extension of mpt

7/30/2019 Extension of Mpt

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Institut for finansiering

Cand.merc.finansiering

Hovedopgave

Forfatter: Elena Kabatchenko Nielsen

Vejleder: Peter Lchte Jrgensen

EFFICIENT PORTFOLIO SELECTION

IN

MEAN-VARIANCE-SKEWNESSSPACE

And additionally:

Can and should structured products be viewed as an independent

asset class in the portfolio of a retail investor?

RHUS HANDELSHJSKOLE, RHUS UNIVERSITET

JANUAR 2008


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Efficient Portfolio Selection in Mean-Variance-Skewness Space

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Table of Contents

1. INTRODUCTION ............................................................................................................ 4

1.1. THESIS STATEMENT..................................................................................................... 5

1.2. DELIMITATION ............................................................................................................ 61.3. STRUCTURE OF THE THESIS ......................................................................................... 6

2. MODERN PORTFOLIO THEORY ................................................................................ 8

2.1. MODERN PORTFOLIO THEORY, SHORTLY..................................................................... 82.2. COMPUTATION OF THE EFFICIENT FRONTIER............................................................... 9

2.2.1. Blacks Model ....................... ......................... ......................... ...................... ..... 12

2.3. DEVELOPMENT OF MODERN PORTFOLIO THEORY...................................................... 12

2.4. JUSTIFICATION FORMEAN AND VARIANCE ................................................................ 142.5. CRITICISM OF THE MODERN PORTFOLIO THEORY ...................................................... 16

3. INTRODUCTION TO SKEWNESS ....................... ...................... ......................... ........ 18

3.1. MEASUREMENT OF SKEWNESS................................................................................... 19

3.2. ECONOMIC IMPORTANCE OF SKEWNESS .................................................................... 213.2.1. Theoretical Justification for Preference for Skewness ........................................ 223.2.2. Behavioral Justification for Preference for Skewness ......................... ................ 23

3.3. PROBLEMS WITH MEASUREMENT OF SKEWNESS ........................................................ 24

4. MEAN-VARIANCE-SKEWNESS PORTFOLIO ANALYSIS ..................... ................ 26

4.1. CHOICE OF THE ARTICLE ........................................................................................... 264.1.1. Short Introduction ............................................................................................. 27

4.2. NOTATION AND SOME FORMULAE.............................................................................. 28

4.3. DUALITY RESULTS .................................................................................................... 31

4.4. MINIMUM VARIANCE PORTFOLIO .............................................................................. 324.5. FINDING MAXIMUM SKEWNESS PORTFOLIO............................................................... 38

5. SKEWNESS OF STOCK RETURNS ........................ ......................... ...................... ..... 45

5.1. DATA ........................................................................................................................ 455.2. SENSITIVITY OF SKEWNESS ....................................................................................... 46

5.2.1. Choice of Differencing Interval ......................... ......................... ........................ 46

5.2.2. Initialization Point ........................ ......................... ......................... ................... 475.2.3. Calculation of the Returns ................................................................................. 48

5.3. ANALYSIS OF SKEWNESS ........................................................................................... 50

5.3.1. Persistence of Skewness ......................... ......................... ......................... .......... 525.3.2. Note on Skewness ......................... ......................... ......................... ................... 53

6. EFFICIENT FRONTIER ............................................................................................... 54

6.1. DATA AND METHODS ................................................................................................ 54

6.2. STOCKCHARACTERISTICS......................................................................................... 556.3. EFFICIENT FRONTIER IN MEAN-VARIANCE FRAMEWORK........................................... 56


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6.4. EFFICIENT FRONTIER IN MEAN-VARIANCE-SKEWNESS SPACE ................................... 596.4.1. Minimum Variance Portfolio ................................. ......................... ................... 59

6.4.2. Maximum Skewness Portfolios ....................... ......................... ...................... ..... 616.4.3. Efficient Frontier in Mean-Variance-Skewness Space ........................ ................ 63

6.5. CONCLUSION............................................................................................................. 65

7. STRUCTURED PRODUCTS .................... ......................... ......................... ................... 66

7.1. DEFINITION OF STRUCTURED PRODUCTS ................................................................... 66

7.2. APPEAL AND RISKS OF STRUCTURED PRODUCTS ....................................................... 677.3. CRITIC ON STRUCTURED PRODUCTS .......................................................................... 687.4. STRUCTURED PRODUCTS AS AN INDEPENDENT ASSET CLASS .................................... 69

8. SKEWNESS OF RETURNS ON STRUCTURED PRODUCTS ......................... .......... 71

8.1. DATA AND EMPIRICAL METHODS .............................................................................. 728.2. THE ALGORITHM IN VISUAL BASIC APPLICATION (VBA) .......................................... 738.3. ANALYSIS ................................................................................................................. 74

8.3.1. OMXC20 ........................................................................................................... 75

8.3.2. Maersk A ...................... ......................... ......................... ......................... .......... 77

8.3.3. Interpretation of the Results ..................... ......................... .......................... ....... 798.4. STRUCTURED PRODUCTS IN THE PORTFOLIO: A COMMENT ........................................ 80

9. CONCLUSION ............................................................................................................... 81

10. FUTURE DIRECTION OF RESEARCH ...................... ......................... ................... 85

11. BIBLIOGRAPHY ....................................................................................................... 87

11.1. ARTICLES.................................................................................................................. 8711.2. BOOKS ...................................................................................................................... 91


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1. IntroductionThe original idea that has motivated this thesis comes from a personal interest for

structured products as financial innovations and strong criticism of these products in

Danish business press. The question that has laid ground for the thesis is whether

structured products can be viewed as a good investment alternative for a retail investor.

In one hand, structured products seem like a very interesting investment class with some

special characteristics. In the other hand, Danish business newspaper Brsen has

published numerous articles (see for example Brsen (2003) and Brsen (2006)) about

the drawbacks of structured products: non-transparency of their characteristics to the

investors and overpricing of these products.

The fact that the price of structured products is higher than it should be in theory is

relatively easy to explain. The market for structured products in Denmark is very new,

and the issuers can take a much higher price because of the lack of competition. The other

reason is that structured products offer some features to retail investors that are not

available to them otherwise, such as participation in the derivative market and

opportunity to monetize their expectations about future price development (Satyajit,

2001). It is fair enough that issuers take a payment for that.

The other branch of criticism: non-transparency of the characteristics of structured

products is more interesting. Characteristics of structured products are different from

those of stocks and bonds, but can be very difficult to analyze because the historical data

of are often not available. There must be something about these products that is veryappealing for an investor, since they enjoyed a very high growth since their introduction

to the financial markets.

The simulations of probability distribution of returns on self-constructed structured

products have shown interesting characteristics towards asymmetry of return distribution.

Skewness, which is the measure of asymmetry of distribution, is much higher for

structured products than it is for stocks. However, if portfolios of retail investors are to be


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constructed according to the traditional portfolio theory, skewness does not play any role,

and therefore structured products will not be of much interest to the investor.

The purpose of this thesis is therefore twofold:

1. To see whether skewness is in fact an important factor for investor, and how it can

be incorporated in portfolio analysis.

2. To see whether structured products can be viewed as an independent asset class in

the portfolio of retail investor.

It is important to notice that this thesis is a theoretical project, which main purpose is totake a look at portfolio analysis, skewness and structured products from a theoretical

standpoint. Empirical part in the thesis should be seen as an application of the introduced

theoretical framework.

1.1. Thesis StatementThe main goal of this thesis is to take a critical look on traditional / Markowitz portfolio

theory and to find out whether other portfolio models can do a better job. This goal will

be achieved by answering the following questions:

What is the reason for including only mean and variance as the main parameters

in Modern Portfolio Theory?

Should higher moments of probability distribution of returns such as skewness be

considered in portfolio analysis?

How should portfolio theory be modified if skewness is included as the third

parameter?

What shape does efficient frontier take in the three-moment portfolio model?

Do Danish stock returns exhibit skewness? Is it persistent?

If Danish stock returns are analyzed in both traditional portfolio framework and

the alternative theory including skewness, what will be the difference between the

results of the two analyses?


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The secondary goal of this thesis is to take a look on structured products as an

independent asset class. The reason for including structured products as a subject to thethesis is their characteristics concerning skewness. Per construction return distribution on

structured products should be skewed, and it is therefore convenient to consider them in

the framework of three-moment portfolio theory. The following questions are sought to

be answered:

Can structured products at all be viewed as an independent asset class?

What are characteristics of structured products? How can they be found when

historical data are most often not available? Can and should structured products be included in portfolio analysis?

1.2. DelimitationA broad subject of portfolio theory is viewed in this thesis. Therefore only the basic

model is considered. It means that I am working with a static portfolio model, i.e. all

dynamic effects of portfolio modeling are not considered. Classic portfolio theory

assumes one-period investment, and it will also be the case for this thesis.

In the empirical part of the thesis, where an empirical study on stocks and structured

products is conducted, dividends and taxes are not taking into account.

Other moments of return distribution higher than skewness, e.g. kurtosis, are not

considered in this thesis, even though some studies indicate importance of kurtosis in the

investment decisions.

In the part where the discussion on structured products takes place, only one kind of

structured products is considered, namely, principle-protected equity-linked note. All

other structures are not analyzed in terms of this thesis.

1.3. Structure of the ThesisThe thesis is divided in two parts:

1. Portfolio theory and skewness: Chapters 2-6.


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Theoretical part is opened with chapter two about Modern Portfolio Theory, where the

reader is presented to the main concept and assumptions behind this theory. Later in thechapter the assumptions are discussed and some criticism of Modern Portfolio Theory is

presented. Chapter three introduces the reader to the concept of skewness, where some

formulae are presented, followed by discussion about why skewness is an important

factor for investors. Here the sensitivity of skewness measure to some parameters is

discussed as well. Theoretical part is finished with chapter four, where three-moment

portfolio model is presented and discussed.

Empirical partof this section has two intentions:

a) Analyze skewness of Danish stock returns

b) To build the efficient frontiers in mean-variance and mean-variance-skewness

space.

Chapter five that opens the empirical part presents the skewness analysis of Danish

stock returns. The following aspects of skewness are analyzed here: Whether returns

on Danish stocks exhibit asymmetry, whether it is persistent and how unstable

skewness measure is with respect to different parameters. Chapter six presents the

second part of empirical analysis that concerns portfolio analysis. Here the efficient

frontier is constructed according to Modern Portfolio Theory using data on four

Danish stocks. Later in the same chapter practical application of three-moment

portfolio model on these four stocks is conducted.

2. Structured Products: Chapters 7-8.Chapter seven discusses structured products and the idea about whether they can beconsidered as an independent asset class in terms of portfolio analysis. Characteristics of

structured products are analyzed in chapter eight. A VBA algorithm is constructed to

simulate probability distribution of returns on structured products and different types of

embedded options are analyzed with respect to their impact on the characteristics of

structured products.


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The common conclusion of the both sections follows after theoretical parts and empirical

analysis. Thesis is finished with some suggestions to a future research in the areaanalyzed in the thesis.

All the empirical work made in this thesis can be found on the enclosed CD.

2. Modern Portfolio TheoryIn this part of the thesis the Modern Portfolio Theory (MPT) will be discussed. It is

assumed that the reader is familiar with the theory. The goal of this chapter is not tosummarize Modern / Markowitz Portfolio Theory, but to discuss the assumptions behind

this theory. In the class the students are taught the basic rules of MPT, i.e. how to

calculate mean return, variance, covariance between assets and how to build up efficient

frontier from a set of probability beliefs about the securities. However, these rules are for

the most accepted without a doubt, and it is intended in this thesis to take a critical look

on this widely accepted portfolio theory. The reader will also be introduced to the

notation in MPT and formulae for mean return, variance etc. This is done, because these

formulae will be used later in the empirical part of the thesis, and because they ground the

mathematical part of MPT.

2.1. Modern Portfolio Theory, shortlyPortfolio analysis consists of two parts: Analysis of securities and combination of the

securities into a portfolio. Modern Portfolio Theory deals with the second part of

portfolio analysis. Modern Portfolio Theory identifies the efficient frontier, i.e. the set of

portfolios that have highest expected return for a given level of risk, and, by duality,

minimum variance for a given expected return. Efficient frontier will be the same for all

investors. Investors risk preferences play role, when investor will pick the combination

of expected return and variance on the efficient frontier that best suites his needs.


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The following assumptions1

are made in Modern Portfolio Theory:

1. The investor seeks to maximize his expected utility of terminal wealth

2. The investor has a single-period investment horizon

3. The investor is risk-averse

4. The investor will choose his optimal portfolios on the basis of means and standard

deviations of returns

5. Markets are perfect: transaction costs and taxes do not exist and securities are

indefinitely divisible

Modern Portfolio Theory states that diversification is the key for combination of

securities in the portfolio because correlation between securities can decrease the overall

variance / risk of the portfolio.

Markowitz builds up the Critical Line Algorithm (Markowitz, 1959, p. 310-312) that

identifies all the feasible portfolios that minimize risk for a given level of expected return.

Graph of this critical line in the expected return standard deviation space is the efficient

frontier. Most of the portfolios on this efficient frontier are well-diversified, because, as

mentioned above, diversification is a powerful tool in terms of risk reduction.

2.2. Computation of the Efficient FrontierThe Markowitz portfolio analysis is built up upon two parameters of return distribution of

assets, namely mean return and variance / standard deviation. These two moments of the

distribution are calculated in the following way2:

Formula 2.1. Expected return on the asset i:

(# Where is expected return on asset i, Rij is actual return on asset i at time j, and N is the number ofobservations.

1Gordon & Francis, 1986, pp. 50-51

2See for example Elton and Gruber, 2003, pp. 45-55


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Formula 2.2. Variance of the returns on asset i:

$ { . $(# Where $ is variance of the returns on asset i, the rest of the notation is as above in formula 2.1..Formula 2.3. Covariance between two assets i and k:

. { . (#

Where is covariance between two assets i and k,Rkj is actual return on asset kat timej, is expectedreturn on asset k. The rest of the notation is as in formula 2.1.In principle, the two moments above should be investors probability beliefs about

performance of a single security. Most often, however, the expected return and variance

are calculated from the historical data. One has to be attentive to the fact that:

When past performances of securities are used as inputs, the outputs of the analysis are

portfolios which performed particularly well in the past.3

Therefore investor should take the past figures as inputs for portfolio analysis carefully,

being attentive of the possible errors of such an approach.

The goal of portfolio analysis is to find efficient combinations, i.e. portfolios of several

securities. The expected return on the portfolio is calculated as weighted average of the

expected returns of respective securities:

Formula 2.4. Expected return on the portfolio:

{ (# Where is expected return on portfolio, iis a weight placed on security i. The rest of the notation is as informula 2.1.

3Markowitz, 1959, p. 3


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In matrix notation:

Formula 2.4.1.: Expected return on the portfolio in matrix notation.

#$ #$

Where is a vector of weights placed on different securities, and is a vector of expected returns onsecurities.

Portfolio variance is a little more complicated than just weighted average of variances of

respective securities, because covariance of the securities enters the formula of portfolio

variance. Portfolio variance is calculated as:

Formula 2.5. Portfolio variance.

$

$

$

(# -

(#

(#

Where $ is portfolio variance, $ is variance of returns on asset i, is covariance between two assets.The rest of notation as is in formulas 2.1. and 2.4.

In matrix notation:

Formula 2.5.1. Portfolio variance in matrix notation.

$ H$ #$ |###$#$#$$$ #$ #$

Where M2 is a variance-covariance matrix and the rest of notation is as in formula 2.4.1.


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2.2.1. Blacks ModelThe Blacks model is used to present the computation of the efficient frontier. TheBlacks model is chosen instead of Markowitz model because the non-negativity

constraint on the portfolio weights is removed in Blacks model, i.e. short sales are

allowed. This model is more convenient to use, as the same notation will be used later in

the thesis.

It is intended to build up the efficient frontier, i.e. to find such portfolios that have

minimum variance for a given level of expected return. This is an optimization problem,

where the objective function variance has to be minimized subject to the two

constraints, i.e. using matrix notation4:

Minimize: H$Subject to given level of expected return: And to the fact that the weights have to sum up to 1: (lis a unit vector).This optimization problem is solved, i.e. weights on risky assets that constitute efficientportfolio are found, by solving the corresponding Lagrangian function:

H$ - # . - ${ . Where #and $ are Lagrangian multipliers.2.3. Development of Modern Portfolio TheoryHicks (1965) claimed in his book that economists were familiar with the idea of portfoliotheory since his article, Hicks (1935), where he argued for diversification of the

investments:

4Gordon & Francis, p. 56


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By dividing up his (investors) capital into small portions, and spreading his risks, he

(investor) would be able to insure himself against any large total risk on the wholeamount.

5

However, Hicks (1935) did not give a proper reason for diversification, i.e. he did not

realize that correlation between assets could lead to elimination of the unsystematic risk

in the portfolio.

Another forerunner for MPT was Dickson Leavens (1945), who used variance as a

measure of risk of the bond portfolio. He showed how risk declines when the number of

bonds is increased in the portfolio, given uncorrelated bond returns.

The era of Modern Portfolio Theory was opened in 1952 with an article Portfolio

Selection by Harry M. Markowitz (Markowitz, 1952). In this article Markowitz argues

for mean-variance rule, i.e. that investor takes an investment decision based on his

probability beliefs about expected or anticipated return of the investment and dispersion /

uncertainty of returns, which is measured by variance. In this article he defines an

efficient set of portfolios calculated on the basis of mean and variance. Markowitz also

introduces the term covariance between assets, which can reduce the overall risk of the

portfolio.

Even though this article of Markowitz was a break-through, Modern Portfolio Theory did

not get a visible interest in the first years after the article was published. It was first after

the works of Tobin (1958), Sharpe (1963, 1964) and Lintner (1965) MPT got its deserved

popularity.6

The efficient frontier when the riskless asset exists was developed first by Tobin (1958).This model is known as Tobin separation theorem, where the set of efficient portfolios

consists of:

Portfolio X*, consisting of only risky assets

Portfolios consisting of partly X* and partly of borrowing or lending.

5Hicks, 1935. The quote is taken from Markowitz, 1987, p. 36



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In this way, the choice of portfolio X* is separate from the choice of borrowing or

lending. The second part of separation theorem: Choice of borrowing and lendingdepends on the risk preferences of the investor.

Later Sharpe (1964) and Linther (1965) used MPT to the problem of equilibrium pricing

of capital assets. They say that because X* does not depend on the investors risk

preferences, i.e. all investors would hold the same portfolio as portfolio of risky assets,

then X* must be a market portfolio. From this assumption, a famous CAPM model for

equilibrium prices of capital assets was developed.

Thus, Markowitz portfolio theory laid the ground for many further developments in the

financial theory and is widely used in practice. However, Modern Portfolio Theory is

based upon some assumptions, where the main assumption is that mean return and

variance of asset returns are necessary and sufficient for conducting portfolio selection.

The next section discusses the reasons for taking only these two parameters in Markowitz

model.

2.4.

Justification for Mean and VarianceIn his work (Markowitz (1952) and Markowitz (1959)), Markowitz assumes that investor

has a set of probability beliefs about return distribution of some securities, i.e. investor

knows the expected return and the standard deviation of the return distribution on these

securities, and proposes a way to how an investor can constitute a portfolio from this set

of securities. Markowitz uses only mean and variance as the only needed characteristics

of the return distribution of securities for constructing portfolio. Markowitz says that

investor should not maximize only the expected return on his portfolio. Explanation for

that he sees in the real world, where investors do diversify. Diversification does not make

any sense, if only expected return is of consideration for the investor, as investor in this

case always would choose to invest his money in the security that gives maximum

expected return.

Markowitz says that investors also consider the dispersion of the return distribution that

most often is measured as variance/standard deviation of the returns. Thus, investors like


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expected return, and dislike variance. Therefore an investor would choose a portfolio that

maximizes expected return for a given variance, or minimizes variance for a given return.Markowitz gives following reasoning for choosing mean and variance as the parameters

for portfolio selection:

Various reasons recommend the use of the expected return-variance of return rule, both

as a hypothesis to explain well-established investment behavior, and as a maxim to guide

ones own action. The rule serves better, we will see, as an explanation of and a guide to

investment as distinguished from speculative behavior.7

The quote above shows that Markowitz chose mean and variance as two important

parameters for portfolio selection because those parameters seemed to him to explain the

investor behavior he could observe at the time.

In his book (Markowitz (1959)) that came after the article in 1952 he says that selection

of the criteria for portfolio analysis depends on the nature of the investor. Thus, the

criteria can be very different, as investors can have different goals with their investment.

However, Markowitz says that his book and his model are useful for all investors, for

which the following characteristics are common:8

1. They prefer more to less, i.e. they want to maximize their return.

2. They want their return to be stable, not subject to uncertainty, i.e. Markowitz

model is not suited for speculative investors.

It is interesting to notice that it is most often said that in order for Markowitz mean-

variance model to hold at least one of the two assumptions has to hold:

1. The return distribution of the assets is normal.

2. The utility function of the investor is quadratic.

Notice that the assumptions stated above were not the first reasoning of Markowitz for

choosing mean and variance as the main criteria for portfolio selection analysis.




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Moreover, the assumption of the normal distribution of asset returns was first suggested

by Tobin (1958), where he stated that

The investor evaluates the future of consols (also, the future of portfolios of bonds and

other debt instruments) only in terms of some two-parameter family of probability

distributions.9

Quadratic utility function was neither the reason to include mean and variance into

portfolio selection. Markowitz explains thoroughly in his book (Markowitz (1959)) why

quadratic utility function can be assumed for the investor, as it can be a good

approximation for other utility functions. So, the choice of quadratic utility function does

not come from empirical observation of investor behavior, but was chosen so it suited the

model.

The main conclusion from the discussion above is that Markowitz has built a theory for

portfolio selection assuming a certain preferences for the investor in his decision making

in the investment process. Even though Modern Portfolio Theory has been widely used

since 1950s, the basic assumptions behind this theory can and should be discussed.

Investor behavior should be analyzed and quantified to reach right conclusions about

investor behavior in the situations of choice.

2.5. Criticism of the Modern Portfolio TheoryIn this section, it is intended to show why, at least in certain circumstances, MPT is not a

good approach to portfolio analysis. Since its origin, Modern Portfolio Theory has been

taken to another level. Nowadays different models of portfolio analysis exist, e.g.

dynamic portfolio analysis, portfolio analysis where the more real assumptions of severalperiods investment horizon are included etc. These models are however not of interest to

the current thesis.

9Tobin, 1958, p. 74


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It is necessary for this thesis to see whether the fundamental assumptions of mean-

variance approach hold, in particular whether the two moments of return distributionindeed are sufficient for conducting a proper portfolio analysis.

According to Tobin (1958), if portfolio theory based on mean and variance of return

distribution should hold, at least one of the following assumptions has to be made:

1. Distribution of asset returns belongs to the two-parameter family, e.g. normal

distribution. If this is the case, the first two moments are sufficient to describe the

whole distribution.

2. Quadratic utility function can be assumed for the investor, i.e. investors goal is to

maximize mean return and minimize variance of return distribution. If quadratic

utility function can be assumed, the higher moments do not play any role in

investors decisions.

Return distribution of stocks in different markets have been investigated by many

researchers. Many studies show that stock returns often are not symmetrical. As an

example, following studies of asymmetry of returns can be mentioned:

a) Study of American equity returns by Beedles (1979), where he finds that the

returns are skewed, even though he finds measure of skewness being quite

unstable over the time;

b) Study of Australian equity returns by Beedles (1986), where he finds significant

positive skewness in the returns.

c) Study of Japanese equity returns by Aggarwal, Rao and Hiraki (1989), where

significant and persistent skewness and kurtosis are found.

These are only a small fraction of similar studies. These findings indicate that assumption

about normality of stock returns is based on the weak ground. It is intended in this thesis

to investigate Danish equity returns in order to see whether they exhibit skewness.

Assumption behind quadratic utility function is that investors do not care about higher

moments of distribution. However, in practice it is known that investors do care for e.g.

skewness / asymmetry of returns and would accept a slightly lower mean return in return


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Efficient Po

for protection against lo

positively skewed returnthree that introduces the re

3. IntroSkewness is a measure of

a distribution is viewed ar

distribution, the shape of

of the shape on the right

always be zero, as thi

distributions are character

right tail, and correspon

negative values, i.e. large

Figure 3.1.: Positive, negative

When distribution is sym

and median11 are the same

In the positively skewed dvalue is larger then the me

estimate up. Correspondin

smaller than the median, b

10Mode defines the most frequ

11Median is the number that s

Positive Skewness

rtfolio Selection in Mean-Variance-Skewness Sp

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sses and possibility of large gain, which

distribution. These issues are discussed inader to third moment of return distribution

uction to Skewness

asymmetry and a third moment of a distrib

ound its mean. Therefore, if the skewness m

istribution on the left side of the mean will

ide of the mean. Thus, skewness of a norm

distribution is perfectly symmetrical.

zed by existence of a few very large positi

ingly negatively skewed distributions have

eft tail. The three situations are illustrated be

nd symmetric distributions. Source: Self-created gra

etrical all three measures of central tende

. However, when distribution is skewed it is

stributions, median lies to the left of the median, because the extreme high positive valu

gly, in the negatively skewed distributions, t

ecause it is pooled down by few extremely la

ent value occurring in the sample

parates the higher half of the sample from the lower

Zero Skewness Neg

ce

is guaranteed by

details in chapterskewness.

tion. Symmetry of

asure is zero for a

be a mirror image

al distribution will

ositively skewed

e values, i.e. large

a few very large

low in figure 3.1.

hs.

cy: mean, mode10

no longer the case.

n value. The means pool the average

e average value is

rge values.

part

tive skewness


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3.1. Measurement of SkewnessIn different sources different measures of skewness have been suggested. Karl Pearson(1895) has been the first to introduce a method for measuring the asymmetry of a

distribution as standardized difference between the average value and mode of

distribution:

Formula 3.1. Skewness measure 1 by Pearson.

J .

Where is mean return and is standard deviation of a distribution.Often the mode is difficult to estimate from the samples, and the median is used instead

in the measurement of skewness:

Formula 3.2. Skewness measure 2 by Pearson.

J { . IJ

The notation is as above in formula 3.1.

As the third moment of distribution, skewness is calculated in the similar way as mean

return and standard deviation12:

Formula 3.3. Skewness as the third moment of return distribution on asset i:

J { . %

Where is expected return on asset i, Rij is actual return on asset i at time j, and N is the number ofobservations.

The problem with this measure of skewness is that it is scale sensitive and cannot be used

in significance testing. Arditti (1971) was the first to suggest a measure of relative

12It is assumed here that all returns are equally likely to occur. Alternatively the probabilities for each

return can be used in calculating skewness


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skewness: the third moment divided by standard deviation raised to the third power. In

this way, the cube of standard deviation is a scaling factor, and thus skewness fordifferent distributions can be compared.

Formula 3.4. Relative Skewness.

IJ { . %9

{ . $9 % $9

The notation is as in formula 3.3.

This is the measure that is most often used in the financial literature, when e.g. stock

returns are analyzed for asymmetry. This measure is also used in statistical packages, e.g.

E-Views and Microsoft Excel. There can be made some modifications to this measure to

get an unbiased estimate of skewness. Thus in Microsoft Excel, the skewness is measured

as:

Formula 3.5. Skewness as it is measured in Microsoft Excel:

J { . { . . %(#

The notation is as in formulas 3.1. and 3.3. above.

The last measure of skewness is accepted and will be used throughout this thesis.

However, it is worth noticing that there exist several measures of skewness. Furthermore,

when different methods applied to a distribution, different values and even the sign of

skewness can appear. It is very important to notice, as skewness is a measure that is much

more sensitive to the way how it is measured, starting point, size of the sample etc. I will

elaborate more on this issue later in this section.


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Notice that if:

J J JI2 J JJI 3.2. Economic Importance of SkewnessIn this chapter it will be shown that skewness in fact is an important factor of return

distribution of the assets and cannot be ignored in the portfolio analysis. The reader will

be introduced to different studies and theories that show and prove that skewness is an

important parameter.

Despite the fact that Markowitz mean-variance portfolio selection theory is still the most

popular and used theory in both universities, banks and investment funds, more and more

authors are agreeing that the traditional portfolio theory is not sufficient. Modern

Portfolio Theory is based on the assumption that only first two central moments of the

return distributions matter. Jean (1971) gives the following three reasons to why

skewness is normally not taken in the portfolio selection analysis:

The form of utility function. If the utility function is quadratic, as assumed in

Markowitz portfolio theory, then the third derivative and all the derivatives of

higher order will be equal to zero. Therefore all the terms beyond the variance

(second derivative) in Taylor series expansion of utility function will be zero.

Distribution of the asset returns. If asset returns are normally distributed as

assumed in Markowitz portfolio theory, then the first two moments will

completely describe the distribution. That is because the distribution of returns

will then be symmetrical, and thus all moments of higher order than the variance

will be zero.

Adequacy of estimation and simplicity. When only two first moments are

considered, the optimization problem is linear and can be solved easily.

It is also important to emphasize that the skewness can be ignored in the portfolio

analysis even if the first two reasons mentioned do not hold in practice, in the case where


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investors empirically do not care about skewness in asset returns. If the (positive)

skewness of the portfolio is not appreciated by investors, it is not necessary to includethis moment in the portfolio analysis.

3.2.1. Theoretical Justification for Preference for SkewnessConsider utility function Uof the investor.

If denotes the wealth of the investor,

denotes investors income

And = 9 is return on the investmentUnder assumption that investors utility function depends only on the sum of his wealth

and income, the utility function can be expressed as13:

l{ - l{ -Letting

{ -be expected value of the investment, the utility function can be

expressed as {. Utility function can be represented by the infinite sum of the termscalculated from the value of the derivatives of this function, i.e. expanded in Taylorseries:

{ l - { $ - -JJJWhere Sk. is skewness of the return distribution.

It is accepted in the financial theory that any utility function has the following properties /

restrictions (e.g., Elton and Gruber, p. 214-215):

Principle of nonsatiation. This principle says that utility of (X+1) dollars will

always be larger than utility of X dollars, i.e. that investors prefer more to less.

Principle of nonsatiation requires positive first derivative of the utility function,

i.e. 2 13

See for example an article of Scott and Horvath (1980)


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Risk aversion. Investors preference for risk is usually discussed in terms of fair

gamble. In fair gamble, the expected value of a gamble is exactly equal to itscost14. Risk aversion means that investors would reject a fair gamble, and would

prefer safe return rather than a risky game with the expected value of the safe

return. It is normally considered that investors would exhibit risk aversion. Risk

aversion require negative second derivative of the utility function: Scott and Horvath (1980, p. 917) show that given the two principles of utility function,

the third derivative of the utility function should be positive. Thus, investors who prefer

more to less and are risk-averse should prefer positively skewed return distributions.

Furthermore, most evidence shows that investors exhibit decreasing absolute risk

aversion, i.e. when the wealth increases, the dollar amount invested in the risky assets

increases, which means decreasing risk aversion. Absolute risk aversion can be expressed

as:

{ . {

{

If investors exhibit decreasing absolute risk aversion, then{ should be less than zero.This means that the third derivative of utility function should be positive:

{ . { { . l { {.{ {$ 7 { 2 If the third derivative is positive, the investors must have positive preference for

skewness.

3.2.2. Behavioral Justification for Preference for SkewnessIt is intuitively easy to see why skewness should be an important factor for investors.

Clearly, investors would have preference for positive skewness, i.e. larger probability for

extremely large gains and limited loss. Preference for positive skewness can also explain

behavior of people that buy lottery tickets. Return distribution of lottery tickets is largely

14See for example, Elton and Gruber, 2003, p. 215


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skewed to the right: there is a possibility of a very large gain, and the loss is limited to the

price of the lottery ticket.

On the other hand, people have aversion towards negative skewness. This aversion

explains why most people buy insurance: expected value of gain on insurance is negative

due to the costs of the insurance company. But insurance protects people from extremely

large losses, and people are willing to pay for that.

As for investment decisions, Alderfer and Bierman (1970) have initiated empirical study

in order to find out whether investors choose the investment alternatives that have ceteris

paribus higher skewness.

The goal of the study by Alderfer and Bierman (1970) was to see whether individuals

behave according to the assumption implicit in the mean-variance framework. Their main

hypothesis was that moments of higher order, in particular skewness, play role in the

investment decisions.

The participants faced several investment alternatives, where some of the alternatives had

similar mean and risk characteristics, but very different skewness of the returndistribution. The study has showed clear preference of the participants for the positive

skewness, even if positive skewness was associated with a lower mean return, i.e. people

were willing to pay for positive skewness.

3.3. Problems with Measurement of SkewnessIt has been shown above that, both theoretically and practically, skewness as the third

moment of the distribution should be considered in portfolio analysis. This means that

skewness should probably be included as a third parameter in the portfolio theory, so that

efficient frontier is built up in mean-variance-skewness space. However, as it will be

shown below, one has to be very careful when working with skewness measure, as it

tends to be much more unstable parameter than mean and variance.

Fogler and Radcliffe (1974) have shown in their study that skewness is a very sample

sensitive measure that depends both on choice of differencing interval and initialization


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point. Fogler and Radcliffe (1974) have investigated data on DJIA15 on the period 1954-

1963, where they have looked at annual, semiannual and quarterly data. Furthermore theyhave looked at skewness measure16 when the initialization points are different, i.e. the

data sample starts at different dates. The results of this study show that relative skewness

differs considerably with choice of differencing intervals. While annual return data in

1954-1963 are positively skewed, the semiannual and quarterly data show quite opposite

result, namely, negatively skewed returns. Skewness varies also with choice of

initialization point, i.e. the results have shown that mean and variance remain more or

less at the same level, while skewness measure is much less stable.

Beedles (1979) discusses further the results of the study by Fogler and Radcliffe (1974).

He provides theoretical explanation for such variations of skewness. In his article Beedles

(1979), where he investigates asymmetry of American equity returns, Beedles says that

the stability of skewness measure depends on whether the process generating Rt, i.e.

returns, is stable and stationary. If the process is not stationary, the skewness measure

will be erratic and highly sample sensitive. Furthermore, he emphasizes that skewness

measure depends on whether returns are calculated as log-returns or simple arithmetic

returns. Skewness of log-returns will always be smaller than skewness of arithmetic

returns, because of the difference of the construction of the returns. However, in the study

of Beedles (1979) he finds out that skewness measure varies much more than expected

for returns calculated in these different ways.

Thus, when the skewness is utilized in some model, e.g. portfolio analysis or asset

pricing, the researcher has to be careful with respect to the choice of the following

factors:

a) Differencing intervals: annual, semiannual, monthly etc. data

b) Log-returns or arithmetic returns

c) Check stability of skewness with respect to starting dates

15Dow Jones Industrial Average

16They used relative skewness as measure of skewness as it shown in section 3.1., formula 3.5.


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4. Mean-Variance-Skewness PortfolioAnalysis

It has been shown above that investors consider mean, variance and skewness in their

investment decisions. Investors will choose those investment alternatives that have higher

mean, lower variance, and higher positive skewness. Thus, investment alternatives have

to be ordered on the basis of all three parameters, as portfolios will not be efficient in

mean-variance-skewness space, if only mean and variance are considered. In this section

the mean-variance-skewness model for portfolio selection is presented.

4.1. Choice of the ArticleThis section of the thesis presents portfolio analysis in mean-variance-skewness17 space

and is based on the article by Athayde and Flres Jr. (2004). This article was chosen

among other articles on the same subject, because it presents a general solution to

portfolio optimization problem in mean-variance-skewness space. The article was written

in 2004, but the model described in the article is becoming more and more

acknowledged. One of the authors, Renato Flres Jr. says18 that the model developed in

this article is mentioned nowadays in any modern book on portfolio theory, i.a. in

Satchell and Scowcroft (2003) and it is also used in i.a. Banque de France. On this basis I

believe that the model in the article has gained some recognition and that its results are

trustworthy.

In this section the reader will be introduced to the findings made in this article. However,

it is not intended to just make a summary of the article, but to explain and understand thetheory and the research made in the article. Some results and propositions of the article

will be cited closely to the text and the references are made, but afterwards each result

and proposition is explained and commented.

17In the article of Athayde and Flres Jr., 2004, the three-moment space is defined as space defined by

mean excess returns, standard deviation and cubic root of skewness. However, I will call mean-variance-

skewness or MVS for convenience purpose.18

From the correspondence with Mr. Renato Jr. Flres


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4.1.1. Short IntroductionThe main assumption in this article is that investors like odd moments (mean, skewness)and dislike even moments (variance, kurtosis). This assumption is based on the article by

Scott and Horvath (1980).

The main result of this paper is calculation of optimal portfolio weights, i.e. calculation of

the efficient frontier in three-dimensional space. The authors assume that the investors

consider the first three moments of the asset returns, that there exists a riskless asset and

short sales are allowed. The authors of this paper focus on the general solution, and

therefore no constraints are imposed on the portfolio weights, which might be necessary

in the real life.

The efficient set of portfolios is presented in the three-dimensional space, where the

dimensions are: excess expected return, standard deviation and cubic root of skewness as

shown in the figure 4.1. below:

Figure 4.1.: Three-moment space as defined by mean excess return, standard deviation and cubicroot of skewness. Source: self-made drawing.

This article presents a general solution to the problem and assumes the existence of the

optimum. Given the highly non-linear nature of the problem, there can be situations,

where the solution is not valid. Those situations are, however, not considered in this

paper. The goal is to find such portfolio set that maximizes skewness for a given return

and variance. It is shown in the paper that the same set presents portfolios that produce

the highest return for given skewness and variance, and the lowest variance for given

return and skewness (see Athayde and Flres Jr., 2004, p. 1337). Therefore this efficient

Excess Return

Standard deviation

J


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set will be efficient in the three-moment space for all three parameters, i.e. investors will

not be able to find portfolio that gets better on one parameter without getting worse onanother.

4.2. Notation and some formulaeWhen the portfolio analysis is extended to include three moments of distribution, the

additional formulae besides those introduced in section 2.2. are needed. Namely the

formula by which the portfolio skewness can be calculated is necessary for such analysis.

Similar to the portfolio variance, portfolio skewness is not just a weighted average of

third moments of asset returns. As assets tend to move together, their returns cannot be

assumed independent. For calculation of portfolio skewness, not only skewness of returns

is needed, but also coskewness between returns. Coskewness is a measure of curvelinear

interaction that occurs in the joint statistical distribution of asset returns19. In this way

intuitive understanding of coskewness is similar to that of covariance between assets.

However, it is important to notice that skewness-coskewness matrix is represented by a

cubic shape as opposite to variance-covariance matrix that has a quadratic shape.

Notice that in portfolio analysis, skewness of asset returns is calculated as the third

moment of distribution and not as relative skewness. Calculation of third moment was

shown previously in section 3.1. with formula 3.3.

Coskewness between N assets is calculated as:

Formula 4.1. Coskewness between N assets:

J ?{ . { . { . C { . { . { . Where is expected return on asset i, Rj is actual return on asset i at time j, and N is the number ofobservations.

Portfolio skewness is calculated as:

19See for example, Simonson p. 383


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Formula 4.2. Portfolio Skewness:

? . C% ?{ . { . { . C Where Rp is actual return on the portfolio, i.e. weighted average of returns at time t of all assets entering in

the portfolio, is expected return on the portfolio, IJ are weights with which assets i, j, and kareentering the portfolio. The rest of the notation is as in formula 4.1. above.

In order to estimate portfolio skewness, the coskewness matrix for the assets is needed.

Coskewness matrix is actually a tensor, a cube of size NNN. However, it can be very

cumbersome and time-consuming to work with these dimensions. The authors are

suggesting transforming NNN cube into NN2 matrix, i.e. slice the coskewness cube

and put the matrices together in order to get NN2 matrix.

If there are two assets (1) and (2), the coskewness matrix will be as following (Athayde

and Flres Jr., p. 1338):

#####$$##$#$#$##$$$$#$$$ Where for example, #$# is coskewness between asset 1 and 2 and so on.Notice that only four elements in this matrix are different, as 112= 121= 211 and so on,

i.e. there is a lot of symmetry, and only J - elements (coskewnesses) should beestimated and not n3.

In case of N assets, coskewness matrix is as following (Satchell and Scowcroft, 2003, p.247):

H% B ####$#####$#$$#$###$#$##$$#$#$#$$$$$$$#$$$ ##$###$$$$#$If M31 denotes a coskewness matrix between the first asset (first slice of the cube) and all

the other assets, the overall coskewness matrix can be viewed as follows:


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H%

?H%#

H%$

H%%

H%

C

In the case of small number of assets entering the portfolio, the coskewness matrix can in

principle be calculated in Microsoft Excel (see file MV Optimization, Sheet

Calculation of Coskewnesses in the folder Portfolio Optimization the enclosed CD).

However, when number of assets is large, calculation of coskewness matrix should not be

done manually.

The following notation is used throughout the article:

is a vector of portfolio weightsx is a vector of expected excess returns

M1 is a vector of mean returns

M2 is a variance-covariance matrix

M3 is coskewness matrix

Rp is expected return on portfolio

is portfolio variance is portfolio skewness

rf is the risk-free interest rate

The characteristics of the portfolio are found in the following way, assuming that the

vector of portfolio weights,

, is known:

Formula 4.3. Mean return of the portfolio:

Formula 4.4. Variance of the portfolio:

6 $


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Formula 4.5. Skewness of the portfolio

H%{ Where stands for Kronecker product20.

4.3. Duality ResultsThe goal of portfolio analysis is finding the efficient set of portfolios, i.e. in the mean-

variance framework the portfolio variance is minimized given some constraints on

portfolio weights and desired return on the portfolio. Thus, the goal is optimization

problem. In order to find optimal portfolio weights and efficient frontier in the three-

dimensional space, the authors need some auxiliary results to solve a much more difficult

optimization problem. In this section, these auxiliary results will be introduced, but not

proved as it does not have any influence of the end result. The references to the article or

respective book(s) will be made, if the reader wants to get acquainted with the proofs.

Duality Result 1: This duality result is taken from Panik, 1976 (Theorem 9.12, p. 210)

and concerns finding extremum of a function when there is a single equality constraint.

This result states that the minimum of a function f(x) constrained by a single equality

condition . { is related to the maximum of g(x), constrained by the objectivefunction . { .This duality result is needed to ensure that the portfolios that are found by minimizing

variance for a given level of return are the same that maximize expected return for a

given level of variance.

Duality Result 2: This duality result is proved in the article and concerns finding

extremum of a function when there are two equality constraints. This result is necessary

for further analysis, because the goal is to maximize skewness while there are restrictions

on both mean return and variance of the portfolio. Duality Result 2 is called Duality

Lemma in the article.

20Kronecker product is a special product of two matrices of arbitrary sizes. Thus, if A is (nm) matrix and B

is (pq) matrix, then C=A B is a (npmq) dimensional matrix and is defined as: C=(aijB). For further

references, see for example Bellman, p. 235.


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Duality Lemma (Athayde and Flres Jr., p. 1338): Let f(x), g(x) and h(x) be real,

continuously differentiable functions of class C

2

on an open set A Rn

. If x* A is astrong (local) minimum of f(x), subject to . { and . { , where and are scalars, with corresponding Lagrange multiplier values given by 1 and 2, 120,and strict second-order conditions

THEN

x* A is also a strong (local) maximum of g(x) subject to{ . { and . {

, with respective Lagrange multiplier values

#and

. .

This duality result is needed for finding efficient frontier in the three-moment space, as it

ensures that efficient frontier will simultaneously: a) minimize variance for given return

and skewness; b) maximize return for given variance and skewness and c) maximize

skewness for given variance and return.

4.4. Minimum Variance PortfolioThe first step in constructing efficient frontier in three-dimensional space is rathertheoretical. A solution found to portfolios that minimize variance for given expected

portfolio return and given portfolio skewness. This chapter is not to be used as a guide to

constructing efficient frontier in practice, but merely a part that provides necessary

theoretical results.

The solution of optimization problem in mean-variance framework is also started from

this step. The goal is to find the efficient set, that is the set of portfolio that give the

highest expected return for given variance, or correspondingly the lowest variance for a

given return. By defining expected returns, a set of portfolios that produce minimum

variance for this given return is found. In mean-variance-skewness space the two

parameters: expected return and skewness are fixed from the beginning, and a set of

portfolios with minimum variance is found.

Define expected return of portfolio as E(rp), and portfolio skewness as .


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Method of Lagrangian multipliers is used to solve this optimization problem:

Minimize variance: H$Subject to following constraints:Given return: - , andGiven skewness: H%{ Minimum variance portfolio can be found by minimizing the Lagrangian function:

H$ - #? . { - C - $? .H%{ C

First-order conditions for the solution of this problem require that the derivatives of this

Lagrangian with respect to , 1 and 2 are set to be equal zero.

# IJ $ Vector H# . denotes a vector of mean excess returns.Number . defines the given (predefined) excess return of the portfolio.With the two definitions above the first-order conditions (derivatives set to zero) areexpressed as follows:

Formula 4.3. First-order conditions:

H$ # - $H%{ # $ H%{


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The Lagrange multipliers 1 and 2 can be found from the above expressions as:

Formula 4.4. Lagrangian multipliers:

# & . $"& . {$$$$ " . $"& . {$$Where:

Formula 4.5. Calculation of A0

" H$#

Formula 4.6. Calculation of A2

$ H$#H%{ Formula 4.7. Calculation of A4

& { H%H$#H%{ Thus to find the solution vector of weights, for minimum variance portfolio, theLagrangian multipliers expressed by formula 4.4. have to be substituted into formula 4.3.

Thus, the solution to this optimization problem has to satisfy the following equation

(system of equations):

Formula 4.8. Solution to minimization problem:

H$ & . $"& . {$

$ - " . $"& . {$

$H%{ As it can be seen from (4), the system is highly non-linear in . However, when the is

found for a series of given mean excess return and skewness, the optimal (minimum)

variances can be calculated by pre-multiplying (4) by vector.Formula 4.9. Optimal variance:

H$ & . $

"& . {$$ - " . $

"& . {$$H%{


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As it was stated above: and H%{ . Therefore the followingexpression for variance is simplified by these formulas:

Formula 4.9.1. Modification of formula 4.8.:

&$ . $ - "{$"& . {$$ Thus, the solution for minimum variance portfolio when expected return and skewness

are given, has been found. When the two parameters expected return and skewness

are changed, the minimum variance frontier can be obtained.

To proceed further in the attempt to finding efficient set in three-moment space, the

authors of the article, make the following proposition:

A positive real number k is fixed, and all pairs ( ) such that %% areconsidered. Remember that R is a mean excess return of a portfolio and is portfolioskewness. The first sentence states that from all portfolios considered, the only chosen are

those where

%%for a given k. The numberkcan vary, as it will be shown later.

Three-moment space is considered in this article, where the three axes are: mean excess

return, R, standard deviation, , and standardized skewness, y3 - cubic root of skewness.From the assumption above, the only points considered in the axis of standardized

skewness are those characterized by %% or% Proposition 121: For a given k, let define the minimum variance portfolio22 when

and

%

23. Correspondingly, denote

# IJ$as variance and

Lagrange multipliers of this minimum variance portfolio,

THEN

21Athayde and Flres Jr., 2004, p. 1340

22Weights of this portfolio are calculated from the formula 4.8.

23It means that the minimum variance portfolio will be different when k is changed.


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For all optimal (minimum variance) portfolios related to skewness/return pairs such that

%% or% :24

The solution to (4) will be with corresponding minimum variance $ and Lagrange multipliers (3) # # and$ $ s The two funds separation property is valid.

This proposition is proved in the article. However, this proof is omitted from this paper

because it is the result that has the main importance for understanding this article.

The result of this proposition is that once the numberkis fixed, the direction is settled in

mean return-skewness space. See the visual presentation of this stand below in the figure

4.2.:

Figure 4.2.: Graphical presentation of proposition 1. Source: self-created graph.

As it is illustrated in the figure 4.2. above and presented in proposition 1, all minimum

variance portfolios in a particular direction specified by choice of number k, are

multipliers of the starting portfolio

. In the article it is stated that along these lines the

only thing that changes is the proportion of risky portfolio and a riskless asset. Thus,the authors say that the two fund separation property holds. That is, investors will hold a

combination of riskless asset and a market portfolio, so that the unit sum condition holds.

24The number k is remained fixed, and the expected return, R, changes. Only portfolios with skewness

% are considered, and between them the optimal ones are chosen.

y3

k-direction

1

R

k

R

kR


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When k changes the solution to the minimum variance portfolio also changes. In the

mean-variance-skewness space, the optimal standard deviation as a function of excessreturn is a straight line defined by:

#.Figure 4.3. presents solutions for , i.e. standard deviation, for a single directiondefined by k.

Figure 4.3.: Solutions for standard deviations in three-moment space. Source: Self-created graph.

When the kdirections vary the whole of optimal variance portfolios arises. They can in

principle have different forms illustrated in the article (Athayde and Flres Jr., 2004, p.

1342). The normal and ideal case is taken from the article and presented in the figure 4.4.

below. The other cases are not considered relevant for this thesis.

Figure 4.4.: Ideal shape of the optimal three-moment set in mean-standard deviation-standardized

skewness space. Source: Athayde and Flres Jr., 2004, p. 1342

k-direction

y3

R

y2

y3

Y2


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As it can be seen from the two figures above and figure 2 in the article of Athayde and

Flres Jr., the shape of the optimal set has a homothetic property. Homothecy means suchtransformation of space that enlarges distances with respect to a fixed point. This fixed

point is the point of intersection of the three axes. This is very important as it means that

the directions that give the highest (and lowest) mean return and skewness, are

independent of the level of standard deviation. That is, if the highest return/skewness is

found for some level of standard deviation, the same line/direction will give the highest

return/skewness for all other levels of standard deviation25. The next problem is to find

those directions that give the highest return and skewness. This problem is discussed in

the next section.

4.5. Finding Maximum Skewness PortfolioIn the classical Markowitz portfolio theory the duality result is used as well. The duality

result used there is that the efficient set consists of portfolios that have minimum variance

for a given return, or correspondingly (duality) that have maximum return for a given

level of variance. Construction of the efficient in mean-standard deviation-standardized

skewness set requires understanding of the three following propositions26

:

Proposition 1: Inspired by Markowitz, the authors find the solution to the classical

optimization problem in mean-variance space. The solution to that is the celebrated

Capital Market Line as in Markowitz. However when this line is situated in the three-

moment space, the corresponding skewness and numberkcan be found. If it is assumed

(calculated) that the skewness of optimal portfolios is not zero then the associated

skewness and number k to the solution above (CML) can be found. By finding this

solution, the direction kassociated with the highest return for a given level of standard

deviation has been found. This kis called kR.

25Proposition 2, Athayde and Flres Jr., p. 1343

26Propositions 3 and 4, Athayde and Flres Jr., p. 1343 and p. 1348


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Proof27: Consider a classical optimization problem in mean-variance space: Minimize

variance for a given level of expected return of the portfolio.

Minimize: H$Subject to the return constraint: Notice that no constraints are placed on the weights, i.e. on the vector . In the mean-

variance framework the additional constraint would that the weights add up to 1.

Minimize Lagrangian function:

Formula 4.9. Lagrangian function:

H$ - #{ .First-order conditions are:

Formula 4.11. First-order conditions

IJ # This gives:

Formula 4.12. Derivative of Lagrangian with respect to : H$ . # ; H$ #

And:

Formula 4.13. Derivative of Lagrangian with respect to Lagrangian multiplier:

# . ; 27

Proofs of this and the following two propositions are necessary as they explain the optimization

problem and the formulas in these proofs are used later in the calculation of the efficient frontier


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Using formulas 4.11 and 4.12, theLagrangian multipliercan be found and it is equal to:

Formula 4.14. Lagrangian multiplier:

# " Where A0 is calculated by formula 4.4.

When the multiplier is found, the vector of portfolio weights, R,28 can be calculated from

formula 4.12.:

Formula 4.15. Weights on minimum variance portfolio:

"H$#The variance of this portfolio can be calculated now when the Ris known:

Formula 4.16. Variance on the optimal/efficient portfolio:

H$ $"The standard deviation as a function of expected excess return defines the Capital Market

Line in the mean-variance space as in Markowitz portfolio analysis with unlimited

riskless lending and borrowing. In the Markowitz theoretical framework only

expected/average return and variance were the factors necessary to build up the optimal

portfolio. In this framework, the skewness of the portfolios will be zero.

However, when this portfolio is considered in the three-moment framework and when the

skewness of assets entering the portfolio is not zero, the associated skewness can be

calculated for this portfolio.

The skewness of this optimal portfolio is found as:

Formula 4.17. Skewness of the efficient portfolio in mean-variance space:

28This is the optimal portfolio associated with highest expected excess returns


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%

H%{ %"% H%{ H$

#

The standardized skewness is the cubic root of the expression above. Remember that kis

a ratio of standardized skewness to the expected excess return. Thus, associated kcan be

calculated as:

Formula 4.18. Direction k:

H%

{

"

Where is cubic root of portfolio skewness as calculated by formula 4.15. above.As it can be seen from formula 4.16. the direction kRis invariant, and all maximum mean

returns lie in the same direction in mean-variance space. The situation is illustrated in the

figure 4.5. below:

Figure 4.5.: Optimal portfolios with highest possible excess returns for a given level of variance.Source: self-made drawing.

Proposition 2: In the same way as proposed in proposition 1 the direction k that gives

optimal portfolios with highest skewness for a given level of standard deviation is found.

The minimization problem solved is one where variance is minimized subject to

skewness. This kis denoted kS.

Proof: The similar steps to those in proposition 1 have to be done. Variance has to be

minimized subject to skewness constraint.

R

y3

Capital Market Line in MV space

k- direction in MVS space


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Minimize:

H$

Subject to the skewness constraint: H%{ Minimize Lagrangian function:

Formula 4.19. Lagrangian function.

H$ - #{ .H%{ First-order conditions are:

Formula 4.20. First-order conditions:

IJ # Thus:

Formula 4.21. Derivative of Lagrangian with respect to :

H$ . #H%{ ; H$ #H%{ And:

Formula 4.22. Derivative of Lagrangian to Lagrangian multiplier:

# . { ; H%{ From formulas 4.19 and 4.20 above, the value of Lagrangian multiplier can be found aS

29

:

Formula 4.23. Lagrangian multiplier:

# $% .29

In this part of the paper the intermediate results are omitted. However, I have checked them in the

article and as there are no mistakes in the derivation of the expressions for Lagrangian multipliers etc., the

reader is directed to the article of Athayde and Flres Jr. (2004) to look at the intermediate results.


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When the value of multiplier is known, the vector of weights S30 can be found from

formula 4.19:

Formula 4.24. Solution to the optimization problem:

&H$#H%{As it can be seen from formula 4.22., the solution to this optimization problem is defined

by a highly non-linear system.

When given skewness is equal to 1, the solution to this system will be:

Formula 4.25. Solution to the optimization problem when skewness is equal to 1:

&H$#H%{Homothecy implies that all the other solutions are linearly related to the solution defined

by formula 4.25., i.e.

Formula 4.26. Condition of homothecy:

Optimal variance is thus, also by condition of homothecy:

Formula 4.27. Optimal variance:

{

$

These portfolios are found in variance-skewness space. However, associated expected

return, when skewness is equal to 1 can be found as usual:

Formula 4.28. Expected return when skewness is equal to 1:

30

Optimal portfolio with the highest skewness


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Expected returns of the portfolios associated with different levels of portfolio skewness

are given as: Associated direction k is invariant, i.e. independent of level of variance and is found as:

Proposition 3: It is furthermore proved in the article that if kR is the direction associated

with the highest return line (CML) and kS is the direction associated with the highest

skewness line, which is supposed to be unique

THEN

4 , i.e. the angle of the highest skewness line is greater than the one of the highestreturn line.

In the figure 4.6. below the two lines - efficient frontiers that maximize return and

skewness for the given level of variance are presented.

Figure 4.6.: Illustration of efficient frontier in mean-variance-skewness space. The efficient frontier is

the area between Max Return and Max Skewness lines. Source: Athayde and Flres Jr., 2004, p. 1342

Y3Max Return

Max Skewness

Y2

R


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The efficient frontier in mean-variance-skewness place, i.e. set of all portfolios where

investor cannot get any better at one parameter without getting worse at another, is thesurface limited by these two lines. In the article, it is stated that:

Within the region between the two canonical lines the lemma applies to both possible

inversions, namely, maximize skewness, given the same mean and the optimum variance,

or maximize mean excess return given the same skewness and the optimum variance.31

5. Skewness of Stock ReturnsIn the theoretical part of thesis in section 3.4. it has been shown that skewness is a

parameter that is important to investor, and should be considered in the portfolio analysis.

Studies on stock returns in different countries (e.g. Beedles (1979 and 1986), Aggarwal,

Rao and Hiraki (1989)) showed significant asymmetry in stock returns. The problem with

skewness measure, as it was shown in section 3.5. is, however, that this measure can be

highly sensitive to the different parameters.

It is therefore intended in this section of the thesis to conduct a study on Danish stock

returns. The main goals of this analysis are:

1. To see whether skewness of Danish stock returns is a measure that is sensitive to:

a. Choice of differencing interval

b. Starting dates

c. Calculation of stock returns: log or arithmetic

2. To see whether Danish stock returns exhibit significant asymmetry and in case of

positive result, whether this asymmetry is persistent across time.

5.1. DataTo conduct empirical analysis of skewness, the monthly data on the stocks from

OMXC2032 index in the period 1987-200733 are used. Similar to the study by Beedles

31Athayde and Flres Jr., 2004, p. 1349


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(1986) each stock with data available for at least 25 monthly returns is taken into

analysis, i.e. only TrygVesta is not taken into analysis, as there are only 24 monthlyreturns are available. In the part, where sensitivity of skewness to the initialization point

and persistence of skewness for single stocks are analyzed, only companies with

historical data available for 20 years are taken into consideration.

5.2. Sensitivity of SkewnessIn this section, sensitivity of skewness towards different parameters will be analyzed.

5.2.1. Choice of Differencing IntervalIn section 3.5., the study by Fogler and Radcliffe (1974) has shown that choice of

differencing interval can have an effect on the size and even the sign of skewness. For

this part of analysis, annual, monthly, weekly and daily data of the 18 stock returns have

been analyzed. Here, returns were calculated as simple arithmetic returns:

Formula 5.1. Formula of arithmetic returns:

. ## Where Pi is the price of the stock at date i.

Skewness is calculated as relative skewness from the section 3.1. with formula 3.5. The

results are shown below in table 5.1:

32OMXC20 is the price index of the 20 most sold Danish stocks in Copenhagen Stock Exchange. All stocks

in OMXC20 index can be seen in the Attachment 133

Notice that not all stocks have return data for all the years


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Daily Weekly Monthly Yearly

Carlsberg -0.228 0.210 0.103 0.364

Coloplast 0.309 0.301 0.400 0.531

Danisco -0.376 -0.012 0.033 -0.142

Danske Bank 0.870 1.069 0.766 0.716

D/S Torm 1.432 1.248 1.175 1.325

DSV -0.400 0.092 0.332 -0.211

FLSmidth 0.370 0.295 0.322 0.224

Genmab -3.627 -1.869 -0.200 -0.235

GN Store Nord 0.018 0.779 -0.117 0.281

Lundbeck -0.499 -0.177 0.418 1.116

Maersk A -0.106 0.501 0.630 1.270

Maersk B 0.762 0.658 0.597 1.165

NKT Holding 0.56 0.27 -0.87 -0.8

Nordea -0.027 -0.543 -0.345 -0.602

Novozymes 0.588 0.246 -0.165 0.080

Sydbank -0.137 1.381 1.349 0.202

Topdanmark 0.030 0.327 0.556 0.170

Vestas 1.148 0.572 -0.267 0.429

William Demant 0.482 0.864 0.271 0.708

Average 0.061 0.327 0.262 0.347

Table 5.1.: Analysis of skewness on Danish stock returns with different differencing intervals.

Skewness is calculated in E-Views as relative skewness. The formula used in E-Views is slightly differentfrom that stated in section 3.1., formula 3.5. and is as following: J # %(# , where { . 9 is an unbiased estimator for the standard deviation. This formula gives approximatelythe same results as with formula 3.5. from section 3.1.

From the table 5.1. above, it can be seen that average skewness has at least the same sign

and for all differencing intervals and approximately the same magnitude for weekly,

monthly and annual stock returns.

Thus, although skewness measure does seem to vary across differencing intervals, at leastaverage skewness for the 19 stocks is of the same sign and for the case of weekly,

monthly and annual data is of the same magnitude.

5.2.2. Initialization PointFogler and Radcliffe (1974) have showed in their study that skewness can change

dramatically with change of the initialization point, i.e. starting date of the returns. To see

whether this is the case for Danish stock returns, the ten stocks with monthly return data


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available for ten years have been chosen. Similar to the study of Fogler and Radcliffe

(1974) initialization point was changed quarterly, i.e. four dates were used as startingdates for skewness analysis. The results can be seen in the table 5.2. below, where only

average skewness is presented.

Date Average Skewness

12/31/1987 0.467

3/31/1988 0.476

6/30/1988 0.465

9/30/1988 0.460Table 5.2.: Skewness analysis for ten Danish stock returns, when initialization points are changed

quarterly. Here skewness

extension of mpt

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