Choosing the Portfolio Selection Model for the Ukrainian Stock Market
OTTO-VON-GUERICKE-UNIVERSITY MAGDEBURG Faculty of Economics & Management DONETSK NATIONAL TECHNICAL UNIVERSITY Faculty of Management Department of International Business Activity
Choosing the Portfolio Selection Model for the Ukrainian Stock Market
Daria Zhuravlyova Kozyria Str. 47, 12 83064 Donetsk International Business Administration Semester: 7
E-Mail: [email protected] Date of submission: 18.10.2010
Table of Contents
List of Tables ........................................................................................................ 3 List of Symbols ..................................................................................................... 4 Abstract ................................................................................................................. 5 1. Introduction ...................................................................................................... 6 2. Models of Portfolio Selection .......................................................................... 7 2.1 Markowitz Portfolio Selection Model ....................................................... 8 2.2 Sharpe Single-Index Model...................................................................... 11 3. Determining the Efficient Stock Portfolio at the Ukrainian Market ........ 13 3.1 Applying the Markowitz Portfolio Selection Model .............................. 14 3.2 Applying Sharpe Single-Index Model ..................................................... 15 4. Conclusion ....................................................................................................... 18 References ........................................................................................................... 20
2
List of Tables Table 1. Data on Stock Quotation Variations Table 2. Return and Risk of the Stock under Review Table 3. Linear Correlation Coefficients for the Stocks Table 4. Structure of the Optimal Portfolio according to the Markowitz Model Table 5. Data on the Stock Market Return and Riskless Asset Return Table 6. Stock Parameters according to the Sharpe Model Table 7. Optimal Stock Portfolio according to the Sharpe Model Table 8. Comparison of the Optimal Portfolio Structure according to the Markowitz and Sharpe Models
3
List of Symbols E(r) – portfolio return σp – portfolio risk wi – percentage of the ith asset in the portfolio σi – some characteristic of the risk of ith asset ri – the yield of the ith asset n – a number of assets in the portfolio σreq – the given level of risk E(r)req – the planned expected return ρij – the linear correlation coefficient between ith and jth assets Rf – the return on the risk-free asset αi – the excess return of the ith asset Rm – the expected stock market return βi – the risk of the ith asset σ2ei – the residual risk of the ith asset
4
Abstract The paper contains 19 pages, 1 illustration and 8 tables. It deals with the issue of using the benefits of portfolio diversification and choosing the most appropriate portfolio selection model for the Ukrainian stock market. Definition of the problem is: What portfolio selection model is the most reliable for diversifying stock portfolio at the Ukrainian stock market?
In the introduction the portfolio diversification is defined as spreading the investment across different assets in order to reduce the level of risk of investments. In Section 2, Markowitz portfolio selection model and Sharpe single-index model of portfolio optimization have been chosen among the numerous models of portfolio optimization, and a short review is given of their assumptions and content. Section 3 introduces the results of choosing the optimal portfolio for the stocks of the six Ukrainian public corporations according to the Markowitz and Sharpe models. They show the difference in applying the models. In Section 4 an assumption is made that the Markowitz portfolio selection model is more reliable for the evolving Ukrainian stock market.
Keywords: portfolio, stock, portfolio diversification, expected return, risk, stock market, Markowitz Portfolio Selection Model, Sharpe Single-Index Model
5
1. Introduction According to Corrado and Jordan, portfolio is a ‘group of assets such as stocks and bonds held by an investor’1. Any asset or their combination is characterized through the term ‘risk’ – ‘the possibility that actual cash flows (returns) will be different than forecasted cash flows (returns)’2. Efficient, or optimal, portfolio lets an investor solve the direct problem – to maximize the expected return from their stock when a certain level of acceptable risk is given,3 as Fabozzi, Modigliani and Ferri claim. In case of indirect problem optimal portfolio provides an opportunity to minimize risk when the level of expected return is assigned. However, it is a rather complex task to form such an effective portfolio, as the higher return on assets is usually connected with the higher level of asset risk.
In order to minimize risks of stock investments with the high return investors widely use the principle of diversification. It means ‘spreading an investment across many assets’ in order to eliminate some part of risks4. Due to diversification, individual risky stocks almost always can be combined in such a way that a less risky portfolio (or their combination) is obtained5. An important concept of the portfolio effect is correlation – ‘the extent to which the returns on two assets move together’.6 The assets are positively correlated if their returns tend to change in the same direction, and negatively correlated if their returns move in the opposite directions. Sometimes it can said that the assets are uncorrelated if there is no connection between changes in their returns. The investor will eliminate some risk due to diversification if the assets in the portfolio are negatively correlated,7 but not only in the case of negative correlation. The
1
Corrado/Jordan (2000), p. 491 Moyer/McGuigan/Kretlow (1992), p. 211 3 See Fabozzi/Modigliani/Ferri (1998), p. 261 4 See Corrado/Jordan (2000), p. 496 5 See Ross/Westerfield/Jaffe (1996), p. 239 6 Corrado/Jordan (2000), p. 497 7 See loc. cit., p. 498 2
6
diversification can also be achieved by investing in a combination of securities with different risk-return characteristics8, as Moyer, McGuigan and Kretlow claim.
Since the 1950’s there have been developed a lot of portfolio optimization models, such as Markowitz portfolio selection model, Sharpe single-index model, Capital Asset Pricing Model (CAPM), Tobin portfolio model,9 Black’s model, TobinSharpe-Lintner model, Quasi-Sharpe model, etc. The Ukrainian stock market today is at the stage of its growth and development, and there is no agreement of opinions of the Ukrainian scientists which portfolio selection model is preferable. Borschtschuk is in favour of the Markowitz approach application to maximizing the stock portfolio return10. Vasilenko and Dyba find that CAPM should be used for optimization of the stock portfolio11. Savchuk and Dudka offer a model based on the Sharpe theory12. Kovalenko claims in his works that the approach to choosing the best portfolio diversification model should be individual, and such a model should protect the investor from the stock prices fluctuations13. The variety of opinions shows that this line of research is relevant for the modern Ukrainian economy. Therefore, in order to determine the portfolio selection model, the described models will be checked empirically.
2. Models of Portfolio Selection Let the expected return E(r) from the stock portfolio and its risk σp be determined with the help of the following functions: E(r) = RETURN(wi; σi; ri), σp = RISK(wi; σi; ri), i = 1 to n,14 where wi is percentage of the ith asset in the portfolio, σi is some characteristic of the risk of ith asset,
8
See Moyer/McGuigan/Kretlow (1992), p. 222 See Tobin (1965) 10 See Борщук (2002) 11 See Василенко/Диба (2006), с. 42 12 See Савчук В./Дудка В. (2001) 13 See Коваленко (2010), с. 9 14 Савчук В./Дудка В. (2001) 9
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ri is the yield of the ith asset, n is a number of assets in the portfolio.
In this case the problem of diversifying stock portfolio is to find such a combination of stocks for which the expected return is maximal and the estimated risk is minimal.
The direct problem containing the given level of risk σreq is of the form: E (r ) → max; σ ≤ σ ; req p wi ≥ 0; ∑ wi = 1
(1)
The indirect problem with the planned expected return E(r)req has the next form:
E (r ) ≥ E (r ) req ; σ p → min; wi ≥ 0; w =1 ∑ i
(2)
We will study two classical models of the portfolio selection because of the simplicity of obtaining the necessary data for evaluating these models.
2.1 Markowitz Portfolio Selection Model
The Markowitz portfolio selection theory (1952) is the first basic theory of portfolio optimization. This model of investments diversification claims that level of portfolio risk depends on the risks of its component assets, taking into account the correlation between return on the stocks in this portfolio. The main assumptions of the theory are as follows: •
investors take into account only such factors as the expected return on asset and its risk;
8
•
the expected value of return on asset is determined by a weighted sum of its components and risk is evaluated through a variance of return, or equals to its standard deviation15;
•
the expected return and risk of assets in the future can be predicted on the base of the data on the return and variance from the past time periods;
•
relations between assets in the portfolio are stated through their linear correlation coefficients.
According to the Markowitz model, the expected return E(r) from the stock portfolio is equal to the sum of the expected return from each its stock times the percentage of this stock in the portfolio: n
E (r ) = ∑ wi ri ,16
(3)
i =1
The portfolio risk σp2 equals:
σp =
n
n
∑∑ w σ w σ i
i =1 j =1
i
j
j
ρ i j ,17
(4)
where ρij is the linear correlation coefficient between ith and jth assets. Therefore, the direct problem has the form:
n ∑ wi ri → max; i =1 n n ∑∑ wiσ i w j σ j ρ i j ≤ σ req ; i =1 j =1 wi ≥ 0; ∑ wi = 1 The indirect problem is the following one:
15
See Markowitz (1952), p. 77 Bodie/Kane/Marcus, p. 219 17 Савчук В./Дудка В. (2001) 16
9
(5)
n ∑ wi ri ≥ E (r ) req ; i =1 n n ∑∑ wiσ i w j σ j ρ i j → min; i =1 j =1 wi ≥ 0; ∑ wi = 1
(6)
On the base of these formulas the investor should determine the efficient portfolios with the help of finding the best combinations of the expected return E(r) and its risk σp when given expected yields of assets E(ri) and linear correlation coefficients ρij. Markowitz represented geometrically the set of efficient combinations as a curve line, inside which the attainable combinations lie, and the inappropriate combinations lie outside, as illustrated in the Fig.1. σp
attainable E(r), σp combinations efficient E(r), σp combinations
E(R)
Figure 1. Efficient Portfolio Set18 The Markowitz portfolio theory helps investors to determine and exclude the inefficient combinations of assets. Still, it must be kept in mind that the disadvantage of this model is orientation only at the characteristics of the set of the stocks involved and usage of the historical data which do not provide the stability of the stock quotations variations at the market. Also, the model requires tedious calculations of weights of the stocks taking into account their correlation. Therefore, still the results of Markowitz
18
Markowitz (1952), p. 82 10
portfolio model can be used only in the conditions of a stable stock market, when the return on asset really depends on its past values.
2.2 Sharpe Single-Index Model The next model was developed by W. F. Sharpe in 1970. In contrast to Markowitz portfolio selection, this one considers the relations of each stock not with each other but with the whole market. The model assumptions include the following statements: •
the return on the stock is equal to its mathematical expectation;
•
there exists the linear regressionship between the market return and the return on each stock;19
•
the risk of the stock means the level of dependence of changes in its return on changes in the general market return;20
•
in common with the Markowitz model, the data on the return and variance from the past time periods are assumed to reflect entirely the future return and risk trends;
•
there exists a riskless asset at the market. The notion of the risk-free asset means that the ‘returns from the initial investment are known with certainty’21. The example of such an asset can be a government bond.
According to this model, the expected return on the stock portfolio is described by the following equation:
E (r ) = R f + ∑ α i wi + ( Rm − R f )∑ β i wi ,22 where Rf is the return on the risk-free asset, αi is the excess return of the ith asset, Rm is the expected stock market return, βi is the risk of the ith asset.
19
See Дубровин/Юськив (2008) See Мойсеєнко І. П. (2006) 21 Moyer/McGuigan/Kretlow (1992), p. 211 22 Савчук В./Дудка В. (2001) 20
11
(7)
The portfolio risk in the Sharpe’s model is equal to:
σ p = (∑ ( β i wi )) 2 + ∑ (σ ei2 wi2 ) ,23
(8)
where σ2ei is the residual risk of the ith asset. Some additional variables, namely the residual risk and the excess return of the asset, are introduced in the Sharpe model compared with the Markowitz model. We will not provide the paper with the formulas on calculating them, as they are rather solid. The direct problem of portfolio selection is the next system:
R f + ∑ α i wi + ( Rm − R f )∑ β i wi → max; (∑ ( β i wi )) 2 + ∑ (σ ei2 wi2 ) ≤ σ req ; wi ≥ 0; ∑ wi = 1
(9)
The indirect problem of portfolio optimization is described in the following way:
R f + ∑ α i wi + ( Rm − R f )∑ β i wi ≥ E (r ) req ; (∑ ( β i wi )) 2 + ∑ (σ ei2 wi2 ) → min; wi ≥ 0; ∑ wi = 1
(10)
Sharpe single-index model let an investor estimate the expected return and the risk of his investments with regard to the situation at the stock market. However, the main disadvantage of this model is the necessity to predict the level of the market yield and an expected rate of return on the riskless asset. Moreover, it does not take into account changes in the riskless asset yield. The errors in case of large difference between the return on risk-free asset and the return on the market portfolio may occur while determining the efficient portfolio, too, as Savchuk and Dudka state.
23
Савчук В./Дудка В. (2001) 12
3. Determining the Efficient Stock Portfolio at the Ukrainian Market For optimal stock portfolio modeling we have taken data on stocks of the six Ukrainian joint stock companies for the last 20 weeks. According to Puxty and Dodds, the return that an investor gets from the asset is equal to:
Re turn =
Dividends + ( Market pricet − Market pricet −1 ) 24 Market pricet −1
In the following calculations we take for the expected return only relative weekly variations of the stock quotations because currently most of the joint stock companies do not set dividends to common stock according to the Securities and Stock Market State Commission.25 For our study we have chosen the businesses whose stock prices grew since the beginning of 2010. The input data are introduced in the Table 1. We have made our calculations with the help of MS Excel application.
Date
1 28.05.10 04.06.10 11.06.10 18.06.10 25.06.10 02.07.10 09.07.10 16.07.10 1 23.07.10 30.07.10 06.08.10
Nizhnedneprovsk tuberolling mill NITR 2 7.83 0.11 2.35 7.08 -1.17 -3.64 5.72 -4.83 Table 1. 2 -1.52 4.54 -7.40
Business and its code Ukrnafta Stakhanov Auto- Ukrtelewagon KrAZ com works UNAF SVGZ KRAZ UTLM 3 4 5 6 15.79 40.89 18.01 3.73 15.31 -11.72 -4.21 -1.20 -0.67 8.73 0.00 4.23 1.09 -0.56 3.30 23.67 0.54 -5.38 -4.26 -4.18 -6.06 -8.89 -4.44 -10.94 11.97 2.51 2.33 8.94 -0.09 -0.48 1.70 -2.75 Data on Stock Quotations Variations, %26 3 4 5 6 1.86 4.59 2.79 1.76 -2.63 4.22 -0.54 -0.40 2.29 7.88 0.00 -0.69
24
Motor Sich MSICH 7 20.37 2.73 3.51 8.42 -1.42 -8.09 4.10 -0.39 7 -1.57 1.83 0.35
Puxty/Dodds (1990), p. 141 Государственная комиссия по ценным бумагам и фондовому рынку (2010) 26 Developed by the author on the base of data from ПФТС (2010) and Information Agency Cbonds.ru (2010) 25
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13.08.10 20.08.10 27.08.10 03.09.10 10.09.10 17.09.10 24.09.10 01.10.10 08.10.10 Table 1.
2.34 -0.42 -3.34 0.00 -6.38 1.27 -0.80 1.15 1.14
-3.39 1.81 -0.77 1.42 0.45 0.13 -0.01 4.41 -2.56 Continued
-0.93 -0.39 -1.32 0.86 7.34 -0.23 1.04 5.61 -0.36
-1.64 0.00 -2.22 1.14 1.12 -1.67 0.00 -1.13 -2.86
-1.58 -1.36 -3.94 -2.41 1.40 -0.05 -0.07 -0.77 -3.23
-4.04 0.45 -0.90 1.24 -0.34 -1.70 0.78 -3.06 -7.06
3.1 Applying the Markowitz Portfolio Selection Model Let us use the Markowitz portfolio selection model for determining the effective stock portfolio of the six Ukrainian businesses. On the base of the input data we have evaluated the return and risk for each stock, as shown in Table 2. The expected return of the stocks is fluctuating from 0.2% to 2.67%, their risk is also unequal and belong to the interval from 2.78% to 6.06%. Thereat, the highest risk absolutely objectively occurs for the stock of Stakhanov wagon work with the highest return. Also we have calculated the pairwise coefficients of the linear correlation between the return on the stock which turned out to be positive in all the cases accordingly to Table 2.
JSC Code Return, % Nizhnedneprovsk tube-rolling mill NITR 0.20 Ukrnafta UNAF 2.05 Stakhanov wagon works SVGZ 2.67 Auto-KrAZ KRAZ 0.37 Ukrtelecom UTLM 0.51 Motor Sich MSICH 0.76 Table 2. Return and Risk of the Stock under Review
Risk, % 3.16 3.95 6.06 2.78 4.07 3.69
While evaluating the efficient diversified stock portfolio we have set the acceptable level of risk equal to 0.9% for solving the direct problem and the planned return at the level of 1.2% for solving indirect problem. The calculation results are given in Table 4. By the calculations, both in the solution of the direct and indirect problems the model does not include Nizhnedneprovsk tube-rolling mill and Motor Sich stocks into the efficient portfolio.
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NITR NITR UNAF SVGZ KRAZ UTLM MSICH Table 3.
1.00 0.38 0.36 0.46 0.59 0.58
UNAF
SVGZ
UTLM
MSICH
1.00 0.42 1.00 0.53 0.91 1.00 0.30 0.24 0.42 1.00 0.69 0.74 0.86 0.58 1.00 Linear Correlation Coefficients for the Stocks Direct Problem σp ≤ 0.9% Stock Portfolio Structure
Requirements Nizhnedneprovsk tube-rolling mill Ukrnafta Stakhanov wagon works Auto-KrAZ Ukrtelecom Motor Sich Optimal Portfolio Parameters Table 4.
KRAZ
0%
Indirect Problem E(r) ≥ 1.2% 0%
33.94% 39.78% 2.66% 5.57% 38.34% 30.69% 25.06% 23.96% 0% 0% σp = 0.9 σp = 0.943 E(r) = 1.036 E(r) = 1.2 Structure of the Optimal Stock Portfolio according to the
Markowitz Model
3.2 Applying Sharpe Single-Index Model For evaluating the parameters of the stock portfolio according to the Sharpe model we need the data on the general market return and the riskless asset return for the expectational horizon. To estimate the stock market return we have used the relative variations of the Ukrainian stock index PFTS. For evaluation of the riskless asset return, which is also changing in time, we have used the data on variations of the price on public interest bearing bonds, as the Ukrainian financiers Savchuk and Dudka offer .27 The results are given in Table 5. As one can see from the table, the stock market return differs considerably from the riskless asset return on its absolute values, as well as on variance.
27
Савчук В., Дудка В. (2001) 15
Basing on the data of Tables 1 and 5 we have calculated the parameters for each stock which are necessary for Sharpe model constructing. The results are summarized in Table 6. Date Market Return Riskless Asset Return 28.05.10 12.87 0.21 04.06.10 2.14 0.00 11.06.10 3.11 0.08 18.06.10 5.82 0.00 25.06.10 1.10 0.05 02.07.10 -8.49 5.75 09.07.10 4.41 -9.63 16.07.10 -0.48 0.64 23.07.10 1.45 0.25 30.07.10 1.21 0.17 06.08.10 3.60 0.32 13.08.10 -2.72 -0.59 20.08.10 0.13 1.65 27.08.10 -1.47 0.04 03.09.10 0.04 0.03 10.09.10 -0.21 0.01 17.09.10 -1.40 0.00 24.09.10 -0.50 1.07 01.10.10 -0.97 0.00 08.10.10 -3.69 0.00 Table 5. Data on the Stock Market Return and the Riskless Asset Return
JSC Nizhnedneprovsk tube-rolling mill Ukrnafta Stakhanov wagon works Auto-KrAZ Ukrtelecom Motor Sich Table 6.
Code
Return, %
Risk, %
β-risk
Excess return, α
Residu al risk, σ2ei
NITR
0.20
3.16
0.514
-0.21
2.68
UNAF
2.05
3.95
0.996
1.25
2.61
SVGZ
2.67
6.06
1.786
1.25
4.74
KRAZ 0.37 2.78 0.9 -0.35 UTLM 0.51 4.07 1.006 -0.30 MSICH 0.76 3.69 1.323 -0.29 Stock Parameters according to the Sharpe Model
2.08 3.19 1.52
While modeling the optimal stock portfolio, having predicted the trend line of the general market return and the riskless asset return, we have set the acceptable level of the risk at 0.9% for solving the direct problem and the level of the expected return at 1.2% for solving the indirect problem, and also the general market return equal to 1% 16
and the riskless asset return 0.01%. The results of using Sharpe model can be seen in Table 7. It follows from the table that for maximizing the expected return the Sharpe model does not include Motor Sich stocks into the efficient portfolio, and the model does not include Motor Sich and Stakhanov wagon works into the optimal portfolio. Prediction
Rm=1% Rf=0.01% Direct Problem σp ≤ 0.9% Stock Portfolio Structure
Requirements
Indirect Problem E(r) ≥ 1.2%
Nizhnedneprovsk tube28.46% 38.36% rolling mill Ukrnafta 14.61% 43.54% Stakhanov wagon works 27.09% 0% Auto-KrAZ 28.00% 15.59% Ukrtelecom 1.84% 2.51% Motor Sich 0% 0% Optimal Portfolio σp = 0.9 σp = 1.759 Parameters E(r) = 1.403 E(r) = 1.2 Table 7. Structure of the Optimal Stock Portfolio according to the Sharpe Model In the next section we will analyze the results obtained and make conclusions as for the usage of the Markowitz and Sharpe portfolio selection models for the stock portfolio diversification.
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4. Conclusion This paper examined the Markowitz and Sharpe portfolio selection models. In Section 2 their main principles are stated and the mathematical models for solving the direct and indirect problem of stock portfolio optimization are described. After calculations given in Section 3 there have been obtained two effective portfolios when maximizing the expected return (direct problem) and two optimal portfolios when minimizing the risk (indirect problem) according to the Markowitz and Sharpe models. One can compare these options in Table 8. When solving the direct problem the value of the expected return differs insignificantly (by 0.367%), and when solving the indirect problem the value of the risk differs more than by 1% (in the Markowitz model it equals 0.943%, and in the Sharpe model it is equal to 1.759%). Requirements
Direct Problem σp ≤ 0.9% Stock Portfolio Structure Markowitz Sharpe
Indirect Problem E(r) ≥ 1.2% Markowitz
Sharpe
Nizhnedneprovsk tube0% 28.46% 0% 38.36% rolling mill Ukrnafta 33.94% 14.61% 39.78% 43.54% Stakhanov wagon 2.66% 27.09% 5.57% 0% works Auto-KrAZ 38.34% 28.00% 30.69% 15.59% Ukrtelecom 25.06% 1.84% 23.96% 2.51% Motor Sich 0% 0% 0% 0% Optimal Portfolio σp = 0.9 σp = 0.9 σp = 0.943 σp = 1.759 Parameters E(r) = 1.036 E(r) = 1.403 E(r) = 1.2 E(r) = 1.2 Table 8. Comparison of the Optimal Portfolio Structure according to the Markowitz and Sharpe Models In this particular case, the results of the Sharpe model are suspected to be less precise, as the Ukrainian financier Moiseenko claims that the Sharpe model is to be applied ‘when considering a large amount of the securities that describe the large stock market share’28. In conditions of the developed and relatively stable stock markets of 28
Мойсеєнко І. П. (2006) 18
the Western countries both classical Markowitz and Sharpe models work effectively. Nevertheless, it is rather difficult to predict the market return and the riskless asset return for the evolving Ukrainian stock market. Hence, the Markowitz model is defined as more appropriate one among the two models considered.
It is worth mentioning that both of the portfolio selection models show demonstrably the advantages of diversification. The portfolio consisting of stocks of businesses from such industries as machine building, telecommunications and oil producing, has the lower risk than any of these stocks individually. The risk diversification phenomenon occurs even if there exists positive correlation among these stocks, although its coefficients are rather low in some cases.
Thus, it may be concluded that the Ukrainian investors should use the benefits of stock portfolio diversification and to optimize it applying the classical Markowitz model or any other model developed on its base. However, if the investors view a set of the stocks that make a large share of the national stock market, they should apply the Sharpe single-index model or any other models based on it.
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Ross, S. A./Westerfield R. W./Jaffe, J.F. (1996), Corporate Finance, 4th edition, Irwin/McGraw-Hill: Chicago. Sharpe, W. F. (1964), Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, in: The Journal of Finance, Vol. 19, Issue 3, 425-442. Tobin J. (1965), The Portfolio Selection: Macmillan and Co., London Борщук І. В. (2002), Ризик і дохідність при портфельному інвестуванні комерційних банків, в: Фінанси України, №7, 115-126. Василенко Д., Диба О. (2006), Основи теорії диверсифікації інвестиційної діяльності, в: Ринок цінних паперів України, №9-10, 35-43. Державна Комісія з цінних паперів та фондового ринку (2010), Загальнодоступна інформаційна база даних ДКЦПФР про ринок цінних паперів, . 10.10.2010.
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Дубровин В.И/Юськив О. И. (2008), Модели и методы оптимизации выбора инвестиционного портфеля, в: Радіоелектроніка. Інформатика. Управління, №1, 2008, с. 49-60. Информационное агентство Cbonds.ru (2010), Котировки акций облигаций, . 10.10.2010.
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Коваленко Ю. М. (2010), Портфельні теорії крізь призму сучасних кризових явищ, в: Актуальні проблеми економіки, № 8(110), 5-9. Мойсеєнко І. П. (2006), Інвестування: Навч. посіб. 05.10.2010. Пересада А.А.,/Шевченко О.Г. (2004), Портфельне інвестування, Київ, КНЕУ. Савчук В., Дудка В. (2001), Оптимізація фондового портфелю, 28.11.2001. . 10.10.2010. Фондова біржа (2010), .10.10.2010.
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Choosing the Portfolio Selection Model for the Ukrainian Stock Market
Daria Zhuravlyova Kozyria Str. 47, 12 83064 Donetsk International Business Administration Semester: 7
E-Mail: [email protected] Date of submission: 18.10.2010
Table of Contents
List of Tables ........................................................................................................ 3 List of Symbols ..................................................................................................... 4 Abstract ................................................................................................................. 5 1. Introduction ...................................................................................................... 6 2. Models of Portfolio Selection .......................................................................... 7 2.1 Markowitz Portfolio Selection Model ....................................................... 8 2.2 Sharpe Single-Index Model...................................................................... 11 3. Determining the Efficient Stock Portfolio at the Ukrainian Market ........ 13 3.1 Applying the Markowitz Portfolio Selection Model .............................. 14 3.2 Applying Sharpe Single-Index Model ..................................................... 15 4. Conclusion ....................................................................................................... 18 References ........................................................................................................... 20
2
List of Tables Table 1. Data on Stock Quotation Variations Table 2. Return and Risk of the Stock under Review Table 3. Linear Correlation Coefficients for the Stocks Table 4. Structure of the Optimal Portfolio according to the Markowitz Model Table 5. Data on the Stock Market Return and Riskless Asset Return Table 6. Stock Parameters according to the Sharpe Model Table 7. Optimal Stock Portfolio according to the Sharpe Model Table 8. Comparison of the Optimal Portfolio Structure according to the Markowitz and Sharpe Models
3
List of Symbols E(r) – portfolio return σp – portfolio risk wi – percentage of the ith asset in the portfolio σi – some characteristic of the risk of ith asset ri – the yield of the ith asset n – a number of assets in the portfolio σreq – the given level of risk E(r)req – the planned expected return ρij – the linear correlation coefficient between ith and jth assets Rf – the return on the risk-free asset αi – the excess return of the ith asset Rm – the expected stock market return βi – the risk of the ith asset σ2ei – the residual risk of the ith asset
4
Abstract The paper contains 19 pages, 1 illustration and 8 tables. It deals with the issue of using the benefits of portfolio diversification and choosing the most appropriate portfolio selection model for the Ukrainian stock market. Definition of the problem is: What portfolio selection model is the most reliable for diversifying stock portfolio at the Ukrainian stock market?
In the introduction the portfolio diversification is defined as spreading the investment across different assets in order to reduce the level of risk of investments. In Section 2, Markowitz portfolio selection model and Sharpe single-index model of portfolio optimization have been chosen among the numerous models of portfolio optimization, and a short review is given of their assumptions and content. Section 3 introduces the results of choosing the optimal portfolio for the stocks of the six Ukrainian public corporations according to the Markowitz and Sharpe models. They show the difference in applying the models. In Section 4 an assumption is made that the Markowitz portfolio selection model is more reliable for the evolving Ukrainian stock market.
Keywords: portfolio, stock, portfolio diversification, expected return, risk, stock market, Markowitz Portfolio Selection Model, Sharpe Single-Index Model
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1. Introduction According to Corrado and Jordan, portfolio is a ‘group of assets such as stocks and bonds held by an investor’1. Any asset or their combination is characterized through the term ‘risk’ – ‘the possibility that actual cash flows (returns) will be different than forecasted cash flows (returns)’2. Efficient, or optimal, portfolio lets an investor solve the direct problem – to maximize the expected return from their stock when a certain level of acceptable risk is given,3 as Fabozzi, Modigliani and Ferri claim. In case of indirect problem optimal portfolio provides an opportunity to minimize risk when the level of expected return is assigned. However, it is a rather complex task to form such an effective portfolio, as the higher return on assets is usually connected with the higher level of asset risk.
In order to minimize risks of stock investments with the high return investors widely use the principle of diversification. It means ‘spreading an investment across many assets’ in order to eliminate some part of risks4. Due to diversification, individual risky stocks almost always can be combined in such a way that a less risky portfolio (or their combination) is obtained5. An important concept of the portfolio effect is correlation – ‘the extent to which the returns on two assets move together’.6 The assets are positively correlated if their returns tend to change in the same direction, and negatively correlated if their returns move in the opposite directions. Sometimes it can said that the assets are uncorrelated if there is no connection between changes in their returns. The investor will eliminate some risk due to diversification if the assets in the portfolio are negatively correlated,7 but not only in the case of negative correlation. The
1
Corrado/Jordan (2000), p. 491 Moyer/McGuigan/Kretlow (1992), p. 211 3 See Fabozzi/Modigliani/Ferri (1998), p. 261 4 See Corrado/Jordan (2000), p. 496 5 See Ross/Westerfield/Jaffe (1996), p. 239 6 Corrado/Jordan (2000), p. 497 7 See loc. cit., p. 498 2
6
diversification can also be achieved by investing in a combination of securities with different risk-return characteristics8, as Moyer, McGuigan and Kretlow claim.
Since the 1950’s there have been developed a lot of portfolio optimization models, such as Markowitz portfolio selection model, Sharpe single-index model, Capital Asset Pricing Model (CAPM), Tobin portfolio model,9 Black’s model, TobinSharpe-Lintner model, Quasi-Sharpe model, etc. The Ukrainian stock market today is at the stage of its growth and development, and there is no agreement of opinions of the Ukrainian scientists which portfolio selection model is preferable. Borschtschuk is in favour of the Markowitz approach application to maximizing the stock portfolio return10. Vasilenko and Dyba find that CAPM should be used for optimization of the stock portfolio11. Savchuk and Dudka offer a model based on the Sharpe theory12. Kovalenko claims in his works that the approach to choosing the best portfolio diversification model should be individual, and such a model should protect the investor from the stock prices fluctuations13. The variety of opinions shows that this line of research is relevant for the modern Ukrainian economy. Therefore, in order to determine the portfolio selection model, the described models will be checked empirically.
2. Models of Portfolio Selection Let the expected return E(r) from the stock portfolio and its risk σp be determined with the help of the following functions: E(r) = RETURN(wi; σi; ri), σp = RISK(wi; σi; ri), i = 1 to n,14 where wi is percentage of the ith asset in the portfolio, σi is some characteristic of the risk of ith asset,
8
See Moyer/McGuigan/Kretlow (1992), p. 222 See Tobin (1965) 10 See Борщук (2002) 11 See Василенко/Диба (2006), с. 42 12 See Савчук В./Дудка В. (2001) 13 See Коваленко (2010), с. 9 14 Савчук В./Дудка В. (2001) 9
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ri is the yield of the ith asset, n is a number of assets in the portfolio.
In this case the problem of diversifying stock portfolio is to find such a combination of stocks for which the expected return is maximal and the estimated risk is minimal.
The direct problem containing the given level of risk σreq is of the form: E (r ) → max; σ ≤ σ ; req p wi ≥ 0; ∑ wi = 1
(1)
The indirect problem with the planned expected return E(r)req has the next form:
E (r ) ≥ E (r ) req ; σ p → min; wi ≥ 0; w =1 ∑ i
(2)
We will study two classical models of the portfolio selection because of the simplicity of obtaining the necessary data for evaluating these models.
2.1 Markowitz Portfolio Selection Model
The Markowitz portfolio selection theory (1952) is the first basic theory of portfolio optimization. This model of investments diversification claims that level of portfolio risk depends on the risks of its component assets, taking into account the correlation between return on the stocks in this portfolio. The main assumptions of the theory are as follows: •
investors take into account only such factors as the expected return on asset and its risk;
8
•
the expected value of return on asset is determined by a weighted sum of its components and risk is evaluated through a variance of return, or equals to its standard deviation15;
•
the expected return and risk of assets in the future can be predicted on the base of the data on the return and variance from the past time periods;
•
relations between assets in the portfolio are stated through their linear correlation coefficients.
According to the Markowitz model, the expected return E(r) from the stock portfolio is equal to the sum of the expected return from each its stock times the percentage of this stock in the portfolio: n
E (r ) = ∑ wi ri ,16
(3)
i =1
The portfolio risk σp2 equals:
σp =
n
n
∑∑ w σ w σ i
i =1 j =1
i
j
j
ρ i j ,17
(4)
where ρij is the linear correlation coefficient between ith and jth assets. Therefore, the direct problem has the form:
n ∑ wi ri → max; i =1 n n ∑∑ wiσ i w j σ j ρ i j ≤ σ req ; i =1 j =1 wi ≥ 0; ∑ wi = 1 The indirect problem is the following one:
15
See Markowitz (1952), p. 77 Bodie/Kane/Marcus, p. 219 17 Савчук В./Дудка В. (2001) 16
9
(5)
n ∑ wi ri ≥ E (r ) req ; i =1 n n ∑∑ wiσ i w j σ j ρ i j → min; i =1 j =1 wi ≥ 0; ∑ wi = 1
(6)
On the base of these formulas the investor should determine the efficient portfolios with the help of finding the best combinations of the expected return E(r) and its risk σp when given expected yields of assets E(ri) and linear correlation coefficients ρij. Markowitz represented geometrically the set of efficient combinations as a curve line, inside which the attainable combinations lie, and the inappropriate combinations lie outside, as illustrated in the Fig.1. σp
attainable E(r), σp combinations efficient E(r), σp combinations
E(R)
Figure 1. Efficient Portfolio Set18 The Markowitz portfolio theory helps investors to determine and exclude the inefficient combinations of assets. Still, it must be kept in mind that the disadvantage of this model is orientation only at the characteristics of the set of the stocks involved and usage of the historical data which do not provide the stability of the stock quotations variations at the market. Also, the model requires tedious calculations of weights of the stocks taking into account their correlation. Therefore, still the results of Markowitz
18
Markowitz (1952), p. 82 10
portfolio model can be used only in the conditions of a stable stock market, when the return on asset really depends on its past values.
2.2 Sharpe Single-Index Model The next model was developed by W. F. Sharpe in 1970. In contrast to Markowitz portfolio selection, this one considers the relations of each stock not with each other but with the whole market. The model assumptions include the following statements: •
the return on the stock is equal to its mathematical expectation;
•
there exists the linear regressionship between the market return and the return on each stock;19
•
the risk of the stock means the level of dependence of changes in its return on changes in the general market return;20
•
in common with the Markowitz model, the data on the return and variance from the past time periods are assumed to reflect entirely the future return and risk trends;
•
there exists a riskless asset at the market. The notion of the risk-free asset means that the ‘returns from the initial investment are known with certainty’21. The example of such an asset can be a government bond.
According to this model, the expected return on the stock portfolio is described by the following equation:
E (r ) = R f + ∑ α i wi + ( Rm − R f )∑ β i wi ,22 where Rf is the return on the risk-free asset, αi is the excess return of the ith asset, Rm is the expected stock market return, βi is the risk of the ith asset.
19
See Дубровин/Юськив (2008) See Мойсеєнко І. П. (2006) 21 Moyer/McGuigan/Kretlow (1992), p. 211 22 Савчук В./Дудка В. (2001) 20
11
(7)
The portfolio risk in the Sharpe’s model is equal to:
σ p = (∑ ( β i wi )) 2 + ∑ (σ ei2 wi2 ) ,23
(8)
where σ2ei is the residual risk of the ith asset. Some additional variables, namely the residual risk and the excess return of the asset, are introduced in the Sharpe model compared with the Markowitz model. We will not provide the paper with the formulas on calculating them, as they are rather solid. The direct problem of portfolio selection is the next system:
R f + ∑ α i wi + ( Rm − R f )∑ β i wi → max; (∑ ( β i wi )) 2 + ∑ (σ ei2 wi2 ) ≤ σ req ; wi ≥ 0; ∑ wi = 1
(9)
The indirect problem of portfolio optimization is described in the following way:
R f + ∑ α i wi + ( Rm − R f )∑ β i wi ≥ E (r ) req ; (∑ ( β i wi )) 2 + ∑ (σ ei2 wi2 ) → min; wi ≥ 0; ∑ wi = 1
(10)
Sharpe single-index model let an investor estimate the expected return and the risk of his investments with regard to the situation at the stock market. However, the main disadvantage of this model is the necessity to predict the level of the market yield and an expected rate of return on the riskless asset. Moreover, it does not take into account changes in the riskless asset yield. The errors in case of large difference between the return on risk-free asset and the return on the market portfolio may occur while determining the efficient portfolio, too, as Savchuk and Dudka state.
23
Савчук В./Дудка В. (2001) 12
3. Determining the Efficient Stock Portfolio at the Ukrainian Market For optimal stock portfolio modeling we have taken data on stocks of the six Ukrainian joint stock companies for the last 20 weeks. According to Puxty and Dodds, the return that an investor gets from the asset is equal to:
Re turn =
Dividends + ( Market pricet − Market pricet −1 ) 24 Market pricet −1
In the following calculations we take for the expected return only relative weekly variations of the stock quotations because currently most of the joint stock companies do not set dividends to common stock according to the Securities and Stock Market State Commission.25 For our study we have chosen the businesses whose stock prices grew since the beginning of 2010. The input data are introduced in the Table 1. We have made our calculations with the help of MS Excel application.
Date
1 28.05.10 04.06.10 11.06.10 18.06.10 25.06.10 02.07.10 09.07.10 16.07.10 1 23.07.10 30.07.10 06.08.10
Nizhnedneprovsk tuberolling mill NITR 2 7.83 0.11 2.35 7.08 -1.17 -3.64 5.72 -4.83 Table 1. 2 -1.52 4.54 -7.40
Business and its code Ukrnafta Stakhanov Auto- Ukrtelewagon KrAZ com works UNAF SVGZ KRAZ UTLM 3 4 5 6 15.79 40.89 18.01 3.73 15.31 -11.72 -4.21 -1.20 -0.67 8.73 0.00 4.23 1.09 -0.56 3.30 23.67 0.54 -5.38 -4.26 -4.18 -6.06 -8.89 -4.44 -10.94 11.97 2.51 2.33 8.94 -0.09 -0.48 1.70 -2.75 Data on Stock Quotations Variations, %26 3 4 5 6 1.86 4.59 2.79 1.76 -2.63 4.22 -0.54 -0.40 2.29 7.88 0.00 -0.69
24
Motor Sich MSICH 7 20.37 2.73 3.51 8.42 -1.42 -8.09 4.10 -0.39 7 -1.57 1.83 0.35
Puxty/Dodds (1990), p. 141 Государственная комиссия по ценным бумагам и фондовому рынку (2010) 26 Developed by the author on the base of data from ПФТС (2010) and Information Agency Cbonds.ru (2010) 25
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13.08.10 20.08.10 27.08.10 03.09.10 10.09.10 17.09.10 24.09.10 01.10.10 08.10.10 Table 1.
2.34 -0.42 -3.34 0.00 -6.38 1.27 -0.80 1.15 1.14
-3.39 1.81 -0.77 1.42 0.45 0.13 -0.01 4.41 -2.56 Continued
-0.93 -0.39 -1.32 0.86 7.34 -0.23 1.04 5.61 -0.36
-1.64 0.00 -2.22 1.14 1.12 -1.67 0.00 -1.13 -2.86
-1.58 -1.36 -3.94 -2.41 1.40 -0.05 -0.07 -0.77 -3.23
-4.04 0.45 -0.90 1.24 -0.34 -1.70 0.78 -3.06 -7.06
3.1 Applying the Markowitz Portfolio Selection Model Let us use the Markowitz portfolio selection model for determining the effective stock portfolio of the six Ukrainian businesses. On the base of the input data we have evaluated the return and risk for each stock, as shown in Table 2. The expected return of the stocks is fluctuating from 0.2% to 2.67%, their risk is also unequal and belong to the interval from 2.78% to 6.06%. Thereat, the highest risk absolutely objectively occurs for the stock of Stakhanov wagon work with the highest return. Also we have calculated the pairwise coefficients of the linear correlation between the return on the stock which turned out to be positive in all the cases accordingly to Table 2.
JSC Code Return, % Nizhnedneprovsk tube-rolling mill NITR 0.20 Ukrnafta UNAF 2.05 Stakhanov wagon works SVGZ 2.67 Auto-KrAZ KRAZ 0.37 Ukrtelecom UTLM 0.51 Motor Sich MSICH 0.76 Table 2. Return and Risk of the Stock under Review
Risk, % 3.16 3.95 6.06 2.78 4.07 3.69
While evaluating the efficient diversified stock portfolio we have set the acceptable level of risk equal to 0.9% for solving the direct problem and the planned return at the level of 1.2% for solving indirect problem. The calculation results are given in Table 4. By the calculations, both in the solution of the direct and indirect problems the model does not include Nizhnedneprovsk tube-rolling mill and Motor Sich stocks into the efficient portfolio.
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NITR NITR UNAF SVGZ KRAZ UTLM MSICH Table 3.
1.00 0.38 0.36 0.46 0.59 0.58
UNAF
SVGZ
UTLM
MSICH
1.00 0.42 1.00 0.53 0.91 1.00 0.30 0.24 0.42 1.00 0.69 0.74 0.86 0.58 1.00 Linear Correlation Coefficients for the Stocks Direct Problem σp ≤ 0.9% Stock Portfolio Structure
Requirements Nizhnedneprovsk tube-rolling mill Ukrnafta Stakhanov wagon works Auto-KrAZ Ukrtelecom Motor Sich Optimal Portfolio Parameters Table 4.
KRAZ
0%
Indirect Problem E(r) ≥ 1.2% 0%
33.94% 39.78% 2.66% 5.57% 38.34% 30.69% 25.06% 23.96% 0% 0% σp = 0.9 σp = 0.943 E(r) = 1.036 E(r) = 1.2 Structure of the Optimal Stock Portfolio according to the
Markowitz Model
3.2 Applying Sharpe Single-Index Model For evaluating the parameters of the stock portfolio according to the Sharpe model we need the data on the general market return and the riskless asset return for the expectational horizon. To estimate the stock market return we have used the relative variations of the Ukrainian stock index PFTS. For evaluation of the riskless asset return, which is also changing in time, we have used the data on variations of the price on public interest bearing bonds, as the Ukrainian financiers Savchuk and Dudka offer .27 The results are given in Table 5. As one can see from the table, the stock market return differs considerably from the riskless asset return on its absolute values, as well as on variance.
27
Савчук В., Дудка В. (2001) 15
Basing on the data of Tables 1 and 5 we have calculated the parameters for each stock which are necessary for Sharpe model constructing. The results are summarized in Table 6. Date Market Return Riskless Asset Return 28.05.10 12.87 0.21 04.06.10 2.14 0.00 11.06.10 3.11 0.08 18.06.10 5.82 0.00 25.06.10 1.10 0.05 02.07.10 -8.49 5.75 09.07.10 4.41 -9.63 16.07.10 -0.48 0.64 23.07.10 1.45 0.25 30.07.10 1.21 0.17 06.08.10 3.60 0.32 13.08.10 -2.72 -0.59 20.08.10 0.13 1.65 27.08.10 -1.47 0.04 03.09.10 0.04 0.03 10.09.10 -0.21 0.01 17.09.10 -1.40 0.00 24.09.10 -0.50 1.07 01.10.10 -0.97 0.00 08.10.10 -3.69 0.00 Table 5. Data on the Stock Market Return and the Riskless Asset Return
JSC Nizhnedneprovsk tube-rolling mill Ukrnafta Stakhanov wagon works Auto-KrAZ Ukrtelecom Motor Sich Table 6.
Code
Return, %
Risk, %
β-risk
Excess return, α
Residu al risk, σ2ei
NITR
0.20
3.16
0.514
-0.21
2.68
UNAF
2.05
3.95
0.996
1.25
2.61
SVGZ
2.67
6.06
1.786
1.25
4.74
KRAZ 0.37 2.78 0.9 -0.35 UTLM 0.51 4.07 1.006 -0.30 MSICH 0.76 3.69 1.323 -0.29 Stock Parameters according to the Sharpe Model
2.08 3.19 1.52
While modeling the optimal stock portfolio, having predicted the trend line of the general market return and the riskless asset return, we have set the acceptable level of the risk at 0.9% for solving the direct problem and the level of the expected return at 1.2% for solving the indirect problem, and also the general market return equal to 1% 16
and the riskless asset return 0.01%. The results of using Sharpe model can be seen in Table 7. It follows from the table that for maximizing the expected return the Sharpe model does not include Motor Sich stocks into the efficient portfolio, and the model does not include Motor Sich and Stakhanov wagon works into the optimal portfolio. Prediction
Rm=1% Rf=0.01% Direct Problem σp ≤ 0.9% Stock Portfolio Structure
Requirements
Indirect Problem E(r) ≥ 1.2%
Nizhnedneprovsk tube28.46% 38.36% rolling mill Ukrnafta 14.61% 43.54% Stakhanov wagon works 27.09% 0% Auto-KrAZ 28.00% 15.59% Ukrtelecom 1.84% 2.51% Motor Sich 0% 0% Optimal Portfolio σp = 0.9 σp = 1.759 Parameters E(r) = 1.403 E(r) = 1.2 Table 7. Structure of the Optimal Stock Portfolio according to the Sharpe Model In the next section we will analyze the results obtained and make conclusions as for the usage of the Markowitz and Sharpe portfolio selection models for the stock portfolio diversification.
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4. Conclusion This paper examined the Markowitz and Sharpe portfolio selection models. In Section 2 their main principles are stated and the mathematical models for solving the direct and indirect problem of stock portfolio optimization are described. After calculations given in Section 3 there have been obtained two effective portfolios when maximizing the expected return (direct problem) and two optimal portfolios when minimizing the risk (indirect problem) according to the Markowitz and Sharpe models. One can compare these options in Table 8. When solving the direct problem the value of the expected return differs insignificantly (by 0.367%), and when solving the indirect problem the value of the risk differs more than by 1% (in the Markowitz model it equals 0.943%, and in the Sharpe model it is equal to 1.759%). Requirements
Direct Problem σp ≤ 0.9% Stock Portfolio Structure Markowitz Sharpe
Indirect Problem E(r) ≥ 1.2% Markowitz
Sharpe
Nizhnedneprovsk tube0% 28.46% 0% 38.36% rolling mill Ukrnafta 33.94% 14.61% 39.78% 43.54% Stakhanov wagon 2.66% 27.09% 5.57% 0% works Auto-KrAZ 38.34% 28.00% 30.69% 15.59% Ukrtelecom 25.06% 1.84% 23.96% 2.51% Motor Sich 0% 0% 0% 0% Optimal Portfolio σp = 0.9 σp = 0.9 σp = 0.943 σp = 1.759 Parameters E(r) = 1.036 E(r) = 1.403 E(r) = 1.2 E(r) = 1.2 Table 8. Comparison of the Optimal Portfolio Structure according to the Markowitz and Sharpe Models In this particular case, the results of the Sharpe model are suspected to be less precise, as the Ukrainian financier Moiseenko claims that the Sharpe model is to be applied ‘when considering a large amount of the securities that describe the large stock market share’28. In conditions of the developed and relatively stable stock markets of 28
Мойсеєнко І. П. (2006) 18
the Western countries both classical Markowitz and Sharpe models work effectively. Nevertheless, it is rather difficult to predict the market return and the riskless asset return for the evolving Ukrainian stock market. Hence, the Markowitz model is defined as more appropriate one among the two models considered.
It is worth mentioning that both of the portfolio selection models show demonstrably the advantages of diversification. The portfolio consisting of stocks of businesses from such industries as machine building, telecommunications and oil producing, has the lower risk than any of these stocks individually. The risk diversification phenomenon occurs even if there exists positive correlation among these stocks, although its coefficients are rather low in some cases.
Thus, it may be concluded that the Ukrainian investors should use the benefits of stock portfolio diversification and to optimize it applying the classical Markowitz model or any other model developed on its base. However, if the investors view a set of the stocks that make a large share of the national stock market, they should apply the Sharpe single-index model or any other models based on it.
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