OTTO-VON-GUERICKE-UNIVERSITY MAGDEBURG
Faculty of Economics & Management
DONETSK NATIONAL TECHNICAL UNIVERSITY
Faculty of Management
Department of International Business Activity
The Portfolio Diversification
Svetlana Maltseva
Cumaka 70
86 700 Khartsyzsk
International Economics
specialisation "European Studies"
Semester: 7
E-Mail: svetlanamalt07@mail.ru
Date of submission: 17.10.2010
Table of contents:
List of Figure…………………………………………………………………………..3
List of Tables………………………………………………………………………......4
List of Abbreviations………………………………………………………………......5
Abstract……………………………………………………………………………......6
1. Introduction……………………………………………………………………......7
2. Stock Market of Ukraine and its Players………………………………………......8
3. Forming the Optimal Portfolio of Risky Securities……………………………......9
4. Optimization of portfolio using risk-free asset……………………………………16
5. Conclusion………………………………………………………………………...18
References………………………………………………………………………….....20
Annex ………………………………………………………………………………...21
2
List of Figures:
Figure 1: Dependence of the investment portfolio return on the risk of investing in the
portfoliо……………………………………………………………………………......13
Figure 2: Types of portfolios…………………………………………………………14
Figure 3: Efficient frontier and risk-free asset………………………………………...17
Figure 4: Graphical interpretation of found tangential portfolio……………………...18
3
List of Tables:
Table 1: Total turnover of Ukrainian companies’ shares and their rank in 2008-2009...8
Table 2: Return and risk of the Ukrainian firms’ securities…………………………...11
Table 3: The optimal structure of securities at different levels of expected rate of
return…………………………………………………………………………………...12
Table 4: Portfolios, formed with risky assets and with risk-free asset………………..17
4
List of Abbreviations:
IP – Investment portfolio
SP – Securities portfolio
RR – rate of return
TP – Tangential portfolio
RFA – Risk-free asset
5
ABSTRACT
In this paper was shown the model of optimization of investment portfolio, using the
securities of Ukrainian stock market. Here were described different types of portfolios
depending on the risk and expected return. In this paper was considered a portfolio with
risk-free asset.
Keywords: investment portfolio, optimization of portfolio, risk, expected return, riskfree asset
6
1 INTRODUCTION
In the Western countries the allocation of capital at the stock market is widely practiced.
In Ukraine, the stock market started to develop actively only with the beginning of
privatization. Today the stock market to some extent has been already formed and this
allows us to ponder over investing money in securities. Therefore, due to the
development of the Ukrainian capital market and increasing the number of securities
emission, the portfolio management, that is the creation of portfolio is one of the topical
questions of financial management. Thus, the main problem to be solved is how to
allocate resources to invest in various alternative investments.
The concept of “securities portfolio” has emerged as one of the ways to cope with high
risks, which are inherent in investing in securities of one issuer. The investor can create
the portfolio (collection of assets) that can achieve the same level of expected return as
any single asset but with a less variable pattern of returns.1 Any negative information
can undo all the calculations about the future of the company, which invested funds.
This is especially true for the emerging stock market. Therefore, the main purpose of
portfolio investment is to minimize market risks inherent in securities of one issuer. The
essence of IP ensures sharing the separation of investment potential among the various
assets. Diversifying means investing money in more than one asset. 2
Diversification is spreading a portfolio over many investments to avoid excessive
exposure to any one source of risk.3 The slogan of diversification is an old English
proverb “don’t put all your eggs in one basket”. In designing a portfolio, investors seek
to maximize the expected return from their investments, given some level of risk they
are willing to accept, portfolios, that satisfy this requirement are called efficient (or
optimal) portfolios.
4
Therefore the aim of optimizing the IP is the formation of
securities, which would be consistent with both the requirements of profitability and
risk – and thus would be sufficiently diversified.
1
See Drury (1996),p.361
2
See Sears (1993),p.15
3
Bodie/Kane/Marcus(2004),p.1050
4
See Fabozzi(1997),p. 261
7
The beginning of the modern theory of financial portfolio was included in the Harry
Markowitz’s articles, and then in the works of William Sharpe and John Litnera, and
was based on the concepts of systematic (market) and unsystematic risk of securities.
Risk that remains after extensive diversification is called market risk, i.e. risk that is
attribute to marketwide risk sources. Such risk is also called systematic risk, or
nondiversifiable risk.
5
The purpose of this assay is to show to the private Ukrainian
investors the mechanism for building a portfolio of securities, using some stable rules,
particularly, mathematical, that can greatly facilitate the investment decision.
2 UKRAINIAN STOCK MARKET AND ITS PLAYERS
Analysis of the Ukrainian stock market in general and domestic local companies, whose
shares are proposed for analysis in this work, is presented in Table 1 (Consideration of
the total turnover of securities of this companies and their rank in 2008-2009 ).6
Rank
Company
2008 2009
10
11
Enakievo Steel Plant
14
10
Motor Sich
22
24
Ukrtelekom
24
20
Donbassenergo
30
17 Nizhnedneprovskiy Tube-rolling Plant
39
43
Poltava GOK(PGOK)
46
62
Illych Mariupol Metallurgical Plant
Turnover,UAH
2008
2009
106399560.45 22 397949.51
86 267 530.98 23 743919.61
36 983 169.62
3 834 039.00
36 936 591.88
7 880 042.00
29 170 093.28
9 673 039.50
18 584 535.54
614 565.00
14 438 337.74
227 800.00
Table 1: Total turnover of shares of Ukrainian companies and their rank in
2008-20097
5
See Bodie/Kane/Marcus(2004) p.223-224
6
See Data of UkrAgrInvest Company
7
See Data of UkrAgrInvest Company
8
3 FORMING THE OPTIMAL PORTFOLIO OF RISKY SECURITIES
(MARKOWIZ MODEL)
Multi-asset portfolios are measured by variance (risk) and expected return of a portfolio
consisting of N different components.8 Herewith, Markowiz Model can be used. to
determine the structure of the optimal portfolio.
The most important assumptions of the first group are the following:
- Market consists of a finite number of infinitely divisible liquid assets, whose gain for a
given period are considered random variables (it means, all assets are risky );
- There are open and reliable historical data about returns of assets, allowing the
investor to estimate the expected (average) values of return and their pairwise
covariances;
- Investors in transactions with the asset free of transaction costs and taxes;
- The investor can form any valid (for this model) portfolio, profitability of which are
also random variables.
As for investor’s behavior, there are two hypotheses: the hypothesis of unsaturation and
risk aversion. These hypotheses suggest that:
• investor always prefers a higher level of welfare ( under the same conditions always
chooses portfolio assets with greater profitability);
• investor from two assets with the same income will prefer an asset with less risk.
• investor always prefers a higher level of welfare ( under the same conditions always
chooses portfolio assets with greater profitability);
• investor from two assets with the same income will prefer an asset with less risk9
Marcowiz’s Model provides searching the portfolio, which is characterized by low-risk,
i.e. has the lowest value of variance of portfolio rate of return (σp2 = D (Rp)), at a given
level of expected RR (mp).10 Formal problem statement is:
8
See Steve Lumbey, p..246
9
See Markowiz problem
10
See Sharpe/Gordon/Bailay(1998),p.213-225
9
N N
 p2  D( R p )    x k x j  kj  min
k 1 j 1
N
m  M ( R )  
x j m j  mC
p
p

j 1
N

 x j  1
 j 1

 x j  0, j  1,..., N

(1)
Where σp2 - variance of rate of return of portfolio securities;
mp - expectation rate of return of portfolio securities;
mc – given expected rate of return of portfolio securities;
σkj - covariance of rates of return k-th and j-th types of securities;
xj - the share of the j-th type of securities in the investment portfolio the share of N number of types of securities of which the portfolio is formed.11
Markowiz’s Model also can be written in matrix form :
 p2  D( R p )  xT x  min
m p  M ( R p )  xT m  mC
 T
I x  1
x  0

(2)
where x - matrix of column dimension N × 1, that reflects the contribution of each type
of securities in the portfolio;
хТ - transposed matrix X;
m - the matrix column dimension N × 1, reflecting the expectation rates of return for
each type of securities
Ω- covariations matrix of rates of return of securities that form a portfolio;
I - an auxiliary unit matrix-column dimension N × 1;
ІТ - transposed matrix I .12
11
See Lyuu (2007),p.554
12
See Lyuu (2007),p.560
10
And:
 11  12 ...  1N

 22 ...  2 N
   21
... ... ... ...

 N 1  N 2 ...  NN
x T  x1
x2
... x N ;

 x1 
m1 
1 

x 
m 
 
; x   2 ; m   2 ; I  1 

... 
... 
...

 
 
 
1 
xN 
m N 

(3)
I T  1 1 ... 1.
According to Zvi Bodie , efficient frontier is curve represents the combined of risky
assets that maximizes the expected RR at each level of portfolio risk.13 Using the data of
listed Ukrainian enterprises, the Markowiz Model and MS Excel software we can build
the efficient frontier.
Issuer
Variance(σi2),%2
(mi), %
Standard deviation
(σi), %
Nizhnedneprovskiy
-1,60
38,86
6,23
PGOK
0,36
21,44
4,63
Donbassenergo
-1,03
26,62
5,16
Yenakiyevo Steel
2,51
17,97
4,24
4,73
272,04
16,49
Ukrtelekom
0,06
14,20
3,77
Motor Sich
1,14
16,59
4,07
Tube-rolling Plant
Plant
Mariupol
Metallurgical Illich
Plant
Table 2: Return and risk of securities of Ukrainian firms 14
To optimize the portfolio securities Microsoft Office tools of, namely the “Solver”,
which is a superset of Microsoft Excel, should be used. To find the model name in the
Markowitz-Ins dialog box "Search for Solutions" tab of the Options should check the
13
Bodie(2002), p.320
14
See Data of UkrAgroInvest Company
11
box next to "non-negative values15. (Annex 1). There was determined the optimal
structure of nine portfolios with different levels of expected RR, namely at the expected
RR from 1 to 4.7%. Characteristics of the findings of optimized securities portfolios
calculation with different levels of expected RR that are formed with the shares of
Ukrainian enterprises are listed in Table 3.
Indicators
Portfolio expected
retirn (mp)for a
week,%
Minimum variance
RR of portfolio
(σp2), %2
Minimum standard
deviation of
portfolio (σp), %
Nizhnedneprovskiy
Tube-rolling
Plant,%
PGOK,%
Donbassenergo,%
Yenakiyevo Steel
Plant, %
Mariupol
Metallurgical Illich
Plant,%
Ukrtelekom,%
Motor Sich,%
IP
1
2
3
4
5
6
7
8
9
1
1.5
2
2.5
3
3.5
4
4.5
4.7
11.41
11.50
12.92
17.80
34.80
74.98
138.45
225.21
266,43
3,38
3,39
3,59
4,22
5,90
8,66
11,77
15,01
16,32
0
0
0
0
0
0
0
0
0
18,59
17,45
0,73
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
23,69
39,30
63,41
95,10
77,74
55,22
32,70
10,18
1,17
0
0
0
1,70
22,26
44,78
67,30
89,82
98,83
29,62
3,79
0
0
0
0
0
0
0
28,10
39,46
35,86
3,20
0
0
0
0
0
Table 3: The optimal structure of securities at different levels of expected return16
To get an idea of the set of efficient portfolios we built a graph showing the relationship
between expected income and risk of the built portfolios. The graph is shown in
Figure 1. Thus, it can be seen in Figure 1 that formed from the shares of Ukrainian
companies, characterization and structure of which are drawn in Table, form a set of
efficient portfolios.
15
See Benning(2007),p.342
16
See Data of UkrAgroInvest Company
12
Expected
portfolio RR, %
5,0
4,5
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
0
2
4
6
8
10
12
14
16
18
Risk (standard deviation,%)
Figure 1: Dependence of the investment portfolio return on the risk of investing in the
portfolio
Any other portfolio that does not belong to the set of efficient portfolios is ineffective
because its risk is higher with the same expected rate of return. The investor chooses
one of the efficient portfolios depending on his personal inclination to risk. All investors
can be divided into three groups:

investors who are inclined to take a risk;

investors who are inclined to take a risk

investors who are neutral to risk.17
The investor is not inclined to take a risk if he chooses the less risky asset from the two
assets with equal expected returns, but different risks. The investor is at risk if the two
assets with equal expected returns, but different risks, he chooses a more risky asset.
The investor is
risk neutral if he does not consider if he does not consider it when making investments
decisions18.
17
See Burenin(1998),p.210
18
See Burenin (1998),p.213
13
The investor is not inclined to risk if the two assets with equal expected returns, but
different risks, he will choose less risky asset. The investor is at risk if the two assets
with equal expected returns, but different risks, he chooses a more risky asset. The
investor is risk neutral if he does not consider it when making investment decisions.19
The level of risks taken are three basic types of investment portfolio:
- Aggressive (speculative) portfolio;
-Moderate (compromise, mixed) portfolio;
- Conservative Portfolio.20
Figure 2: Types of portfolios21
Aggressive (speculative) portfolio is an investment portfolio, formed according to the
criterion of maximizing the profit or growth of invested capital, regardless from the
attendant level of Investment Risk. It allows you to get maximum rate of invested return
on Investment Capital, but it is accompanied by the highest level of Investment Risk,
with which the invested capital may be lost entirely or the large part of it. 22
19
See Burenin,p.291
20
See Sharpe(1997)
21
See Sharpe(1997)
22
See Sharpe(2004)
14
Moderate (compromise, mixed) portfolio is formed aggregate financial investment
instruments, that overall portfolio risk close to average. Naturally, in such investment
portfolio Investment rate of return on invested capital is also close to the market
average, such as Corporate Bonds.
Conservative portfolio is the investment portfolio, formed by the criterion of
minimizing the level of Investment Risk. This portfolio is formed by the most cautious
investors, it eliminates the use of financial instruments, the level of Investment Risk of
is higher than the market one.23
Let us make a classification of the portfolios , that are in Figure 1.
Portfolios 1-3 can be attributed to the conservative IP, because the expected growth of
RR of these portfolios over the risk of growth for each of the above portfolios. This is
because that invested in these portfolios are diversified capital. That is, as can be seen in
Table 3, number 1 to the portfolio includes securities of four companies of Ukraine, to
portfolio number 2 the four companies, and to portfolio number 3 includes the three
companies.
The conservative portfolios have little risk, which is characterized by relatively low
expected profitability. Therefore, investors who are risk averse will choose the portfolio
of a group of conservative portfolios and invest in shares of companies of various
industries, which will diversify the portfolio of securities. As it can be seen in Table 3
capital investors invested in mining (Poltava GOK), steel (Yenakiyevo Steel Plant),
telecommunications (Ukrtelekom) and aircraft engineering (Motor Sich) industries.
The mixed portfolio that provides relatively proportional increase of return on
investment while increasing the level of risk includes portfolio number 4 and number 5.
These portfolios will be attractive to risk-neutral investor. Portfolios are formed from
the group acquired shares Enakiev’s Factory (steel production), OJSC Mariupol
metallurgical Plant, Illich (ferrous metals) and Motor Sich "(aircraft). It should be noted
that when level of risk is growing, the investor do not invest in shares of Motor Sich,
the proportion of shares of Illich Mariupol metallurgical combine starts to grow, and the
proportion of shares of Enakievo plant starts to decrease.
23
School of Investing(2003)
15
Portfolios № 6-9 form a group of aggressive portfolios. The growth of income for these
portfolios is accompanied by rapid growth of risk. However, these portfolios allow
investors to expect the largest growth capital. It should be noted that these portfolios can
not be called diversified as OJSC Illich Mariupol metallurgical combine and Enakievo
MOH share of both activities, namely the production of ferrous metals. The constant
growth of the shares of Illich Mariupol metallurgical combine (number 9 in the
portfolio, their share is 98.83%) should be noted, but different and the biggest one-week
rate of return. That portfolio number 9 is the riskiest, while at the same time is most
profitable.
4 PORTFOLIO OPTIMIZATION USING RISK-FREE ASSET
Solving the problem of optimal portfolio is changing, given the existence of the market
as risky and risk-free securities. Under the risk-free asset we understand such security,
which is characterized entirely predictable rate of return. In this case, if the investor
buys a risk-free asset at the beginning of investment period, he knows exactly what will
be its price at the end of period. Since the uncertainty of the final price risk-free asset is
missing, the standard deviation of its RR equal to zero. In addition, the covariance
between RR of the risk-free asset and any risky asset is also equal to zero: 24
 ij   ij i j , if  j   F  0, то  iF   iF i  0  0
(4)
Since all corporate bonds have a probability of non-payment of income, the risk-free
asset can not be issued by the corporation. Risk-free asset can only be a security issued
by the government. However, not everyone is a security risk-free government in terms
of portfolio management, but only one whose maturity coincides with the period of
ownership of that asset is determined by investor. It is subject to interest rate risk and
the risk of refinancing rate25. The difference between the portfolios of investors will be
only in the proportions of distribution of capital between the tangential portfolio riskfree assets. Tangential portfolio is determined by the point of contact line that goes from
the point corresponding to a risk-free RR to the set of efficient portfolios.
24
Lyuu(2007) ,p.574
25
Lyuu(2007),p.575
16
Indicators
IP
1
2
3
Expected return (mp) for the week, %
1
1,5
2
Share of investments in risk-free asset
0,70
0,47
0,23
The share of investments in tangential portfolio
0,30
0,53
0,77
Standard deviation (σp), %
1,27
2,28
3,29
Nizhnedneprovskiy Tube-Rolling Plant (Х1),%
0
0
0
PGOK,%
0
0
0
Donbassenergo,%
0
0
0
Yenakiyevo Steel Plant %
0
0
0
Mariupol Metallurgical Illich Plant,%
23,60
42,49
61,38
Ukrtelekom),%
2,11
3,79
5,48
Motor Sich,%
0
0
0,00%
Table 3: Portfolios formed with risky assets and with risk-free asset26
.
mp
1
Set of efficient
portfolios
mT
RF
0
σT
σp
Figure 3: Efficient frontier and risk-free asset27
26
See UkrAgroInvest Company
27
See Lumbey(1998), p.251
17
5,0
4,5
Portfolio
4,0 return, %
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
0
2
4
6
8
10
12
14
16
18
Portfolio deviation , %
Figure 4: Graphical interpretation of found tangential portfolio28
5 CONCLUSION
Thus, as a summary of the above work, the following conclusions can be made. Increase
of expected return increases the risk. And the portfolio of the lowest income, and,
therefore the lowest risk level, is the most diversified. Given their individual preferences
for risk and expected return, investors should choose the appropriate mix of investments
that will enable them to achieve their started goal.29
Portfolios №1-3 can be called the conservative IP, due to the fact that the increase of
expected rate of return of these portfolios prevail over the risk of growth for each of the
above portfolios. An investor that risk averse, will choose the portfolio of a group of
conservative portfolios and invest in shares of companies of various industries, that’s
will diversify the portfolio of securities.
28
See Data of UkrAgroInvest Company
29
See Sears(1993),p.15
18
In a mixed portfolio, providing relatively proportional increase of return on investment
while increasing the level of risk include portfolio number 4 and number 5. These
portfolios will be attractive to risk-neutral investor. Portfolios №6-9 form a group of
aggressive portfolios. The growth of income for these portfolios is accompanied by
rapid growth of risk. However, these portfolios allow investors to expect the largest
growth of capital. These portfolios generally very difficult to be denominated as
diversified since they include securities of firms that have a common field of activities.
An optimal decision for the investor to form a tangential portfolio in order to minimize
risk with the same level of expected return. It should be noted that with increasing of
expected rate of return of portfolio the share of risky securities in the tangential
portfolio grows with increased risk for the investor. Therefore, every investor
depending on his propensity for risk can choose a set of portfolios formed with the risky
and risk-free assets that suits his expectations of investment. Therefore, investors should
first invest in a portfolio that contains risk-free asset, as it greatly helps to reduce risk at
the same level of expected return, which is inherent in the portfolio that is formed solely
of risky securities.
19
REFERENCES
Benning, S, (2007) Financial Modeling using Excel, published by “Publishing House
Williams”,592 p.
Bodie/ Kane/ Marcus(2004), INVESTMENTS,6edition,published by McGrawHill/Irwin; 6 edition, 1090 pages
Bodie/ Kane/ Marcus (2002),Investment Principles, forth edition, published by
Publishing House Williams, 984 pages
Burenin A.N.,(2008) Portfolio Management, published by Scientific-Technical Society
of the Academician S.I.Vavilov”, 440 pages
Colin Drury(1996), Management and Cost Accounting, forth edition, published by
International Thomson Business Press, 928 pages
Data of UkrAgroInvest Company(2009),The Report on the liquidity of the shares in the
Ukrainian stock market
Fabozzi F.J., Modigliani F., Ferri M.G.(1994), Foundations of Financial Markets and
Institutions, published by Prentice-Hall, Inc., 564 pages
Lumbey S.(1995),Investment Appraisal and Financial Decisions, fifth edition, published
by Chapman & Hall, 668 pages
Lyuu Y.D.(2007), Methods and Algorithms of Financial Mathematics, publishrd by
BINOM. Knowledge Lab, 751 pages
Markowiz problem, is available at: http://www.wikiznanie.ru/ruwz/index.php/Задача_Марковица_(пример)
School of investing(2003), Something about portfolios : good and different ones” is
available at: http://www.ua.all-biz.info/publications/?pubid=41,13.02.2003
Sears S.,Trennepohl L.(1993),Investment Management, published by Dryden Press, 992
pages
Sharpe,W.F.(1997), Financial Planning in Fantasyland, available at:
http://www.stanford.edu/~wfsharpe/art/fantasy/fantasy.htm
Sharpe Sharpe,W.F.(2004)Investment strategy for a long term”, available at:
http://www.stanford.edu/~wfsharpe/art/UBSArticle.pdf
Sharpe W.F., Gordon J.A., Bailey J.(1998),INVESTMENTS: translated in RussianM.:INFRA-M. ,1050 pages
20
Nizhnedneprovskiy
Tube-Rolling Plant
PGOK
Donbassenergo
Yenakiyevo Steel
Plant,
Mariupol
Metallurgical Illich
Plant
Ukrtelekom
Motor Sich
Motor Sich
Ukrtelekom
Mariupol
Metallurgical Illich
Plant
Yenakiyevo Steel
Plant,
Donbassenergo
PGOK
Nizhnedneprovskiy
Tube-Rolling Plant
ANNEX
38,863
10,936
23,344 18,100
-0,084
11,697 15,810
10,936
23,344
18,100
21,442
16,846
9,398
16,846 9,398
26,616 18,523
18,523 17,969
0,430
11,927
30,230
11,011
11,132
9,267
7,705
8,923
7,550
-0,084
0,430
11,927 30,230 272,043
8,541
15,432
11,697
15,810
11,011
7,705
11,132
8,923
14,199 11,737
11,737 16,594
9,267
7,550
8,541
15,432
Nizhnedneprovskiy
1,000
0,379
0,726 0,685 -0,001 0,498
Tube-Rolling Plant
PGOK
0,379
1,000
0,705 0,479 0,006 0,631
Donbassenergo
0,726
0,705
1,000 0,847 0,140 0,573
Yenakiyevo Steel
0,685
0,479
0,847 1,000 0,432 0,580
Plant
Маріупольський
-0,001
0,006
0,140 0,432 1,000 0,137
металургійний
завод ім. Ілліча
Ukrtelekom
0,498
0,631
0,573 0,580 0,137 1,000
Motor Sich
0,623
0,408
0,425 0,437 0,230 0,765
Table: Correlation coefficients of RR of Ukrainian shares
21
Мотор Січ
Укртелеком
Єнакіївський
металургійний
завод
Маріупольський
металургійний
завод ім. Ілліча
Донбасенерго
Полтавський ГЗК
Нижньодніпровськи
й трубопрокатний
завод
Table: Covariance RR of Ukrainian shares
0,623
0,408
0,425
0,437
0,230
0,765
1,000