doi: 10.3934/dcdsb.2021116

Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs

1. 

Polytechnic University of Marche, Department of Management, Piazzale R. Martelli 8, 60121, Ancona (AN), Italy

2. 

University of Modena and Reggio Emilia, Department of Economics Marco Biagi, Via Jacopo Berengario 51, 41121, Modena, Italy

* Corresponding author: giovanni.campisi@unimore.it

Received  October 2020 Revised  February 2021 Early access  April 2021

It is well known that regulation and efficiency are two important issues on banking literature. The goal of the paper is to analyse them through a banking duopoly model with heterogeneous expectations. To this purpose, we consider two scenarios. In the first one, we focus on regulation effects. In particular, empirical literature on Italian banks finds evidence on the asymmetry of the costs of regulation that penalize small banks with respect to the large ones. In this direction, we analyse a duopoly model where small banks and large banks have different forecasting rules and we capture the differences of the regulations' effects assuming asymmetry in the cost functions. We introduce linear cost function for small banks and quadratic cost function for large banks. In the second scenario, we study the relation between regulation and bank efficiency highlighting empirical results showing that large banks register higher level of inefficiency than small banks. Moreover, in order to stress new evidences and to confirm empirical results on banking regulation and efficiency, we conduct an analytical and numerical analysis.

Citation: Serena Brianzoni, Giovanni Campisi. Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021116
References:
[1]

H. N. Agiza and A. A. Elsadany, Nonlinear dynamics in the Cournot duopoly game with heterogeneous players, Physica A: Statistical Mechanics and its Applications, 320 (2003), 512-524.  doi: 10.1016/S0378-4371(02)01648-5.  Google Scholar

[2]

A. AgliariC. Chiarella and L. Gardini, A re-evaluation of adaptive expectations in light of global nonlinear dynamic analysis, Journal of Economic Behavior & Organization, 60 (2006), 526-552.  doi: 10.1016/j.jebo.2004.07.006.  Google Scholar

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F. Aiello and G. Bonanno, Profit and cost efficiency in the italian banking industry (2006-2011), Economics and Business Letters, 2 (2013), 190-205.  doi: 10.17811/ebl.2.4.2013.190-205.  Google Scholar

[4]

P. Alessandrini, M. Croci and A. Zazzaro, The geography of banking power: The role of functional distance, in The Banks and the Italian Economy, Springer, 2009, 93-123. Google Scholar

[5]

P. Alessandrini, M. Fratianni, L. Papi and A. Zazzaro, The asymmetric burden of regulation: Will local banks survive, Money and Finance Research group (Mo. Fi. R. ) Working Papers No, 125, 1-19. Google Scholar

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P. Alessandrini and L. Papi, L'impatto della "bolla regolamentare" sulle banche: Alcune valutazioni, Money and Finance Research group (Mo. Fi. R. ) Working Papers No, 150, 1-35. Google Scholar

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A. N. BergerA. SaundersJ. M. Scalise and G. F. Udell, The effects of bank mergers and acquisitions on small business lending, Journal of Financial Economics, 50 (1998), 187-229.   Google Scholar

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G. -I. Bischi, C. Chiarella, M. Kopel, F. Szidarovszky et al., Nonlinear Oligopolies, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-02106-0.  Google Scholar

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G.-I. BischiL. Stefanini and L. Gardini, Synchronization, intermittency and critical curves in a duopoly game, Mathematics and Computers in Simulation, 44 (1998), 559-585.  doi: 10.1016/S0378-4754(97)00100-6.  Google Scholar

[12]

S. BrianzoniG. Campisi and A. Russo, Corruption and economic growth with non constant labor force growth, Communications in Nonlinear Science and Numerical Simulation, 58 (2018), 202-219.  doi: 10.1016/j.cnsns.2017.07.007.  Google Scholar

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[14]

S. BrianzoniC. Mammana and E. Michetti, Nonlinear dynamics in a business-cycle model with logistic population growth, Chaos, Solitons & Fractals, 40 (2009), 717-730.  doi: 10.1016/j.chaos.2007.08.041.  Google Scholar

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L. FantiL. GoriC. Mammana and E. Michetti, The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics, Chaos, Solitons & Fractals, 52 (2013), 73-86.  doi: 10.1016/j.chaos.2013.04.002.  Google Scholar

[21]

L. FantiL. Gori and M. Sodini, Nonlinear dynamics in a Cournot duopoly with relative profit delegation, Chaos, Solitons & Fractals, 45 (2012), 1469-1478.  doi: 10.1016/j.chaos.2012.08.008.  Google Scholar

[22]

C. A. Favero and L. Papi, Technical efficiency and scale efficiency in the Italian banking sector: a non-parametric approach, Applied Economics, 27 (1995), 385-395.  doi: 10.1080/00036849500000123.  Google Scholar

[23]

L. GardiniF. TramontanaV. Avrutin and M. Schanz, Border-collision bifurcations in 1d piecewise-linear maps and Leonov's approach, International Journal of Bifurcation and Chaos, 20 (2010), 3085-3104.  doi: 10.1142/S021812741002757X.  Google Scholar

[24]

A. Giannola, A. Lopes, C. Ricci and G. Scarfiglieri, Divari territoriali ed efficienza del sistema bancario italiano, Quintieri B. (a cura di), Finanza Istituzioni e Sviluppo Regionale, il Mulino, Bologna, 219-253. Google Scholar

[25]

L. Giordano and A. Lopes, Preferenza al rischio e qualità degli impieghi come determinanti dell'efficienza del sistema bancario italiano, Giannola, A. (a cura di), Riforme istituzionali e Mutamento Strutturale, Mercati, imprese e istituzioni in un sistema dualisticoi, Carocci, Roma. Google Scholar

[26]

L. Giordano and A. Lopes, Le banche italiane tra consolidamento e recuperi di efficienza (1999-2008). una promessa mantenuta?, Giannola, A. Lopes, A. Sarno, D. (a cura di), I Problemi Dello Sviluppo Economico e del suo Finanziamento Nelle Aree Deboli, Carocci, Roma, 169-202. Google Scholar

[27]

C. GirardoneP. Molyneux and E. P. Gardener, Analysing the determinants of bank efficiency: The case of Italian banks, Applied Economics, 36 (2004), 215-227.  doi: 10.1080/0003684042000175334.  Google Scholar

[28]

M. A. Klein, A theory of the banking firm, Journal of Money, Credit and Banking, 3 (1971), 205-218.  doi: 10.2307/1991279.  Google Scholar

[29]

F. LamantiaM. Pezzino and F. Tramontana, Dynamic analysis of discontinuous best response with innovation, Journal of Economic Dynamics and Control, 91 (2018), 120-133.  doi: 10.1016/j.jedc.2018.01.024.  Google Scholar

[30] A. Medio and M. Lines, Nonlinear Dynamics: A Primer, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511754050.  Google Scholar
[31]

M. Monti et al., Deposit, Credit and Interest Rate Determination Under Alternative Bank Objective Function, . Karl Shell, Giorgio P. Szegö (Eds. ), Mathematical methods in investment and finance, North-Holland/American Elsevier, 1972, 431-454. Google Scholar

[32]

A. Resti, Evaluating the cost-efficiency of the Italian banking system: What can be learned from the joint application of parametric and non-parametric techniques, Journal of Banking & Finance, 21 (1997), 221-250.  doi: 10.1016/S0378-4266(96)00036-2.  Google Scholar

[33]

E. Saltari and G. Travaglini, L'economia Italiana Del Nuovo Millennio, (Vol. 282). Carocci, Roma, 2009. Google Scholar

[34]

M. L. Stefani, V. Vacca, D. Coin, S. Del Prete, C. Demma, M. Galardo, I. Garri, S. Mocetti and D. Pellegrino, Le Banche Locali e il Finanziamento Dei Territori: Evidenze Per l'Italia (2007-2014), 324, Banca d'Italia, 2016. Google Scholar

[35]

F. Tramontana, Heterogeneous duopoly with isoelastic demand function, Economic Modelling, 27 (2010), 350-357.  doi: 10.1016/j.econmod.2009.09.014.  Google Scholar

[36]

F. Tramontana, L. Gardini, V. Avrutin and M. Schanz, Period adding in piecewise linear maps with two discontinuities, International Journal of Bifurcation and Chaos, 22 (2012), 1250068, 1-30. doi: 10.1142/S021812741250068X.  Google Scholar

[37]

F. Tramontana, L. Gardini, R. Dieci and F. Westerhoff, The emergence of bull and bear dynamics in a nonlinear model of interacting markets, Discrete Dynamics in Nature and Society, 2009 (2009), Art. ID 310471, 1-30. doi: 10.1155/2009/310471.  Google Scholar

[38]

F. TramontanaF. Westerhoff and L. Gardini, On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders, Journal of Economic Behavior & Organization, 74 (2010), 187-205.  doi: 10.1016/j.jebo.2010.02.008.  Google Scholar

[39]

F. TramontanaF. Westerhoff and L. Gardini, A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities, Mathematics and Computers in Simulation, 108 (2015), 16-40.  doi: 10.1016/j.matcom.2013.06.002.  Google Scholar

show all references

References:
[1]

H. N. Agiza and A. A. Elsadany, Nonlinear dynamics in the Cournot duopoly game with heterogeneous players, Physica A: Statistical Mechanics and its Applications, 320 (2003), 512-524.  doi: 10.1016/S0378-4371(02)01648-5.  Google Scholar

[2]

A. AgliariC. Chiarella and L. Gardini, A re-evaluation of adaptive expectations in light of global nonlinear dynamic analysis, Journal of Economic Behavior & Organization, 60 (2006), 526-552.  doi: 10.1016/j.jebo.2004.07.006.  Google Scholar

[3]

F. Aiello and G. Bonanno, Profit and cost efficiency in the italian banking industry (2006-2011), Economics and Business Letters, 2 (2013), 190-205.  doi: 10.17811/ebl.2.4.2013.190-205.  Google Scholar

[4]

P. Alessandrini, M. Croci and A. Zazzaro, The geography of banking power: The role of functional distance, in The Banks and the Italian Economy, Springer, 2009, 93-123. Google Scholar

[5]

P. Alessandrini, M. Fratianni, L. Papi and A. Zazzaro, The asymmetric burden of regulation: Will local banks survive, Money and Finance Research group (Mo. Fi. R. ) Working Papers No, 125, 1-19. Google Scholar

[6]

P. Alessandrini and L. Papi, Banche locali e piccole imprese dopo la crisi tra nuove regole e innovazioni digitali, Money and Finance Research group (Mo. Fi. R. ) Working Papers No, 148, 1-25. Google Scholar

[7]

P. Alessandrini and L. Papi, L'impatto della "bolla regolamentare" sulle banche: Alcune valutazioni, Money and Finance Research group (Mo. Fi. R. ) Working Papers No, 150, 1-35. Google Scholar

[8]

A. N. BergerA. SaundersJ. M. Scalise and G. F. Udell, The effects of bank mergers and acquisitions on small business lending, Journal of Financial Economics, 50 (1998), 187-229.   Google Scholar

[9]

G. -I. Bischi, C. Chiarella, M. Kopel, F. Szidarovszky et al., Nonlinear Oligopolies, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-02106-0.  Google Scholar

[10]

G. I. Bischi and F. Lamantia, Coexisting attractors and complex basins in discrete-time economic models, in Nonlinear Dynamical Systems in Economics, Springer, 2005, 187-231. doi: 10.1007/3-211-38043-4_7.  Google Scholar

[11]

G.-I. BischiL. Stefanini and L. Gardini, Synchronization, intermittency and critical curves in a duopoly game, Mathematics and Computers in Simulation, 44 (1998), 559-585.  doi: 10.1016/S0378-4754(97)00100-6.  Google Scholar

[12]

S. BrianzoniG. Campisi and A. Russo, Corruption and economic growth with non constant labor force growth, Communications in Nonlinear Science and Numerical Simulation, 58 (2018), 202-219.  doi: 10.1016/j.cnsns.2017.07.007.  Google Scholar

[13]

S. BrianzoniL. Gori and E. Michetti, Dynamics of a Bertrand duopoly with differentiated products and nonlinear costs: Analysis, comparisons and new evidences, Chaos, Solitons & Fractals, 79 (2015), 191-203.  doi: 10.1016/j.chaos.2015.05.014.  Google Scholar

[14]

S. BrianzoniC. Mammana and E. Michetti, Nonlinear dynamics in a business-cycle model with logistic population growth, Chaos, Solitons & Fractals, 40 (2009), 717-730.  doi: 10.1016/j.chaos.2007.08.041.  Google Scholar

[15]

S. Brianzoni, C. Mammana and E. Michetti, Local and global dynamics in a discrete time growth model with nonconcave production function, Discrete Dynamics in Nature and Society, 2012 (2012), Art. ID 536570, 1-22. doi: 10.1155/2012/536570.  Google Scholar

[16]

E. Dia and M. Giuliodori, Portfolio separation and the dynamics of bank interest rates, Scottish Journal of Political Economy, 59 (2012), 28-46.  doi: 10.1111/j.1467-9485.2011.0567.x.  Google Scholar

[17]

R. Dieci and F. Westerhoff, Heterogeneous speculators, endogenous fluctuations and interacting markets: A model of stock prices and exchange rates, Journal of Economic Dynamics and Control, 34 (2010), 743-764.  doi: 10.1016/j.jedc.2009.11.002.  Google Scholar

[18]

G. FagioloD. Giachini and A. Roventini, Innovation, finance, and economic growth: An agent-based approach, Journal of Economic Interaction and Coordination, 15 (2020), 703-736.  doi: 10.1007/s11403-019-00258-1.  Google Scholar

[19]

L. Fanti, The dynamics of a banking duopoly with capital regulations, Economic Modelling, 37 (2014), 340-349.  doi: 10.1016/j.econmod.2013.11.010.  Google Scholar

[20]

L. FantiL. GoriC. Mammana and E. Michetti, The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics, Chaos, Solitons & Fractals, 52 (2013), 73-86.  doi: 10.1016/j.chaos.2013.04.002.  Google Scholar

[21]

L. FantiL. Gori and M. Sodini, Nonlinear dynamics in a Cournot duopoly with relative profit delegation, Chaos, Solitons & Fractals, 45 (2012), 1469-1478.  doi: 10.1016/j.chaos.2012.08.008.  Google Scholar

[22]

C. A. Favero and L. Papi, Technical efficiency and scale efficiency in the Italian banking sector: a non-parametric approach, Applied Economics, 27 (1995), 385-395.  doi: 10.1080/00036849500000123.  Google Scholar

[23]

L. GardiniF. TramontanaV. Avrutin and M. Schanz, Border-collision bifurcations in 1d piecewise-linear maps and Leonov's approach, International Journal of Bifurcation and Chaos, 20 (2010), 3085-3104.  doi: 10.1142/S021812741002757X.  Google Scholar

[24]

A. Giannola, A. Lopes, C. Ricci and G. Scarfiglieri, Divari territoriali ed efficienza del sistema bancario italiano, Quintieri B. (a cura di), Finanza Istituzioni e Sviluppo Regionale, il Mulino, Bologna, 219-253. Google Scholar

[25]

L. Giordano and A. Lopes, Preferenza al rischio e qualità degli impieghi come determinanti dell'efficienza del sistema bancario italiano, Giannola, A. (a cura di), Riforme istituzionali e Mutamento Strutturale, Mercati, imprese e istituzioni in un sistema dualisticoi, Carocci, Roma. Google Scholar

[26]

L. Giordano and A. Lopes, Le banche italiane tra consolidamento e recuperi di efficienza (1999-2008). una promessa mantenuta?, Giannola, A. Lopes, A. Sarno, D. (a cura di), I Problemi Dello Sviluppo Economico e del suo Finanziamento Nelle Aree Deboli, Carocci, Roma, 169-202. Google Scholar

[27]

C. GirardoneP. Molyneux and E. P. Gardener, Analysing the determinants of bank efficiency: The case of Italian banks, Applied Economics, 36 (2004), 215-227.  doi: 10.1080/0003684042000175334.  Google Scholar

[28]

M. A. Klein, A theory of the banking firm, Journal of Money, Credit and Banking, 3 (1971), 205-218.  doi: 10.2307/1991279.  Google Scholar

[29]

F. LamantiaM. Pezzino and F. Tramontana, Dynamic analysis of discontinuous best response with innovation, Journal of Economic Dynamics and Control, 91 (2018), 120-133.  doi: 10.1016/j.jedc.2018.01.024.  Google Scholar

[30] A. Medio and M. Lines, Nonlinear Dynamics: A Primer, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511754050.  Google Scholar
[31]

M. Monti et al., Deposit, Credit and Interest Rate Determination Under Alternative Bank Objective Function, . Karl Shell, Giorgio P. Szegö (Eds. ), Mathematical methods in investment and finance, North-Holland/American Elsevier, 1972, 431-454. Google Scholar

[32]

A. Resti, Evaluating the cost-efficiency of the Italian banking system: What can be learned from the joint application of parametric and non-parametric techniques, Journal of Banking & Finance, 21 (1997), 221-250.  doi: 10.1016/S0378-4266(96)00036-2.  Google Scholar

[33]

E. Saltari and G. Travaglini, L'economia Italiana Del Nuovo Millennio, (Vol. 282). Carocci, Roma, 2009. Google Scholar

[34]

M. L. Stefani, V. Vacca, D. Coin, S. Del Prete, C. Demma, M. Galardo, I. Garri, S. Mocetti and D. Pellegrino, Le Banche Locali e il Finanziamento Dei Territori: Evidenze Per l'Italia (2007-2014), 324, Banca d'Italia, 2016. Google Scholar

[35]

F. Tramontana, Heterogeneous duopoly with isoelastic demand function, Economic Modelling, 27 (2010), 350-357.  doi: 10.1016/j.econmod.2009.09.014.  Google Scholar

[36]

F. Tramontana, L. Gardini, V. Avrutin and M. Schanz, Period adding in piecewise linear maps with two discontinuities, International Journal of Bifurcation and Chaos, 22 (2012), 1250068, 1-30. doi: 10.1142/S021812741250068X.  Google Scholar

[37]

F. Tramontana, L. Gardini, R. Dieci and F. Westerhoff, The emergence of bull and bear dynamics in a nonlinear model of interacting markets, Discrete Dynamics in Nature and Society, 2009 (2009), Art. ID 310471, 1-30. doi: 10.1155/2009/310471.  Google Scholar

[38]

F. TramontanaF. Westerhoff and L. Gardini, On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders, Journal of Economic Behavior & Organization, 74 (2010), 187-205.  doi: 10.1016/j.jebo.2010.02.008.  Google Scholar

[39]

F. TramontanaF. Westerhoff and L. Gardini, A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities, Mathematics and Computers in Simulation, 108 (2015), 16-40.  doi: 10.1016/j.matcom.2013.06.002.  Google Scholar

Figure 1.  In the $ (c_1,c_2)- $plane the set of points between the red and blue curve ensures positive equilibrium loans, while the yellow curve represents the parameter values $ (c_1,c_2) $ for which a flip bifurcation occurs. In Panel (a) the parameters are: $ a = 3.2 $, $ \alpha = 1.5 $, $ b = 1.5 $, $ \gamma = 0.7 $, $ r_k = 2.8 $. In panel (b) $ \gamma = 0.1 $ while the other parameters as in Panel (a)
Figure 2.  Two interesting economic scenarios through bifurcation diagrams. In Panel (a) the parameters are: $ a = 3 $, $ \alpha = 1.38 $, $ b = 0.12 $, $ \gamma = 0.2 $, $ r_k = 2.8 $, $ c_{1} = 0.15 $ and $ c_{2}\in (0.01,0.6) $. In Panel (b) $ c_{1} = 0.1 $ and $ c_{2}\in (0.01,0.6) $ while the other parameters as in panel (a)
Figure 3.  On the left is depicted the attractor of the 4-cycle for $ \alpha = 1.25 $, $ \gamma = 0.35 $, $ r_{k} = 2.8 $, $ a = 3.2 $, $ b = 0.1 $, $ c_{1} = 0.08 $, $ c_{2} = 0.7 $. On the right panel a bifurcation diagram for $ c_{1}\in (1.4,2.1) $ showing the period-doubling of the 2-cycle for $ \alpha = 2 $, $ \gamma = 0.25 $, $ r_{k} = 2.3 $, $ a = 3.2 $, $ b = 0.4 $, $ c_{2} = 1.1 $
Figure 4.  On the left, basins of attraction of the two-cycle, given the following values of the parameters, $ \alpha = 1.35 $, $ \gamma = 0.7 $, $ r_{k} = 1.5 $, $ a = 3 $, $ b = 1.9 $, $ c_{1} = 0.01 $, $ c_{2} = 0.27 $. On the right, basins of attraction showing the birth and the stability of a 4-cycle for $ a = 3.27 $ and the other parameters as in the left panel
Figure 5.  In (a), four pieces chaotic attractor for $ \alpha = 1.38 $, $ \gamma = 0.35 $, $ r_{k} = 2.8 $, $ a = 3 $, $ b = 0.12 $, $ c_{1} = 0.13 $, $ c_{2} = 0.78 $. In (b), two pieces chaotic attractor for $ c_{2} = 0.8 $ and the other parameters as in Panel (a)
Figure 6.  In (a), connected chaotic attractor for $ c_{2} = 0.84 $ and the other parameters as in Panel (a) of Figure 5. On the right, cycle cartogram in the ($ c_{1},c_{2} $)-plane for parameter values as in Panel (a) of Figure 5
Figure 7.  The time series of the loans of large (Panel (a)) and small (Panel (b)) Italian banks. The real time series are depicted in blue, while in red the time series of the loans resulting from the stochastic model
Figure 8.  Monte Carlo simulations for analyse differences in loans' demand of the two banks for several combinations of costs and regulation parameters. In (a), $ c_{1},c_{2}\in [0.01,0.4] $ and the other parameters as in Table 1. In (b) and (c), joint effect of the costs and regulation keeping fixed all other parameters as in Table 1. In (b), the loans' demand are plotted with respect to the costs of the two banks when $ c_{1},c_{2}\in [0.01,0.3] $ and $ \gamma\in[0.05,0.4] $. In (c) the loans' demand are plotted with respect to the parameter of regulation $ \gamma $ when $ c_{1},c_{2}\in [0.01,0.3] $ and $ \gamma\in[0.05,0.4] $. In (d), the effects of regulation on the loans' demand when $ \gamma\in [0.05,0.4] $ and the other parameters as in Table 1
Table 1.  Parameter setting and initial values
$ \alpha $ $ \gamma $ $ r_{k} $ $ a $ $ b $ $ c_{1} $ $ c_{2} $ $ \sigma_{1} $ $ \sigma_{1} $
1.04 0.18 3.6 1.68 0.043 0.08 0.38 0.12 0.06
$ \alpha $ $ \gamma $ $ r_{k} $ $ a $ $ b $ $ c_{1} $ $ c_{2} $ $ \sigma_{1} $ $ \sigma_{1} $
1.04 0.18 3.6 1.68 0.043 0.08 0.38 0.12 0.06
Table 2.  Summary statistics of banks' loans including mean, standard deviation (sd), skewness, minimum and maximum value for real time series of the loans of large banks ($ L_{1}^{real} $), real time series of the loans of small banks ($ L_{2}^{real} $) and the simulated stochastic time series of the loans for large ($ L_{1}^{SS} $) and small ($ L_{2}^{SS} $) banks
Mean sd Min Max Skewness Kurtosis
$ L_{1}^{real} $ 291, 440, 000 64, 858, 000 192, 020, 000 356, 490, 000 -0.4471 1.6507
$ L_{2}^{real} $ 224, 430, 000 25, 598, 000 186, 240, 000 263, 480, 000 -0.0867 1.7266
$ L_{1}^{SS} $ 345, 330, 000 7, 315, 000 331, 320, 000 355, 280, 000 -0.4075 2.2085
$ L_{2}^{SS} $ 224, 080, 000 3, 652, 000 216, 660, 000 230, 890, 000 -0.1410 2.9930
Mean sd Min Max Skewness Kurtosis
$ L_{1}^{real} $ 291, 440, 000 64, 858, 000 192, 020, 000 356, 490, 000 -0.4471 1.6507
$ L_{2}^{real} $ 224, 430, 000 25, 598, 000 186, 240, 000 263, 480, 000 -0.0867 1.7266
$ L_{1}^{SS} $ 345, 330, 000 7, 315, 000 331, 320, 000 355, 280, 000 -0.4075 2.2085
$ L_{2}^{SS} $ 224, 080, 000 3, 652, 000 216, 660, 000 230, 890, 000 -0.1410 2.9930
[1]

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