CC BY
47
УДК 336.76(477)
L. V. Shabalina,
PhD (Economics), ^ss. Prof., D. V. Zhuravlyova,
Student majoring, Donetsk National Technical University
CHOOSING THE PORTFOLIO SELECTION MODEL FOR THE UKRAINIAN STOCK MARKET
Definition of the Problem. According to Corrado and Jordan, portfolio is a ‘group of assets such as stocks and bonds held by an investor’ [1, p. 491]. 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, p. 211]. 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, as Fabozzi, Modigliani and Ferri claim [3, p. 261]. 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, due to which, individual risky stocks almost always can be combined in such a way that a less risky portfolio (or their combination) is obtained [4, p. 239].
Since the 1950’s there have been developed a lot of portfolio optimization theories, such as Markowitz portfolio selection model, Sharpe single-index model, Capital Asset Pricing Model (CAPM), Tobin portfolio model [5], Black’s model, Tobin-Sharpe-Lintner model, Quasi-Sharpe model, etc. that still are in demand. The Ukrainian stock market is at its early evolving stage, and the national investors do not have much experience in managing their portfolios, especially individual investors. Thus, the issue of the reliability of portfolio selection models is up to date in the current circumstances of the frequent fluctuations of the stock quotations and instability of the whole Ukrainian stock market development.
Analysis of the Latest Researches and Publications. The Ukrainian economists show their disagreement about the appropriate models for the portfolio optimization. Borschtschuk is in favour of the Markowitz approach application to maximizing the stock portfolio return [6]. Vasilenko and Dyba find that CAPM should be used for optimization of the stock portfolio [7]. Savchuk and Dudka offer a model based on the Sharpe theory [8]. 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 fluctuations [9]. The variety of opinions shows that this line of research is relevant for the modern Ukrainian economy.
The object of the paper is, therefore, to determine the most reliable portfolio selection model through their empirical study.
Research Results. Let the expected return E(r) from the stock portfolio and its risk cp be determined with the help of the following functions:
E(r) = RETURN^; ^; r), sp = RISK(wi; si; rX i = 1 to n, [8]
where wi is percentage of the ith asset in the portfolio,
s is some characteristic of the risk of ith asset, 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 s is of the form:
req
E(r) ® max; s < s ;
p req ’
w >0;
X w =1
(1)
The indirect problem with the planned expected return E(r) has the next form:
req
E (r) > E (r) r
(2)
wi > 0;
X wi =1
We will study two classical models of the portfolio
selection—the Markowitz portfolio selection model and the Sharpe single-index model—because of the simplicity of obtaining the necessary data for evaluating these models.
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:
„ Dividends + (Market price, - Market price, ,)
Re turn =--------------------- t----------------^ [10 p 141]
Market pricet-1 [10, p. 141]
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 [11]. 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. The calculations have been made with the help of MS Excel application.
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 according to Table 3.
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.
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 [8]. 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.
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.
Table 1
Data on Stock Quotations Variations, %
Date Business and its code
Nizhnedne-provsk tube-rolling mill Ukrnafta Stakhanov wagon works Auto-KrAZ Ukrtele-com Motor Sich
28.05.10 7.83 15.79 40.89 18.01 3.73 20.37
04.06.10 0.11 15.31 -11.72 -4.21 -1.20 2.73
11.06.10 2.35 -0.67 8.73 0.00 4.23 3.51
18.06.10 7.08 1.09 -0.56 3.30 23.67 8.42
25.06.10 -1.17 0.54 -5.38 -4.26 -4.18 -1.42
02.07.10 -3.64 -6.06 -8.89 -4.44 -10.94 -8.09
16.07.10 -4.83 -0.09 -0.48 1.70 -2.75 -0.39
23.07.10 -1.52 1.86 4.59 2.79 1.76 -1.57
30.07.10 4.54 -2.63 4.22 -0.54 -0.40 1.83
06.08.10 -7.40 2.29 7.88 0.00 -0.69 0.35
13.08.10 2.34 -3.39 -0.93 -1.64 -1.58 -4.04
20.08.10 -0.42 1.81 -0.39 0.00 -1.36 0.45
27.08.10 -3.34 -0.77 -1.32 -2.22 -3.94 -0.90
03.09.10 0.00 1.42 0.86 1.14 -2.41 1.24
10.09.10 -6.38 0.45 7.34 1.12 1.40 -0.34
17.09.10 1.27 0.13 -0.23 -1.67 -0.05 -1.70
24.09.10 -0.80 -0.01 1.04 0.00 -0.07 0.78
01.10.10 1.15 4.41 5.61 -1.13 -0.77 -3.06
08.10.10 1.14 -2.56 -0.36 -2.86 -3.23 -7.06
Remark : Developed by the author on the base of the data from [11-12] ----------------------------------------------------------- 128 ------
Table 2
Return and Risk of the Stock under Review
JSC Code Return, % Risk, %
Nizhnedneprovsk tube-rolling mill NITR 0.20 3.16
Ukrnafta UNAF 2.05 3.95
Stakhanov wagon works SVGZ 2.67 6.06
Auto-KrAZ KRAZ 0.37 2.78
Ukrtelecom UTLM 0.51 4.07
Motor Sich MSICH 0.76 3.69
Table 3
Linear Correlation Coefficients for the Stocks
NITR UNAF SVGZ KRAZ UTLM MSICH
NITR 1.00
UNAF 0.38 1.00
SVGZ 0.36 0.42 1.00
KRAZ 0.46 0.53 0.91 1.00
UTLM 0.59 0.30 0.24 0.42 1.00
MSICH 0.58 0.69 0.74 0.86 0.58 1.00
Table 4
Structure of the Optimal Stock Portfolio according to the Markowitz Model
Requirements Direct Problem Indirect Problem
on <0.9% E{r)> i.2%
Stock Portfolio Structure
Nizhnedneprovsk lube-rolling in ill 0% 0%
Ukmafta 33.94% 39.78%
Stakhonov wanon works 2.66% 5.57%
Auto-KrAZ 38.34% 30.69%
Ukrlekcom 25.06% 23 M%
Motor SicEi 0% 0%
Optimal Portfolio Parameters cr,, = 0.9 = 0.943
£3 11 o tm , II to
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% 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 for minimizing the risk.
Thus, after calculations 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%).
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’ [13]. In conditions of the developed and relatively stable stock markets of 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.
Table 5 It is worth mentioning that both of the portfolio
Data on the Stock Market Return selection models demonstrate the advantages of
and the Riskless Asset Return 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.
Conclusion. Basing on the results of the study, 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. Also, investors may include risk-free assets, e. g. treasury bonds, into their portfolio, and use of the classical models for may be the issue of determining the optimal portfolio structure in this case demands further investigations for the existing optimization models.
References
1. Corrado Ch. J. Fundamentals of Investments: Valuation and Management / Ch. J. Corrado, B. D. Jordan.
— NY : Irwin/McGraw-Hill: McGraw-Hill Companies,
Table 6
Stock Parameters according to the Sharpe Model
JSC Code Return, % Risk, % P-risk Excess return, a Residual risk, 2 O ei
Nizhnedneprovsk tube-rolling mill NITR 0.20 3.16 0.514 -0.21 2.68
Ukrnafta UNAF 2.05 3.95 0.996 1.25 2.61
Stakhanov wagon works SVGZ 2.67 6.06 1.786 1.25 4.74
Auto-KrAZ KRAZ 0.37 2.78 0.9 -0.35 2.08
Ukrtelecom UTLM 0.51 4.07 1.006 -0.30 3.19
Motor Sich MSICH 0.76 3.69 1.323 -0.29 1.52
Table 7
Structure of the Optimal Stock Portfolio according to the Sharpe Model
Prediction Rm=1% Rf=0.01%
Requirements Direct Problem Indirect Problem
Op < 0.9% E(r) > 1.2%
Stock Portfolio Structure
Nizhnedneprovsk tube-rolling mill 28.46% 38.36%
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 Parameters O II 0. 9 Op = 1.759
E(r) = 1.403 E(r) = 1.2
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 8
Comparison of the Optimal Portfolio Structure according to the Markowitz and Sharpe Models
Requirements Direct Problem Indirect Problem
Op < 0.9% E(r) > 1.2%
Stock Portfolio Structure
Markowitz Sharpe Markowitz Sharpe
Nizhnedneprovsk tube-rolling mill 0% 28.46% 0% 38.36%
Ukrnafta 33.94% 14.61% 39.78% 43.54%
Stakhanov wagon works 2.66% 27.09% 5.57% 0%
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 Parameters Op = 0.9 pO II 0. 9 Op = 0.943 Op = 1.759
E(r) = 1.036 E(r) = 1.403 .2 II (r) pE E(r) = 1.2
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05.10.2010.
Shabalina L. V., Zhuravlyova D. V. Choosing the Portfolio Selection Model for the Ukrainian Stock Market
The paper deals with the issue of choosing the most reliable portfolio diversification model for the Ukrainian stock market. The optimal portfolio structure developed on the base of the Markowitz portfolio selection model and the Sharpe single-index model has been analyzed. The conclusions about reliability of use of these two models for the Ukrainian investors have been made.
Key words: portfolio, stock, portfolio diversification, expected return, risk, stock market, markowitz portfolio selection model, sharpe single-index model.
Шабаліна Л. В., Журавльова Д. В. Вибір моделі оптимізації портфеля цінних паперів для українського фондового ринку
Проаналізовано структуру оптимального портфеля цінних паперів, побудованого на основі оптиміза-ційних моделей Марковиця й Шарпа. Зроблено висновок щодо надійності результатів цих моделей для використання українськими інвесторами.
Ключові слова: акція, портфель цінних паперів, фондовий ринок, диверсифікація, ризик, прибутковість, модель Марковиця, модель Шарпа.
Шабалина Л. В., Журавлева Д. В. Выбор модели оптимизации портфеля ценных бумаг для украинского фондового рынка
Проанализирована структура оптимального портфеля ценных бумаг, составленного на основе оптимизационных моделей Марковица и Шарпа. Сделан вывод относительно надежности результатов этих моделей для использования украинскими инвесторами.
Ключевые слова: акция, портфель ценных бумаг, фондовый рынок, диверсификация, риск, доходность, модель Марковица, модель Шарпа.
Received by the editors: 15.04.2011
and final form in 25.11.2011
