Modelling and Forecasting the Net Income Dynamics Tatiana I. Chinaeva 1,2[0000-0001-7441-8061], Elena I. Larionova 1[0000-0001-7335-4481], and Viktoriya V. Narbut 1[0000-0003-1551-5114] 1 Financial University under the Government of the Russian Federation, 49 Leningradsky Ave., Moscow, 125993, Russia Larionova_len@mail.ru, vika-30@yandex.ru 2 Institute for the Study of Science of the Russian Academy of Sciences, 32 Nakhimovsky Ave., Moscow, 117218, Russia t.chinaeva@yandex.ru Abstract. This paper addresses modelling and forecasting the net income, as one of the key indicators for characterizing the banking system of a country. The article discusses the net income time series for the United States from 2010 to 2018. The objective of this work is to select the most appropriate model for modelling trends in net income. The result of this study actually established that the change in net income is due to two main factors, the nature of the dynamics of each of which differ significantly. It is concluded that the resulting model can be used to forecast the value of the net income in the United States. The pa- per presents the forecast for the considered indicator for four quarters in ad- vance. The authors used methods for generalization and comparative analysis of alternative approaches to modelling and forecasting net interest income, as well as logical, mathematical and economic and statistical methods and techniques using Statistica 10.0 to make the necessary calculations. Keywords: net income, net interest income, noninterest income, trend-seasonal model, modelling and forecasting. 1 Introduction The authors of the article examined the theoretical and practical aspects of research and development of methods for predicting the net interest income of banking sys- tems. The analysis revealed that despite a large number of scientific works devoted to this issue, a unified and generally accepted methodology for modelling and forecast- ing net income, net interest income and noninterest income has yet to be devel- oped.The article substantiates the choice of a model for modelling trends in net in- come, identifies factors affecting the change in net income (net interest income and noninterest income). The paper presents the results of net income forecasting using a multiplicative trend-seasonal model, measures the correlation of fluctuations of series characterizing the dynamics income and net interest income, based on the measure- ment of the correlation between deviations from trends. The essence of modelling using economic and statistical methods is that the predicted indicator is determined based on specific models that show its functional dependence on certain factors. Proceedings of the 10th International Scientific and Practical Conference named after A. I. Kitov "Information Technologies and Mathematical Methods in Economics and Management (IT&MM-2020)", October 15-16, 2020, Moscow, Russia © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) As objects of the empirical base of the study, the authors selected a database of the International Monetary Fund. Section 8 deals with open interfaces, and Section 9 deals with training issues for the digital economy. 2 Literature Review Financial development, economic growth and bank efficiency are inextricably inter- connected. Several studies have provided empirical evidence of the nonlinear impact of financial development on economic growth [1-9]. The banking sector is a key seg- ment of the country's financial system, while in some works the authors have repeat- edly noted that the prerequisites for dynamic economic growth are provided by an efficiently functioning financial system [10]. In his article, N.A. L'vova and I.A. Darushin disclose the content of the emerging financial market in conjunction with the category “emerging financial system”. They reveal the functional and institutional signs characteristic of the emerging financial systems and analyses the criteria for assessing the level of financial development [11]. The authors refer to the opinion of B.B. Rubtsov that the emerging financial markets are more characterized by a model focused on the banking system [12], which, like any economic system, must be managed. In this regard, it is necessary to take into account the forecast of its development for the future [13]. In [14] it is noted that the banking sector as a whole is highly sensitive to changes in economic development. During financial crises, this relationship is particularly pro- nounced [15]. In publications [16-18], the authors note that the activities of banks are associated with credit risks, solvency, interest rate, liquidity and other financial trans- actions. The indicator that most clearly reflects the efficiency of banking is net interest in- come, which is one of the most stable and sustainable components of the bank's profit. The use of econometric and economic-statistical methods in predicting the results of the banking system has specific features. The dynamics of the net income of commer- cial banks, as a financial result of banking activities, can vary within a fairly wide range, bringing the bank both a significant profit and a considerable loss. The analysis of the models for forecasting the income of commercial banks showed that they are often based on the extrapolation method, i.e. the studied indicator is put in depend- ence on the time factor, without reflecting the connection of the studied indicator with internal and external factors that can affect its value. For specific purposes, this ap- proach can be considered quite justified. The first attempts to build predictive models of the profit of the Russian banking sec- tor are given in the works [19-23]. In the work of V.M. Gumerov [19], was proposed a mathematical model for forecasting the net income from foreign exchange transac- tions of a commercial bank, the modelled indicator in which is the average conversion income of the bank, pairwise correlation coefficients were calculated between the modelled indicator and specific factors, as well as in pairs the factors themselves. The author concluded that it is inexpedient to model the dependence of the studied indica- tor on any internal or external factors. It is proposed to include in the final model for forecasting the net income of a particular bank the average value of the adjustment factor, which is equal to the ratio of the net convertible income of a specific bank to the average value of the indicator. A similar scheme for constructing the forecast model is used for the indicator net income of banks from loan and deposit operations in foreign currency. However, as a result, the author concludes that in both variants of the proposed final models for predicting the overall financial result from the bank's activities, there are quite significant deviations, i.e. theoretical values when testing a model for significance exceed empirical ones. At the same time, the model is focused on assessing the activities of individual banks, and not the banking system as a whole. One of the first attempts to construct a factor-second model for forecasting the profit of the Russian banking sector was proposed by M.E. Mamonov [20]. In the work of O.V. Radeva [21], diffuse indices of the Bank of Russia were used to predict the vol- umes of the corporate and retail portfolios of the domestic banking system. D.V. Shimanovsky, in his works [13, 24], devoted to the methods of forecasting the total net interest income using the example of the banking system of the Russian Federa- tion, proposes to include in the forecast scenarios corresponding to the most likely shocks for the domestic economy. The econometric model proposed by the author allows for short-term forecasting of the dynamics of the net interest income of the banking system, based on which the study concludes that the main exogenous sources of the dynamics of net interest income are exchange rate fluctuations. In his research, the author analyzes the macro-financial modelling of trends in the national banking system, considered in the domestic scientific literature and notes that the works de- voted to forecasting the indicators of the banking system of Russia are somewhat scattered and do not take into account the previous experience. The works of foreign authors have a much wider variety of approaches to modelling and forecasting banking indicators. Publication [25] considers a vector autoregression model or VAR model with four variables: the rate of GDP growth and the volume of the loan portfolio of commercial banks, the rate of federal funds and supply in the credit market. The authors of the article [26] analyze panel data using estimates of both demand and supply of credit resources. The work [27] proposes a forecasting approach for multiple yield curves that use the characteristics of modern interest rate markets, represented by cross-tenor dependencies. The article by S.V. Shchurina and M.A. Vorobyeva [28] considers the importance of financial forecasting for ensuring the activities of banks. In it, the authors note that the development of econometric research contributes to forecasting various statistical indicators of banks. Analyzing the existing research methods of various aspects of banking, the author of publication [29] concludes that even the most avant-garde re- search in this area using complex economic and mathematical methods requires fur- ther refinement. Furthermore, there are problems related to external risks, credit risks and liquidity risks [30-34]. It would be advisable to take them into account in fore- casting models, but this is associated with some difficulties, in particular, the fact that it is hard to separate factors of credit risk and liquidity risk using purely statistical methods [35]. A.A. Shirov in his work [36] draws attention to the fact that one should not overesti- mate the capabilities of the forecasting and analytical tools. Strictly speaking, no econometric model can serve as a criterion for the correctness of certain actions and inactions. Rather, the task of economic modeling is to transform theses into a more or less coherent set of arguments supported by quantitative estimates. The problems of constructing predictive models, assessing their quality, adequacy and accuracy are the subject of discussion, primarily in foreign literature [37-42], where their open discus- sion contributes to improving models and eliminating shortcomings. In Russian prac- tice, the methodology of most predictive models remains closed, and therefore, as noted in [43], a comparative analysis of their quality is difficult or impossible. 3 Methodological framework, data and model specifications The main forecasting methods include trend extrapolation methods, methods of analy- sis of cause-and-effect relationships and their modelling [44]. The seasonality index is most commonly used to measure seasonal fluctuations, the calculation procedure of which depends on the type of dynamic series [45]. Graphic analysis of changes in quarterly values of the net income for the period from the first quarter of 2010 to the first quarter of 2018 attests to trend and seasonal com- ponent (see Fig. 1). There is a steady recurring increase in net income in the second quarter and a decrease in its value in the fourth quarter of each year. 70 000,0 Net income, mln. US dollars 60 000,0 50 000,0 40 000,0 30 000,0 20 000,0 2016Q1 2016Q3 2010Q1 2010Q3 2011Q1 2011Q3 2012Q1 2012Q3 2013Q1 2013Q3 2014Q1 2014Q3 2015Q1 2015Q3 2017Q1 2017Q3 2018Q1 Fig. 1. Dynamics of US interest income for the period from the first quarter of 2010 to the first quarter of 2018 The amplitude of seasonal fluctuations during the period under consideration increas- es; therefore, a model with multiplicative seasonality is best suited for modelling the trend of net income (see Table 1). Table 1. Multiplicative model of net income dynamicsfor the period from the first quarter of 2010 to the first quarter of 2018. Net income, Moving Seasonal Seasonal Dezonalized Period mln. average component index row US dollars 2010Q1 25302,25 - - 101,1 25022,65 2010Q2 33029,05 - - 104,1 31738,40 2010Q3 28475,05 30035,14 94,8 101,7 27996,47 2010Q4 33334,20 34553,34 96,5 93,1 35802,17 2011Q1 43375,05 37307,18 116,3 101,1 42895,74 2011Q2 44044,41 42774,55 103,0 104,1 42323,33 2011Q3 50344,54 44031,68 114,3 101,7 49498,39 2011Q4 38362,70 45649,64 84,0 93,1 41202,97 2012Q1 49846,92 46767,31 106,6 101,1 49296,09 2012Q2 48515,07 47612,64 101,9 104,1 46619,29 2012Q3 53725,85 49763,55 108,0 101,7 52822,87 2012Q4 46966,36 52009,04 90,3 93,1 50443,62 2013Q1 58828,90 55120,26 106,7 101,1 58178,82 2013Q2 60959,93 54528,19 111,8 104,1 58577,85 2013Q3 51357,57 56935,90 90,2 101,7 50494,40 2013Q4 56597,21 55806,92 101,4 93,1 60787,51 2014Q1 54312,97 54968,43 98,8 101,1 53712,79 2014Q2 57605,99 56071,42 102,7 104,1 55354,97 2014Q3 55769,50 54687,76 102,0 101,7 54832,17 2014Q4 51062,59 55525,95 92,0 93,1 54843,12 2015Q1 57665,71 56535,41 102,0 101,1 57028,48 2015Q2 61643,84 57181,63 107,8 104,1 59235,04 2015Q3 58354,37 58705,67 99,4 101,7 57373,60 2015Q4 57158,77 58444,96 97,8 93,1 61390,64 2016Q1 56622,85 58991,91 96,0 101,1 55997,14 Net income, Moving Seasonal Seasonal Dezonalized Period mln. average component index row US dollars 2016Q2 63831,64 60745,07 105,1 104,1 61337,35 2016Q3 65367,01 61936,23 105,5 101,7 64268,38 2016Q4 61923,41 63253,46 97,9 93,1 66508,04 2017Q1 61891,80 64788,00 95,5 101,1 61207,87 2017Q2 69969,76 65766,47 106,4 104,1 67235,61 2017Q3 69280,92 65345,67 106,0 101,7 68116,51 2017Q4 60240,21 67536,20 89,2 93,1 64700,22 2018Q1 70653,89 - - 101,1 69873,13 Source: For the calculations, the authors used Statistica 10.0. Fig. 2 presents the trend and seasonal component of the multiplicative model of net income. 80000,00 106,0 104,0 70000,00 102,0 60000,00 Net income, mln. US dollars 100,0 Seasonal index, % 50000,00 98,0 40000,00 96,0 94,0 30000,00 92,0 20000,00 90,0 10000,00 88,0 Tren Seasonal component 0,00 86,0 d 2010Q1 2011Q1 2011Q3 2012Q1 2012Q3 2013Q1 2013Q3 2014Q1 2014Q3 2015Q1 2015Q3 2016Q1 2016Q3 2017Q1 2017Q3 2018Q1 2010Q3 Тренд Сезонная компонента Fig. 2. Components of the multiplicative net income model: trend and seasonality indices The quadratic trend was best suited to describe the trend line of net income: Both the parabola equation itself and its parameters were statistically significant. The verification of the model adequacy involved testing for independence, normality and randomness of the distribution of the residual component. The equation and its parameters are statistically significant. The model is adequate. The Shapiro-Wilk test showed that the remnants of the model are distributed normal- ly. The Durbin-Watson test confirmed the independence of the residues. Using the turning peak criterion was revealed the randomness of the distribution of the model residuals. The values of the obtained seasonality indices (see Fig.3). Seasonalкомпонента Сезонная component 106,0 106,0 104,1 104,1 104,0 104,0 101,7 101,7 102,0 102,0 101,1 101,1 сезонности, % 100,0 100,0 Seasonal index, % 98,0 98,0 96,0 96,0 Индексы 94,0 94,0 92,0 92,0 93,1 93,1 90,0 90,0 88,0 88,0 86,0 86,0 I quarter квартал I IIquarter кварталII IIIquarter кварталIII IVquarter кварталIV Fig. 3. Seasonality indices 4 Empirical results and discussion The results of net income forecasting using a multiplicative trend-seasonal model are presented in the following Table 2. Table 2. Net income forecasting results usingmultiplicative trend-seasonal model Estimated values of interest Estimated values of interest income, with an account Period income from the parabolic to seasonality, mln. US Dol- model, mln. US Dollars lars 2010Q1 30668,53 31011,22 2010Q2 32751,93 34083,79 2010Q3 34770,20 35364,56 2010Q4 36723,33 34191,84 2011Q1 38611,32 39042,77 2011Q2 40434,18 42078,44 2011Q3 42191,91 42913,14 2011Q4 43884,51 40859,37 2012Q1 45511,96 46020,52 2012Q2 47074,29 48988,57 2012Q3 48571,48 49401,76 2012Q4 50003,54 46556,59 2013Q1 51370,46 51944,47 2013Q2 52672,25 54814,17 2013Q3 53908,90 54830,42 2013Q4 55080,42 51283,51 2014Q1 56186,81 56814,64 2014Q2 57228,06 59555,24 2014Q3 58204,18 59199,12 2014Q4 59115,16 55040,12 2015Q1 59961,01 60631,01 2015Q2 60741,72 63211,79 2015Q3 61457,30 62507,86 2015Q4 62107,75 57826,41 2016Q1 62693,06 63393,59 2016Q2 63213,24 65783,81 2016Q3 63668,28 64756,63 2016Q4 64058,19 59642,41 Estimated values of interest Estimated values of interest income, with an account Period income from the parabolic to seasonality, mln. US Dol- model, mln. US Dollars lars 2017Q1 64382,97 65102,38 2017Q2 64642,61 67271,30 2017Q3 64837,12 65945,44 2017Q4 64966,49 60488,09 2018Q1 65030,73 65757,38 2018Q2 65029,83 67674,27 2018Q3 64963,80 66074,29 2018Q4 64832,64 60363,47 2019Q1 64636,34 65358,59 70 000,0 Net income, mln. US dollars 60 000,0 50 000,0 40 000,0 30 000,0 20 000,0 2010Q1 2010Q3 2011Q1 2011Q3 2012Q1 2012Q3 2013Q1 2013Q3 2014Q1 2014Q3 2015Q1 2015Q3 2016Q1 2016Q3 2017Q1 2017Q3 2018Q1 2019Q1 2018Q3 Исходные данные Прогноз Source: For the calculations, the authors used Statistica 10.0. Fig. 4. The forecast of net income using a multiplicative trend-seasonal model The forecast accuracy was calculated using the mean absolute percentage er- ror (MAPE). The error value was 6.6%, which indicates its high accuracy. According to the completed forecast, there can be expected an increase in net income in the second quarter of 2018 compared with the first quarter by 1916.89 million US dollars (or 2.9%), a decrease in the third and fourth quarters by 1599, 98 and 5710.83 million US dollars, respectively, then again growth in the first quarter of 2019 to 4995.12 million US dollars (or 8.3%). In the first quarter of 2019, net income will be 65,358.59 million US dollars. Net income is made from the sum of net interest income and noninterest in- come less interest expenses and reserves. The nature of the dynamics of net interest and noninterest income for the period from the first quarter of 2010 to the first quarter of 2018 varies greatly (see Fig. 5). Net interest income, mln. US dollars 140 000,0 130 000,0 120 000,0 110 000,0 100 000,0 90 000,0 80 000,0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 Noninterest income, mln. US dollars 75 000,0 70 000,0 65 000,0 60 000,0 55 000,0 50 000,0 45 000,0 40 000,0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 Fig. 5. Dynamics of net interest and noninterest income for the period from the first quarter of 2010 to the first quarter of 2018 In the dynamics of net interest income, there is a trend; there are no seasonal fluctuations. The quadratic trend was best suited to describe the trend line. Both the equation itself and its parameters are statis- tically significant (see Fig. 6). Regression statistics Multiple R 0,961 R Square 0,923 Adjusted R Square 0,917 Standard Error 2360 Observations 33 ANOVA Signifi- df SS MS F cance F Regres- sion 2 1990675820 995337910 178,7 0,0000 Residual 30 167074198 5569140 Total 32 2157750018 Equation coefficients Coeffi- t P- Upper cients Standard Error Stat value Lower 95% 95% Y- inter- 8 cept 117559,0 1311,1 9,7 0,0000 114881,4 120236,5 Variable X - 1 -1947,9 177,8 11,0 0,0000 -2311,0 -1584,8 Variable X 1 2 72,9 5,1 4,4 0,0000 62,5 83,2 Fig. 6. Characteristics of the statistical significance of the parabola equation for de- scribing the net interest income trend Verification of the independence, normal distribution and randomness of the residual component of the model showed that the model is adequate and can be used for forecasting. In contrast to the dynamics of net interest income, there was no trend in the dynamics of noninterest income, the levels of the series are independent (see Fig. 7). Час тная автокорреляционная ф ункция Private autocorrelation function Lag Corr. S.E. 1 +,124 ,1741 2 +,074 ,1741 3 -,028 ,1741 4 -,027 ,1741 5 +,020 ,1741 6 -,213 ,1741 7 +,199 ,1741 8 +,065 ,1741 9 +,028 ,1741 10 -,037 ,1741 11 -,073 ,1741 12 +,035 ,1741 13 +,006 ,1741 14 -,002 ,1741 15 +,117 ,1741 0 -1,0 -0,5 0,0 0,5 1,0 Conf. Limit Fig. 7. Private autocorrelation function of noninterest income dynamics To test the hypothesis about the existence of a trend, the original series was di- vided into equal parts, and the hypotheses about equality of dispersions were tested using a two-sample F-test for dispersions (see Table 3) and equality of means using the paired two-sample t-test for means (see Table 4 ). Table 3. Two-sample F-test for the dispersion of the series "Noninterest income" Series 1 Series 2 Mean 63656 64972 Variance 10868264 5079056 Observation 16 16 df 15 15 F 2,140 P(F<=f) one-tail 0,076 F critical one-tail 2,403 Table 4. Paired two-sample t-test for the mean of series "Noninterest income" Series 1 Series 2 Mean 64972 63656 Variance 5079056 10868264 Observation 16 16 Pearson correlation 0,042 Hypothesized mean difference 0 df 15 t-stat 1,346 P(T<=t) one-tail 0,099 t critical one-tail 1,753 P(T<=t) two-tail 0,198 t critical two-tail 2,131 The results of hypothesis testing confirmed the initial assumption about the ab- sence of a trend in the dynamics of noninterest income. The change in net income is determined by two main indicators, the nature of the dynamics of which varies greatly, and therefore there are differences in their mod- elling and forecasting. The first factor is the net interest income, in which there is a trend, described by a quadratic trend, and the dynamics of which can be forecasted. The second factor is non-interest income, in which there is no trend; its dynamics is difficult to model and forecast. To continue the analysis, a measurement of the correlation of fluctuations of se- ries characterizing the dynamics of net income and net interest income was made. It was based on the measurement of the correlation between trend deviation (see Fig. 8). Regression Statistics Multiple R 0,992 R Square 0,984 Adjusted R Square 0,953 Standard Error 491 Observations 33 ANOVA Signifi- df SS MS F cance F Regression 1 485774272 485774272 2013,2 0,0000 Residual 32 7721419 241294 Total 33 493495692 Coeffi- Standard P- Lower Up- cients Error t-stat value 95% per 95% Variable X 1 0,989 0,022 44,87 0,0000 0,944 1,034 Fig. 8. The results of correlation and regression analysis of the dynamics of net in- come and net interest income Net income fluctuations, therefore, are almost entirely (by 98.4 percent) related to net interest income fluctuations. On average, the deviation of net income from its trend is 0.989 of the deviation of net interest income from its trend. To measure the correlation of the fluctuations in series, the differences in the levels of the two series were correlated as well. Both the regression models and the regres- sion equation coefficients, however, turned out to be statistically insignificant and not interpretable. The fact that the correlation of deviations from trends showed statisti- cally significant results indicates the correct selection of trend models and their good quality. Fluctuations in non-interest income, despite the absence of a trend, also have a statistically significant effect on net income fluctuations (see Fig. 9). Regression Statistics Multiple R 0,400 R Square 0,160 Adjusted R Square 0,133 Standard Error 3657 Observations 33 ANOVA Signifi- df SS MS F cance F Regres- sion 1 79016430 79016430 5,91 0,021 Residu- al 31 414479262 13370299 Total 32 493495692 Coeffi- Standard P- Lower Upper cients Error t-stat value 95% 95% - Y- intercept -34481,99 14198,45 -2,43 0,021 63439,93 -5524,05 Variable X 1 0,53 0,22 2,43 0,021 0,09 0,98 Fig. 9. The results of the correlation and regression analysis of the dynamics of net income and non-interest income Sixteen percent of net income fluctuations are associated with fluctuations in non- interest income. On average, the deviation of net income from its trend is 0.53 times the deviation of non-interest income from its trend. The above results suggest that the models perform reasonably well with satisfactory predictive performance. However, the problems of building predictive models in gen- eral, as well as financial results of commercial banks as a whole, based on economet- ric modelling and a system-analytical approach, taking into account the entire set of factors affecting the process of their formation, is a subject of discussion, requires further development and coverage in the economic literature. 5 Conclusions: Net income modelling and forecasting, therefore, allows to draw the following con- clusions:  To describe the net income trend, multiplicative seasonality model was best suited. In it, the trend is described by a quadratic trend , and seasonality indices 101 1 percent for the first quarter, 104.1 percent for the second quarter, 101.7 percent for the third quarter, 93.1 percent for the fourth quarter.  Net income is formed from the sum of net interest income and noninterest income. The dynamics of net interest income has a trend. It is described by the quadratic trend , with no seasonal fluctuations. There is no trend in the dynamics of noninterest income; the levels of the series are independent.  Net income fluctuations are almost entirely (by 98.4 percent) related to net interest income fluctuations. On average, the deviation of net income from its trend is 0.989 of the deviation of net interest income from its trend. The impact of noninterest in- come is reflected in the fact that the net income deviation from its trend is 0.53 times the non-interest income deviation from its.  According to the completed forecast, there can be expected an increase in net in- come in the second quarter of 2018 compared with the first quarter by 1916.89 mil- lion US dollars (or 2.9 percent), a decrease in the third and fourth quarters by 1599, 98 and 5710.83 million US dollars, respectively, then again growth in the first quarter of 2019 to 4995.12 million US dollars (or 8.3 percent). 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