Application of the Divisia Index with Interconnected Factors in the

Folia Oeconomica Acta Universitatis Lodziensis ISSN 0208-6018 e-ISSN 2353-7663

www.czasopisma.uni.lodz.pl/foe/

4(330) 2017 DOI: http://dx.doi.org/10.18778/0208-6018.330.09

Jacek Białek University of Lodz, Faculty of Economics and Sociology, Department of Statistical Methods, jbialek@ uni.lodz.pl

Radosław Pietrzyk Wrocław University of Economics, Faculty of Management, Information Systems and Finance, Department of Financial Investment and Risk Management, [email protected]

Application of the Divisia Index with Interconnected Factors in the Warsaw Stock Exchange Index (WIG) fluctuation analysis Abstract: This paper presents a method of economic factorial analysis based on the Divisia index extended to interconnected factors. We verify the applicability of the presented method to financial market research by examining fluctuations of the Warsaw Stock Exchange WIG Index (WIG). We con‑ sider four main factors of WIG changes: the GDP growth, the PLN/EUR rate, the S&P500 and the un‑ employment rate. Due to computational reasons we apply the transformation that produces variables in the bigger the better form. We use quarterly data from the time interval between 2003 and 2014 divided into periods of bull and bear market. All considered variables are assumed to change linearly between quarters. The main conclusion is that during market prosperity, GDP and S&P500 changes exhibit the strongest influence on WIG changes. Keywords: Factorial analysis, Divisia index, interconnected factors, Warsaw Stock Exchange Index analysis JEL: C43, G10

[129]

130  Jacek Białek, Radosław Pietrzyk

1. Introduction The main aim of the paper is to verify the utility of the extended Divisia price in‑ dex in financial data analysis. On one hand Divisia’s methodology is connected with continuous time price models and its application is naturally limited. On the other hand the analysed economic processes can be approximated by continuous functions between observation moments (linear, exponential and others) and thus, Divisia’s method seems to be applicable. As the title of the paper suggests, our work is an application of Divisia’s method on the Warsaw Stock Exchange and it can be treated as its illustration. Nevertheless, the added value of the paper is the mathematical formula for WIG changes derived from Divisia’s general formula for eight factors. In particular, we intend to examine fluctuations of the Warsaw Stock Exchange WIG Index (WIG) by considering four main factors of WIG changes: the GDP growth, the PLN/EUR rate, the S&P500 and the unemployment rate. Having linear approximations of these processes for quarterly data, we are going to describe WIG changes using the extended Divisia index with interconnected factors, i.e. assuming that the above‑mentioned factors are related. Obviously, the problem of WIG fluctuations measurement is already known and it is considered in many studies by many authors. Nevertheless, the presented Divisia approach is quite unique: firstly, it is a continuous time approach, secondly, it takes into ac‑ count connections between indicators (factors). According to our best knowledge, no paper about the application of this method in finance currently exists. International studies have examined the problem of the impact of macroeco‑ nomic factors on stock market changes for many years. In particular, these stud‑ ies involved well‑developed capital markets: United States, Japan, Great Britain, Germany and so on. They usually concentrated on key determinants like infla‑ tion, interest rate, industrial output, GDP growth, crude oil prices, money supply, exchange rate or consumption. Nelson (1976) analysed the impact of the inflation rate (both expected and unexpected) on stock market prices in the American mar‑ ket and indicated that inflation growth negatively influences the whole market. Similar conclusions can also be found in (Humpe, Macmillan, 2009). The authors additionally showed a positive impact of the industrial output both on the Amer‑ ican and Japanese market. Chen, Roll and Ross (1986) and Fama (1981) present‑ ed similar findings. Studies show that the largest influence from macroeconomic data is typical for the GDP, and the GDP growth stimulates upward trends in stock markets. Such conclusions come from, among others, Fama (1981) and McMillan (2010). A positive impact of the exchange rate was shown in the Canadian mar‑ ket research (Dadgostar, Moazzami, 2003) and the Japanese study (Mukherjee, Naka, 1995). A comprehensive review of the aforementioned research, including the direction of the relation between economic factors and the market behaviour was presented, among others, in the doctoral thesis by Wiśniewski (2014). FOE 4(330) 2017  www.czasopisma.uni.lodz.pl/foe/

others, in the doctoral thesis by Wiśniewski (2014). 2. Divisia Application index approach and its extended version of the Divisia Index with Interconnected Factors in the Warsaw Stock Exchange Index… 131 A  key  element  of the Divisia  index  approach  was  to  introduce  time  as  a  continuous 

2. Divisia index approach and its extended version

variable t (Divisia, 1925). In the original work, Divisia considers two deterministic functions  Divisia index was to introduce time as a continuous connected A key with element prices of the and quantities of approach the considered commodities (factors P(t) and Q(t) variable t (Divisia, 1925). In the original work, Divisia considers two determinis‑

respectively) and he assumes that these continuous functions exist at any point in time. Thus, tic functions connected with prices and quantities of the considered commodities (factors P(t) and Q(t) respectively) and he assumes that these continuous functions exist at any point in time. Thus, the value function for the considered set of com‑ modities can be expressed as V (t )  P(t )Q(t ) .

the value function for the considered set of commodities can be expressed as

(1)

The next step in Divisia’s approach differential changes (1) of the value V (t )wasP (to t )Qconsider (t ) .

function and that leads to the following formula:

The next step in Divisia’s approach was to consider differential changes of the value function and that leads  dV (tto the ) Q(following t )dP(t )  formula: P(t )dQ(t )

= dV (t ) Q(t )dP (t ) + P (t )dQ (t )

or equivalently:

or equivalently:

V (t ) θˆGREG =

 dVd y(t)+   Qx(t )−dP(t )dx P(Btˆ)dQ(t ), ∑ ∑  ∑ 

(2)

T

L

i∈s

i

L

i

i

i∈Ω

i∈s

i Li

(2)

(3)

(3)

where L denotes the curve of the factors’ change. where L denotes the curve of the factors’ change. The above‑mentioned publication suggests consideration of the two factors (variables) as  The above‑mentioned publication suggests consideration of the two factors

functions of time t. It can be easilycase with n generalizedfactors influencing  to a case continuous(variables) functionsas continuous of time t. It can be easily generalized to a with n factors influencing the value of V (time notation is omitted for conveni‑

the value of V (time notation is omitted for convenience), namely when ence), namely when V  X 1 X 2 ... X n , ,



where 

(4)

(4)

(5)

(5)

where

= ∆V  V

+ X X ... X dX ∫ XX XX ......XX dX dX  ∫ X X ...X dX L L

2 2

3 3

n n

1 1

L L

1 1

3 3

n n

2 2

+ ...∫ X 1 X 2 ... X n −1dX n .  ... X 1 X 2 ...X n1dX n . L L

Divisia’s idea was developed by Sheremet et al. (1971) and Vaninsky (1987).

Divisia’s was developed et al. (1971)described and Vaninsky  These Theseidea publications extendedby theSheremet factorial decomposition in (4) and(1987). (5)

to the case with continuously differential function of factors, i.e. and (5) to  the  case  with  publications extended the factorial decomposition described in (4)

V = f (X continuously i.e. differential function of factors, 1 X 2 ... X n ) , with

with



(6)

V  f ( X 1 X 2 ... X n ) ,

∆V = ∫ ∇V  dX , V  L V dX ,

(6) (7)

(7)

L

[ f1 , f 2 ,..., f n ] is a gradient vector of the function f(X1, X2, …, Xn), dX(dX1, dX2, where  V  www.czasopisma.uni.lodz.pl/foe/  FOE 4(330) 2017 …, dXn) and the symbol “  ” denotes the inner product of two vectors. Vaninsky  and 

132  Jacek Białek, Radosław Pietrzyk where ∇V = [ f1, f2, …, fn] is a gradient vector of the function f(X1, X2, …, Xn), dX(dX1, dX2, …, dXn) and the symbol “  ” denotes the inner product of two vectors. Vaninsky and Meerovich (1978) extended the case described in (6) and (7) by add‑ Meeroviching(1978) extended the case described in (6) and (7) by  adding  equations  of  the  equations of the factors’ interconnection

factors’ interconnection Meerovich (1978) extended the case described in (6) and (7) by  adding  equations  of  the  factors’ interconnection for jj =1,1,2,..., (8) 2,...,m m. .  j ( X1, X 2 ,..., X n )  0 , for (8)

 j ( X1, X 2 ,..., X n )  0 , for j  1, 2,..., m . As a result, the following formula was obtained As a result, the following formula was obtained As a result, the following formula was obtained V  V(I  ΦΦ ) dX,

 (9) V   V(I  ΦΦ ) dX, where I denotes the n × n identity matrix, Φ is a Jacobian matrix described as follows L





(8)

(9) (9)

L

where I denotes the n × nthe identity matrix,matrix, Φ is a Jacobian matrix described as follows where I denotes n × n identity Φ is a Jacobian matrix described as fol‑  j lows , i  1, 2,...n , j  1, 2,..., m , (10) ij   X ij , i  1, 2,...n , j  1, 2,..., m , (10) ij   X i and the upper index “+” denotes the inverse matrix. If the columns of the matrix ∂Φgeneralized j , i = 1, 2,...n , j = 1, 2,..., m , (10) Φ ij = and the upper index “+” denotes the inverse matrix. If the columns of the matrix ∂Xgeneralized Φ are linearly independent, then we obtain (Vaninsky, 2013): i

Φ are linearly independent, then we obtain (Vaninsky, 2013): and the upper index “+” denotes matrix. If the columns Φ the  (generalized ΦT Φ) 1 ΦT inverse . (11)  T  T 1 of the matrix Φ are linearly independent, then we obtain (Vaninsky, 2013): Φ  (Φ Φ ) Φ . (11) It  is  worth  adding that index decompositions based on Divisia’s idea concern both + T −1 T continuous  . following  (11) Öbe = found  (Ö Ö )in Öthe  It  is and  worth  adding that decompositions based on Divisia’s concern discrete time index and  can  sample idea papers:  Ang, both Liu, discrete and  continuous  time  and  can  found  in  the  following  sample  papers:  Ang, Liu, Chew (2003), Ang (2004), Choi and Ohbe  (2014). It is worth adding that index decompositions based on Divisia’s idea concern Chew (2003), Ang (2004), Choi and Oh both discrete and continuous time(2014). and can be found in the following sample pa‑ pers: Ang, Liu, Chew (2003), Ang (2004), Choi and Oh (2014).

3. Choice of variables 3. Choice3.ofChoice variablesof variables Referring  to  previous  studies  from  well‑developed  capital  markets,  we  chose  initial  Referring to previous studiesfrom  from well‑developed capital markets, we chose Referring  studies  well‑developed  capital  markets,  we  ini‑ chose  initial  determinants ofto theprevious  return from stock market indexes in Poland. GDB growth (GDP), industry  tial determinants of the return from stock market indexes in Poland. GDB growth

determinants of the return from stock market indexes in Poland. GDB growth (GDP), industry  production,  oil  prices,  money  supply,  exchange  expenditure,  (GDP), industry production, oil prices, moneyrates,  supply,consumption  exchange rates, consumptionindices  of  expenditure, indices of international stock exchanges were classifiedexpenditure,  as stimulatingindices  of  production,  prices,  money  supply,  exchange  rates,  consumption  international oil  stock  exchanges were  classified  as stimulating factors.  In turn, inflation (CPI),  factors. In turn, inflation (CPI), interest rate (IR) and unemployment (UNEMPL)

international  stock  exchanges were  as stimulating factors.  In turn, inflation (CPI),  should have negative impacts classified  on returns form stock indices. Initial selection, whichon returns interest  rate  (IR)  and  unemployment  (UNEMPL)  should  have  negative  impacts was(IR)  aimed at reducing the number of explanatory variables model,impacts excludedon returns interest  rate  and  unemployment  (UNEMPL)  should  have in the negative  form stock indices. Initial selection, which was aimed at reducing the number of explanatory  the industry production, oil prices, money supply, consumption expenditure and

form stock indices. Initial selection, which was aimed at reducing the number of explanatory  from further consideration. The main argument for in‑ supply, variables interest in therate model, excluded the industry production, oilrejecting prices,these money dicators was the redundancy of part of the information. Since the majority of the variables in the model, excluded the industry production, oil prices, money supply, consumption expenditure and interest rate from further consideration. The main argument for  FOE 4(330) 2017  www.czasopisma.uni.lodz.pl/foe/ consumption expenditure and interest rate from further consideration. The main argument for  rejecting these indicators was the redundancy of part of the information. Since the majority of  rejecting these indicators was the redundancy of part of the information. Since the majority of  the Polish international trade is settled in euro, the EUR/PLN rate was chosen as the exchange 

Application of the Divisia Index with Interconnected Factors in the Warsaw Stock Exchange Index… 133

Polish international trade is settled in euro, the EUR/PLN rate was chosen as the exchange rate. The American stock exchange index S&P500, having the strong‑ est correlation with the Polish WIG, was set as the foreign market indicator. We estimated correlations between the WIG index returns and chosen indi‑ cators in the whole time interval. First we checked if the time series were station‑ ary, and in the case of WIG, EUR/PLN, GDP, CPI, UNEMPL the processes were transformed into stationary processes. In all cases the results were in line with the assumed direction of impact on the WIG. The largest, in absolute terms, positive impact was observed for the S&P500 (Table 1) and for the GDP and CPI while the WIG. The largest, in absolute terms, positive impact was observed for the S&P500 (Table 1) UNEMPL and EUR/PLN exhibited negative correlation. and for the GDP and CPI while the UNEMPL and EUR/PLN exhibited negative correlation. Table 1. Correlation coefficients between WIG index returns and selected variables Table 1. Correlation coefficients between WIG index returns and selected variables in the in the whole period whole period EUR/PLN GDP EUR/PLN –0,3632 0,2949 Source: own work –0,3632

CPI

UNEMPL

GDP 0.0002 0,2949

S&P500

CPI –0.0228 UNEMPL 0.7026 S&P500 0.0002 –0.0228 0.7026

The CPI  index  was  rejected  from Source: furtherown study workdue to the correlation coefficient being  very close to zero for the whole period. 

The CPI index was rejected from further study due to the correlation coeffi‑ cient being very close to zero for the whole period. Figure 1 presents the WIG index and the other main indicators which were se‑ igure  1 presents the WIG index and the other main indicators which were selected to show  lected to show the influence of economic factors on the Polish capital market. the influence of economic factors on the Polish capital market.  70000

70000

5

9

4,8

8

60000

60000 4,6 50000

7 50000

4,4

6

4,2

40000

4 30000

3,8

40000

EUR/PLN

WIG 4

30000

20000 2

3,4 10000

10000

1

3,2

0

3

0

0 Q1 2003 Q2 2003 Q3 2003 Q4 2003 Q1 2004 Q2 2004 Q3 2004 Q4 2004 Q1 2005 Q2 2005 Q3 2005 Q4 2005 Q1 2006 Q2 2006 Q3 2006 Q4 2006 Q1 2007 Q2 2007 Q3 2007 Q4 2007 Q1 2008 Q2 2008 Q3 2008 Q4 2008 Q1 2009 Q2 2009 Q3 2009 Q4 2009 Q1 2010 Q2 2010 Q3 2010 Q4 2010 Q1 2011 Q2 2011 Q3 2011 Q4 2011 Q1 2012 Q2 2012 Q3 2012 Q4 2012 Q1 2013 Q2 2013 Q3 2013 Q4 2013 Q1 2014 Q2 2014 Q3 2014

Q1 2003 Q2 2003 Q3 2003 Q4 2003 Q1 2004 Q2 2004 Q3 2004 Q4 2004 Q1 2005 Q2 2005 Q3 2005 Q4 2005 Q1 2006 Q2 2006 Q3 2006 Q4 2006 Q1 2007 Q2 2007 Q3 2007 Q4 2007 Q1 2008 Q2 2008 Q3 2008 Q4 2008 Q1 2009 Q2 2009 Q3 2009 Q4 2009 Q1 2010 Q2 2010 Q3 2010 Q4 2010 Q1 2011 Q2 2011 Q3 2011 Q4 2011 Q1 2012 Q2 2012 Q3 2012 Q4 2012 Q1 2013 Q2 2013 Q3 2013 Q4 2013 Q1 2014 Q2 2014 Q3 2014

70000

25

70000

2 200,00

2 000,00

23 60000

60000

1 800,00

21 50000

19

1 600,00

50000

1 400,00

17

40000

15

WIG

40000

1 200,00

UNEMPL

30000

1 000,00

13 30000

9

400,00

7 10000

200,00 Q1 2003 Q2 2003 Q3 2003 Q4 2003 Q1 2004 Q2 2004 Q3 2004 Q4 2004 Q1 2005 Q2 2005 Q3 2005 Q4 2005 Q1 2006 Q2 2006 Q3 2006 Q4 2006 Q1 2007 Q2 2007 Q3 2007 Q4 2007 Q1 2008 Q2 2008 Q3 2008 Q4 2008 Q1 2009 Q2 2009 Q3 2009 Q4 2009 Q1 2010 Q2 2010 Q3 2010 Q4 2010 Q1 2011 Q2 2011 Q3 2011 Q4 2011 Q1 2012 Q2 2012 Q3 2012 Q4 2012 Q1 2013 Q2 2013 Q3 2013 Q4 2013 Q1 2014 Q2 2014 Q3 2014

Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 2002 2003 2003 2004 2004 2005 2005 2006 2006 2007 2007 2008 2008 2009 2009 2010 2010 2011 2011 2012 2012 2013 2013 2014 2014 2015 2015

S&P500

600,00

20000

10000

5

WIG

800,00

11

20000

0

GDP

3

3,6

20000

5

WIG

Figure 1. Quotations of the WIG Index and selected indicators igure 1. Quotations of the WIG Index and selected indicators Source: own work

Source: own work

The figure suggests that there is a strong dependence between the considered factors and  www.czasopisma.uni.lodz.pl/foe/  FOE 4(330) 2017 the WIG index. In the subsequent study we examined the influence of economic factors with a  lag of 1, 2 or 3 quarters. The results showed that the WIG responds mainly to GDP changes 

134  Jacek Białek, Radosław Pietrzyk The figure suggests that there is a strong dependence between the considered factors and the WIG index. In the subsequent study we examined the influence of economic factors with a lag of 1, 2 or 3 quarters. The results showed that the WIG responds mainly to GDP changes with a lag of 3 quarters (the largest value of the correlation coefficient), however the differences are subtle. For other varia‑ bles, such a shift was not observed.

4. Factorial decomposition of changes in Warsaw Stock Exchange index fluctuations In the empirical part of this paper we refer to the generalized Divisia index method (GDIM) with interconnected factors described in (8)–(11). Many au‑ thors apply the presented approach or its discrete version to the factorial de‑ composition of CO2 emission (Fernández González, Presno, Landajo, 2015; Vaninsky, 2013), changes in the production level (Vaninsky, 1986) or chang‑ es in energy intensity (Choi, Oh, 2014). According to our best knowledge, there are no financial applications of the discussed methodology. We veri‑ fy the applicability of the GDIM to financial market research by examining f luctuations of the Warsaw Stock Exchange WIG Index (WIG). We consider four main factors of WIG changes: X1 = PLN/EUR rate, X 3 = GDP growth, X 5  =  S&P500 and the unemployment rate (UNEMPL) transformed into a the‑bigger‑the‑better variable, i.e. X7 = 1 – UNEMPL. Let us also denote by  V  =  WIG and we  add four artificial variables: X 2  =  WIG/(PLN/EUR), X4 = WIG/GDP growth, X6 = WIG/S&P500 and X8 = WIG/(1 – UNEMPL). Thus we can write:

= V X= X= X= X 7 X 8 . 1X2 3X4 5X6

(12)

In other words we have V = X1 X2 with the factors’ interconnections written as follows:

ϕ1 ( X 1 ,..., X 8 ) = X 1 X 2 − X 3 X 4 = 0 ,

(13)



ϕ2 ( X 1 ,..., X 8 ) = X 3 X 4 − X 5 X 6 = 0 ,

(14)



ϕ3 ( X 1 ,..., X 8 ) = X 5 X 6 − X 7 X 8 = 0 .

(15)

FOE 4(330) 2017  www.czasopisma.uni.lodz.pl/foe/

Application of the Divisia Index with Interconnected Factors in the Warsaw Stock Exchange Index… 135

The form of the function V and relations (13)–(15) allow us to write:

∇Z = [ X 2 , X 1 , 0, 0, 0, 0, 0, 0]



(16)

and

 X2   X1 −X4  −X Ö = 3  0   0  0   0 



0 0 X4 X3 −X6 −X5 0 0

0   0  0   0  X6   X5  −X8   − X 7  .

(17)

As a consequence we obtain: 8

where:

where:

(18)

(18)

VVX1 = XX 2((XX232 + XX242)( )(XX252 + XX262)( )(XX272 + XX282))// D D, , ∇ X1 2 3 4 5 6 7 8

(19)

(19)

L

k 1 L

VX 2 X1 ( X232  X242 )( X252  X262 )( X272  X282 ) / D , ∇VX 2 =X 1 ( X 3 + X 4 )( X 5 + X 6 )( X 7 + X 8 ) / D , VX3 X 4 ( X12  X 22 )( X 52  X 62 )( X 72  X 82 ) / D , ∇VX 3 =X 4 ( X 12 + X 22 )( X 52 + X 62 )( X 72 + X 82 ) / D , VX 4 X 3 ( X12  X 22 )( X 52  X 62 )( X 72  X 82 ) / D , ∇VX 4 =X 3 ( X 122 + X 222)( X 522 + X 622)( X 722 + X 822) / D , VX5 X 6 ( X1  X 2 )( X 3  X 4 )( X 7  X 8 ) / D ,

(21)

2 2 2 2 2 2 ∇ VVXX56 = XX65((XX112 + XX222)( )(XX332 + XX442)( )(XX772 + XX882)) // D D , ,

(23)

(24)

VVXX67 = XX58((XX1122 + XX2222)( )(XX3232 + XX4242)( )(XX7252 + XX8262)) // D D , , ∇

(24)

(25)



VVX8 = XX 7((XX212 + XX222)()(XX232 + XX242)( )(XX252 + XX262))// D D, , ∇ X7 8 1 2 3 4 5 6

(25)

(26)



∇V2X 8 =X2 7 ( X 122 + X 222 )( X 232 + X 242 )( X 52 + X 62 ) / D , D ( X 3  X 4 )( X 5  X 6 )( X 7  X 8 ) 

(26)



with 

V   V(I  ΦΦ ) dX   VX k dX k , ,

 X 12 (( X 52  X 62 )( X 72  X 82 )  X 32 ( X 52  X 62  X 72  X 82 ) 

(20)

(22)

www.czasopisma.uni.lodz.pl/foe/  FOE 4(330) 2017

 X 42 ( X 52  X 62  X 72  X 82 ))  X 22 (( X 52  X 62 )( X 72  X 82 )   X 2 ( X 2  X 2  X 2  X 2 )  X 2 ( X 2  X 2  X 2  X 2 )).

(20) (21) (22) (23)

(27)

136  Jacek Białek, Radosław Pietrzyk with

( X 32 + X 42 )( X 52 + X 62 )( X 72 + X 82 ) + D= + X 12 (( X 52 + X 62 )( X 72 + X 82 ) + X 32 ( X 52 + X 62 + X 72 + X 82 ) + + X 42 ( X 52 + X 62 + X 72 + X 82 )) + X 22 (( X 52 + X 62 )( X 72 + X 82 ) +

+ X 32 ( X 52 + X 62 + X 72 + X 82 ) + X 42 ( X 52 + X 62 + X 72 + X 82 )).

(27)

In order to make calculations, we need to parameterize the curve of the fac‑ tors’ dynamics. It is typical for the analytical purposes to assume a linear or an exponential change in the quantitative indicators in time (Vaninsky, 2013) and it seems to be justified by relatively short time subintervals (here quarters) con‑ sidered in our research. For the purpose of illustration, we assume linear changes of indicators in each quarter. In particular, we use a model time t that varies in the time interval [0, 1] for each quarter. It may be shown that the length of the time in‑ terval does not affect the final results. We assume the WIG, PLN/EUR, GDP and UNEMPL have linear dynamics (between quarters), i.e.:

V (t ) =V (0) + (V (1) − V (0)) ⋅ t ,

(28)



X k (t ) = X k (0) + ( X k (1) − X k (0)) ⋅ t , k = 1,3,5, 7 ,

(29)



= dV V (1) − V (0) ,

(30)



= dX k ( X k (1) − X k (0)) .

(31)

where:

The results presented in Section 4 are obtained in terms of the relative change in the value of Warsaw Stock Exchange Index (WIG). All calculations were made in Mathematica 6.0.

5. Empirical study 5.1. Data set description and characteristics of time intervals In this section we apply the discussed approach to examine fluctuations of the War‑ saw Stock Exchange WIG index (WIG). As it was mentioned before, we take four main factors of WIG changes into consideration: X1 = PLN/EUR rate, X3 = GDP, FOE 4(330) 2017  www.czasopisma.uni.lodz.pl/foe/

Application of the Divisia Index with Interconnected Factors in the Warsaw Stock Exchange Index… 137

X5 = S&P500, the unemployment rate (UNEMPL) transformed into a the‑big‑ ger‑the‑better variable, i.e. X7 = 1 – UNEMPL and some other, artificial variables presented in Section 3. As a representative of the Polish capital market, we used the broadest stock market index (WIG), which offers the most complete description of the condition of the Polish economy. We conducted the study for the period from the 2nd quarter of 2003 (Q1 2003) to the 3rd quarter of 2014 (Q3 2014), which was divided into 5 subperiods characterised by subsequent downward and up‑ ward trends. As a result we were able to study various market conditions. The choice of such a long study period (over 10 years) guarantees that the data is neither specific nor unrepresentative. The period includes the bull market connected with Polish accession to the European Union as well as the time of the global financial crisis. Figure 2 presents the whole study period and its subperiods. WIG 70000

60000

50000

40000

30000

20000

10000

Q3 2014

Q2 2014

Q1 2014

Q4 2013

Q3 2013

Q2 2013

Q1 2013

Q4 2012

Q3 2012

Q2 2012

Q1 2012

Q4 2011

Q3 2011

Q2 2011

Q1 2011

Q4 2010

Q3 2010

Q2 2010

Q1 2010

Q4 2009

Q3 2009

Q2 2009

Q1 2009

Q4 2008

Q3 2008

Q2 2008

Q1 2008

Q4 2007

Q3 2007

Q2 2007

Q1 2007

Q4 2006

Q3 2006

Q2 2006

Q1 2006

Q4 2005

Q3 2005

Q2 2005

Q1 2005

Q4 2004

Q3 2004

Q2 2004

Q1 2004

Q4 2003

Q3 2003

Q2 2003

Q1 2003

0

Figure 2. WIG index between Q1 2003 and Q3 2014 Source: own work

The chosen periods have various lengths (from 4 to 18 quarters) and substan‑ tial differences regarding returns. In the whole study period the WIG index growth amounted to 292.6%. This was the result of upward trends in the bull‑market pe‑ riods (respectively 331.88%, 102.69% and 32.98%) and downward movements (re‑ spectively –63.71% and –15.1%). Relevant data is presented in detail in Table 2. The choice of such subperiods allowed us to check the performance of our model in various market conditions. www.czasopisma.uni.lodz.pl/foe/  FOE 4(330) 2017

138  Jacek Białek, Radosław Pietrzyk Table 2. Rate of returns of WIG Index in selected periods Period No. The initial quarter The final quarter 1 Q1 2003 Q2 2007 2 Q2 2007 Q1 2009 3 Q1 2009 Q1 2011 4 Q1 2011 Q4 2011 5 Q4 2011 Q3 2014 The whole interval Q1 2003 Q3 2014

% WIG 372.72 –63.71 102.69 –15.10 32.98 292.60

Source: own work

Quarterly data was used for all considered variables, mainly due to the availabili‑ ty of the most important factor determining changes in the stock market index, which is the GDP growth. Other indicators were transformed into quarterly data (PLN/ EUR, S&P500) or their final values for relevant quarters were used (UNEMPL).

5.2. Results We investigated to what extent the changes in the PLN/EUR, GDP, S&P500 and unemployment rate affected the WIG value increase (or decrease). For each time interval we measured the contribution of the considered variables to the relative change in WIG value (see Table 3) by using the following formula: ∆VX dX k k (see (18)–(27)). Our results are presented in Table 3. L

∫

Table 3. Factors’ contribution to the change in WIG value depending on time interval Factors X1 X2 X3 X4 X5 X6 X7 X8 Total percentage change

Period 1 3.948 77.289 79.469 9.913 45.621 78.921 8.512 79.050 372.720

Contribution to the relative change in WIG value Period 2 Period 3 Period 4 Period 5 The whole interval –3.949 5.669 –3.531 –0.181 2.602 –12.124 17.4028 –5.178 15.517 68.948 –16.392 34.610 3.332 0.000 47.005 8.728 –17.209 –11.699 15.349 57.926 –11.386 17.917 –1.681 17.893 62.286 –4.903 5.086 –6.992 –2.766 40.186 0.092 –1.209 –0.303 1.441 9.026 –14.668 23.137 –8.318 14.001 69.476 –63.330 102.620 –22.673 45.906 299.532 Source: own work

We can observe that as a rule (especially during the market prosperity) GDP and S&P500 movements exhibit the strongest influence on the WIG changes. The FOE 4(330) 2017  www.czasopisma.uni.lodz.pl/foe/

Application of the Divisia Index with Interconnected Factors in the Warsaw Stock Exchange Index… 139

PLN/EUR value and the unemployment rate seem to have marginal influence on the WIG fluctuations, only 2.6% and 9%, respectively. Also, in all subperi‑ ods, the influence of PLN/EUR changes seems to be minor. In the first two peri‑ ods of the bear market and the first two periods of the bull market, the EUR/PLN strengthened the WIG index trend. Only in the period 5 did the EUR/PLN chang‑ es result in slowing down the index growth. The level of the unemployment rate positively influenced the growth of the index, but in subperiods 2 and 3 chang‑ es in the unemployment rates adversely affected changes of the WIG index. The studies showed that the GDP growth and the S&P500 index had signif‑ icant impacts on the change of the WIG index over the whole period. In subperi‑ ods 1–3 the GDP influenced the WIG strengthening its trend. In the subperiod 4 the GDP slowed down the downward trend. In turn, in the subperiod 5 there was no impact on the index. The American index had the greatest impact on the level of the WIG index. Movements in the Warsaw Stock Exchange followed the trends in the US. These results confirm a very high correlation between the Polish and the American capital markets. In addition, in all subperiods the changes in the Polish market were con‑ sistent with changes in the US market. As a consequence the US market strength‑ ened the observed trends.

6. Conclusions This paper presents a tool for factorial decomposition of WIG changes by inter‑ connected factors: the GDP growth, PLN/EUR rate, S&P500 index, and unem‑ ployment rate. Due to computational reasons we applied the transformation that produces variables in the form the bigger the better. For analytical purposes only, we also used some artificial factors (like the WIG/GDP or WIG/S&P500), which have poor interpretation. As it was mentioned above, our main conclusion is that GDP and S&P500 fluctuations have the strongest influence on WIG changes. An‑ other observation is that drops in the S&P500 index may have the result that, in spite of positive signals from the economy, the WIG index also goes down. This means that international trends and global investors’ decisions have the largest in‑ fluence on market behavior in the Warsaw Stock Exchange and fundamental fac‑ tors from the Polish market are of secondary importance. The presented method cannot serve as  a  prognostic tool, however it  may be helpful in initial variable selection in econometric regression models. Its draw‑ back is computational complexity. On the other hand its main advantage is the pos‑ sibility to include interconnections between the factors, which are not necessarily linear. Further studies, which would give more detailed results, require a nonlin‑ ear approximation of the considered processes in quarterly time intervals and in‑ www.czasopisma.uni.lodz.pl/foe/  FOE 4(330) 2017

140  Jacek Białek, Radosław Pietrzyk clusion of a higher frequency of the data. There are also some limitations of the presented method in the area of interpretation, i.e. we do not have any information indicating whether the considered interconnected factors are statistically signifi‑ cant. All considered processes have a deterministic character and must be approx‑ imated by continuous functions. Since the model is deterministic we cannot use the inferential statistics for the WIG change description and, in particular, we can‑ not generalize conclusions. However, after the application of the Divisia method, we obtain the knowledge about the indicator influence on the dependent variable changes in the considered time interval. It is worth adding that the stochastic gen‑ eralization of the Divisia price index can be found in literature (Białek, 2015) but, according to our best knowledge, there no stochastic generalizations for the extend‑ ed Divisia price index with interconnected factors. Finally, we would like to stress that the aim of the paper is to present some interesting method with interconnect‑ ed factors rather than to evaluate it in competition with existing approaches. This paper does not play the role of a method proposition since the method is already known and thus we do not compare it to other existing methods of WIG changes analysis, like econometric ones. As a rule, Divisia’s approach is treated as some theoretically perfect method without practical applications (von der Lippe, 2007). We merely hope to change this opinion by showing its possible application for WIG fluctuations analysis. References Ang B.W. (2004), Decomposition analysis for policymaking in energy: which is the preferred method?, “Energy Policy”, vol. 32, pp. 1131–1139. Ang B.W., Liu F.L., Chew E.P. (2003), Perfect decomposition techniques in energy and environmental analysis, “Energy Policy”, vol. 31, pp. 1561–1566. Białek J. (2015), Generalization of the Divisia price and quantity indices in a stochastic model with continuous time, “Communications in Statistics: Theory and Methods”, vol. 44, no. 2, pp. 309–328. Chen N.F., Roll R., Ross S.A. (1986), Economic Forces and the Stock Market, “The Journal of Busi‑ ness”, vol. 59, no. 3, pp. 383–403. Choi K.H., Oh W. (2014), Extended Divisia index decomposition of changes in energy intensity: A case of Korean manufacturing industry, “Energy Policy”, vol. 65, pp. 275–283. Dadgostar B., Moazzami B. (2003), Dynamic Relationship Between Macroeconomic Variables and the Canadian Stock Market, “Journal of Applied Business and Economics”, vol. 2, no. 1, pp. 7–14. Divisia F. (1925), L’indice Monetaire et la Theorie de la Monnaie, “Revue d’Economic Politique”, vol. 39, no. 5, pp. 980–1020. Fama E.F. (1981), Stock Returns, Real Activity, Inflation, and Money, “The American Economic Review”, vol. 71, no. 4, pp. 545–565. Fernández González P., Presno M.J., Landajo M. (2015), Regional and sectoral attribution to percentage changes in the European Divisia carbonization index, “Renewable and Sustainable Energy Reviews”, vol. 52, pp. 1437–1452. FOE 4(330) 2017  www.czasopisma.uni.lodz.pl/foe/

Application of the Divisia Index with Interconnected Factors in the Warsaw Stock Exchange Index… 141 Humpe A., Macmillan P. (2009), Can macroeconomic variables explain long‑term stock market movements? A comparison of the US and Japan, “Applied Financial Economics”, vol. 19, no. 2, pp. 111–119. Lippe P. von der (2007), Index Theory and Price Statistics, Peter Lang Publishing, Frankfurt. McMillan D.G. (2010), Stock Market Fundamentals and Bubbles: Implications for Prices, GDP and Consumption, “SSRN Electronic Journal”, http://www.ssrn.com/abstract=1695443 [ac‑ cessed: 18.12.2016]. Mukherjee T.K., Naka A. (1995), Dynamic Relations between Macroeconomic Variables and Japanese Stock Market: An Application Of A Vector Error‑Correction Model, “The Journal of Fi‑ nancial Research”, vol. 18, no. 2, pp. 223–237. Nelson C.R. (1976), Inflation and Rates of Return on Common Stocks, “The Journal of Finance”, vol. 31, no. 2, pp. 471–483. Sheremet A., Dei G., Shapovalov V. (1971), Method of the chain substitutions and development of the factorial analysis of the economic indicators, “Vestnik Moskovskogo Universiteta, Ser. Ekonomika”, vol. 4, pp. 62–69. Vaninsky A.Y. (1986), Analysis of production efficiency based on the generalized integral method, [in:] A. Aksenenko (ed.), Accounting and analysis of production efficiency (Uchet i analiz effectivnosti proizvodstva), Financy i Statistika, Moscow. Vaninsky A.Y. (1987), Factorial Analysis of Economic Activity (Factornyi Analiz Khozyaistbennoi Deyatel’nosti), Financy i Statistika, Moscow. Vaninsky A.Y. (2013), Economic Factorial Analysis of CO2 Emissions: The Divisia Index with Interconnected Factors Approach, “International Journal of Social, Behavioral, Educational, Economic, Business and Industrial Engineering”, vol. 7, no. 10, pp. 2772–2777. Vaninsky A.Y., Meerovich V. (1978), Problems of the methodology of analysis of the impact of structural change on the indicators of production efficiency (Voprosy metodologii analiza vliyaniya strukturnykh sdvigov na pokazateli effectivnosti proizvodstva), “Proceedings of the National Scientific Conference ‘Economic Leverages of the Efficiency of Using Material, Labor, Finan‑ cial, and Natural Resourses’”, Central Economic‑Mathematical Institute, Moscow. Wiśniewski H. (2014), Wpływ zmiennych makroekonomicznych na indeksy giełdowe, Uniwersytet Warszawski, Warszawa. Zastosowanie indeksu Divisia z powiązanymi czynnikami do analizy fluktuacji indeksu WIG Streszczenie: W artykule zaprezentowano metodę analizy ekonomicznej opartej na indeksie Divisia z powiązanymi czynnikami. Zweryfikowano możliwości aplikacyjne wymienionej metody do bada‑ nia zmienności indeksu WIG. W analizie uwzględniono cztery główne zmienne wpływające na indeks warszawski: GDP, kurs PLN/EUR, indeks S&P500 oraz stopę bezrobocia, przy czym dokonano (w razie konieczności) transformacji zmiennych na stymulanty. Analizą objęto lata 2003–2014 i uwzględnio‑ no dane kwartalne, przy czym interwał czasowy podzielono na podokresy związane z hossą i bessą na giełdzie warszawskiej. Przyjęto również model czasu ciągłego z założeniem, że między kwarta‑ łami wartości zmiennych zmieniają się liniowo. Głównym wnioskiem z przeprowadzonego bada‑ nia jest wyodrębnienie najbardziej wpływowych zmiennych objaśniających w postaci GDP i indek‑ su S&P500. Słowa kluczowe: analiza czynnikowa, indeks Divisia, czynniki powiązane, indeks WIG JEL: C43, G10

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142  Jacek Białek, Radosław Pietrzyk © by the author, licensee Łódź University – Łódź University Press, Łódź, Poland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license CC‑BY (http://creativecommons.org/licenses/by/3.0/) Received: 2017-01-14; verified: 2017-04-27. Accepted: 2017-09-12

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