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In this paper, we used the Singular Value Decomposition (SVD) to find the relationships in the fluctuation of the six market indexes CAC 40, DAX, DOW JONES 30, FTSE 100, IBEX35 and NIKKEI 225 during the year 2018. This technique allows relating several indexes in a very similar way the classical Principal Component Analysis (PCA). In fact, we will just use the statistical software to confirm some results.

It is assumed that there are six indexes: CAC 40, DAX, DOW JONES 30, FTSE 100, IBEX35 and NIKKEI 225 with n = 254 trading days, in fact not all indexes have the same number of days, when a value was missing the value of the index has been repeated. In this order, let be

Q = ( q i j ) = ( q 1 , q 2 , q 3 , q 4 , q 5 , q 6 ) ∈ R 254 × 6 . (1)

In

Q i j = q i j − q ¯ j , with q ¯ j = 1 254 ∑ i = 1 254 q i j , (2)

and the second step is to scale the values in each column using a characteristic value S j for that column, in this case we used the maximum entry in the jth column in absolute value, so

P = ( p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ) = ( p i j ) with p i j = Q i j S j , (3)

for i = 1 , ⋯ , 254 ; j = 1 , ⋯ , 6 . In

The question considered is, is there a connection between the movements in the indexes of the markets? and if there are, which are the relationships and which are not?

This paper is organized as follows. In Section 2, we describe the principal components and possible linear approaches. In Sections 2 and 5, we study the one and two dimensional approximations respectively and we compare our results with those obtained with the software in Matlab. In Section 8, we consider the five dimensional approximation with a result maybe a little surprising. Finally in Section 11, we analyze the numerical results and draw the main conclusions.

Our numerical methods were implemented in Matlab, the codes are available on request. The experiments were carried out in an Intel(R) Core(TM)2 Duo CPU U9300 @ 1.18 GHz, 1.91 GB of RAM.

The goal of Principal Component Analysis is to find the principal directions of the normalized data matrix P ∈ R 254 × 6 . This technique has widely be used in computer science for data reduction as it enables to summarise the main directions of the data set. However, we will use an alternative option to Component Analysis called singular value decomposition. Before properly entering data

processing we split our data set into a training and testing set, respectively P 1 ∈ R 127 × 6 and P 2 ∈ R 127 × 6 . P 1 is used to find the principal components while P 2 is used to test the accuracy of the approximations. In practice, we select the even rows of the dataset for training and the odd for testing.

Singular Values Composition (SVD) gives: P 1 = U Σ V T where U is an 127 × 127 orthogonal matrix, V is an 6 × 6 orthogonal matrix and Σ = diag ( σ 1 , ⋯ , σ 6 ) is an 127 × 6 diagonal matrix. The columns of V are called the principal components. This technique has widely been applied in applied mathematics. For further analysis we suggest some literature references [

σ 1 = 8.0164 , σ 2 = 3.3344 , σ 3 = 1.9863 ,

σ 4 = 1.2209 , σ 5 = 0.9312 , σ 6 = 0.5284.

and

V = ( v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) = ( 0.4129 − 0.0473 0.1955 0.3034 − 0.7363 − 0.3934 0.4820 0.1892 − 0.1485 0.1407 − 0.1590 0.8153 0.1561 − 0.7115 − 0.2788 0.5518 0.2950 − 0.0157 0.5053 − 0.0008 0.7078 − 0.0484 0.4868 − 0.0663 0.4869 0.3991 − 0.5855 − 0.0815 0.2817 − 0.4181 0.2837 − 0.5445 − 0.1345 − 0.7581 − 0.1709 0.0316 )

Because we want to determine the best linear fit of the normalized data

P = ( p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ) ,

one need select one of the following linear functions

p = α 1 v 1 , (4)

p = α 1 v 1 + α 2 v 2 , (5)

p = α 1 v 1 + α 2 v 2 + α 3 v 3 , (6)

p = α 1 v 1 + α 2 v 2 + α 3 v 3 + α 4 v 4 , (7)

p = α 1 v 1 + α 2 v 2 + α 3 v 3 + α 4 v 4 + α 5 v 5 , (8)

the first choice (4) corresponds a one-dimensional approximation or linear case, (5) is a two-dimensional approximation, etc.

In this section, we only focus in the first linear relation defined in the previous section: p = α v 1 that written by components is

p j = α v j 1 , for j = 1 , ⋯ , 6.

Our goal is to assess the predictive power of this first approximation, in other words, if we knew an index, could we predict another index? For example, if we select the French market index (Cac) ( j = 1 ) we take α = p 1 / v 11 , then the best fits for the other indexes are

p j = α v j 1 = v j 1 v 11 p 1 , for j = 2 , ⋯ , 6.

This procedure applied in each day i = 1 , ⋯ , 127 is the prediction

p i j = v j 1 v 11 p i 1 , for j = 2 , ⋯ , 6. (9)

The resulting lines are shown in

On the other hand, the Principal Component Analysis (PCA) is a classical technique with a very wide bibliography, see for example ( [

· >> [coefs,score] = pca(data);

· >> vbls = {‘Cac’,’Dax’,’Dow’,’Ftse’,’Ibex’,’Nikkei’}

· >> biplot(coefs(:,1:2),’Scores’,score(:,1:2),’VarLabels’,vbls)

where the each column of the matrix

coefs = ( v 1 , − v 2 , v 3 , − v 4 , − v 5 , v 6 )

contains coefficients for one principal component in descending order of component variance, and the matrix score ∈ R 127 × 6 correspond to observations. All six variables are represented in the biplot

In this subsection we will try to answer the following question: knowing two indices are we able to predict the other four ones? For simplification, let assume we only have the first two variables; the French CAC index and the German DAX index. We would like to predict the other indices: Dow Jones ( j = 3 ), Ftse ( j = 4 ), Ibex ( j = 5 ) or Nikkei with ( j = 6 ). For each index j, because p = α 1 v 1 + α 2 v 2 , in components

p j = α 1 v j 1 + α 2 v j 2 , for j = 2 , ⋯ , 6.

and we must compute the parameters α 1 , α 2 using the data, i.e., for each i = 1 , ⋯ , 127 solving the two dimensional system

( p i 1 p i 2 ) = ( v 11 v 12 v 21 v 22 ) ( α i 1 α i 2 ) , (10)

and the i-th prediction for the j index is

p i j = α i 1 v j 1 + α i 2 v j 2 . (11)

In

On the other hand, using the before software in Matlab we can represent three components by typing:

· >> biplot(coefs(:,1:3),’Scores’,score(:,1:3),’VarLabels’,vbls)

with the result in

The question that we now consider is whether known five indices, how well can predict the sixth.

Now the fit is (8) i.e.

p = α 1 v 1 + α 2 v 2 + α 3 v 3 + α 4 v 4 + α 5 v 5 , (12)

which in matrix form is

( p 1 p 2 ⋮ p 6 ) = ( v 11 v 12 ⋯ v 15 v 21 v 22 ⋯ v 25 ⋮ ⋮ ⋱ ⋮ v 61 v 62 ⋯ v 65 ) ( α 1 α 2 ⋮ α 5 ) . (13)

In order to make the mathematical formulation easier, we assume that the interesting variable is p 1 , then we need to find the α k ’s in terms of p 2 , ⋯ , p 6 what is achieved by solving the linear system

( p 2 p 3 ⋮ p 6 ) = ( v 21 v 22 ⋯ v 25 v 31 v 32 ⋯ v 35 ⋮ ⋮ ⋱ ⋮ v 61 v 62 ⋯ v 65 ) ( α 1 α 2 ⋮ α 5 ) , (14)

If we write α k = ∑ j = 2 6 a k j p j for k = 1 , ⋯ , 5

p 1 = α 1 v 11 + α 2 v 12 + α 3 v 13 + α 4 v 14 + α 5 v 15 = a 2 p 2 + a 3 p 3 + a 4 p 4 + a 5 p 5 + a 6 p 6 ,

with a k = ∑ j = 1 5 v 1 j a j k .

Finally, this procedure applied in each data i = 1 , ⋯ , 127 result prediction

p i 1 = a 2 p i 2 + a 3 p i 3 + a 4 p i 4 + a 5 p i 5 + a 6 p i 6 . (15)

In Figures 9-11 we have represented the six cases. Similarly to the previous

There are two different behaviors, while for the indexes Dax, Cac and Ibex the predictions are reasonable, the other three indexes have quite bad predictions.

In this paper, we have considered six indexes: CAC 40, DAX, DOW JONES 30, FTSE 100, IBEX35 and NIKKEI 225 in the year 2018, and we have tried to link their movements using the principal components of the matrix of their fluctuations. Our numerical results are very close to the numerical software with Matlab, however, the result of the last Section 8 might be a little surprising, especially because of the deviation in the English index Ftse.

In this point, a question we might ask is: how many columns of V is needed for our analysis? ( [

R k = σ k + 1 2 + ⋯ + σ 6 2 σ 1 2 + ⋯ + σ 6 2 , (16)

for k = 1 , ⋯ , 5 representing in

This small academic exercise does not conclude important results, however, we believe that it is quite easy to extend this kind of analysis to longer series of data or also apply it to other indexes, for example in Investing.com there are 45 indexes. This would need to prepare the data files, a routine but important job, but with little academic interest. The main philosophy of all this is written by Yuval Noah Harari in the introduction of [

This work was supported by Spanish Ministry of Sciences Innovation and Universities with the project PGC2018-094522-B-100 and by the Basque Government with the project IT1247-19.

The authors declare no conflicts of interest regarding the publication of this paper.

Vadillo, F. and Vadillo, N. (2021) Singular Valued Decomposition and Principal Component Analysis to Compare Market Indexes. Journal of Mathematical Finance, 11, 484-494. https://doi.org/10.4236/jmf.2021.113027