BscB, 6 semester
Bachelor Thesis
Department of business studies
GROUP: S11-13,72
Authors:
Anders G. Nielsen
Gestur Z. Valdimarsson
Supervisor:
Michael Christensen
How to create portfolios for
different risk groups and what
to consider
Aarhus School of Business and Social Sciences
Aarhus University
Spring 2011
TABLE OF CONTENTS
Abstract .............................................................................................................................................................................. 3
1. Introduction .................................................................................................................................................................... 5
1.1. Problem statement .................................................................................................................................................. 5
1.2. Delimitations ............................................................................................................................................................ 6
1.3. Research method ..................................................................................................................................................... 6
1.4. Assumptions ............................................................................................................................................................ 8
2. Markowitz Portfolio Selection Theory ............................................................................................................................ 9
2.1. Weak point in Markowitz’s theory ........................................................................................................................ 11
2.2. Sub conclusion ....................................................................................................................................................... 12
3. Diversification ............................................................................................................................................................... 13
3.1 Expected return and risk for an asset ..................................................................................................................... 13
3.2 Covariance and Correlation .................................................................................................................................... 15
3.2.1 Covariance ....................................................................................................................................................... 15
3.2.2 Coefficient of correlation ................................................................................................................................. 16
3.3 Expected return and risk for a portfolio ................................................................................................................. 19
3.4 The efficient frontier ............................................................................................................................................... 20
3.4.1 Calculations of the efficient frontier without short selling .............................................................................. 21
3.5 Diversification strategy ........................................................................................................................................... 23
3.6 Expected utility; the reason for different risk groups ............................................................................................. 25
3.7. Subconclusion ........................................................................................................................................................ 27
4. Data............................................................................................................................................................................... 29
4.1. Risk free return ...................................................................................................................................................... 29
4.2. Test for normality .................................................................................................................................................. 29
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5. Performance evaluation ............................................................................................................................................... 32
5.1. Creating a Benchmark ............................................................................................................................................ 32
5.2. Performance indicators ......................................................................................................................................... 33
5.2.1. Sharpe ............................................................................................................................................................. 33
5.2.2. Treynor............................................................................................................................................................ 34
5.2.3. Jensen ............................................................................................................................................................. 34
5.3. Evaluating the performance .................................................................................................................................. 34
5.3.1. Performance evaluation using benchmarking technique #1 .......................................................................... 35
5.3.2. Performance evaluation using benchmarking technique #2 .......................................................................... 37
5.4. Subconclusion ........................................................................................................................................................ 39
6. Conclusion..................................................................................................................................................................... 41
7. Discussion ..................................................................................................................................................................... 44
7.1. Black-Litterman model ........................................................................................................................................... 44
7.2. Regression analysis ................................................................................................................................................ 46
7.3. Forecasting problems for expected values ............................................................................................................ 48
8. References .................................................................................................................................................................... 49
Appendix 1: Eview’s output. ............................................................................................................................................. 50
Appendix 2: Jarque-Bera Table. ........................................................................................................................................ 60
Appendix 3: Fund information. ......................................................................................................................................... 61
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ABSTRACT
This thesis will try to evaluate the usefulness of Harry Markowitz’s theory and see if it is as relevant today
as it was in 1952 when his article portfolio selection first came out. To do this the paper has been divided
into two parts, a theoretical review of portfolio selection and an empirical study.
The theoretical part aims to look at Markowitz’s theory on portfolio optimization and go through the
conditions that need to be fulfilled in order to apply this theory optimally. The paper shows that the
condition of normal distribution on the returns is important, and also that incorrect expected returns may
result in a far from optimal portfolio. Calculations of the covariance and correlation are performed to see
what effect they have on a portfolios variance. These calculations show that the covariance and correlation
have a large impact when trying to reduce the risk of a portfolio. It is because of these covariances that it is
possible to construct a mathematical model for the actual portfolio variance.
By the use of Markowitz mean-variance optimization, it is possible to create the efficient frontier, which is a
line of optimal portfolios, where the expected returns are maximized for any given risk.
This paper uses historical data for the calculation of expected returns, standard deviations and correlation
for and between assets. For the expected returns, a geometric approach has been taken, as this paper will
show the geometric returns are more accurate than the arithmetic returns, which tends to overestimate
the return values. It will also be discussed why historical data is not very reliable and not an accurate
measure of the future returns.
Because investors have different preferences about risk, the empirical study will include portfolios with
different risk. The thesis quickly goes through what it will say to be a risk-averse investor and what it will
mean to be risk-affine.
The empirical study will consist of a portfolio simulation which will show how well Markowitz’s theory
performs in a “real-world” example. Three different portfolios are created to test his theory. The simulation
is running from 01.01.2008 to 31.12.2010 and is based on 26 Danish stocks and 3 Danish government
bonds. A test for normality will have to be conducted before the simulation can be conducted, this is to see
if the returns meet the requirement of Markowitz that they need to be normally distributed. This condition
was however not met since none of the assets turned out to be normally distributed.
After the portfolio simulation a performance evaluation will show how well the portfolios did. According to
the performance indicators only one of the portfolios created is able to beat the benchmarks. Another
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portfolio suffered major losses, mainly due to inaccurate return forecasts and partly lack of diversification.
This outlines one of the main problems of Markowitz’s theory.
After the portfolio simulation and performance evaluation it seems like Markowitz’s theory is as relevant
today as it was more than 50 years ago. However it has some pitfalls as it puts too much faith on exact
forecasting and has conditions that are hard to fulfil in reality.
Finally it will be discussed how the expected returns could have been more accurately forecasted. This
discussion will focus on Black-Litterman and regression analysis, which are being explained in a simple
manner. It will also be discussed why expected returns can never be predicted with 100% accuracy.
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1. INTRODUCTION
Beating the market is something investors always have been trying to do using different financial
techniques. For many companies it is essential to be able to make the best portfolio possible. Companies
such as pension funds, hedge funds, banks and investment firms, all of which can lose customers and a lot
of money on failed investments. If a company could find a best practise, that as a rule can beat the market
every time and thereby create better returns with certainty, it could drastically improve their
competitiveness within the financial markets. But is it possible to create such a best practise?
In this thesis we will see if one of the most well-known portfolio selection theories can be used to create
efficient portfolios which could beat the market.
In March 1952 the article Portfolio selection by Harry Markowitz was published in the Journal of Finance.
This article would later become a cornerstone in modern portfolio theory, mostly because of the
introduction of mathematical models on how to spread the risk in a portfolio. According to Rubinstein
(2002) Markowitz “...can boast that he found the field of finance awash in the imprecision of English and left
it with the scientific precision and insight made possible only by mathematics.”, a sentence that can only be
considered high praise. But since then a lot has happened in the financial markets, things move faster and
have become more dynamic. So is Markowitz’s portfolio selection method still applicable today? Can his
way of risk diversification still be used to pick an effective portfolio and can it beat the market?
1.1. PROBLEM STATEMENT
The main objective of this assignment will be to find out if we can create three portfolios that can
outperform the market over three years. This will be done using Markowitz’s mean variance portfolio
theory. By doing this we will try to see if Markowitz’s theory is still applicable today. When dealing with
Markowitz, we will also try to identify weak and strong points in his theory, this is important in order to find
out what potential pitfalls should be considered when creating a portfolio.
How can diversification help the investor create a low risk portfolio, which can provide the investor with a
more reliable return?
Another problem we are going to look into is the use of benchmark portfolios. Benchmarking is an
important part of the performance evaluating process. A process that is crucial for the investor since it will
tell him how well the applied investment strategy has done compared to others. It is therefore important
for the investor to know what to consider when creating a benchmark.
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1.2. DELIMITATIONS
To be able to conduct our research we will have to establish a couple of delimitations:
•
We will only use Danish stocks and bond. This is to simplify the simulation; we don’t think that this
choice will weaken our conclusion since the benchmark portfolios used when conducting our
performance evaluation also will consist exclusively of Danish stocks and bonds. We are aware of
the fact that only including Danish stocks and bonds will limit our possibility of diversification,
which could have decreased the risk. However we avoid the foreign exchange risk by limiting our
self’s to domestic investments.
•
We will not use active portfolio management strategies, these strategies are used to gain small
pick-ups in the ROI. The only interaction we will have with the portfolio will be the monthly asset
adjustment.
•
Tax and trade costs are excluded from our calculations.
•
In our portfolio simulation we will not short sell or have any leverage.
•
There will be no cash inflow or outflow for the three years our simulation runs. This means that no
funds will be taken out of the portfolio and no further investments will be made in the portfolio for
the three years it runs. Dividend will however be reinvested in its asset, dividend calculations are
included in the return index numbers we use for our portfolio.
1.3. RESEARCH METHOD
To tackle the problem at hand we have chosen to split this thesis into two parts. The first part will be a
theoretical review. This review will be made to see which assumptions and conditions have to be fulfilled in
order to apply Markowitz’s theory. We will then look at whether these assumptions and conditions can be
fulfilled in the real world. Diversification theory will be evaluated and it will be shown under which
conditions it is most effective.
The second part of our thesis will be an empirical study, where we will apply the theory reviewed in part
one to a portfolio simulation. Our main focus in this simulation will be on Markowitz’s portfolio selection
theory from 1952. We will use Markowitz theory to find the optimal portfolio for our different risk groups.
As mentioned above we only chose Danish stocks and bonds. The stocks we are going to use are the stocks
on OMXC20 and all large- and mid- cap stocks on the Copenhagen stock exchange. However we exclude
Maersk A, this was done due to the high correlation between Maersk A and Maersk B. The same was the
case for Rockwool A. B&O, Torm and Thrane where included to have stocks from all industries represented.
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Industries were defined according to the definitions used on the Copenhagen stock exchange
(www.nasdaqomxnordic.com). On the bond side we will use three government bonds one short, one
medium and one long. All stock and bond prices are based on the reinvestment of dividend and interest
(Also called the return index).
Before using these assets in our portfolio selection process we will have to test if their returns are normally
distributed. We will do this by conduction a Jarque-Bera test of our assets in Eviews. When all assets have
been tested for normal distribution we can start putting the simulation together in Excel. To forecast the
expected return and standard deviation (risk) we will use historical data one year before the investment
date. For example on 01-10-2009 we will use data from 01-10-2008 to 30-09-2009 to forecast next year’s
expected return and standard deviation. By doing this we assume that the only factor determining future
investment returns and risk, are historical returns and historical risk. This is however a wild assumption
because it is common knowledge that historical return is not the best indicator for future return. We will
however use this forecasting method in our simulation to simplify our calculations. Later in the paper we
will discuss and explain how forecasting models can be used to create a better estimation of future returns.
After completing the above steps we can begin to select the assets in our portfolio based on the Markowitz
portfolio selection theory. The three portfolios will be the basis of our simulation. These portfolios will be
updated once a month. The reason for this choice is because we need to change the portfolio constantly to
fit the changes in the market, but it would be unwise to change it every day or every week because of the
transaction cost that would be involved in the real world. We have set the duration of our simulation to be
three years, running from 01.01.2008 to 31.12.2010.
After the three years have passed we will conduct a performance evaluation to see if any of our portfolios
were able to beat the market. In that part of our assignment we will also look at how performance
evaluation should be conducted and discuss what makes a good benchmark portfolio.
After the performance evaluation we will be able to draw some conclusions on the effectiveness of
Markowitz’s mean variance portfolio theory. Here both the theoretical review and the results gained from
our empirical study will be taken into consideration.
In the end we will discuss what we could have done differently and what effect it might have had on our
results.
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1.4. ASSUMPTIONS
To conduct our simulation and apply the relevant theory the following assumptions have been made:
•
In this example we have the role of price takers. This means that we assume that we will not trade
a stock or a bond at a high enough volume to affect the pricing of that asset.
•
Investors are rational.
•
Investors see an increase in risk as a negative thing.
•
Investors see an increase in return as a positive thing.
•
We assume that returns are normally distributed. This assumption will be tested later.
•
As mentioned above we assume that historical risk and return can be used to forecast future risk
and return. The validity of this assumption will be discusses later.
•
Risk and return are the only things considered when taking an investment decision.
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2. MARKOWITZ PORTFOLIO SELECTION THEORY
Markowitz is considered the father of modern portfolio theory, mainly because of his article “Portfolio
Selection” from 1952 where he as the first person gave a mathematical model for portfolio optimization
and diversification. He later gave a more detailed description of his theory in the book Portfolio Selection:
Efficient Diversification of Investments from 1959. Before Markowitz’s theory, diversification was well
known, as he stated himself in his historical review of portfolio theory from 1999: “Diversification of
investments was a well-established practice long before I published my paper on portfolio selection in
1952.” (Markowitz 1999). However no-one had made a scientific model explaining it. The thing that made
Markowitz’s theory revolutionizing was that it was the financial version of “The whole is greater than the
sum of its parts” (Rubinstein 2002). In Markowitz article (1952) he stated that it is not the variance of the
individual asset ( ) that is important, but the contribution that the assets variance makes to the variance
of the entire portfolio ( ). This means that one of most important things to consider when creating a
portfolio; is the correlation between the different assets, and the covariance of the portfolio (Markowitz
1952). The formula given by Markowitz to calculate the variance of the entire portfolio looks like this
(Rubinstein, 2002):
= + Where is the proportion of the portfolio held in asset j, and is the correlation between the returns of
asset j and k. According to Rubinstein (2002) Markowitz’s paper is the first to contain this equation in a
published paper. We will have a closer look at this formula in a later chapter on risk diversification. Since
we don’t deal with short selling in our assignment, & ≥ 0.
Markowitz’s portfolio selection theory can however not stand alone when creating a portfolio. As the name
of the theory indicates it focuses only on portfolio selection, in another way, what assets a portfolio should
consist of. In his article Markowitz divides the process of portfolio creation into two stages, saying that the
first step focuses on forecasting the expected return of assets and the second stage focuses on the choice
of portfolio and how the assets should be combined. After describing this two-step process, he states that
he will only be concerned with the second stage (Markowitz 1952).
Another important aspect of Markowitz’s theory is that risk, although reduced by diversification, cannot be
completely removed by it (Markowitz 1952). In his article he states that: “The portfolio with maximum
expected returns is not necessarily the one with minimum variance.” (Markowitz 1952). Here he also
dismisses the validity of the law of large numbers in finance. The law of large numbers is that if you would
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diversify your investment across a very large number of assets your risk will be close to zero. This does
however not take the correlation of stocks into account, since it assumes that each asset is independent
from each other. Markowitz says that the investor will at some point not will be able to gain a higher
expected return by accepting higher risk. To illustrate the notion that an increased risk not always gives a
higher return Markowitz uses Expected return – Variance. This concept illustrates, in a simple manner, that
the investor will not always gain a higher expected return by taking a higher risk. If we look at figure 2.1 it is
clear to see that taking on additional risk in some cases decreases expected return. This will be illustrated in
a more mathematical manner in the chapter on diversification.
Figure 2.1.: Expected return – Variance
Source: Markowitz, 1952, Portfolio Selection.
Michaud (1989) listed some of the strong points about Markowitz’s theory in this article The Markowitz
optimization Enigma: Is “Optimized” Optimal?
•
Satisfaction of client objectives and constraints.
Markowitz can provide a convenient framework for integrating a list of important client constraints. These
could for example be a certain risk level or certain expected return.
•
Control of portfolio risk.
Markowitz’s portfolio optimizer can be used to control the portfolio’s exposure to various components of
risk.
•
Implementation of style objectives and market outlook.
This means that it is easy for the investor to choose the appropriate exposure to different risk levels, the
stocks of interests and the benchmark.
•
Efficient use of investment information.
Markowitz’s model is designed to optimize portfolios from the investment information given. So
information in a portfolio context will be used optimally.
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•
Timely portfolio changes.
Markowitz can process large amounts of information quickly. According to Michaud (1989) this is
particularly important for large institutions which quickly need to determine the impact of new information
on its entire portfolio. This aspect of Markowitz’s theory could have become even more important, since
the markets move a lot faster now than they did in 1989.
2.1. WEAK POINT IN MARKOWITZ’S THEORY
Although Markowitz’s portfolio selection is broadly recognised it has also meet some criticism, which has
proven that the selection method has its weak sides. One of the critics is Richard O. Michaud which in his
1989 article “The Markowitz optimization Enigma: Is “Optimized” Optimal?” pointed out some weaknesses
in Markowitz’s theory. Michaud (1989) states that the main reason Markowitz’s theory is not being applied
in investment firms is for political reasons. He argues that because portfolio optimization puts its
emphasize on qualitative techniques, it would move the power of investment decisions from senior
managers to qualitative experts and thereby changing the firm’s organizational structure. This reasoning is
however not perfect. We can easily assume that if these changes in organizational structure would lead to
higher returns, they would probably be implemented. If not, new companies would be formed that applied
this strategy. This argument by Michaud could however be obsolete since the article is from 1989. We will
put more focus on the other critic points in his article. His other critic points were:
•
Michaud (1989) state that Markowitz’s theory is “estimation-error maximization”. He says this
because of the importance of correctly estimated expected returns, variance and covariance play in
Markowitz’s theory. Even though these factors can be estimated with some precision they will
never be 100% correct and an investor will therefore not be able to create an optimal portfolio
using Markowitz’s theory. He criticizes that the optimization process over weight’s assets that have
large estimated returns, negative correlations and small variances. While it underweight’s assets
with small estimated returns, positive correlations and large variances. Stating that these
estimations are those which are most likely to have estimation errors (Michaud, 1989).
•
To replace the mean with the expected return of a given asset, when using historical data, is
according to Michaud (1989) also a drawback. More powerful statistical tools will have to be used
to gain a better estimation of expected return.
•
An aspect that is also quickly mentioned by Michaud (1989) is the low level of stability when using
Markowitz’s portfolio selection theory. A small change in the input can have a huge impact on the
output (portfolio). For some investors this could reduce the profitability of Markowitz’s model, due
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to the costs of trading, and there by rule out some of its usefulness. Especially private investors will
find that large changes in the portfolio on a monthly or weekly basis will lead to high transaction
costs; they should therefore not use Markowitz’s portfolio optimization, unless they invest very
high amounts.
Another concern that might arise when applying the Markowitz model is the possibility of a lack of
diversification. The optimal portfolio will only take expected returns, variance and covariance into account
when finding the best possible combination for a given expected return or a given expected risk. Therefore
it could invest a very large part of the portfolio in a single asset not considering diversification. It would
require a lot of trust in the accuracy of the expected returns and variances forecasted, since an estimation
error in an over weighted asset could have devastating effect on the investment. To avoid this problem we
have in our simulation put in some constrains that we believe will guarantee some level of diversification.
The first constraint is that there cannot be invested more than 10% in a single stock, the second constraint
is that the portfolio will have to consist of minimum 20% government bonds. We will have more on the
constraints we used when running our simulation later.
2.2. SUB CONCLUSION
In this chapter we have gone through the main aspects for Markowitz’s theory, and also looked at some of
the main critic points. Markowitz introduced the mathematical portfolio optimization process; he did this
by putting focus on the correlation of assets and by creating a formula for the variance of a portfolio. This
theory also dismissed the principal that a higher risk always meant a higher return. However some critic has
been pointed at Markowitz’s theory due to the high confidence put in exact estimates on future risk and
return. If the estimates are off, the portfolio optimization process could lead to a portfolio that is far from
optimal. Most of the critic is based around the fact that Markowitz’s theory only looks at one part of the
portfolio creating process and puts way to much faith in the first part being conducted to a precision that
will ensure exact values for future risk and return. We will later in our assignment look into these
forecasting problems and discuss the validity of Markowitz when historical values are used to predict future
stock earnings and future risk.
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3. DIVERSIFICATION
3.1 EXPECTED RETURN AND RISK FOR AN ASSET
A portfolio is a combination of several assets; we are weighting these assets by a simulation to create our
portfolios where, among other, their expected returns and standard deviations are being considered.
According to Moffett, Stonehill and Eiteman (2009) the return of an asset can be calculated in one of two
ways as arithmetic and geometric returns. There is no rule when to use which approach, but it is often seen
in research projects that the geometric approach have been used. The arithmetic method is although the
most intuitive approach, where you simply take the average of the historical returns. This is however not
the most precise method, because if we had a stock giving 100% return one year, and -50% the next year,
then the arithmetic return would be:
) =
100% − 50%
= 25%
2
But the actual return has been 0% and the arithmetic return has therefore overestimated the expected
return. If the expected return should be accurate, then it requires that the investor will take out his
winnings or replace his loses every year. In such a case he would double his investment the first year, but
lose half of his original investment the second year, and hereby he would have made a profit of the 25%.
Robert Ferguson et Al (2009), state that the expected arithmetic return is a poor long-term indicator of the
return, and that the mean-variance efficient portfolios with arithmetic returns always will reward additional
risk. Ferguson et Al on the other hand argues that a too aggressive investment will drive the expected
return negative, opposite of what is happening with use of the arithmetic returns. Instead they are
recommending the geometric approach, where the natural logarithm is being used to calculate the returns.
If we take the same example as before, with 100% and -50% returns, then the geometric approach will give
an expected return of zero:
) = 1 + 1) + 1 − 0,5) = 0,693 − 0,693 = 0%
The zero percentage return is equal to the expected return in the long run, if the two outcomes each have a
probability of 50%.
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In our simulation we have used the geometric approach to calculate our expected returns from historical
data. The expected return and standard deviations have been calculated using daily returns one year
before the investment date. We have calculated the daily return for a specific asset as:
= !
"
%
" #$
rt is the return at date t, Pt is the price at date t, and Pt-1 is the price the day before Pt. We have then used
the assumption that there are 250 trading days in one year, and to get the yearly expected return, we have
taken the average of the return for one year before investment date, and then multiplied with the amount
of trading days (250):
1
) = 250 ∗ (
'
()$
t is the number of observations one year before the investment date, ri is the return on the observation at
date i, and the 250 is the number of trading days in one year. This is possible because the geometric return
has a special feature to stretch over several periods. The return can over n periods be calculated as the sum
of all the returns during this period, which is what we are doing when we take the average of the returns
and then multiply with the amount of trading days. One concern about the geometric method is that for
extreme negative values, the numbers will be inaccurate. Let us say that we have one stock which
decreases with 70% in our observation period, in our case with daily returns this is though pretty unlikely,
then the geometric return will be ln(1-0,7) = -120,4%, a return lower than what is actually possible, since a
stock will never be able to drop below a value of zero, and therefore cannot lose more than 100% of its
actual value. Another concern about our expected returns is that we assume that if an asset has increased
20% in value for a given year, it will also increase with an equal amount the next year.
For the individual assets’ standard deviations, we are using the same values as we used to calculate our
expected returns. The mathematical formula to calculate the standard deviation is:
,
1
=*
( − +)
−1
()$
We skipped this calculation step and instead we simply used the stdev() formula in excel, which calculates
the standard deviation for a sample. This calculation only gives us, as it also did with the expected return,
the standard deviation for a single day, and what we need is the standard deviation for a year (250 days as
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stated previously). Because we are dealing with the standard deviation and not the variation, we cannot
just multiply with the 250 days, but we need to multiply the daily standard deviation with the square root
of 250 to get the yearly standard deviation.
3.2 COVARIANCE AND CORRELATION
Beside the risk of the assets independently, Markowitz is also taking the coefficient of correlation between
the assets into account. By taking the correlation into account it will be possible to diversify the portfolio
and hereby bring down the risk of the portfolio, it will even be possible for the portfolio to obtain a lower
risk than the one of the least risky asset. We will talk about how it this possible to obtain such lower risk in
section 3.2.2.
3.2.1 COVARIANCE
The combined risk of the portfolio is not only calculated by the included assets standard deviation, but also
by the use of the covariance between the assets that has been included in the portfolio. The covariance is a
measure of how the assets’ variations are between each other. We can take A.P. Møller (APM) as an
example, a company that both have an A and a B stock. This A and B stock vary much alike on the market, if
the A is increasing with 1,5%, the B stock will most likely increase with a percentage close to that of A. The
reason for this is that the A and B stock are affected by the same factors, all positive and negative factors
that affect APM will have a positive/negative impact on both the A and B stock, because if it is going good
for APM, then both stocks will go up, and the opposite if it is going bad. If we look at two different
companies in the same industry, it could also be that a surprisingly good result for one of the companies
could affect the other, because investors may see a positive trend in the industry.
The covariance between two assets can be calculated with the following formula (Keller, 2005):
-.
,
1
= ∗ ( − +- )01( − +. 2
/
()$
Where n is the number of observations, xi is the return of asset x at day i, µx and µy are the mean values for
the return of x and y respectively. We can from this formula see that the covariance can take any value
from negative infinite to positive infinite. The two assets will have a positive covariance if both assets are
either higher or lower than their respective mean value at a specific day. The assets would have a negative
covariance if one of the assets had a higher return than its mean while the second asset had a lower return
than its mean. The covariance does not give a good indication of how much the two assets actually
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correlate with each other, because it is affected by both of the number of observations and the size of the
gap between the actual return and the mean at the time of the observations. It is because of this difficult to
say whether two assets have a strong relationship with each other, if their returns are varying together, or
if their trends are just barely connected.
3.2.2 COEFFICIENT OF CORRELATION
Another way to give a better impression of how much two assets correlate with each other, is by using the
coefficient of correlation, which will have a value between -1 and 1. Two assets that have a correlation
coefficient of 1 are perfect correlated, meaning that when one of them increases with 1% the other asset
will increase with 1% as well. On the other hand if the coefficient is -1, then when one of the assets are
increasing by 1% the other asset will decrease by the same amount, so they will go in opposite directions.
The coefficients can be calculated using the following formula (Keller, 2005):
=
-.
- .
We took a more simple approach in our calculations, by using the excel formula “Correl” which
automatically calculates the correlation coefficient between two assets. The table below shows how two
assets correlate with each other with different correlation coefficients:
if
•
0 < ≤ 1 then the assets will have a positive correlation, which stocks generally do among
themselves, and so does bonds.
•
•
= 0 the assets will not correlate with each other
−1 ≤ < 0 then the assets will correlate negative with each other, which is often the case
between stocks and bonds.
We will here show how different correlation coefficients between two assets will have their impact on a
portfolio only consisting of the same two assets, the formula we use to calculate the variance of a weighted
portfolio is the one used by Markowitz (1952):
=
,
5(
()$
∗
(
,
,
+ 2 5( ∗ 5 ∗ (
()$ 6$
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And as we showed previously we have that:
=
-.
↔ -. = ∗ - ∗ .
- .
By inserting this in Markowitz formula we get that the variance of two weighted assets will be:
= 5( ∗ ( + 5 ∗ + 2 ∗ 5( ∗ 5 ∗ ( ∗ ( ∗ We will now show the calculations of the portfolio standard deviation for ( = 1
= 5( ∗ ( + 5 ∗ + 2 ∗ 5( ∗ 5 ∗ ( ∗ ( ∗ = 5( ∗ ( + 5 ∗ + 2 ∗ 5( ∗ 5 ∗ ( ∗ = 05( ∗ ( + 5 ∗ 2 = 05( ∗ ( + 5 ∗ 2
We can from this formula see that if two assets are correlating perfectly with each other, then the risk of
the portfolio will be a weighted average of the two assets standard deviations, and it will therefore not be
possible to reduce the risk more than investing everything in the asset with the lowest standard deviation.
This makes sense, because as we mentioned before, perfect correlation means that when one asset
increases with 1% the other asset will increase with 1% as well, and there will be no risk reducing benefits
from combining the two assets. Perfect correlation is of course basically impossible in the real world, where
not only the company’s performance is influencing the stock value, but also subjective views from investors
and other factors have influence.
For( = 0
The mathematical calculations will start from the same formula as previous with two assets in the portfolio,
but will later be expanded to contain m assets.
In these calculations we assume that all assets are weighted equally and also that their standard deviations
are equal.
= 5( ∗ ( + 5 ∗ + 2 ∗ 5( ∗ 5 ∗ ( ∗ ( ∗ = 5( ∗ ( + 5 ∗ = 5$ ∗ $ + 5 ∗ + ⋯ + 59
∗ 9
1 1 1 = ! % ∗ $ + ! % ∗ + ⋯ + ! % ∗ 9
:
:
:
Page | 17
1 = : ∗ ! % ∗ :
=
=
:
√:
We can see from this formula that when we add 3 extra assets for each asset we already have in our
portfolio (we multiply our assets with 4), our risk will be halved, if the assumptions mentioned above are
fulfilled. If we keep adding assets the risk will almost completely be removed. This is the law of large
numbers. However this is not possible due to the fact that assets will always have some level of correlation.
This means that some things can affect the entire market and every asset in it. This is what is also called the
systematic risk, the risk that cannot be removed by diversification.
For ( = −1
= 5( ∗ ( + 5 ∗ + 2 ∗ 5( ∗ 5 ∗ ( ∗ ( ∗ = 5( ∗ ( + 5 ∗ − 2 ∗ 5( ∗ 5 ∗ ( ∗ = 05( ∗ ( − 5 ∗ 2 = 05( ∗ ( − 5 ∗ 2
Here we can see that the risk of the portfolio can be completely eliminated by diversification if:
5( ∗ ( = 5 ∗ 5( =
5 ∗ (
If we insert this in the equation for the portfolio standard deviation, we get;
5 ∗ = 05( ∗ ( − 5 ∗ 2 = !
∗ ( − 5 ∗ % = 5 ∗ − 5 ∗ = 0
(
Where we with perfect correlation weren’t able to reduce the risk of the portfolio to less than the asset
with the lowest risk, then we can in this formula see that if two assets have a correlation coefficient of -1,
then it is possible to completely eliminate the risk, if we choose a specific weight between the two assets. It
is of course, as with the perfect correlation, not realistic to find to assets which have a correlation of exactly
-1. The only two assets that will be perfectly uncorrelated will be a stock and the short selling of the same
stock, which would give no risk, but the winnings/loses would also outtake each other, and you would
therefore be sure to lose nothing more or less than the transaction costs.
Page | 18
Figure 3.1: Illustration of how the combined standard deviation between two assets can be reduced with
different correlations.
Source: http://invisibleeconomists.splinder.com/archive/2009-08
3.3 EXPECTED RETURN AND RISK FOR A PORTFOLIO
The expected return for a portfolio, combined by multiple assets, can by the expected return of the
individual assets be calculated using the following formula (Markowitz, 1952):
,
0 2 = 5( ∗ ( )
()$
Where 0 2 is the expected return of the portfolio combined with N assets available, 5( is the weight of
asset i, and ( ) is the expected return of asset i. From this formula we can see that the portfolios
expected return is the weighted average of the expected returns for the individual assets. If all assets were
weighted equally, then we could replace 5( with ,, so if we had 10 assets, each asset would have a weight
of
$
$<
= 10%.
$
The risk of a portfolio cannot be accumulated as easy as the expected return, this is due to the correlation
between different assets. As mentioned earlier, the correlation also has an influence on the accumulated
standard deviation and it will therefore be necessary to include the correlation in the calculation of a
portfolios risk. As we are using excel to create our portfolios, a matrix formula will be very useful, we are
going to use a matrix calculation of the variance for the portfolio (Markowitz, 1998, p. 172):
Page | 19
… $@ >$
⋱
⋮ EB ⋮ E
⋯ @@ >@
$$
= =>$ , … >@ A B ⋮
@$
But as shown, we can instead of the covariance use the correlation coefficients:
=
-.
↔ -. = ∗ - ∗ .
- .
By inserting this in the previous matrix we get that:
$
,
…
>
=>
A
= $
@ B ⋮
0
… 0
1
⋱ ⋮ EB ⋮
⋯ @ @$
… $@ $
⋱
⋮ EB ⋮
⋯ 1
0
1
= =>$ ∗ $ , … >@ ∗ @ A B ⋮
@$
… 0 >$
⋱ ⋮ EB ⋮ E →
⋯ @ >@
… $@ >$ ∗ $
⋱
⋮ EB ⋮ E
⋯ 1 >@ ∗ @
And finally we will have to take the square root of the variance to obtain the portfolio risk/standard
deviation.
3.4 THE EFFICIENT FRONTIER
The efficient frontier is the optimal portfolio combination of the included assets, which provides the lowest
possible risk for a given expected return, or as we use to create our portfolio, the highest possible expected
return for a given risk.
Figure 3.2: An example of how the efficient frontier and MVP could look like.
12,0%
10,0%
8,0%
6,0%
4,0%
MVP
2,0%
0,0%
0,0%
5,0%
10,0%
15,0%
20,0%
25,0%
30,0%
Source: Own creation
Page | 20
If you follow the curve of the efficient frontier, you will eventually reach the point with the lowest possible
risk, which is called the minimum variance portfolio. All combinations which will give an expected return
lower than the minimum variance portfolio, are not included in the efficient frontier. This is because it for
these risks will be possible to gain a higher expected return. This is illustrated in figure 3.2.
3.4.1 CALCULATIONS OF THE EFFICIENT FRONTIER WITHOUT SHORT SELLING
If we look at the lines in figure 3.2, then every point on the line could actually be different portfolios which
we are able to create with the available assets. The way we created our efficient frontier, and hereby our
portfolios, were to choose a static standard deviation for two of our portfolios, and then use the excel
solver to find the maximum expected return for the given standard deviation. So in this case we wanted to
optimize the following formula:
I
( ) = 5( ∗ HG
()$
We have some specific conditions we also want the solver to fulfill when solving this optimization problem:
I
5( = 1
1.
()$
1$ …$9
5$ ∗ $
1
…
$
9
⋮
= =5$ ∗ … 59 ∗ 9 A> J
K > B
E
⋮⋮⋱⋮
59 ∗ 9
9$ …… 1
2.
5( ≥ 0,
3.
4.
L = 1, 2, … :
A fourth condition we use in our optimization problem is that none of the stocks can have a
weight higher than 10%.
5.
The fifth and last condition is that we as a minimum for all our portfolios will have at least 20%
bonds.
Condition 1 makes sure that the sum of our different asset weights adds up to 100%, so we have invested
our full capacity in assets. Condition 2 will only be included in two of our portfolios, 5% and 15% risk
portfolios, where we are able to set the standard deviation of our portfolio to a specific value, for which the
solver will then find the highest possible expected return. Condition 3 states that the weight of an asset
cannot be negative, which eliminates the possibility of short selling any of the assets. As stated in our
Page | 21
delimitations we have chosen not to include short selling, this is partly because there has been made new
rules about short selling certain stocks (National Bank of Denmark, 2008) and partly due to our rather
unreliable expected returns. Condition 4 makes sure that none of the stocks in our portfolios have a weight
higher than 10%. Since our expected returns are not very reliable, we made this rule to make sure that we
have some diversification in our portfolios. We are more interested in a broad diversification than an
extremely high expected return we cannot be certain of. Condition 5 says that at least 20% of a given
portfolio should be invested in the three included bonds. This is also because of our not completely reliable
expected returns. Estimation errors could make the portfolio end up with 100% of the investment in stocks,
given that the stock market had a very high increase the year before. As we can see our calculated
expected returns for many stocks are higher than 50%.
For our third and last portfolio we remove condition 2, and instead of optimizing our expected return, we
are going to optimize the ratio between our expected return and our standard deviation. The reason we are
choosing this third way to create a portfolio, is because our portfolio will then be able to have different risk
and adapt to the market. If the stock market for example is experiencing a positive trend, then our portfolio
could adapt to this trend, and hereby increase the risk so the ratio between stocks and bonds would
increase on the stock side. Figure 3.3 illustrates where the third portfolio is located on the efficient frontier.
It is located at the tangent point of a line going through 0.0 (no risk and no return) and the efficient
frontier.
Figure 3.3: shows how we find the third optimal portfolio (E(r) / Std. Dev.).
Source: Own creation
Page | 22
3.5 DIVERSIFICATION STRATEGY
Diversification is an important aspect of portfolio theory because as we showed above, a diversification
strategy can decrease the risk and uncertainty for investors. We also showed that a requirement for making
diversification possible is that the assets do not have perfect correlation between each other. By reducing
the risk, an investor will be able to rely much more on the expected return of his investments.
When we are talking about risk on assets, then it’s important to recognize between two different types of
risk; Unsystematic and systematic risk (Christensen & Pedersen, 2003, p. 49).
The systematic risk is a risk which does not occur for a single asset, for example due to internal disturbance,
but is a market specific risk which has an impact on all stocks in the market. The systematic risk is not
possible to reduce by diversification as it is with the unsystematic risk. The reason it is not possible to
remove the systematic risk by diversification, is because no matter if you invest in one or ten stocks, then
they will all be affected by the systematic risk. It is though possible to reduce the systematic risk by shortselling, this way if your stocks is decreasing/increasing due to some outcome which has an effect on the
entire market, then the stocks you have borrowed and sold, will also have decreased/increased in value
and therefore you buy them back cheaper/more expensive (Christensen & Pedersen, 2003).
An example of systematic risk could for example be, as we are experiencing now, increasing oil prices. Since
all companies are affected by the oil prices in some amount, either just by the increase in energy prices, or
in worst case that the company is depending on fossil fuels for example for the shipping industry. In such a
case the entire market would be hit, and therefore it wouldn’t matter if you had invested in one company
or another, some companies would of course be hit harder than others, but this extra loss is under the
category of unsystematic risk.
Unsystematic risk is a risk for a single company, it could be the dividend policy, or if essential raw material
should either increase or decrease in prices. We can say that the unsystematic risk is a risk or chance that a
certain event takes place, whether that event is negative or positive, which only affect one or more
companies or an entire industry, but the entire market will not be affected by this event (Christensen &
Pedersen, 2003). An example of this could be a company, whose managers are taking some very bad/good
decisions that decreases/increases the value of the company and will therefore also affect the stock price
for the company, but it will in general not have any direct effect on other companies in the market. A
crucial difference between the systematic and unsystematic risk is, that the unsystematic risk can be almost
completely eliminated by spreading your investments on many different assets, this have been showed
Page | 23
possible previously in this paper and is only an option for assets which do not have perfect correlation. Such
a spread would of course minimize the impact of the managers’ decision in a single company and also the
impact of how raw material may affect different industries. This risk spread is working because your
investment in a company may now be only 10% instead of having invested 100%, so if the stock decreased
with 10%, then you would have lost 10% of your investment if you had invested 100% in the company, but
after the spread you will only lose 1% of your total investment on the same decrease. This also shows
another assumption about the diversification strategy. You won’t diversify your investment if you have for
$100 shares in one company and then invest another $100 in another company. This would actually
increase your risk, because you are now investing $200 which you risk to lose instead of $100. So you do
not lower your risk by investing more money. The way the diversification strategy works is that if you have
$100 worth of share in one company, then you could for example sell shares worth $50, which you would
then spend on buying shares in another company. Figure 3.4 shows how the systematic and unsystematic
risk changes as more assets (which are equally weighted) are added to the portfolio, where all assets have a
standard deviation of 15, of which 10% is unsystematic risk and 5% is systematic risk, and the unsystematic
risk has a correlation of zero between the assets:
Figure 3.4, unsystematic and systematic risk with different amount of assets
16%
14%
12%
10%
Total risk
Systematic
8%
total risk
6%
4%
Systematic risk
2%
0%
0
10
20
30
40
Number of
assets
Source: Own creation
This figure shows that if we assume that the assets all have the same expected return, we are able to
reduce the risk on our portfolios, while maintaining the same high expected return. If all assets in the
market for example gives 12% in return and had, as in this case, a 15% standard deviation (10%
unsystematic, which have zero correlation between the assets, and 5% systematic), then we could by
Page | 24
including 32 different assets, be able to make a portfolio with an expected return of 12% and a standard
deviation of: =
$<%
+
√M
5% = 6,77%, which is much lower than if we only invested in a single asset.
3.6 EXPECTED UTILITY; THE REASON FOR DIFFERENT RISK GROUPS
An investors’ view on risk is of course important to consider when creating a portfolio, some investors will
settle for a low expected return, if they in return are given a low risk and hereby a higher probability of not
losing their investment. Other investors would much rather have a portfolio with very high expected return
and an equally high risk, than settle with the more certain low expected return, these different behaviors
are described in the expected utility theory which talks about risk aversion (people trying to avoid risk) and
risk affine (risk-seeking people). Christensen & Pedersen (2003) explains on page 66 that an investors’ risk
profile determines how much he will invest in the risky portfolio and how much he will put in the risk-free
asset. In our situation we do not include a risk-free asset in our model; we will create portfolios with
different risk levels. Then it will be up to the investors to determine which portfolio matches their risk
profile and how much of their wealth they will put in this portfolio.
It will, for each investor, be possible to have a different risk profile, which should then be used to optimize
the portfolio to match this exact investor. This risk profile can be quantified by a utility function, which
calculates how much utility a certain winning or loss will give this investor. If we say we have two investors,
they each have a different utility function:
Investor 1: O) = Investor 2: O) = ln)
U is the utility function of x, which is the investor’s wealth, and as we can see, x cannot be negative in this
example. Figure 3.5 and 3.6 on the next page illustrate how the two investors’ utility functions behave.
Page | 25
Figure 3.5
Figure 3.6
x^2
ln(x)
Source: Own creations
As we can see investor 1 prefers to earn $1 extra rather than avoid losing $1, so in a game where he is able
to bet $1 with a 50-50 percentage chance of winning $2 or lose his bet, he will take the bet, because if he
should win the extra $1 it would give him more utility than he would lose if he lost the bet and the dollar.
Investor 1’s utility, if he did not take the bet, would be: 12 = 1, if he on the other hand took the bet and lost,
his utility would drop down to: 02 = 0, but if he won the bet his utility would increase to: 22 = 4. We can see
that if the investor does not take the bet, he will have a utility of 1, but if he takes the bet, his expected
utility will be O) = 0 ∗ 0.5 + 4 ∗ 0.5 = 2 so his expected utility will be higher if he takes the bet. He will
actually be willing to pay more than $1 for the bet; the highest amount investor 1 would be willing to pay
for the bet is √2 = $1.41 which is $0.41 more than the expected return. From this we can see that investor
1 is risk-seeking and will be willing to take risk of losing more, if he is at least able to have the same chance
of winning the same extra amount. Investor 1 would in this case be a candidate for our portfolio with a risk
of 15%, since he will have the risk of losing more money, but he would at the same time be able to increase
his return even more.
Investor 2 is just opposite to investor 1; investor 2 will be happier not losing $1 than he will be if he wins $1.
For mathematical reasons the same bet cannot be used in this example. If investor 2 is being offered a bet
where he can either win $3 or $1 each with a probability of 50%, he will have an expected return of $2, but
since this investor is risk averse, he will much rather just have the $2 with no bet, instead of having the risk
that he is only getting $1. The utility for the investor will in each outcome of the bet be; O1) = ln1) = 0,
O3) = ln3) ≈ 1.1. Investor 2 will therefore have an expected utility of: O) = 0.5 ∗ 0 + 0.5 ∗ 1.1 =
0.55, and from this we can calculate how much he will be willing to pay for taking the bet: V <.WW = $1.73,
which is less than the expected return, so he is willing to take the bet if he is getting a risk premium of
$2 − $1.73 = $0.27. So we can say that investor 2 will only take a bet if he statistically is going to make a
Page | 26
profit of such a bet. This kind of investor will most likely be putting his money in our 5% risk portfolio, or if
he is very risk-averse, then he may invest in our E(r)/STD portfolio, because this portfolio is trying to
minimize the risk of losing any money, since the value of the expected return divided by the standard
deviation just have to be 3, before we with 99.9% certainty can say that the return will be higher or equal
to 0%. This is of course only true under the assumption that our estimates of the expected returns and
standard deviations are true and normally distributed.
Because we have these different risk profiles on investors, it could also be interesting to have a portfolio for
these different risk classes, so we may come up with an idea if there is a specific risk group that are doing
better than another, although this would require a longer period of observations than just three years to
actually be able to make such an conclusion. If we look at it hypothetically, then in the long run, the
portfolios with the highest expected returns, and hereby also higher risk, should intuitively be the portfolios
that are doing best. What is though important to think about is that with high risk, there is also a higher
probability of losing a lot, so you may lose 50% one year, which will take an increase of 100% the other year
to neutralize. This shows that if the risk is too high, we might lose so much that we need a higher increase
than what is possible.
3.7. SUBCONCLUSION
In this chapter we started by showing why the arithmetic returns were not very useful to calculate our
expected returns, as they in the long run overestimated the actual return. Instead we used the geometric
approach, which was very handy to apply, since we were able to sum up the daily historical returns to get
the expected yearly returns. We did also state that extreme negative values, when calculating geometric
returns, could give a negative expected return of more than 100%. We found out how covariance between
assets were able to bring a portfolios risk to a point lower than the asset with the lowest risk, but we also
stated that the coefficients of correlation was a better indicator as it would give a value that showed how
much the assets were correlating with each other. By calculations we saw that if two assets had perfect
correlated, then it would not be possible to decrease a portfolios risk to a point lower than the least risky
asset, but if they on the other hand were perfectly uncorrelated, it would actually be possible to remove
the risk completely. Later on we showed how the expected return of a portfolio could be calculated by
adding the sum of the different weights of the specific assets multiplied by the assets expected return. We
also showed how it was possible to calculate the risk of a portfolio by the use of the assets weight, standard
deviation and the correlation between each other. The calculations for Markowitz mean-variance portfolio
model was showed and we argued why our portfolios had to be restricted by some constraints, which
Page | 27
made sure that we spent all our capital (100%) in assets, that we had a specific risk, it did not short-sell any
asset, and some constraints for how many percentage of the investment could be invested in a single stock
and that at least 20% of the investment was invested in any of the included bonds. Most of these
constraints are made to make sure our portfolios are fairly diversified. We are then explaining why
diversification is important in a portfolio, because when diversifying a portfolio it will lead to less
unsystematic risk, and the predictions of the portfolios future return will then be much more accurate and
reliable. We describe the difference between systematic and unsystematic risk, where the systematic risk is
something that affects all assets, and it will therefore not be possible to diversify this away, whereas the
unsystematic risk can, because it is a risk which is only affecting a few companies or sector. Finally we
explained how two investors may have completely different investment preferences, because one of them
may not be willing to lose anything, and therefore will accept lower returns if he believes he is not losing
anything. The other investor may be willing to lose 30% of his investment for the opportunity to earn 70%.
Page | 28
4. DATA
In this chapter we will find the risk free return and conduct a Jarque-Bera test to find out if our data is
normally distributed. Both of those things will have to be done before we can conduct our simulation and
our performance evaluation.
4.1. RISK FREE RETURN
The risk free return is a theoretical term for the return an investor should expect from an investment with
zero risk. This is however hard to apply to the real world since all investments, even government bonds,
carry with them a small amount of risk. In our assignment we will use 4% as our risk free return; this is
based on the short government bond that gives a 4% interest rate.
4.2. TEST FOR NORMALITY
Before it is possible to use Markowitz theory, we will have to make sure that the returns of our selected
assets are normally distributed. The reason for this is the use of variance and mean when applying
Markowitz’s portfolio selection theory, these statistical values cannot be used with accuracy unless we
know that the sample is normally distributed. As seen above, one of Markowitz’s assumptions is also that
stock returns are normally distributed. To know if our returns are normally distributed we will have to
conduct a Jarque-Bera test on each of the assets. As mentioned in our research method section, we did this
in Eviews. The Jarque-Bera test is a goodness-of-fit test for normal distribution, with two degrees of
freedom, one for the skewness and one for the kurtosis (Thadewald & Büning, 2007). When conducting a
Jarque-Bera test we will have to set up the hypothesis that our returns are normally distributed.
X< : ZℎVV'\]^_`]]V' _^^5`^:`aL]'Lb\'L^
X$ :ZℎVV'\]^_`]]V' a^^'_^^5`^:`aL]'Lb\'L^
We will have to reject H0 if cd ≥ >$#∝;
(Thadewald & Büning, 2007). A significance level of 0,05 will be
used. So H0 will be rejected if cd ≥ 5,99 (Madsen, p. 29). To find the Jarque-Bera value we use this formula
(Thadewald & Büning, 2007):
Ml
cd =
i − 3)
gh +
j
6
4
Where S is the skewness h = +̂ M ⁄+̂ and K is the kurtosis i = +̂ n ⁄+̂ . Skewness is an indicator that shows
if the mean differs from the median, if returns are perfectly normally distributed the skewness should be 0.
Page | 29
The Kurtosis tests if returns centre around the mean, for a perfectly normally distributed sample the
kurtosis should be 3 (Thadewald & Büning, 2007). To find S and K we will need µ. µ is an estimation for the
theoretical central movement and can be calculated as shown below (Thadewald & Büning, 2007):
+̂ =
@
1
( − ̅ )
()$
We can insert the formula for µ into our skewness and kurtosis formulas to make them look like this:
Skewness
h=
Kurtosis
1 @
∑ )M
()$ ( − ̅
1 @
∑()$( − ̅ )n
i=
1
q ∑@()$( − ̅ ) r
1
q ∑@()$( − ̅ ) r
Ml
Now when we know how to test for normality using the Jarque-Bera test, we will try testing one of our
assets. The test will be conducted on the AP Møller B stock. To get the statistical values we needed we used
Eviews. After uploading our sample into Eviews we got histograms, descriptive statistics and Jarque-Bera
test for all our assets as output. Below we can see a histogram of the AP Møller B stock returns, with
descriptive statistics and Jarque-Bera test on the right side of the histogram. We have done this for all our
assets; please see appendix 1 for Eview’s output for all of our assets.
Figure 4.1: Histogram and descriptive statistics for AP Møller B, stock returns.
240
Series: APM
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.000393
0.000000
12.29000
-13.92000
2.600359
0.103010
5.791794
Jarque-Bera
Probability
340.8902
0.000000
40
0
-10
-5
0
5
10
Source: Own computing in Eview, please see appendix
By looking at the histogram we can see that the stock returns for the AP Møller B seem to follow a normal
distribution, however we will conduct the Jarque-Bera test to make sure. Looking at the statistical data on
Page | 30
the right side of the histogram we can see that the skewness is close to 0, which indicates a normally
distributed sample. The kurtosis however is 5.791794, which is relatively far away from the ideal kurtosis
value of 3. The reason for the high kurtosis could be some of the outlying values; values like -13.92 and
12.29 are far away from the histograms peak. From the skewness and kurtosis we can calculate the JarqueBera value using the above formula.
cd =
1044
5,791794 − 3)
g0,103010 +
j
4
6
cd = 174 ∗ 1,9591
cd = 340,8902
This fits perfectly with the value for Jarque-Bera given by Eviews. We can now use this value to test our
hypothesis that our sample is normally distributed. Since our Jarque-Bera value is much higher than the
acceptable 5.99, we will have to reject our Ho, this means that the returns on this asset is not normally
distributed. In appendix 2 we have put a table that shows the Jarque-Bera values for all our assets. As one
can see, no asset was a Jarque-Bera value below 5.99. This means that the return of our assets does not
follow a normal distribution. The reason for the high Jarque-Bera values is mostly due to the high kurtosis.
Stocks like for example KBH Lufthavn have experienced very high losses and very high gains on some days,
with extremes like 15.82% and -28.55%. Many of these outliers help push the kurtosis of KBH Lufthavn to
28.01287 which is very high.
Even though our returns do not follow a normal distribution, we will still try to apply Markowitz’s theory
when creating our portfolio. We will assume that the returns are close to normally distributed, this
assumption we backup by the histograms from Eview’s that seem to be close to a normal distribution.
Page | 31
5. PERFORMANCE EVALUATION
At the end of an investment period it is important to conduct a performance evaluation to see how well the
investment strategy applied have done. In this chapter we will discuss different ways to evaluate the
performance of our three portfolios. We will create a number of different benchmarks to hold our
portfolios up against, and in the end we will find out how well our portfolios have done. When conducting a
performance evaluation it is important to think about the fine line between skill and luck. When do you
know if your portfolio selection strategy is a success or if you’re just lucky in the portfolios lifetime?
In this chapter the portfolio with a risk level of 5% will be referred to as portfolio A, the portfolio with a risk
level of 15% will be referred to as portfolio B and the portfolio where r/std is maximized will be referred to
as portfolio C.
5.1. CREATING A BENCHMARK
When choosing a benchmark it is important to find a benchmark that is as close to the investment strategy,
applied by the investor, as possible (Christensen & Pedersen, 2003). Stock & Bond mix should be taken into
consideration and it should also be evaluated how diversified the portfolio in question is (Elton & Gruber, p.
630-631). To evaluate our performance as good as possible, we use two different benchmarking
techniques. The first benchmark portfolio we will create consists of two parts, a Danish investment fund
that focuses on Danish stocks and a long Danish government bond. This benchmark has been chosen to see
how well our investment strategy has done compared to other investors that focus on Danish stocks. We
included government bonds to get the risk of the benchmark down to a risk level that can be compared to
our portfolios. To make this benchmark as good a performance evaluator as possible the stock-bond-mix of
the benchmark has to be close to the stock-bond-mix of our portfolios. Since the stock-bond-mix is
different in the different portfolios, due to the different risk level, we have decided to take the average
stock-bond-mix of our portfolios and use it for the benchmark. The stock-bond-mix average will be
calculated on a monthly basis, this is because the stock-bond-mix of our portfolios changes every month
due to rebalancing.
The investment fund chosen for the benchmark is Danske Invest Danmark – Akkumulerende. This fund
consists almost entirely of Danish stocks, on February 28 2011 the fund consisted of 93.93% Danish stocks
(see figure 4.1). Even though this fund also consists of some foreign stocks we believe that it can be used as
a part of our benchmark, due to its high share of Danish stocks. Danske Invest Danmark – Akkumulerende
Page | 32
does not pay out any form of dividend, but reinvests the earnings in its portfolio; this is a good match to
our investment strategy where dividend is reinvested. For full fund info, please see appendix 3.
The other benchmarking technique we have chosen to apply is a bit more alternative. We will here make a
comparison between our portfolios and that portfolio’s first asset allocation. Meaning that we will compare
the results we got when we rebalanced our portfolios once a month to the results gained from holding the
portfolio created in January 2008 throughout the entire period. The reason for this choice of benchmark is
because our original portfolio has the same risk level and applies the same portfolio selection strategy as
the portfolio it will be compared to. We believe this is a good way to evaluate the performance of our
portfolios, mainly because of the commonalties between the portfolios and its benchmarks. A problem that
could occur when using this method of benchmarking, is that the stock-bond-mix and expected risk could
be very different between the benchmark and our portfolio throughout the investment period. This
technique is meant to see if extra return is gained from actively managing the portfolio.
Figure 5.1: Distribution of stocks
Source: http://www.nasdaqomxnordic.com/
5.2. PERFORMANCE INDICATORS
To find out if our portfolios have outperformed our benchmarks we will use three performance indicators:
Sharpe, Treynor and Jensen, all three are based on the risk adjusted return of the portfolios.
5.2.1. SHARPE
The Sharpe index was created in 1966 and is one of the first measures used for performance evaluation. It
shows the excess return gained above the risk-free rate per variability measure (Elton & Gruber, p. 638639). The Sharpe ratio is calculated in the following way:
Page | 33
Sharpe: z{x
t-uvwwxv yx@
|{}({~x((}( .
=
x #x

(Elton & Gruber, p. 639)
As we can see the Sharpe index is based on the return and risk level of a portfolio and uses therefore,
unlike Treynor and Jensen, the CML line.
5.2.2. TREYNOR
Unlike the Sharpe index the Treynor index looks at the SML-line. It does however look like the sharpe index,
looks at the excess return, but puts it up against the β. β is the non-diversifiable risk. By using the Treynor
model we get the excess return per non-diversifiable risk unit. Treynors value is calculated like this (Elton &
Gruber, p. 644-645).
Treynor:
t-uvwwxv yx@

=
x #x

(Elton & Gruber, p. 645)
5.2.3. JENSEN
Jensen’s model finds the difference between the returned gain by actively managing the portfolio,
compared to investing in a combination of the market portfolio and a riskless asset to gain the same risk
level (Elton & Gruber, p. 645). In other words it finds the difference between the SML-line and the
portfolio’s return. The weakness of this index is that is does not take diversification into account, only the
ability of a portfolio to gain additional expected return. This can however be explained by the fact that
Jensen (1967) originally used his index to evaluate the performance of mutual funds, and might have
assumed that they for that reason where well diversified. In our case it will be the difference between
actively managing our portfolios compared to investing in the benchmark portfolios. Jensen’s alpha is
calculated like this:
Jensen: = − | + − | ) (Jensen, 1967)
In the formulas given above for Sharpe, Treynor and Jensen, rp is the return on our portfolio, rb is the return
from the benchmark portfolio and rf is the risk free return.
5.3. EVALUATING THE PERFORMANCE
After we have chosen our benchmarks and performance indicators it is time to look at the performance of
our portfolios. As mentioned above we will use the performance indicators of Sharpe, Treynor and Jensen.
But first let’s have a look at how the portfolios did during the three year investment period. Figure 4.2
shows the development of the portfolios with an initial investment of 100 DKK. As we can see portfolio C is
Page | 34
the one that gained the highest return with its 18.75% over 3 years. Our lowest performer was portfolio B,
it lost 52.02% of its value during those 3 years. In the middle lies portfolio A with a return on investment of
6.16%. A possible reason for the huge loss obtained by portfolio B could be that this portfolio contains a
higher percentage of stocks then the two other portfolios, and stocks had a turbulent time around the
financial crisis. It could also be seen as a sign on the poor accuracy of the expected returns, when using
historical returns to forecast future returns. If the expected returns found were to hold the portfolio should
have had a positive ROI, instead it ended up with a negative ROI of 52.02%. This brings back the criticism
directed at Markowitz’s theory, that it puts too much faith in the precise forecasting of expected returns.
To see how well our investment strategy did we will now compare it to our benchmarks. First let us look at
our investment fund benchmark.
Figure 5.2: Portfolio development
120,00
110,00
100,00
90,00
80,00
Port A
Port B
70,00
Port C
60,00
50,00
40,00
Source: Own creation, see excel file: Portfolio simulation
5.3.1. PERFORMANCE EVALUATION USING BENCHMARKING TECHNIQUE #1
As mentioned above our first benchmark consisted of a Danish investment fund and a long government
bond. To make our comparison as good as possible we used the average stock-bond-mix of our portfolios.
Figure 4.3 shows the development in the amount of bonds our portfolios and our benchmark consist of.
Here we can see that the stock-bond-mix of A and C is very different from the one of B, in that A and C have
a very high bond percentage. The benchmark lies in the middle with its average bond-ratio.
Page | 35
Looking only on the return of investment, no portfolio was able to outperform the benchmark that gained
an ROI of 19.08% over three years. With a return of 18.75%, portfolio C came closest to the benchmark and
only gained a 0.33 percentage point’s lower return then the benchmark. Portfolio A had a 12.92 percentage
point lower return than the benchmark, while portfolio B is a staggering 71.10 percentage points behind
the return gained from our benchmark.
Figure 5.3.: Bond percentage
100,00%
90,00%
80,00%
70,00%
60,00%
Port 5%
50,00%
Port 15%
40,00%
Port r/std
30,00%
Benchmark
20,00%
10,00%
nov-10
sep-10
jul-10
maj-10
mar-10
jan-10
nov-09
sep-09
jul-09
maj-09
mar-09
jan-09
nov-08
sep-08
jul-08
maj-08
mar-08
jan-08
0,00%
Source: Own creation, see excel file: Portfolio simulation
However as we mentioned in the chapter on performance indicators, will we have to take the risk into
account. For that we have used the Sharpe, Treynor and Jensen indexes. Figure 5.4 shows the results from
calculations done in excel to find the Sharpe, Treynor and Jensen values. The calculations can be seen in the
excel file “portfolio simulation”. For the performance indexes green indicates a superior performance while
red indicates that the portfolio has been outperformed.
As we can see, portfolio B was in no performance measure able to outperform the benchmark. Meanwhile
portfolio C was, according to the Sharpe, Treynor and Jensen indexes, able to outperform the benchmark.
This makes a lot of sense since we can see that portfolio C was close to the return of the benchmark, at a
notable lower risk. Portfolio A is an interesting case however, since it according to Treynor and Jensen was
able to outperform the benchmark, but was according to the Sharpe index it could not. The Sharpe index is
the only one, of our three performance indexes, that directly takes the standard deviation into account. As
Page | 36
we have seen in our previous chapter on diversification, a well-diversified portfolio has a lower standard
deviation and would therefore indirectly get a higher Sharpe value. This could be the reason why, according
to the Sharpe index, portfolio A is outperformed by its benchmark. Because while portfolio A for long
periods mostly consists of government bonds, it’s benchmark has a lower bond share and also partly
consist of the investment fund which is much more diversified.
Figure 5.4.: Sharpe, Treynor and Jensen indexes for benchmarking technique #1
Risk free return:
4,00%
Port A
Bench
Port B
Bench
Port C
Bench
Return
6,16%
19,08%
-52,02%
19,08%
18,75%
19,08%
Risk
5,68%
8,26%
18,58%
8,26%
4,26%
8,26%
Sharpe
0,380
1,826
-3,014
1,826
3,466
1,826
Treynor
0,972
0,151
-7,722
0,151
11,160
0,151
Jensen
0,018
0,000
-0,571
0,000
0,146
0,000
Own creation, see excel file: Portfolio simulation.
Another reason could be the use of different market lines, as mentioned Sharpe uses the CML-line while
Treynor and Jensen use the SML-line.
From the above we can conclude that the only portfolio that could outperform the benchmark was the C
portfolio. This could be due to the high percentage of bonds in the portfolio, because while stock prices
decreased during the financial crisis, bonds were able to give a stable and low risk, return on investment.
We can also see that the investments in stocks are diversified much better than for the other two
portfolios.
So for now the winner is portfolio C, now let’s look at the performance of the three portfolios compared to
the alternative benchmarks.
5.3.2. PERFORMANCE EVALUATION USING BENCHMARKING TECHNIQUE #2
On figure 5.5 it can be seen that portfolio C is the highest performer here with its 18.75%, its benchmark
comes close with it 15.33%. As mentioned above risk should be taken into account. Figure 5.6 shows the
return, risk, Sharpe-, Treynor- and Jensen index for our portfolios and benchmarks.
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Figure 5.5.: Portfolio and benchmark performance, using benchmarking technique #2.
120,00
110,00
100,00
Port A
90,00
Port B
80,00
Port C
Bench A
70,00
Bench B
60,00
Bench C
50,00
40,00
Source: Own creation, see excel file: Portfolio simulation
Figure: 5.6.: Sharpe, Treynor and Jensen indexes for benchmarking technique #2
Risk free return:
4,00%
Port A
Bench A
Port B
Bench B
Port C
Bench C
Return
6,16%
11,28%
-52,02%
1,55%
18,75%
15,33%
Risk
5,68%
6,60%
18,58%
20,54%
4,26%
4,66%
Sharpe
0,380
1,103
-3,014
-0,119
3,466
2,434
Treynor
1,166
0,073
-22,425
-0,025
6,970
0,113
Jensen
0,020
0,000
-0,560
0,000
0,145
0,000
Own creation, see excel file: Portfolio simulation.
Again portfolio C is the only portfolio that clearly outperforms its benchmark. Portfolio A fails to
outperform according to the Sharpe index, while portfolio B fails to outperform on all three performance
measures. Portfolio B has a staggering 53.57 percentage point’s lower return than benchmark B. The
reason for the huge difference between portfolio B and benchmark B could be because we used historical
returns in our forecasting for expected returns. All the stocks which price had skyrocket in 2007 where
picked by the solver when creating this portfolio and many of these stocks experienced a huge dive in
prices during the financial crises. Another thing that could have influenced this portfolio is the high risk
Page | 38
level, because we set the expected risk to be 15% the solver only invested a small portion of the portfolio in
government bonds.
5.4. SUBCONCLUSION
In this chapter we have discussed the essentials of a good benchmark, what performance indicators to use
and why performance indicators are important. Finally a performance evaluation of the three portfolios
where conducted using two different benchmarking techniques.
As mentioned above the essentials of a good benchmark have been shortly discussed. Those essentials
were a benchmark that was as close to the investment strategy applied as possible, a similar level of
diversification and a similar stock & bond mix. The importance of performance estimators was underlined
due to the fact that both risk and return should be considered when doing a performance evaluation.
Sharpe, Treynor and Jensen were used to make the risk adjusted performance evaluations.
As we could see in this chapter, portfolio C has been the best performer beating both of the other
portfolios and the benchmarks. It had a better performance according to all three performance measures,
which indicates that this has been our optimal portfolio during the three year period. The reasons for this
are believed to be its large share of bonds obtained, while trying to maximize return per standard
deviation. Bonds have performed better mainly because of two things:
1) During the financial crisis stock prices plummeted while bonds gave a somewhat secure return on
investment.
2) Expected returns for bonds where more precise than the expected returns for stocks, when
historical data was used as the only variable in forecasting.
3) Interest rates dropped during the financial crisis, making the old bonds with higher interest rates
more attractive and therefore more expensive.
By looking at reason number two, we are reminded on one of the pitfalls of Markowitz’s theory. As
mentioned in the theoretical review, Markowitz’s model can be an estimation error maximizer because it
assumes that the expected returns are highly accurate. In this case expected returns were only based on
historical data, therefore estimation errors were bound to happen. For portfolio B, a portfolio with a high
percentage of stocks, these estimation errors meant that during the simulation the solver overweighed
stocks that had skyrocketed right before the crisis, many of which plummeted in 2008.
Page | 39
It can easily be said that we need better estimates to be able to apply Markowitz with maximum efficiency.
The fact, that only portfolio C was able to outperform the benchmarks, emphasizes this point. This also
raises questions for the real world applicability of Markowitz’s theory.
Page | 40
6. CONCLUSION
This paper has taken its starting point in Harry Markowitz’s original portfolio selection theory from 1952.
We started by exploring the theoretical aspects of the mean variance portfolio theory, and saw how
Markowitz as the first person demonstrated the importance of covariance between assets. Stating that the
variance of the individual asset is not important, only the contribution that the asset’s variance has on the
portfolio’s variance. His theory also provides a good framework for different investors, since it easily can be
modified to fit different risk and return preferences, as we also showed this can be a huge benefit because
investors may have different preferences. The mean variance portfolio theory also dismissed the notion
that a higher risk is always awarded with a higher return, at some point the investor will not be able to gain
additional return by taking on additional risk.
Even though the theoretical concept is well defined, its use in practice is somewhat more difficult. There
are some assumptions that are critical for the effectiveness of the mean variance portfolio theory which
have been criticized. One of its main weaknesses was its reliance on well forecasted expected return and
expected risk. Forecasting errors could lead to a portfolio selection that is far from optimal. In our
simulation we used the historical returns to forecast future returns, this however proved not to be a very
effective forecasting method, and therefore reduced the usefulness of Markowitz’s theory. Markowitz also
assumes that returns are normally distributed, and as we showed this is not the case for any of our 29
assets. Although it could be argued that the assets were close to a normal distribution.
We showed in the paper that the geometric return is giving a better estimate of the expected return,
whereas the arithmetic return provides overestimated returns. The geometric returns gave on the other
hand also some problems, as we showed an extreme decrease in the asset value would give a negative
expected return of more than 100%, which is not a realistic return. It was shown how it is possible to
reduce the portfolio risk, to a risk lower than that of the least risky asset, by the use of the covariances
between all included assets. We also showed why the correlations were better, to give an intuitively
indication of how much two assets are varying with each other, than the covariances.
The paper also showed how easily specific constraints can be implemented into the simulation of the mean
variance portfolio, so an investor is able to make his own preferences about stocks and bonds, and the
individual assets by themselves, if he likes.
It was also stated why diversification is an important strategy for investors, as it can help minimizing the
risk and still obtain the same expected return. We explained how systematic risk could not be diversified,
Page | 41
because it was affecting the entire market. On the other hand unsystematic risk is something that only
affects one or a few companies or industries. Because of this is it possible to diversify the unsystematic risk
by spread an investment over multiple assets, which are located in different industries, and also invest in
other securities.
We used the skewness and kurtosis in a Jarque-Bara test, to test whether the return of our included assets
were normally distributed. As mentioned above these calculations showed that none of the assets were
normally distributed. This result contradicts the condition in Markowitz’s portfolio selection theory, that
returns have to be normally distributed. It therefore questions if the mean and variance can be used as
precise measures for risk and return. However as we mentioned above we assumed that the returns were
approximately normally distributed and therefore that the use of mean and variance could be defended,
although it is necessary to keep in mind that the results may not be completely accurate.
Before conducting our performance evaluation we discussed who to create the optimal benchmark. As we
found out a benchmark will have to have a similar investment strategy, stock-&-bond-mix and should focus
on the same market as the portfolio that is to be evaluated. It also makes no sense to compare a portfolio
to a benchmark based on a short time period, because it will be almost impossible to distinguish between
what is luck and what is skill. In our case the simulation had a lifespan of three years; we could therefore
assume that our performance evaluation was creditable. To evaluate the performance we chose to use
Sharpe, Treynor and Jensen as performance indicators.
To make our performance evaluation as effective as possible we applied two different benchmarking
techniques; one where we compared to a Danish investment fund and another where we compared our
portfolios to their first portfolio selection. We found out that the benchmark which consisted of the
investment fund gave the highest return, but was according to Sharpe, Treynor and Jensen outperformed
by the portfolio where we optimized return per standard deviation. The reason why the r/std portfolio
could outperformance the benchmark even though it gain a lower return on investment, its risk was much
lower compared to the benchmark. Having an only 0.33 percentage point lower return, but a 4 percentage
point lower risk made the r/std portfolio a better choice. The r/std portfolio was the best performer,
outperforming both of its benchmarks and gaining a return of 18.75% over the three year period. We
believe that the r/std portfolio performed so well because it had a high percentage of bonds, which
compared to stocks, were extremely profitable during the financial crisis, and that the stocks investment
part was more diversified than the other portfolios.
Page | 42
The worst performer was our 15% risk portfolio, which lost 52.02% of its investment, which we do not need
a benchmark to tell us that this is bad. We assumed that the reason for this extreme loss was mainly due to
our poorly forecasted returns, and partly due to low diversification. The 15% portfolio is a good example of
how bad it can go when poorly forecasted returns are used to optimize a portfolio, using the mean-variance
portfolio model. This showed the previously mentioned weakness of Markowitz’s theory and also that
historical returns are a bad indication of future returns.
In the end we can say that the mean-variance model by Harry Markowitz has a good theoretical concept. It
is able to explain how diversification reduces portfolio variances, but as we showed in our portfolios, it may
not always prioritize diversification. Instead the model is good at optimizing expected returns, if you are in
possession of reliable data. It does however have some conditions that are hard to fulfil, an example of this
is the condition that returns should be normally distributed and that the forecasts should be highly
accurate. If this is not the case, we saw with our portfolio 15%, that the model will overweight the wrong
assets and in that way maximize estimation error. All in all we believe that Markowitz’s theory is as
applicable today as it was when he introduced it, you could even say that with the advancement in
technology it has become even easier to apply. However investors will have to keep its pitfalls in mind
when using his model for portfolio optimization.
Page | 43
7. DISCUSSION
As we have shown in our paper, it is very unreliable to use historical data to estimate our expected returns;
on the other hand we also showed that it is far more precise to use historical data to estimate standard
deviations for assets. We also talked about how Markowitz’s Mean-variance portfolio model relies on
accurate estimations of expected return, standard deviation and correlations between assets. As we were
able to see from our portfolio creations, it is a critical assumption, because if one or a few stocks have an
extreme expected return, then the Mean-variance model will tend to overweight these stocks a lot and
thereby undermine the importance of diversification. Especially when using historical values as expected
return, we undermine diversification for stocks which have had a great return earlier, which could maybe
be an indication of that the stocks was undervalued previously and is now either at true value or even
overvalued. In this case, the mean-variance model would suggest we invest in these stocks, and if they
show to be overvalued then we may lose even more, because we also minimized our diversification we do
not have a high weight of other stocks pulling in different directions and hopefully towards a positive
return. No matter what, the mean-variance theory does not, with historical values, provide a way to say
whether a stock is over- or undervalued, and we are therefore not able to recognize which stocks may have
a strong positive return in the future. Since the Mean-variance portfolio model by Harry Markowitz, is
relaying on expected returns, standard deviations and correlation between assets, it will be necessary to
find another more precise way, to predict how returns on assets will be in the future. In this chapter we will
show some different models which may be better to forecast the expected returns.
7.1. BLACK-LITTERMAN MODEL
This problem is taken up by Fischer Black and Robert Litterman in their article “Global portfolio
optimization” from 1992. In this article they are considering the mean-variance portfolio model with a
global Capital Asset Pricing Model (CAPM) equilibrium, where they use the CAPM to come up with
expected values for all the assets. They are also making room for the investor to have his own views on an
asset if he believes this asset is going to under- or outperform the market, then the investor will have to
state how confident he is on this view, and then the model will take its starting point from the CAPM value
and regulate it based on the investors view and confidence. This is effective because as Black and Litterman
(1992) state, an investor is not able to have a reliable view on all assets in the market, so in the case where
the investor does not have a specific view of how an asset will perform in the future, he can use, what Black
and Litterman calls, the neutral view of CAPM. This approach should give a much better result than using
historical values to calculate the expected return, because we won’t see the same extreme values as with
Page | 44
the historical ones and the mean-variance portfolio model will therefore give a more diversified result,
which we have also seen in our portfolios, can be important. In “global portfolio optimization” Black and
Litterman are considering a global CAPM equilibrium, because they are creating a global portfolio, in our
case we are basically only considering large and mid cap and three government bonds, and it will therefore
be from these assets our global equilibrium should be found.
This approach could have been used in our simulations, and maybe given our portfolios a better return,
especially on the high risk portfolio. Black and Litterman are doing this in a much more sophisticated way,
and we would probably have to simplify it some. What we could have done was to relate all assets to our
expected market returns as in SML and make a regression model. This way we could have calculated our
expected returns as (Christensen & Pedersen, 2003, p. 129):
) = | +
9 )
∗ 09 ) − | 2 = | + 09 ) − | 2
9 )
In this case we do need to have an expectation of the future return of the market, but to use the historical
returns to calculate the expected future return of the market, is most likely more reliable than to use
historical values on individual assets. An individual asset can of course run into some very good years for
some period, but in the long run, it will most likely experience some ups and downs due to internal
disturbance or an industry specific crisis or boom, both systematic and unsystematic. The market on the
other hand will only experience systematic crisis and booms. We know that the market is influenced by
macroeconomics, and if we assume that the GDP is relatively stable, which of course would need to be
checked in such a study, then we could say that the market returns in the long run also would be relatively
stable. In our simulation, we would then need to take the historical return over maybe 25 years of the
market, in our case 29 assets, and then calculate the annual return, which we could use as the expected
future market return. The historical return should of course be compared to the inflation and interest rates.
This model will in most cases be more precise than by the use of historical returns, but it will still not be
able to identify the actual future returns of the assets. We will have an error on the expected market
return, since we assume that the market will have a fixed return (taking inflation and interest rates into
consideration), but our estimation of the beta will also be inaccurate since it is calculated from the
historical covariance between the market and an asset, and the variance of the market return, as stated in
the previous formula. Another point is that, as with many other statistical calculations, it has to be assumed
that the returns are normally distributed, which we have shown is not the case, at least not on a daily basis.
Page | 45
The use of the mean-variance portfolio model can be extremely efficient, but it relies on accurate estimates
of the expected return, standard deviations and the correlations for all assets, if just one of these estimates
are way off, then it will not be possible for the model to create an optimal portfolio. As we can see from our
portfolios, our expected returns are way off the actual future returns. Therefore our 15% portfolio, which
consists mainly of stocks, ends up by having a really bad return. An investor would have lost more than half
his investment in just three years if he had invested in this portfolio.
7.2. REGRESSION ANALYSIS
As it has been shown in financial literature, macro environment has an impact on the market return. We
sometimes hear in the news that the stock exchange has reacted in a positive or negative way on the
American employment rate reports, the reason for these reactions are because this report is used as a good
indicator of how the financial situation is progressing. Blanchard (2009) uses a dividend model, to show
how stock prices are calculated, where he states that the stock price is equal to the discounted value of all
future dividends:
"< =
v$
v
+
+⋯
1 + $
1 + $ 01 + ( 2
If we look at this model without any other factors, then we could argue that the dividend is the
unsystematic risk, since the dividend is individual for each company, and that the interest rates are the
systematic risk, where an increase or decrease will have an effect on all stocks’ discounted dividends. We
do know that there are many more factors influencing the stock price, and Blanchard (2009) also state that
government spending have influence on dividends, and that investor optimism or the inverse can have a
great impact on share prices. What we do need to figure out, before making our regression, is which factors
may have an influence on stock prices, and what is important to remember is that some stocks are being
influenced by factors which other stocks may not react to. An example of this could be a Danish company
who is taking orders in American dollars, while the other Danish companies may take orders in Danish
currency. In this case a strengthening of the American dollar would have a positive impact on the company
which is taking orders in dollars, because when they are then going to exchange them into Danish kroner,
then they would make an extra profit on the exchange rate as well. All the companies which are taking
orders in Danish kroner, would not be directly exposed to the exchange rate, and their stocks would
therefore not increase or decrease because of a strengthening dollar.
Page | 46
A regression analysis makes a function with a dependent variable, which in our case is the stock prices, and
then gives some coefficients on how independent variables affect the dependent one. If we look at the
regression from our Black-Litterman section:
) = | + 09 ) − | 2
Then we have that the beta coefficient tells us how much asset k, increases if the expected market return
increases with 1 and the risk-free rate remains constant. This is a regression with k’s expected stock return
as dependent variable and expected market return minus the risk-free rate as independent variable. If we
instead wanted to use variables from the macroeconomic environment as our independent variables, such
as for example GDP, the interest rate and the unemployment rate, then we would get a beta value for each
of these variables, which would tell how the expected return of asset k would react to changes in the
independent variables. The calculated beta values would, beside an expected value, also have a standard
deviation. This is because they are not solely explaining the decreases and increases in the stock return. It
would then be possible to see if this beta value is significantly different from zero, if it is not, then we
cannot say that the variable has an impact on the dependent variable. These regressions are therefore
neither completely reliable, since the beta values in most cases will not be exact. Another example of when
such a regression will be very unreliable is if the company, you have just collected data from to make a
regression, suddenly starts to hedge on currencies and raw material prices. Such a hedge would mean that
even though the prices on for example oil for Mærsk shipping increased rapidly, then if Mærsk had made a
hedge on the oil price the increase would not have an effect on Mærsk’s financial statement, at least not on
the short run. A hedge is if a company or investor is trying to minimize its exposure to a risk, by for example
make sure that you are able to sell a stock at a minimum price even if the stock should have fallen below
this price. In the Mærsk example, it would be because they had already made an agreement with an oil
producer/supplier to supply them with oil at a fixed price for a year or two.
There is still another problem with using the regression analysis to find expected returns of an asset from
macroeconomic variables, because we will then need to predict how those independent variables will be in
the future, which may be as difficult as it is to predict the expected returns of the stock itself. There is one
option where it would be very useful, and that is if the markets were not efficient, then it would maybe be
possible to use data of the independent variables at time t-1, meaning that what happened with e.g. the
interest rate yesterday, will have an effect on the stocks today. This way you would be able to get a better
picture of how the expected return would be, but the market is too efficient for that to happen.
Page | 47
7.3. FORECASTING PROBLEMS FOR EXPECTED VALUES
There is so far no forecasting model that can give a precise forecast for expected returns of assets, and we
do not believe that there will ever be a precise way to predict how future stock returns will be, this is for
several reasons. First of all stock prices are always going to vary a lot, because investors will have different
opinions about how risky and profitable a stock will be, due to this believe they will have different believes
on how the market will be in the future. Even if a model existed, that could predict how future returns
would look like as the market is right now, then this model would, unless only one or a few investors knew
about it, change the market return itself. If all investors could use a model to predict how the market would
be tomorrow, they would be able to react to this prediction already today, and since all investors knew how
the market would be tomorrow the stocks would already have made these changes today. In this case the
returns would again be unpredictable, because if the model could predict future returns, investors would
then be able to predict 1 day, 5 days or several months ahead and the market would then be randomly
affected by how far into the future different investors would try to predict.
Page | 48
8. REFERENCES
Black, F. & R. Litterman (1992): Global portfolio optimization. Financial Analysts Journal, Vol. 48 Issue 5, p
28-43.
Blanchard, Oliver (2009). Macroeconomics, Fifth Edition. USA: Pearson Prentice Hall.
Christensen, Michael & Pedersen, Frank (2003). Aktieinvestering: Teori og praktisk anvendelse, Second
Edition. Denmark: Jurist- og Økonomforbundets forlag.
Elton, Edwin J. & Gruber, Martin J. (1995): Modern Portfolio Theory and Investment Analysis, Fifth Edition.
USA: John Wiley & Sons.
Ferguson, R.; Leistikow, D. & Yu, Susana (2009). Arithmetic and Continuous Return Mean-Variance Efficient
Frontiers. Journal of Investing, Vol. 18, Issue 3, p62-69.
Jensen, Michael C. (1968): The performance of mutual funds in the period 1945-1964. Journal of Finance,
Vol. 23 Issue 2, p389-416
Keller, Gerald (2005). Statistics for management and economics, Seventh Edition. China: Thomson Learning.
Madsen, H. (2007): Statistical Tables, First Edition. Aarhus: Academica.
Markowitz, Harry M. (1952): Portfolio Selection. Journal of Finance, Vol. 7 Issue 1, p77-91.
Markowitz, Harry M. (1998): Portfolio Selection: Efficient Diversification of Investments, Second Edition.
USA: Blackwell Publishers.
Markowitz, Harry M. (1999): The Early History of Portfolio Theory: 1600-1960. Financial Analysts Journal,
Vol. 55 Issue 4, p5-16
Michaud, Richard O. (1989): The Markowitz optimization Enigma: Is “Optimized” Optimal? Financial
Analysts Journal, Vol. 45 Issue 1, p31
Moffett, Stonehill and Eiteman (2009). Fundamentals of Multinational finance, Third Edition. USA: Pearson
Prentice Hall.
Nasdaq OMX: Fund info, Danske Invest Danmark - Akkumulerende, Aktier Danmark.
http://www.nasdaqomxnordic.com/Investeringsbeviser/FundInfo/?Instrument=CSE39046
National bank of Denmark (2008). Finansiel stabilitet 2008 2. Halvår.
http://www.nationalbanken.dk/C1256BE2005737D3/side/Finansiel_stabilitet_2008_2_halvaar/$file/kap03.
htm
Rubinstein, M. (2002): Markowitz’s “Portfolio Selection”: A Fifty-Year Retrospective. Journal of Finance, Vol.
57 Issue 3, p1041-1045
Thadewald, T. & H. Büning (2007): Jarque–Bera Test and its Competitors for Testing Normality – A Power
Comparison. Journal of Applied Statistics, Vol. 34 Issue 1/2, p87-105.
Page | 49
APPENDIX 1: EVIEW’S OUTPUT.
240
Series: APM
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.000393
0.000000
12.29000
-13.92000
2.600359
0.103010
5.791794
Jarque-Bera
Probability
340.8902
0.000000
40
0
-10
-5
0
5
10
500
Series: BO
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
400
300
200
100
0
-30
-20
-10
0
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.169531
0.000000
25.98000
-33.60000
3.293627
-0.674172
19.67639
Jarque-Bera
Probability
12176.52
0.000000
20
300
Series: CARLSBERG
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.024330
0.000000
14.55000
-16.66000
2.722508
-0.098139
8.952039
Jarque-Bera
Probability
1542.740
0.000000
50
0
-15
-10
-5
0
5
10
15
Page | 50
350
Series: COLOPLAST
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
300
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.044023
0.000000
12.96000
-18.48000
1.862057
-0.335042
17.58313
Jarque-Bera
Probability
9270.577
0.000000
50
0
-15
-10
-5
0
5
10
200
Series: NORDEN
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
160
120
80
40
0
-20
-15
-10
-5
0
5
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.000939
0.000000
17.77000
-21.69000
3.493960
-0.367700
6.855862
Jarque-Bera
Probability
670.2690
0.000000
15
280
Series: DSV
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
240
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.017816
0.000000
17.49000
-12.69000
2.656744
0.458835
7.771834
Jarque-Bera
Probability
1027.144
0.000000
40
0
-10
-5
0
5
10
15
Page | 51
300
Series: DANISCO
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.015987
0.000000
12.90000
-20.27000
2.191709
-0.160832
13.21657
Jarque-Bera
Probability
4544.958
0.000000
50
0
-20
-15
-10
-5
0
5
10
240
Series: DANSKE
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.046360
0.000000
13.98000
-17.19000
2.686351
0.019026
7.323871
Jarque-Bera
Probability
813.3328
0.000000
40
0
-15
-10
-5
0
5
10
240
Series: FLS
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.042500
0.000000
13.18000
-14.94000
2.926523
-0.090241
5.833125
Jarque-Bera
Probability
350.5739
0.000000
40
0
-15
-10
-5
0
5
10
Page | 52
400
Series: G4S
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
350
300
250
200
150
100
50
0
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.018860
0.000000
7.150000
-13.51000
1.751436
-0.386452
7.471320
Jarque-Bera
Probability
895.6686
0.000000
8
500
Series: GN
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
400
300
200
100
0
-30
-20
-10
0
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.047481
0.000000
18.89000
-27.90000
2.972031
-0.762044
16.39826
Jarque-Bera
Probability
7909.870
0.000000
20
300
Series: JYSKE
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.034904
0.000000
11.11000
-12.53000
2.433788
-0.052037
6.426056
Jarque-Bera
Probability
511.0682
0.000000
50
0
-10
-5
0
5
10
Page | 53
600
Series: KBHL
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
500
400
300
200
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.017835
0.000000
15.82000
-28.55000
2.355183
-1.350912
28.01287
Jarque-Bera
Probability
27533.04
0.000000
100
0
-30
-20
-10
0
10
350
Series: LUNDBECK
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
300
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.028391
0.000000
11.10000
-18.17000
2.216530
-1.153993
13.85670
Jarque-Bera
Probability
5358.974
0.000000
50
0
-15
-10
-5
0
5
10
280
Series: NORDEA
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
240
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.004090
0.000000
16.62000
-11.25000
2.732645
0.700869
8.576797
Jarque-Bera
Probability
1438.351
0.000000
40
0
-10
-5
0
5
10
15
Page | 54
300
Series: NOVO
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.100776
0.000000
9.270000
-14.74000
1.845864
-0.250398
9.282351
Jarque-Bera
Probability
1727.765
0.000000
50
0
-15
-10
-5
0
5
10
280
Series: NOVOZYMES
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
240
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.049224
0.000000
12.14000
-9.930000
2.233693
0.205190
6.244139
Jarque-Bera
Probability
465.1389
0.000000
40
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
240
Series: ROCKWOOL
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.015862
0.000000
19.05000
-19.25000
3.194660
0.215463
9.050304
Jarque-Bera
Probability
1600.446
0.000000
40
0
-20
-15
-10
-5
0
5
10
15
20
Page | 55
300
Series: SYDBANK
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.052797
0.000000
16.30000
-15.63000
2.520425
-0.462647
10.91885
Jarque-Bera
Probability
2765.049
0.000000
50
0
-15
-10
-5
0
5
10
15
500
Series: TDC
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
400
300
200
100
0
-15
-10
-5
0
5
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.032261
0.000000
15.20000
-18.32000
1.737746
-0.486882
24.37224
Jarque-Bera
Probability
19910.85
0.000000
15
400
Series: TORM
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
300
200
100
0
-20
-10
0
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.117835
0.000000
15.94000
-24.33000
3.068724
-0.195027
9.828782
Jarque-Bera
Probability
2035.122
0.000000
10
Page | 56
320
Series: THRANE
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
280
240
200
160
120
80
40
0
-10
-5
0
5
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.015795
0.000000
16.61000
-12.69000
2.493461
0.186398
8.447315
Jarque-Bera
Probability
1296.831
0.000000
15
320
Series: TOPDK
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
280
240
200
160
120
80
40
0
-10
-5
0
5
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.022730
0.000000
14.07000
-9.960000
1.992457
0.115396
8.233861
Jarque-Bera
Probability
1193.925
0.000000
15
350
Series: TRYG
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
300
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.032682
0.000000
7.140000
-13.61000
1.804599
-0.682533
9.339605
Jarque-Bera
Probability
1829.349
0.000000
50
0
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
Page | 57
400
Series: VESTAS
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
350
300
250
200
150
100
50
0
-20
-10
0
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.029215
0.000000
17.80000
-26.20000
3.697651
-0.484747
11.17545
Jarque-Bera
Probability
2948.337
0.000000
20
320
Series: WILLIAM
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
280
240
200
160
120
80
40
0
-15
-10
-5
0
5
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.010144
0.000000
16.25000
-14.56000
2.336994
-0.287168
10.25437
Jarque-Bera
Probability
2303.572
0.000000
15
500
Series: STAT7
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
400
300
200
100
0
-2
-1
0
1
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.022730
0.010000
2.250000
-2.460000
0.455033
0.049914
6.624317
Jarque-Bera
Probability
571.8352
0.000000
2
Page | 58
700
Series: STAT6
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
600
500
400
300
200
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.016485
0.010000
1.210000
-1.100000
0.165130
0.253505
16.97442
Jarque-Bera
Probability
8506.058
0.000000
100
0
-1.0
-0.5
-0.0
0.5
1.0
240
Series: STAT4
Sample 1/01/2007 12/31/2010
IF 1/2/2007 - 12/31/2010
Observations 1044
200
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.022117
0.010000
1.840000
-1.600000
0.332180
0.149124
6.334966
Jarque-Bera
Probability
487.6762
0.000000
40
0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Page | 59
APPENDIX 2: JARQUE-BERA TABLE.
From 02.01.2007 to 31.12.2010
Skewness
Kurtosis
Jarque-Bera
APM B
0,103010
5,791794
340,89
B&O
-0,674172
19,676390
12176,52
Carlsberg B
-0,098139
8,952039
1542,74
Coloplast B
-0,335042
17,583130
9270,58
Norden
-0,367700
6,855862
670,27
DSV
0,458835
7,771834
1027,44
Danisco
-0,160832
13,216570
4544,96
Danske Bank
0,019026
7,323871
813,33
FLSMIDTH
-0,090241
5,833125
350,57
G4S
-0,386452
7,471320
895,67
GN Store Nord
-0,762044
16,398260
7909,87
Jyske Bank
-0,052037
6,426056
511,07
KBH Lufthavn
-1,350912
28,012870
27533,04
H Lundbeck
-1,153993
13,856700
5358,97
Nordea
0,700869
8,576797
1438,35
Novo Nordisk
-0,250398
9,282351
1727,77
Novozymes
0,205190
6,244139
465,14
Rockwool B
0,215463
9,050304
1600,45
Sydbank
-0,462647
10,918850
2765,05
TDC
-0,486882
24,372240
19910,85
Torm
-0,195027
9,828782
2035,12
Thrane&Thrane
0,186398
8,447315
1296,83
TopDanmark
0,115396
8,233861
1193,93
Tryg
-0,068253
9,339605
1829,35
Vestas
-0,484747
11,175450
2948,34
William Demant
-0,287168
10,254370
2303,57
Stat 7%
0,049914
6,624317
571,84
Stat 6%
0,253505
16,974420
8506,06
Stat 4%
0,149124
6,334966
487,68
Source: Own calculations done in Eviews, please see appendix for full output.
Page | 60
APPENDIX 3: FUND INFORMATION.
Page | 61