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91. ПРОБЛЕМНЫЕ СТАТЬИ
УДК 519.862.6
ECONOMETRIC MODELING OF GROWTH OF SBERBANK'S
CAPINALIZATION D.S. Batishchev
Financial University under the Government of the Russian Federation, Moscow, Russia
dima.batishchev@yandex.ru V.P. Nevezhin
Financial University under the Government of the Russian Federation, Moscow, Russia
VPNevezhin@fa. ru
Abstract: The article is about an explanation of the capitalization growth of the Russian bank Sberbank based on construction of an econometric model. During the study, it is suggested that Sberbank bought its own shares and thereby reduced the share of free float and increased the EPS, which had a positive effect on the prices of its general stock. A description of all the conducted study procedures of the pair-wise linear econometric model and the obtained results are given. The statistical and econometric analysis procedures were performed with special functions of the MS Excel.
Keywords: Free float, capitalization, shares, stock exchange, modeling, econometric model, pairwise linear regression, correlation, model adequacy.
JEL Classification: C01, C2, C51, E44, G21. 1. Introduction
Prior to consider the activities of PJSC Sberbank on the stock exchange, we present some data about this bank.
Sberbank of Russia (further referred as Sberbank) is a Russian financial conglomerate, the largest transnational and universal bank in Russia, Central and Eastern Europe. It is controlled by the Central Bank of the Russian Federation, which owns 50% of the share capital plus one voting share.
It provides a wide range of banking services. As of January 1, 2016, Sberbank's share in the total assets of the Russian banking sector was 28.7%, and 46% in the market of private deposits. The loan portfolio takes 38.7% of all loans issued to the public.
In 2018, the value of the Sberbank brand was 670.4 billion rubles, which was Russia's most expensive brand (wikipedia.org, 2019).
To understand the phenomenal growth of Sberbank's capitalization over the past few years, it is necessary to understand the reasons for the growth of its shares on the Moscow Stock Exchange. Thus, since the beginning of 2015, the shares of this bank have grown by an average of 3.5 times, rising from a mark of 61 rubles in March 2015 per share to 221 rubles in November 2017. In our opinion this
phenomenon is somewhat ambiguous, since, for example, shares of other large banks operating in Russia, such as VTB and Rosbank, were either in the trend channel or had a down trend during this period of time (see fig. 1).
Fig. 1 - The growth of Sberbank / VTB / Rosbank shares for the last 3 (compiled from [5]) These facts show that the growth of Sberbank shares may be partly artificial. For example, Sberbank buys part of its own shares on the Moscow Stock Exchange. Thus, it forwards the reduction of its free-rotating shares on the market and leads to an artificial increase in earnings per share (EPS), see fig. 2. This phenomenon is called "Buyback".
Fig. 1 - EPS per share of Sberbank by year (J.H. Stock, M.W. Watson, 2011), (company "Finam", 2019)
However, this is only our hypothesis and we don't have actual confirmation of Sberbank's operations. But we did an attempt to try to validate this hypothesis with constructing a pairwise linear econometric regression reflecting the dependence of stock prices on Sberbank's investment in securities.
2. Correlation analysis
To analyze the dynamics of changes in the stock price (Y) and the impact on it, several Sberbank financial indicators were selected (further referred as "factors") for 30 months. These included the following factors: net assets (X\), net income (X2), deposits of individuals (X3), loan portfolio (X4), arrears (X5), investments in securities (X6). The correlation analysis of explanatory factors with explainable (Y) for the specified period was calculated with the CORREL function of the MS Excel. It showed the following results, see table 1.
Table 1 - Correlation depencence
Correlation coefficient between (ryx) Value
Y/X1 0.613
Y/X2 0.452
Y/X3 0.362
Y/X4 0.503
Y/X5 0.438
Y/X6 0.727
On the basis of obtained results, it can be confirmed that the greatest influence on the stock price was done by "investment in securities" (X6). For further research, a paired linear econometric model has been specified as "share price" (Y) dependence on "investment in securities" (X6), having the form: y = ao + Q6-X6 + £ (1)
with selected baseline data for 30 months, see table. 2
Table 2 - Source dataset (banki.ru,2017), (company "Finam", 2019)
№ Y X6
1 165.2 2113390072
2 159.8 2075396135
3 156.0 2054659285
4 172.2 2138913007
5 173.25 2183359187
6 158.7 2309730038
7 147.4 2379242608
8 145.34 2452223475
9 143.5 2357959614
10 139.15 2388249582
11 133.0 2350026225
12 132.56 2368961532
13 123.55 2358005053
14 109.9 2343325650
15 107.0 2287225576
16 96.5 2022010108
17 101.26 1849204759
18 102.9 1782743494
19 90.53 1733496920
20 75.3 1683207266
21 74.5 1708901351
22 72.3 1770923427
23 72.35 1750222951
24 73.5 1836358932
25 76.9 1841353261
26 62.88 1886565087
27 75.91 1926871190
28 61.5 1880202307
29 54.9 1886019899
30 72.25 1843840173
In the specified paired linear econometric model, it is necessary to obtain estimates of its parameters aO and a1.
3. The Gauss-Markov theorem
conditions verification
To estimate the parameters of econometric models the Method of Least Squares (OLS) is used, but its application requires the Gauss - Markov theorem's conditions fulfillment.
The condition of the presence of heteroscedasticity of random perturbations of the model has been verified using the Goldfeld-Quandt
test. The values of the Goldfeld-Quandt statistic GQ and GQ_1 are calculated using the formulas:
ESS,
GQ =
ESS
GQ- =
1
and reversed function
ESS,
GQ ESSj
(2)
As well as the value of the Fcrit with a degree of freedom equal to 10 (2 of the sorted array are selected from the same number of observations from 12 subarray values) and the probability of trust equals to 95%. The results of this tests are presented in table 3 and figure 3.
Table 3 - A comparison of Fcrit with GQ/GQ-1
GQ 0.564 < Fcrit 2.98
GQ-1 1.773 < Fcrit 2.98
100
rs 3 50 Ill II
'w 01 ce 0 .....111
-50 x6
Fig. 3 - The random perturbations of the pair model
Due to the condition GQ < Fcrit and GQ"1 < Fcrit is satisfied during the test, the random perturbations in the paired econometric model are assumed to be homoscedastic (Nevezhin, V.P., Nevezhin, Yu.V., 2017). The plot in figure 3 also shows the homoscedasticity of random perturbations.
To determine the presence or absence of autocorrelation, a Durbin-Watson test was applied with Durbin-Watson (DW) statistics calculation using the formula:
DW = -
(3)
Also, table values (Nevezhin, V.P., Nevezhin, Yu.V., 2017) of dL and du was defined. The results
shown at table 4.
Table 4 - DW/ dL / du values
DW dL du
0.203 1.35 1.49
The value of the DW statistics belongs to interval 0 < DW < dL, which confirms the presence of positive autocorrelation of random residues. The result obtained in this case is connected with the fact that not all explanatory factors influencing the explained factor are considered in the chosen model. This conclusion has a business case - Sberbank's stock prices depend on many other factors besides its investment in securities. For example, investor mood or informational news connected to Sberbank may also affect stock prices. We want to use the econometric study to show the fact that the current growth in Sberbank's stock in some way may be associated with its own stock's buyback.
Checking that random errors of the econometric model are distributed according to normal law was carried out using the Helving agreement test. As a result of the test, the hypothesis that random evaluations are of a normal nature is accepted.
4. Model parameters and their
statistical significance evaluation
The conducted testing of the Gauss-Markov conditions showed that the OLS can be used to estimate the parameters of the paired linear econometric model, which will calculate unbiased and effective estimates. The following results were obtained using the LINEST function of the MS Excel, see table 5.
Table 5 - LINEST calculation results
/V aix 1.094E-07 -113.542 ao
ai 1.955E-08 40.406 <7 ao
R2 0.528 26.705 Ou
F 31.338 28 V2
ESS 22350.127 19969.09 RSS
d0 and d1 - the obtained pair regression estimates
Gn and <7„ - standard errors of estimated
a0 a1
parameters
R2 - coefficient of determination
2
i=1
Ou - standard regression error
F - obtained regression Fisher statistics
V2 - degree of freedom
RSS - squared residual obtained regression
Thus, the following values of the parameters of the econometric model were obtained:
a0= - 113.54; a:=1.09E-07
The statistical significance of the parameters of the model obtained was calculated using the t-test. For this, the value of tcrit was calculated using the function TINV, and the values of ta0 and ta1 are calculated by the formula:
fa =■
(4)
a„
As a result of the calculation obtained values ta0 = -2.81, ta\ = 5.598 and tcnt = 2.048. Due to modulus of tai, i = 1.2 greater than tcrit, it is argued that they are both statistically significant.
5. A regression for statistical
significance and adequacy check
Validation of the resulting econometric regression type
y= - 113.64 + 1.09E-07-x6
(40.406) (1.95E-08) (26.71) R2=0.53 for statistical significance was performed using the F-test. For this, the value of Fmod = 31.34 was compared with the value of Fdt = 2.403. So Fmod > Fcrit, then this regression is declared statistically significant.
The coefficient of determination R2»0.53. This corresponds to the fact that the stock prices of the resulting regression by 53% are explained by investments in securities.
Calculated average approximation error (A) is 20%, which corresponds to a satisfactory value.
The obtained regression adequacy check was carried out using interval estimation. The entire available sample of 30 values was divided into a training (27 values) and controlling (3 values). The model parameters were estimated on the basis of the data from the training sample. Each of the three values of the explained factor (yQ) of the control
sample was calculated to determine the confidence interval
Jo = fo — tcritay0 (5)
70+ = fo + tcritGy 0 (6)
Also, to determine the interval of each value (y 0
), the average standard error of the forecast was calculated using the formula:
_ (7)
1 + - + n
(*0 - *)
n
X(* - *)2
n - the number of training sample values; (x0 — x)2 - squared difference of the x0 values of the control sample and x training sample;
Ei=1(xi —x)2- the sum of the squares of the difference xj and x training sample; ou- standard error of the resulting training set regression.
The results of the regression adequacy test are shown in table 6.
Table 6 - Confidence intervals
= f0 — tcritay0 low bound Y (control set) = f + tCrit<Jy0 upper bound
42.045 61.5 152.243
42.659 54.9 152.792
38.184 72.25 148.835
Since all statistical observations belong to confidence intervals, the resulting pairing regression model can be defined as adequate.
The resulting regression is presented in fig. 4 -plot Yn shows changes in investments in securities on changes in the price of stocks of Sberbank.
Fig. 4 - Yn regression plot 5. Conclusions
The resulting econometric pair regression is adequate and statistically significant, its parameters are also statistically significant and adequate. Thus,
i=i
a
the conducted econometric studies have shown the possibility of the existence of an economic-mathematical model reflecting a certain dependence of the Sberbank's stocks price change on its investments in securities. The resulting model indirectly confirms the fact that the current rise in Sberbank's stocks prices and its capitalization is partially artificial, due to the fact that since the beginning of 2015 this bank has been buybacking its own shares to increase its market value. As a result of this operation, net profit per share increases, and this contributes to the growth of investor's confidence and has a positive effect on exchange rate indicators.
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