market values and the return
on stocks with low book-to-market values, the “value” factor (HML).
The three factors are tested with the time series regression
-
Rit – Rft = αit + bi1(I1t-Rft) + ... + bij(Ijt-Rft) + eit
(19)
The APT says that αi = 0 for all i, which can be tested by individual t-test or multivariate Wald –
and Gibbons-Ross-Schanked (GRS) test.
2.4.2 The F&F three factor model
Under the frames of APT, F&F specified the following model
49
50
see also Elton, 2003, table 16.1
Elton (2003) pp 278-279
28
-
Ri-RF = αit + bit*(Rm-Rf) + si*SMB + hi*HML + eit 51
(20)
The first beta term characterizes the CAPM. The two other terms are defined in the chapter above.
The parameters αi, bi, si and hi are estimated by OLS. If the CAPM holds, then αi = si = hi = 0 for
all i´s. If the three factor model holds within the frames of the APT, then αi = 0 for all i´s.
Fama and French (Fama 1996) tests the above written three factor model on US monthly data from
1963 to 1993, where the following data series are constructed.
1. Rf is the 1-month T-bill rate
2. Rm is the value weighted return on a broad portfolio of NYSE, AMEX and NASDAQ
stocks.
3. SMB and HML are found by once every year spreading the stocks into two groups after size
(market value), S and B, and into three groups after the book-to-market values: L, M, H.
This combined gives six groups; S-L, S-M, S-H, B-L, B-M, B-H.
SMB is then defined as the difference between the average monthly return of the three S groups and
the average monthly return of the three B groups. Likewise, HML is defined as the difference
between the average monthly return on the two H groups and the monthly average return of the two
L groups. The monthly returns are then calculated over the following year.
25 portfolios are constructed based on the size of the stock (5 groups) and book-to-market values (5
groups). Ri is then estimated as the average monthly return on the portfolios, that is i= 1, …, 25.
(19) is then estimated for each of the 25 portfolios over the period 1963-1993 and the GRS test is
used to test the H0 hypothesis that
-
H0: αi = 0 for all i
(21)
It is analyzed whether the three factor model can explain the anomalies the CAPM could not
explain (e.g. the returns dependency of E/P, C/P and revenue; long term reversal; short term
momentum).The positive relation between return and E/P and the negative relation between return
and earlier growth in revenue can both be explained by the three factor model ( Fama, 1996, table 25). Further, the three factor model can explain the long term momentum, but not the short term
momentum. Portfolios with relative good returns in the last five years gives on average relative bad
returns the next month (long term momentum), and portfolios with relative good returns in the last
years gives on average relative good returns the next month (short term momentum) (F&F (1996)
table 7).
51
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html
29
For the US data over the period 1964-1993 F&F found the following52
-
RM – RF = 5,94% (std.dev = 16,33%)
-
SMB = 4,92% (std. dev = 15,44%)
-
HML = 6,22% (std. dev = 13,11%)
-
bi ≈1 for all 25 portfolios
-
si > 0 for all portfolios except the largest stocks
-
hi > 0 for all portfolios except those stocks with the lowest Book-to-Market values
-
ai ≠ 0, but ai < 0 for bi > 1 and ai>0 for bi<1 which is consistent with the zero-beta three
factor model.
They found a positive relation between return and E/P and a negative relation between return and
growth in revenue. F&F conclude53 that the three-factor relation is a good model for the returns on
portfolios formed on size and book-to-market equity. However (20) is just a model, which surely
does not explain expected returns on all securities and portfolios.
Fama and French’s three factor model today rule out the CAPM as the main price asset model. The
model is good empirically. However the model has been criticised for being too ad hoc and without
any concrete financial theoretical arguments.
2.5 Long-term vs. short-term investors
From several perspectives one has to state whether the data material and model concern the short or
long term investor. In most literature however, the basic CAPM and the three factor model mainly
relates to the short term investor. The same goes for this project.
Empirically it is more interesting and relevant to assume that investor has a long investment horizon
due to several observations. Pension funds invest long term, mutual funds mainly approach the long
term investor and many of the well known investment advice and rules of thumb attend the long
term investor54. Intuitively a short investment horizon with short State bonds (T-bills) or cash are
relatively risk free while stocks and long bonds are riskier. In the classical short run MV diagram
52
Fama, Table 1
Fama, pp 82
54
In the long run stocks have a higher return and less risk than bonds. It is assumed that the younger, the bigger the
share of stocks, and finally the more risk averse the more bonds and less stocks.
53
30
the efficient frontier consist of both stocks and long bonds, while the intersect with the y-axe
indicate the rate of the short risk free bond or cash55
With a long investment horizon short bonds or cash intuitively are risky and the risk free asset
becomes a long bond. Focusing on the long term investor two factors are important to put under
consideration. First the possibility of rebalancing portfolios, since it is possible to change the
portfolio weights after each period. Secondly an investor saves to hold a given level of consumption
at an old age. Any given investors utility function should therefore be defined over consumption
rather than wealth56. When dealing with a long term investor, the main purpose is to analyze how
mean reversion and human capital affects the optimal portfolio choice. Models have been developed
to take those considerations into account, like C-CAPM, models for the equity premium puzzle and
asset allocation puzzle etc. Those models will not be dealt with here.
One of the main results in the static MV-analysis is the two fund separation theorem that states that
the division of risky assets and long bonds are independent of the degree of risk aversion. However,
investors and investment advisers does not act according to this theorem57. Risky investors are often
recommended to hold more stocks than long bonds compared to less risky investors. This asset
allocation puzzle can be solved by explicitly distinguishing between short and long term investors 58.
Another classical result in portfolio analyses states that if return has zero autocorrelation and
investor only hold financial wealth (e.g. no human capital) then the optimal portfolio for the short
term investor will also be optimal for the long term investor (Engsted, 2004).
This time
diversification is the opposite of what investors actually do and what investment advisers intuitively
recommend. Like the risk by diversification across stocks59 can be reduced, one can also reduce the
risk by diversifying across time. One way to solve this problem between short and long term
portfolios, is by assuming that the return on stocks is negatively auto correlated (mean reverting)
and by adding human capital.
55
Elton (2003)
Mean value and variance of return
57
Campbell, 2002, Table 1.1 ”Asset allocation puzzle”.
58
Campbell, 2002, chapter 3
59
See (6)
56
31
The long viewed investor saves for consumption for a longer period (e.g. pension), so utility has to
be defined over consumption, and it is assumed that the risk adverse investor will smooth out
consumption over time by selling and buying financial assets.
The C-CAPM is a special case of the CAP model developed on the basis of among other Rolls
Critique. The model incorporates consumption as a part of an assets wealth and attends the long
term investor60. The C-CAPM has proved to give a better fit than the original basic CAPM and the
F&F three factor model, however, there are still huge and important problems concerning the
availability and liability of the data material. Next to that several statistical failures still disturb the
results and fit of the model. The C-CAPM will not be explored further here, and the focus will
continue to be the short term investor.
2.6 Conditional CAPM and Unconditional CAPM
When Betas vary over time, standard OLS is disturbed and cannot be used to assess the fit of a
conditional CAPM. Also, when betas vary over time and are correlated with time-varying market
risk premium, OLS alphas and betas provide inconsistent estimates of conditional alphas and
conditional betas61
The unconditional CAPM was the first model of the popular CAP model – tested and used by
Black, Jensen and Scholes (1972), Fama and Macbeth (1973), Fama and French (1992, 1993) and
many others. It differs from the conditional CAPM by specifying a constant beta over the entire
sample period. Basically, the model is mostly tested by estimating the basic regression 62 on
portfolios of stocks sorted by book-to-market ratios. Under the null hypothesis of the unconditional
CAPM, the unconditional alpha is zero and the systematic risk, which is represented by Beta,
determine the expected returns. If the null hypothesis that alpha is equal to zero is rejected, it may
be concluded that the unconditional CAPM cannot price the average excess returns of asset i. In
many historical empirical studies the CAPM has been rejected.
(Ang, 2005) shows how the OLS alpha and beta estimates are unreliable in the presence of time
varying factor loadings, e.g. over the long horizon63, as CAPM can explain the appearance of B/M
60
Ang, 2005
Ang, 2005, pp1
62
Ri,t = α+ βt*Rm,t + σ *εi,t
63
1926-2001
61
32
effect from OLS alphas in the post-1963 sub sample, but not in the pre-1963sample. When
conditional betas vary over time, OLS cannot provide consistent estimates of either conditional
betas or conditional alphas.
The conditional CAPM differs from the unconditional version in several ways, is more realistic and
adapts fantastic to reality and the empirical tests. (Ang, 2005) use rolling OLS betas to show that
the true conditional betas vary over time. However it turns out, that the rolling OLS betas is not
directly the true betas since OLS estimates of conditional betas are misspecified 64. Never the less, it
is concluded that they “… can provide some rough characterizations of the true conditional beta
process”.
The perfect model would include rolling OLS betas. However, with a relative short sample period
from the Danish Capital Market this will not be conducted. We hold the assumption of a short
investment period and rely to the unconditional CAPM.
2.7 Resume
Private as well as institutional investors consider questions about how risk averse they are. To
eliminate individual risk, it is necessary for the investor to diversify optimally. The MV laid the
ground for the development of CAPM, explaining the diversification and two-separation theorem.
According to the CAPM single assets and portfolios are priced according to the covariation with the
market portfolio. The single index model gives a static description of the assets sensitivity towards
variation in an arbitrary market index. The CAPM Beta on the other side provides an economic
equilibrium description of the assets price determination from the true market portfolio that in
theory includes all assets, both financial and non-financial. As any other financial theory CAPM is
built on simplified assumptions that do not hold exact in the real world, for which reason it
becomes an empirical question how well CAPM describes the financial markets in reality. This lead
to other econometric tests of the CAPM and the APT model, as an extension of the CAPM under
which frame the popular F&F three factor model and C-CAPM are built.
64
Ang (2005), pp 6
33
Part 3
The Danish Capital Market.
This chapter is of relevance for the tests, the empirical results and the conclusions. Bias in the data
collected might explain the errors and the extreme results found in econometric tests. Denmark
unfortunately does not share the same access to data bases and series of adjusted prices reflecting
the capital market for years back as for example the US. Some attempts have been done in Denmark
to estimate a consistent time series of return data. However, the quality should be discussed before
the series are used, due to failures and problems characterising the Danish capital market.
34
3. Data series in Denmark
In the US one can find historical returns on stock data of good quality. These data have been used in
several studies with high risk premiums on stocks. Over the years several analyses and studies have
been conducted in Denmark on historical stock returns and bonds. However, most of these attempts
have now aged, only covers short periods, focus on prices or does not include the newest finance
theory65.
In this project data from Data Stream will be used. However, several data has been excluded due to
errors, missing values etc. and also the chosen period for the sample has been shortened.
3.1 Problems
Stock index price from StatBank Denmark since 1914 has been found on DataStream. This is a
unique example, since most data on the Danish capital market is not consistent and the rules of
reporting have been changing over the years. Three main problems are noticed by Parum (1998)
when estimating returns from stock index data. Each problem area might have an influence on the
validity of the result of the econometric tests:
1. Random sampling representatives before 1983
A problem when collecting data in Denmark is the definition of stocks on the Danish stock
exchange. Up till 1972 it was custom to distinguish between ordinary registered, extra ordinary
registered and not registered stocks which makes it troublesome to collect representative data. Next
to that most calculations until 1983 was based on random samples. After 1983 the Danish stock
exchange calculated a stock index based on all registered stocks.
Parum (1998) mentions that return calculations based on the Danish stock index up until 1983 is
based on random samples of relatively large and liquid companies, which would undervalue the
return from a representative portfolio of stocks. Calculations of return data on the Danish capital
market could show an eventual “small firm effect” that is so weak that it won’t affect the index66.
2. Bankruptcy problems
Common knowledge says that when a company goes bankrupt the value of the assets reduces to 0.
Parum (1998) points out problems with the presentation of Danish stock index data, since it seem as
if there are too big rises over time67. Often a company’s stocks are suspended, then the company is
65
Parum (aug 1998)
Empirical analyses based on American stock data points toward the existence of a “small firm effect”, that the
average stock return is much higher for small than large companies.
67
See Excel 1.
66
35
declared bankrupt and finally the stocks are deleted. While the company stocks are suspended they
still enter into the calculations of the indices with the price at the time of suspension. A central point
for the number of biases becomes the adjustment of bankruptcy.
3. Consistency problems before January 1924
It is not possible to find representative data from the Danish capital market before January 1924.
This is not relevant for this paper, since the projects goal is not to go that far back in time.
Denmark is covered in taxes, so would it make a difference if taxes were included in the
calculations?
Parum (1998) states that if the CAPM is used for estimating return on equity, then it is a
consistency criterion to use a CAPM which is derived with respect to as well company taxes as
investor taxes. In this project it is assumed that investors are taxed equally on all financial claims.
As explained in part 2, Investors will only claim a price for taking on systematic risk. In equilibrium
the expected return on a zero-beta stock will be equal to the risk free rate after corporate tax. Other
studies of the Danish capital market mark the Danish risk premium as the difference between the
return of the stock market portfolio estimated after corporate taxes but before investor tax. When
investor taxes as well as corporate taxes are present, different assets return will be compared on an
after-investor tax basis. A necessary assumption hereof is to do meaningful comparisons on a before
investor tax basis (but after corporate tax basis) so that the assets get the same tax treatment on
investor level.
The tax rules for Danish stock companies and Danish private investors have varied considerably
over the last couple of years. The Danish tax system have varied between being a so called classical
tax system with straight forward elements of double taxation and a so called neutral tax system. For
example defines Parum (1998) the Danish tax system from 1983-1995 as a neutral tax system where
income from stocks and bonds are taxed equally.
Taxes, rules, registration etc makes it difficult to create a consistent data set from the Danish Capital
Market. None the less, material from DataStream has been downloaded and analyzed.
3.2 The Sample period
A sample for a period of more than 10 years will not be optimal due to among others rolls critique
and the structure of the Danish capital market. However, this project will cover 17 years of the
Danish capital market. The following chapters will introduce the material collected.
36
Ang (2005) tests the CAPM over the long run for the period 1926 – 2001, where as Fama (1996)
tests the period 1963-1995. Opposite to the US, representative data from the Danish capital market
is not valid or achievable for that long a time period. To increase the validity of the data and avoid
further bias and complexities, a shorter but more reliable time period is chosen.
A time period more than 20 years back in time is difficult to create on the Danish capital market and
at the same time reassuring that the data is valid and representative (Parum 1998, Zikmund 2000).
In 1989 the KFX68 was introduced, representing a new period of data registration on the Danish
capital market. This marked the beginning of a better and more extensive registration of Danish
stock data.
The chosen period for this project is influenced by several factors which could complicate the
validity of the data. However, a period of 17 years is somewhat reasonable since it will increase the
validity of the data with fewer firms eliminated from the test. This will give a good time frame
combined with a relatively good representation of the market. The problem could have been solved
if the author had created a whole new index based on all of the corporation´s yearly reports. This
would be time consuming and according to Parum (1998) it is not worth the energy to spend the
time and economy on estimating indices that goes further back in time due to the complexity of the
Danish capital market.
3.3 Data
One of the most important and demanding tasks of the project is the data process. Working with
empirical real life data, one cannot fully eliminate empirical biases that, if not corrected or
accounted for, might cause wrong results or conclusions. Though accounting for and knowing the
biases, it cannot fully outweigh the negative influence here of, since some of them cannot be
explained.
Data selection and research is therefore a critical underlying task for the results of the relative
simple and straight forward basic theory. Empirical evidence however shows, that the theory fails
due to strict assumptions and biases from the data material. Related to this subject is the widely
known and well accepted Rolls Critique.
68
Today the OMX
37
3.3.1 Data description
Lots of empirical tests are made on American stock market data recent years. Databases like
COMPUSTAT and CRSP provides material covering years back calculated with different
frequencies and within different sectors in the US.
Choice of data and database is of huge importance for the direction of this project. Four different
databases have been under consideration; the Danish stock exchange, Compustat, Datastream and
Accountdatabase. However, though fully completed databases, biases do still interrupt and distort
empirical modelling.
A student-worker at the Aarhus Business School finished the task of collecting and gathering the
material from Datastream69 in the second half of 2006. This may have caused biases, since no
personal control of the data collection has been conducted. Zikmund (2000, pp244) define two
important sample errors, Sample selection error and Random sampling error. Sample selection error
is “An administrative procedural error caused by improper selection of a sample, thus introducing
error”, and random sampling error is: “A statistical fluctuation that occurs because of chance
variation in the elements for a sample”.
As a consequence it was necessary to go over the information and further delete errors and
companies from the Data material. The data covers a period of monthly observations from
august1989 to august 2006.
3.3.2 Data screening
This project is based on secondary data downloaded from the internet database, DataStream.
Though the data base is found representative, the process of adjusting price and collecting the data
might cause complications for the final tests. According to Zikmund (2000) using a secondary
source has advantages and disadvantages:
Advantages (Zikmund, 2000, pp 125):
-
69
Time and money saved
www.datastream.dk.
38
-
For some reports it may not be possible to obtain the needed reports due to deleted,
restrictions or financial matters.
Disadvantages (Zikmund, 2000, pp 125)
-
The data is not designed especially to meet needs; Tax, dividends, registrations, correct units
of measurement, variation in definition of terms, time period, user has no control over
accuracy etc.
When secondary data does not exactly meet the researcher’s needs, it often has to be transformed
into a more suitable frame. Researchers should verify the accuracy of the data whenever possible.
The data found on DataStream was in prices, and for that reason return was calculated70. If the
accuracy of the data cannot be established, the researcher must determine if using the data is worth
the risk. In this case, since no better material is found the tests will be conducted knowing the biases
found in the material. Certain criteria are important for the accuracy and understanding of the
material.
Timeperiod:
-
This is one of the critical decisions for the project. Due to the problems of data collection in
Denmark, a period of august 1989 to august 2006 is chosen. A longer time period could
have been favourable. The reason for this was the introduction of the KFX and a better
registration of stocks.
-
The data can be divided into three sub periods. 1989 to 1999 is the period before the stock
crash, primo 2000 to ultimo 2002 we have the stock crash and primo 2003 to 2006 is the
period after the stock crash. This may have an effect for the size and amount of return data71.
Riskfree rate:
-
As for the Risk free rate, Rf, Carsten Tanggard72 calculated the spot zero coupon rate for
1976-1991. For 1991 to October 1995 the 1 year bond yield is used and for 1995- 2006 the
10 year bond yield is used. The Rf is in p.a. and is divided by 12 to fit the monthly data73.
Market portfolio Return, RM:
-
The market return is an important discussion, since the outcome of the project could be
biased towards what proxy to use. One should consider what would be the best market
Rt = (Pt – Pt-1) / Pt-1
Note that for the datamaterial in Excel, green squares is a ”jump” in size, yellow square is negative BE/ME which has
been deleted and red squares are unlikely numbers.
72
Carsten Tanggard homepage: http://www.tanggaard.com/
73
For simplicity compound interests are not taken into account.
70
71
39
proxy and secondly if the data available are statistical representative; KFX, KVX, OMX or
new Nordic market index registration. In general, portfolios should not be limited to a
national market, like e.g. OMX. Adding foreign assets will often give better spread of risks
and diversification. If one observe the Finish, Swedish or Danish market its interesting to
note that the finish market is relatively big due to Nokia that covers more than 50% of the
finish market, Sweden has a more developed stock culture and is 4 times bigger than the
Danish market and the Danish market is a relatively large bond-market mainly due to the
Danish “realkreditobligationer”. Also note (Elton, 2003, table 12.1) that US has been 50%
of the world’s total stock market, Europe 33% and Japan 13% (the massive fall in the
Japanese stock market since 1990 has made Japans share of the world stock market smaller).
This makes one consider new elements, as
o Should one consider currency risk? It is assumed that currency rate Sek/dkk = 1 in
this project.
o Is it more realistic with a market portfolio consisting of a world-market-index instead
of a national index?
Historically, correlations between stocks across countries have been less than for stocks within
national borders. Before the financial crisis in mid 2008 the international stock markets became
more correlated due to globalisation, free capital moves and a higher degree of integration of the
international financial markets.
It could be argued to use the Nordic Market index, since the Scandinavian countries is somewhat an
integrated institution. However with the goal to analyse the Danish capital market the best
representative statistically and in practice would be the OMX. The sensitivity of the choice of
market proxy to the result of the project will not be dealt with here74. However, the OMX consist of
20 stocks, where 2 stocks make up 40% of the index; Novo Nordisk and A.P Moeller. These stocks
are also very sensitive to conjunctures, especially A.P Moeller stocks. Next to that the companies
that represents the OMX is changing all the time, which does not make the index consistent. In this
project the market return is represented by the OMX.
Other considerations to take into account:
o Dividend adjusted. The total return index found on Datastream for the period is
Dividend adjusted.
74
(Fama, 1992) pp 449 shows that the choice of market proxy is not sensitive to the SLB model.
40
o C-CAPM; What to be included to account for Rolls Critique.
o Value weighted stock index or value weighted Return.
o Since stock prices vary over the business cycle, it could be argued that the Risk
premium, may also vary over time (see Parum, 1998)
Stocks:
Market prices are observed. However, prices have not so nice statistical properties, so a scale free
measure of return is calculated
-
Return = Rt = [Pt-Pt-1]/Pt-1
-
Dividend adjusted index will be used, assuming that all dividends are reinvested. Stocks are
measured in excess of the risk free rate, Rf, and available on the Danish Stock Exchange. 13
stocks were deleted due to “error” or “no data available”.
Characteristics and deciles
-
The data will be manually split into 5 different portfolios characterized by BE/ME and
branch respectively.
o ME is stock prices times shares outstanding, and BE is DataStream book value of
stockholders equity, plus balance sheet deferred taxes and investment tax credit,
minus the book value of preferred stocks. Firms with negative BE/ME will not be
used in the portfolios. However, they will be a part of the proxy for the market
return.
o 5 Portfolios are calculated based on BE/ME. Deciles are 0-04; 0,4-0,8;0,8-1,2; 1,21,4 and 1,4 - ∞
o 5 portfolios are calculated based on branch; Financial, Material, Consumption,
Health and Energy.
o Few interpolations in the data set were done, but a necessity for optimizing the
material.
Each period new portfolios are made, new firms entered and firms deleted due to missing data,
bankruptcy etc75. In note 2 each of the 171 companies is described with available data, codes,
currency and branch.
3.4 Resume
In Denmark databases and indices to the same extent and quality as found in the US are not
available. According to Parum (1998) attempts have been made to create return series for the
75
See Excel 1
41
Danish Stock market (CSE, ministerial matters etc). However, it is noted that there are only three
real attempts in Denmark to set up continuous time series for the Danish Stock markets total
return76 .
From January 1924 it is possible to find Danish stock indices on a monthly basis. Three main
problems arise though, when estimating returns from Danish stock index data; First, problems
concerning the representative of the random sampling, Secondly bankruptcy, and finally
consistency problems. These are dealt with in (Parum, 1998), and will not be discussed further here,
but had the purpose to remind the reader about what problems arises in the data processing and
what biases influences the end conclusion. It should as well be noted, that the dividend policy as
well as taxes has changed over the years.
Five portfolios on branch and five portfolios on BE/ME is made for 1989-2006 on monthly data
downloaded from DataStream. The sample is with error and one has to take account of these errors
when it comes to the empirical tests. 171 companies are included in the portfolios and due to
missing data and bankruptcy etc not all will be representative in the portfolios over the entire
period. Despite sample errors and biases, the tests will be presented in part 4.
76
Christiansen og Lystbæk (1994), Lund og Engsted (1996) and Nielsen og Risager (1997)
42
Part 4
Empirical test
Ang (2005) perform econometric tests of the CAPM over the long run, 1926 to 2005. It is showed
that “… under a conditional CAPM with time-varying betas, predictable market risk premium, and
stochastic systematic volatility, there is little evidence that the conditional alpha for a book-tomarket trading strategy is statistically different from zero”77.
Value stocks have market values that are small relative to the accountant’s book value. This
category of stocks has large average returns. Growth stocks are the opposite of value and have had
low average returns78. High average returns are consistent with CAPM if these portfolios of stocks
have high sensitivities to the market as in high betas. Small and value stocks have abnormally high
returns, and conversely growth stocks seem to do systematically worse.
This project will not use the conditional approach. However, for a generation portfolios with high
average returns also had high betas. This project will test and illustrate the unconditional CAPM on
Danish capital market data based on BE/ME and industry portfolios respectively.
77
78
Ang(2005) pp.1
Cochrane (2001), chapter 20.2
43
4. Econometric tests of the Danish capital market
The major task behind this project was to sort the data material. However the statistical tests clearly
show that either this process has not been to its fullest, the models have been incomplete or the
material is of bad quality. Up till now this paper has presented a thorough presentation of the theory
behind the CAPM and APT and sample biases in the Danish capital market. This section presents
the process behind the tests and the results hereof.
4.1 The portfolios
Early empirical tests show that if one plot the average returns versus betas of individual stocks,
betas are measured with error. Fama (1973) addressed the problem by grouping stocks into
portfolios, since individual stock betas vary over time as the size, leverage and risks of the business
change. Portfolio betas are better measured because the portfolio has lower residual variance and
may be more stable over time and hence easier to measure accurately. Intuitively individual stock
returns are so volatile that you cannot reject the hypothesis that average returns are the same79. By
grouping stocks into portfolios based on some characteristic, you, all things being equal, reduce the
portfolio variance and it is more realistic what an investor would do.
Fama and Macbeth and Black, Jensen and Scholes formed their portfolios on betas. More recently
size, book/market, industry and many other characteristics have been used to form portfolios80.
In this project 5 portfolios are formed on BE/ME and 5 on industry as well. All calculations are
done in Excel, even though programs as Eviews have better features with regard to statistical
descriptive tests. To use Excel takes more manual calculations which require an understanding of
vector and statistical calculations. However, this method may also increase the chances for bias and
mistakes. Basically Excel fulfils the demands for the regression tests in this project, and is a very
easy tool to work with.
Correlation matrixes for the portfolios are shown in table 1 and table 281.
Cochrane (2001), σ/sqr(T) is big if σ = 40% - 80%.
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html
81
Excel 2
79
80
44
Table 1 Correlation Matrix for branch
Fin
Mat
Cons
Health
Ener
Fin
1,00
-0,64
0,53
0,43
0,22
Mat
1,00
-0,56
-0,55
-0,20
Cons
Health
1,00
0,42
0,13
Ener
1,00
0,15
1,00
Table 2 Correlation Matrix for BE/ME
Decile1
Decile2
Decile3
Decile4
Decile5
Decile1 Decile2 Decile3 Decile4
1
-0,29716
1
-0,04518 -0,29229
1
-0,07482 -0,06379 0,002561
1
0,031055 0,177309 -0,16737 -0,20752
Decile5
1
Material portfolio is negative correlated with the other portfolios. This is not very surprising, since
it can be argued that this portfolio is not as service minded a portfolio as health or energy and reacts
to different cycles. In Table 2 deciles are negative correlated or very weak correlated. Adrem (1999,
pp 74) refer to Gujarati, who argue that correlation between two variables should be above 0,8 to
cause a problem. This does not seem to cause a problem in this project.
Table 3 shows descriptive statistics for the portfolios.
Table 3 Descriptive statistic
Average
Median
Std.dev
curtosis
Skewness
#
Rm
714,88
587,93
763,54
6,02
2,25
207,00
Fin
1,05
1,18
3,82
0,65
0,01
207,00
mat
-0,68
-0,97
5,95
1,15
0,56
207,00
cons
0,72
0,63
4,61
-0,02
0,03
207,00
health
1,23
1,36
6,70
1,17
-0,57
207,00
ener
1,95
0,00
20,45
85,08
7,63
207,00
45
Average
Median
Std.dev
curtosis
Skewness
#
Decile1
3,67
0,21
28,33
24,05
3,39
207,00
Decile2
2,42
0,96
21,59
11,94
2,41
207,00
Decile3
3,58
0,31
31,60
10,66
2,02
207,00
Decile4
10,41
-0,56
66,71
37,27
5,04
207,00
Decile5
3,19
1,33
24,55
6,34
1,52
207,00
Characteristics for the Normal curve can be explained by Skewness, Kurtosis and a Jarque-Bera test
as follows82:
-
Skewness: S ~ N ( 0,6/T)
-
Curtosis: K ~ N ( 3, 24/T)
-
JB = (T-K) [ 1/6γ2 + 1/24 γ2] ~ χ2, where T = # observations and K = # regressor if applied
to residuals and otherwise its 0
For the Normal Curve then S = 0 and K = 3. In Table 3 Skewness is positive except for health
which is -0,57, whereby the portfolios are right skewed except for health which is skewed to the
left. Kurtosis is large and positive for deciles and health which show that the portfolios have fat
tails. Fat tails may cause a problem for autocorrelation, since fat tail makes it difficult to estimate
the variance. However, despite some statistical issues, the following tests were applied83.
4.2 The tests
For a long time a strategy or characteristic that seemed to give high average returns also turned out
to have high betas. But strategies that one might have thought gave high average returns (holding
volatile stocks) turned out not to have high average returns when they did not have high betas.
The results from the tests in this project are far from consistent with the CAPM or earlier tests,
found by the author, as can be seen from Figure 2 and table 484.
The following vector model was applied for the 5 portfolios on BE/ME85
-
Zt = σ + β*Zomx,t + et
Figure 2 and table 4 show the results from the test.
82
Campbell (1997) pp 17
See enclosure 5.
84
Excel 1
85
See (15), Excel 1 and Enclosure 2.
83
46
Figure 2 CAPM and BE/ME portfolios
Y: Excess Average return, X: Beta
The figure plots the Excess average returns against market betas. The solid line draws the CAPM
prediction by fitting the market proxy and Rf86. The dots show the CAPM estimation for the five
portfolios87. You can see how the CAPM prediction should fit: Portfolios with higher average
returns have higher betas. However as can be seen from the figure, the estimates does not fit to the
CAPM. The lines should all lie on a 45o line if the model is correct. There may be several reasons
for this; calculation error, data error, RF is not considered the risk free rate, wrong market proxy
(Rolls critique), a non linear model would suit better or a more general asset pricing model as F&F
three factor model is needed. As can be seen from Table 4, several errors come up.
Table 4 Beta & t-tests
Beta
se(beta)
t-stat
alpha
se(alpha) t-stat
0,004924 0,065587 0,075069 -10,8083 23,95667 -0,45116
-0,03136 0,064982 -0,48259 -20,6374 #NUM! #NUM!
0,102039 0,101646 1,003866 40,45027 8,209645 4,927165
86
87
CAPM: (RM-RF)*β
(Rp-RF) against Beta
47
0,065674 0,142839 0,459776 -6,8512 14,30816 -0,47883
0,022353 0,053124 0,420769 -4,42874 9,153836 -0,48381
Beta in portfolio 2 is negative with negative alpha. Portfolio 3 is positive beta with positive alpha
and finally for portfolio 1, 4 and 5 we have positive beta with negative alpha. For
-
H0 : α = 0 (The model holds)
-
H1 : α ≠ 0
the model is rejected with WALD ( ≈ 3731) and GRS ( ≈ 724 ) test much higher than the critical
values.
Calculating and writing the data material in percentage, affects the estimated nominal size of alpha
and the nominal variance of alpha. Since Beta is the slope and a relative number, the size of Beta
will not be affected88. However, the numbers in Table 1 still looks wrong and with error.
One of the first significant failures of the CAPM was the small-firm effect. The smallest firms
(should have been plotted to the far right and above the straight line in Figure 2) earn an average
return a few percent too high given their betas. Book market sorted portfolios in earlier tests, had a
tendency to show a large variation in average returns unrelated to market betas.
The portfolios created in this project have different stock sizes (ME) within the same Book/Market
(BE/ME) category. Cochrane (2001) shows that variation in size produces a variation in average
returns which is positively related to variation in market betas, and that variation in Book/Market
ratio produces a variation in average return that is negatively related to market beta. He concludes
“Because of this value effect, the CAPM is a disaster when confronted with these portfolios”. What
F&F does is to create a multifactor model, and show that variation in average returns of size and
BE/ME portfolios can be explained by the betas on the SMB and the HML. As a result, their
portfolios end up with betas close to one on the market portfolio. Betas in table 1 are close to zero.
As can be seen from figure 2 there is no systematic trend between the estimated portfolio returns
and beta. From table 4 it become clear that the model is rejected and there is no statistical
acceptance. A model solution could be the F&F three factor model as explained by Cochrane
(2001).
88
See Excel 1
48
The material was split into 5 portfolios based on the respective branch. The diversification of the
data material was based on a subjective view regarding what branch the company would fit into.
For the hypothesis:
H0: αbranch = 0 ; CAPM holds
H1: αbranch ≠ 0
The following model was tested by simple OLS regression (12) :
-
Rbranch,t - RFt = αbranch+ βbranch(ROMX,t – RFt) + ebranch,t
(22)
Table 5 Estimating the CAPM
P-value
R2
Conc.89
Branch
Obs
Alpha
p-value Beta
Avr.
Return
-1,17
Materials,
Industry
Consumer
goods
Health
Finance,
IT,
Telecomm
unication
Utilities,
Energy
206
0,7461
0,4563
-0,8842
0
0,1180
Accept
206
1,6571
0,1082
0,6552
0,004
0,0408
Accept
0,23
206
206
1,0411
1,2332
0,2670
0,2287
1,0262
1,005
0
0,00015
0,2085
0,063
Accept
Accept
0,74
0,55
206
1,7070
0,1047
0,0534
0,2971
0,000
Accept
1,46
H0 : α = 0 ; CAPM accepted
H1: α≠0 ; CAPM rejected
Significance level = 5%
The tests of (22) were conducted with a significance level of 5%. As table 5 shows, H1 was rejected
for all branches. In figure 3, beta for each branch is plotted against the excess average return. The
dots are the estimated points and the straight line is the Market CAPM first shown in figure 2.
89
H0 : α = 0
49
Figure 3 Estimating Branch portfolios
X : Beta , Y: Average return.
Fin
Mat
Cons
Health
Ener
Beta
1,0045
-0,8842
0,6552
1,0262
0,0534
Avr.return
0,55
-1,17
0,23
0,74
1,46
As can be seen from figure 3, four out of five of the estimates from all branch portfolios are lower
than the predicted CAPM line. Cochrane (2001) points out that it is typical that the estimated
market line through the stock portfolios is steeper than predicted, while measurement errors in betas
usually mean that the estimated line is too flat. In table 5, statistically the models for portfolio
branches is accepted with α = 0 conducted with a simple t-test and a significance level of 5%.
However as can be seen from figure 3 the estimated portfolios do not combine fully with the
predicted CAPM.
The tests so far in this project have resulted in extreme values and estimated models have been
rejected. However, the confusing part is that none of the results is to combine with what has been
found in other articles or tests so far. Though the simple CAPM was accepted for each of the
portfolios created on branch, the tests were weak and intuitively the sample was small and very
little representative.
50
4.2.1 So where did it go wrong.
In part 2 the econometric theory was covered, and initially since we assume that investors can
borrow and lend at a risk free rate of return the S&L version of the CAPM was used.
Zt was defined as a (5x1) vector of excess returns for 5 portfolios of assets. The following model
applied for the 5 portfolios, see (15):
-
Zt = α + β*Zomx,t + et
(23)
Where
-
E[et] = 0
-
E[et,et´] = ∑ , the variance-covariance matrix of residuals.
-
E[Zomx,t] = uomx and E[(Zomxt – uomx)2] = σomx2
-
Cov[Zomx,t,et] = 0
Beta is the (5x1) vector of betas, Zomxt is the time period t market portfolio excess return, and alfa
and et are (5x1) vectors of asset return intercepts and disturbances, respectively90.
Two approaches can be used for the estimation procedure; Ordinary least squares (OLS) and
Maximum Likelihood (ML). Campbell (1997) uses the ML approach to get estimates of the
unconstrained model and argue that OLS regressions lead to the same estimators for alfa and beta.
Ang and Chen (2005) pp 1 prove that “… OLS inference produces inconsistent estimates of
conditional alphas and betas”.
One can suspect that the model used should not be a linear model, tested on ML, maybe a
conditional model or maybe something is missing in the model that is left out.
If only a 5-10 years sample was used to estimate Beta, the estimate would become very noisy.
However, using a much longer estimation period of say 80 years won’t work due to the amount of
available material and since Beta changes over time. One solution is to form portfolios. According
to Campbell (1997) table 5.2, the more the number of portfolios the less the power of the tests.
Though it would have been optimal with a larger number of data, N = 5 is an optimal number of
portfolios. Campbell (1997) concludes that increasing the length of the time series can lead to a
significant payoff in terms of power. A longer time period would have been preferable.
90
Excel 1
51
It is worth to notice that if portfolios are formed randomly, they might lack power, and it could have
been optimal to have formed portfolios on estimated Betas as Black-Jensen-Schole, which might
have reduced the errors in the tests.
Table 6 show the results from this project and Campbell (1997). Campbell rejects the hypothesis of
CAPM. Especially two important biases come up for why the model is rejected:
Table 6 Project sum up
Project
Campbell (1997)
Time Period
1989-2006
1965-1994
Portfolio Characteristic
BE/ME
ME
Data frequency
Monthly data
Monthly data
Number of PF
5
10
RF
Tanggaard, 1Y Swap and 10Y One month US Treasury Bill
Bond Yield
Market Proxy
OMX
CRSP value Weighted index
Tests
Wald and GRS
Wald, GRS etc
Conclusion
Reject
Reject
-
Is it the true market portfolio proxy
-
Selection bias
These are the same important biases realised on the Danish capital market in this project.
Assuming that the OMX does not fully reflect the market proxy, Fama and French has calculated
the monthly market return91 and published it on their webpage. However, it is not clear what
companies that make up the market return in the calculations.
Roll (1977) emphasizes that tests of the CAPM really only reject the mean variance efficiency of
the proxy and that the model might not be rejected if the return on the true market portfolio were
used. Then we are back to the question about validity and second hand data, and whether it was
optimal for the author to calculate the market proxy manually or if the F&F market proxy should
have been used. However, Campbell refer to Kandel and Stambaugh (1987) who find that as long as
91
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#International
52
the correlation between the true market and the proxy is 0,70 or above, then the rejections of CAPM
remain intact.
It could be interesting to test F&F three factor model or change the market proxy to cover more than
the OMX. The Danish stock market is a small market, approximate 40% is made up by cyclical
stocks like Novo Nordisk and A.P. Moeller and due to among other things globalisation the market
is affected by countries with a more advanced stock culture, like e.g. US and Sweden. Next to that
many companies are not represented in the sample due to missing data. Manually split of the data
could have caused errors as well, and as Zikmund (2000) argue the validity would be better if a
second person had covered the data process as well.
Fama and French (1992, 1993) find that beta cannot explain the difference in return between
portfolios formed on the BE/ME. In fact, firms with high BE/ME have higher average returns than
is predicted by the CAPM. In general, the results in the anomalies literature (e.g. size and value
effect) may signal important deviations from the CAPM on a theoretical basis. However, this also
opens up for the possibility that the evidence against the CAPM is overstated because of
datasnooping and sample selection bias92. Lo and MacKinlay (1990) illustrate the potential
magnitude of datasnooping biases, which shows that the influence of the biases can be immense.
The conclusion is that the biases should be considered as a potential explanation for the model
deviations in this project.
Campbell (1997, pp 212) refer to Kothari, Shanken and Sloan (1995) who argue that “data
requirements for studies looking at book-market ratios lead to failing stocks being excluded and a
resulting survivorship bias”. It is concluded93 that one should be aware of the potential problems
that can arise from sample selection bias94 - and the same goes for this project.
4.3 Resume
Despite statistical errors and problems, 2 * 5 portfolios were created on BE/ME and branch
respectively. The tests showed extreme values and lacked power and empirical comparison. Based
92
Campbell (1997) pp 212
Campbell (1997) pp 212
94
Zikmund (2000)
93
53
solely on the above test results it is hard to conclude on what characterizes the Danish capital
market.
Despite the evidence against the CAPM on the Danish capital market, the CAPM remains a widely
used tool in finance. Different sources may explain the evidence against the model. Some may
argue that the model should be replaced by multifactor models or it may be argued that the evidence
is overstated due to mismeasurement of the market portfolio, improper data information, data
snooping and selection bias or maybe. A second theory is that one simply cannot explain the
anomalies of stock market behaviour.
54
Part 5
Conclusion and Interpretation
For years Capital markets have been tested and descriptive models have been constructed, based on
intuitive ideas or statistical evidence. The aim of this project was to test the Danish capital market
and model what factors based on the CAPM that can describe the Danish market. The final part of
the project will interpret the results and material covered.
55
5. Conclusion, interpretation and perspectives
For years F&F among others tested and modelled the US capital market and came up with several
impressive results. In Denmark data registration, collection and processing has complicated the
processes and procedures behind this project. Based on this and model failures, the final tests did
not come up with valid impressive results.
5.1 Conclusion
Going through relevant theory and literature, regarding financial models to describe the capital
market showed different models that varied in factors included, purpose and test methods. No
unique test method or definition is present that fully valid explain capital markets, due to biases in
material, test methods and factors included in the model. The general impression is that APT is the
best acknowledged frame work, and especially the F&F three factor model has shown good simple
results.
The aim of this project was to test and model the Danish capital market on the CAPM. The
complications with regard to available material and econometric reliable results make it difficult to
accept the validity of the tests. No conclusion on how to model the Danish capital market was
made.
The single index model is a simple static description of single assets correlation with the whole
market. However intuitively and fundamentally other things besides the market index affect stock
prices. Following the multi index model was presented. The multi index model adds other relevant
factors to the model besides the market index. In the static single and multi index model no
equilibrium assumptions are made and it is assumed that the unsystematic risk is uncorrelated.
Price formation models describes the creation of equilibrium return and risk premium in the market.
CAPM, C-CAPM, APT and F&F three factor model are some of the most well known models. The
CAPM implies that the expected return of an asset must be linearly related to the covariance of its
return with the return of the market portfolio. The Beta becomes an economic equilibrium
description
As any other financial theory CAPM is built on simplified assumptions that do not hold exact in the
real world. Despite Rolls critique, what defines the true market portfolio, and stringent assumptions
56
it is amazing how well the model has worked for several years. The model has been expanded with
the introduction of APT, conditional versus unconditional tests, consumption, the long term investor
and other relevant factors.
Anomalies like the size effect, which cannot be explained by the CAPM, turned out with positive
results with the F&F three factor model. What F&F does is to create a multifactor model, and show
that variation in average returns of size and BE/ME portfolios can be explained by the betas on the
SMB and the HML. As a result, their portfolios end up with betas close to one on the market
portfolio
Under the CAPM, Branch and BE/ME portfolios were tested on the Danish capital market, with
OMX as a proxy for the market return. An important part of the project was the data collection. For
this, DataStream was used. The materials collected were not fully representative, and due to among
others collection and sample bias the process and result of the tests were not found fully valid.
The tests resulted in extreme values and the estimated model on BE/ME portfolios were rejected.
The CAPM t-tests for each portfolio made on branch were accepted, though the assumptions behind
the model were not fully fulfilled.
It is concluded that the tests conducted and the materials found were not valid and descriptive for
the Danish capital market. Based on the material found not much is to learn about the Danish capital
market.
5.2 Perspectives
This project contributes to an understanding of financial descriptive models found in literature and
factors, materials and tests related to the Danish capital market. However the tests conducted were
not fully representative and accepted. The results give a reason to expand the project further.
It was stated that several models are found valid in literature when it comes to model the capital
market. However, most assumptions are not reliable and valid in a real world context. It would be
interesting to test what influence each assumption have on the tests and the significance of the tests.
The Danish capital market is small compared to a more adapted stock culture in e.g. Sweden or US.
Due to globalisation – especially in the Nordic region – it would be worthy to conduct the tests on a
Scandinavian market. This would increase the size of the sample and make the tests more reliable
57
ad valid. In this context a weighted analyses would be interesting since e.g. Nokia and Statoil make
up most of the Finish and Norwegian stock market respectively.
Due to the lack of availability of material in Denmark and the changing registration rules it would
be interesting to collect and calculate reliable and valid series of data. This would obviously affect
the outcome of the tests.
This project tests the unconditional version of the CAPM. However, the results were not valid and
found to be full of error. Ang (2005) tests the conditional version of the CAPM. This test method
could help to increase the validity of the test results on the Danish capital market.
Several factors influence and describe the way capital markets move. The tests performed in this
project don’t cover the amount of factors found in literature and what intuitively could describe the
market. More factors can be found on Fama and Frenchs homepage.
The stock culture is not as big in Denmark as one finds it in Sweden. Since a very large share of the
Danish capital market is made up by the Danish “realkreditobligationer”, it would have been
obvious to include these in the project. This would increase the amount of material to be analysed
and would illustrate a more advanced and valid market proxy than the OMX.
During a financial crisis, intuitively people buy the safe bonds and stay away from the more risky
stocks95. This analysis covers three sub-periods as mentioned above – before, during and after the
financial crisis in 2000. It could be interesting to include bonds and secondly to perform the
analysis split on the above mentioned sub-periods.
Intuitively, since Denmark does not have as an advanced stock culture as found in many other
countries, it would have been interesting to include the long term investor. In a real world context,
people save and invest for future consumption – this can be described by the long term investor.
Literature has several contributions to how the long term investor should be included, and further
new anomalies have become apparent. None the less this is an important question, and is presumed
to be closer to a real world context in Denmark.
95
This, however, does not seem to be the picture in the crisis of 2009
58
Finally it would be interesting to test F&F famous three factor model in Denmark and compare the
results to the tests on the US capital market. In a world of globalisation it would be interesting to
see whether similarities could be found.
More tests and factors would definitely contribute to a more complete picture of the Danish Capital
Market.
59