=Paper=
{{Paper
|id=None
|storemode=property
|title=Mathematical Model of Banking Firm as Tool for Analysis, Management and Learning
|pdfUrl=https://ceur-ws.org/Vol-1000/ICTERI-2013-p-401-408-ITER.pdf
|volume=Vol-1000
|dblpUrl=https://dblp.org/rec/conf/icteri/SelyutinR13
}}
==Mathematical Model of Banking Firm as Tool for Analysis, Management and Learning==
https://ceur-ws.org/Vol-1000/ICTERI-2013-p-401-408-ITER.pdf
Mathematical Model of Banking Firm as Tool
for Analysis, Management and Learning
Victor Selyutin1,2 and Margarita Rudenko1
1
Research Institute of Mechanics and Applied Mathematics,
vvs1812@gmail.com, ritusik@mail.ru
2
Economic Faculty of Southern Federal University
220/1, av. Stachki, 344090, Rostov-on-Don, Russia
Abstract. An essential concern for banking firms is the problem of assets and
liabilities managing (ALM). Over last years a lot of model tools were offered
for solving this problem. We offer the novel approach to ALM based on trans-
port equations for loan and deposit dynamics. Given the bank's initial state, and
various deposit inflow scenarios the model allows provide simulations includ-
ing stress-testing, and can be used for assessment of liquidity risk, for examine
loan issue decisions to choose reasonable solution, and in the learning purposes.
Keywords. Asset- liability management, Differential equations, Liquidity risk,
Duration
Key terms. Banking, Mathematical Modelling, Decision making
1 Introduction
A banking firm is rather a complex system within the context of management prob-
lem. It is caused by a considerable number of financial flows and the funds, having a
various origin and differing by dynamic and probabilistic characteristics, and at the
same time forming the unified system. Stable functioning of the system is provided
due to hierarchy, external (prudential supervision) and internal regulators and restric-
tions, and feedbacks.
Among the mathematical models of banking firms it is possible to separate two ba-
sic groups. There are models of optimization of assets portfolio (static, single and
multi-period) using linear and dynamic programming mainly [1-2], and models of
assets and liability management (ALM), using methodology and the technique of the
stochastic differential equations [3-5].
One of the problem solved by models ALM is management of various risks (espe-
cially credit risk and interest-rate risk), including the problem of default probability
decrease.
402 V. Selyutin and M. Rudenko
In connection with computer engineering development, from the middle of 70th
years of the last century the computer models of banks focused on problems of plan-
ning and decision-making support systems began to appear. However such projects
had no further development [6-8].
Then we will turn our attention to one of the possible approaches to bank model-
ling as a dynamic system, which can be called hybrid. The main tasks which the
model developed must solve are the analysis and management of liquidity and stress-
testing of a bank. In addition, it can be used for optimization of assets profile.
Aggregation of elements of balance sheet can be varied according to the objectives
of modelling and principles developing of state variables vector. We will use the fol-
lowing simplified schematic (Tab. 1).
Fixed assets of bank we will ignore, taking into account only financial flows. Ob-
viously, balance sheet equation takes place:
A= S+B+Q+X=Y+C+M=L , (1)
where equity (capital) of a bank C is a balancing variable.
For detailed modelling of credit risks, loans issued can be divided by categories of
the debtors having various reliabilities. Division of deposits on time and demand is
necessary for calculation of instant liquidity. It is ignored in considered below version
of the model for simplicity.
Table 1. The aggregated balance sheet of commercial bank.
Assets (A) Liabilities (L)
Loans issued Business Debt (Y) Time deposits
(X): Private customers On-demand deposits
(buyer`s credits, mort- and current ac-
gage etc.) counts
Other banks Inter-bank credits (M)
Securities Shares (Q)
Bonds (B)
Reserves (S) Cash Equity, including retained profit
Rest fund, loan loss re- of last periods (C)
serves etc.
Formally it is possible to mark three groups of operations in the balance-sheet ta-
ble:
Reallocation of assets between separate items
Reallocation of liabilities between separate items
Identical change of assets and liabilities at one period
Though the bank opens a position in liabilities with grant of a loan (opening of a
credit line) at one time, from the formal point of view this operation is resolved into
reallocation of asset`s items.
Mathematical Model of Banking Firm … 403
Similarly, if the deposit remains unclaimed in maturity date it either is prolonged,
or is transferred in demand deposits (with no interest accruing or with the minimum
percentage) according to contact conditions. Actually, in this case there is a realloca-
tion of liability`s items.
At last, when interest on loans (or other types of income or expenses) are received
(or repaid), at one time it is changed both assets, and liabilities, own capital of bank
increases or decreases.
2 Model with Certain Terms of Loans and Attracted Funds
The main difficulty in modelling of assets and liabilities dynamics is concerned with
necessity taking into account terms of loans and deposits. Due to these the state vari-
ables must depend on two parameters - current time (t) and current "age" () or the
remained term to maturity (T-). That is why dynamics of the issued loans can be
described by following transport equation:
x x
u ( t , ) (2)
t
T
In addition X ( t ) x ( t , ) d - total amount loans issued,
0
T
X * ( t ) x ( t , ) e d - present value of loans, Т – term of loans.
0
Movement of time deposits is described similarly:
y y
v ( t , )
t (3)
T
Y ( t ) y ( t , ) d - total amount of time deposits,
0
T
Y * ( t ) y ( t , ) e d - present value of time deposits, T – term of deposits.
0
Variables u(t,) and v(t,) denote the flows of issued loans (temporary outflow of
financial resources of bank) and deposits (temporary inflow) distributed by time tak-
ing into account amortization (interest payment or installment credits). Accordingly,
total inputs of loans U(t) and deposits V(t) (or its present values U*(t) and V*(t)) is
described as:
T T
U (t ) u (t , ) d , U * ( t ) u ( t , ) e d
0 0
T T
V (t ) v (t , ) d , V * ( t ) v ( t , ) e d
0 0
Solution of the equations (2-3) can be represented in the closed form:
404 V. Selyutin and M. Rudenko
t
x ( t , ) u ( , t ) d ( t )
0
t
y (t , ) v ( , t ) d ( t )
0
where () and ()- initial distributions of loans and deposits by "age", or may be
obtained by use corresponding equations with finite differences.
Dynamics of reserves (S) and equity (C) is described by the equations including
stochastic members which consider random nature of change in value of shares and
possible loans losses:
dS U t V t X X Y Y B B M M Z t dt Qdt
QdWt xt dJ t
dC X X Y Y B B M M Z t dt Qdt QdW t
xT dJ t
where dWt – increment of Wiener stochastic process, dJt – increment of compound
Poisson process with exponential distributed size of jumps (loan losses), Z(t) – opera-
tion expenses and payment for dividends; xT(t) – repayment of a loans in maturity
date, X, Y, B, M – accordingly interest on loans, deposits, bonds income, cost of
credits; - average portfolio return of trading securities, - volatility of securities
portfolio.
Investments in liquid assets - shares Q(t) and bonds B(t) can be considered as some
parameters of management and to be calculated, proceeding from structure of assets
chosen or planned by bank taking into account loan demand. Similarly, the volume of
received loans M(t) can be select depending on bank`s requirement in financial re-
sources.
It is necessary to add the equations of dynamics of duration to the equations of
movement of assets and liabilities to model liquidity risk taking into account change
of interest rates
If r – the annual interest rate, so in this case Macaulay duration for an asset x(t,) is
defined by expression:
T
1
D x t T x t , e d ,
X * t 0
where = ln(1+r).
Similarly duration of another financial flows y(t,), u(t,), v(t,) are calculated. It is
possible to show that dynamics of duration is described by any of presented below the
equations which is chosen according to liquidity research tasks.
dD x U * (t )
D u (t ) 1 ( t ) D x D x
dt X * ( t )
dD x U * (t ) xT (t )
( D u (t ) D x ) 1 D x D x
dt X * (t ) X * (t )
Mathematical Model of Banking Firm … 405
dD x x (t )
( t )[ D u ( t ) D x ] 1 D u ( t ) T D x
dt X * (t )
where dX * 1
(t )
dt X *
Model (2-3) has been transformed to system of difference equations and realized as
computer program [9]. The user independently chooses one of two operating modes
of the program: calculation in case of predefined planning horizon, or calculation with
possible correction of parameters, setting physical speed of calculation.
The program is interactive as the user can change values of some key parameters in
the process of calculation, without interrupting its work. As key parameters are cho-
sen: a fraction of cash invested in various kinds of assets, revenues (interest rates), a
duration of demand deposits, credit demand, inflow of deposits, crediting scenarios
(distribution of loans by time).
Dynamics of inflows and outflows of cashes; diagram of change of durations of as-
sets and liabilities; distributions of loans and deposits, and also input flow by time are
displayed on the screen of computer.
A stress-testing is provided in the program. The user can choose the period of
stress-testing and such stresses-scenarios as decrease in inflow of deposits, decrease
in duration of deposits (the scenario of outflow of deposits); decrease in accessible
volume of attracted funds on the interbank market.
3 Model with Fixed Terms of Lending and Borrowing
Model (2-3) presented above is rather difficult in numerical realization and does not
allow to consider some important facts, for example, dependence on interest rates
from different terms of lending or borrowing. Therefore we will consider simplified
modification of previous model under supposing that terms of loans and deposits are
fixed.
It is possible to fix the most typical terms according to the classification used in the
bank reporting, in spite of the fact that terms of loans (or deposits) can be arbitrary.
Both loans and time deposits are structured by terms as follows: till 30 days, from 31
till 90 days, from 91 till 180 days, from 181 days till 1 year, from 1 year till 3 years,
over 3 years.
Accordingly, it is possible to establish several typical periods Tk for each of them
time transactions are described by the partial differential equation of the first order
x x
a( , x) (4)
t
with a boundary condition x(t,0)=u(t) and the initial condition x(0,)=(). Initial and
boundary condition should be consistent, that is u(0)=(0).
Here t - current time, 0t<, - elapsed time since the moment of settlement of
transaction ("age" of an loan or deposit), 0<T, a(,x)- value of "amortization" of an
loan or deposit (inflation, installment credit etc.).
406 V. Selyutin and M. Rudenko
Similarly (2-3), the variable x(t,) is the allocated variable characterizing some
credit tools, accounted in assets or in liabilities (loans for limited period, time depos-
its, interbank lending or borrowing, coupon bonds or other assets and liabilities with
the fixed term of repayment).
Further it will be assumed that
a ( , x ) x (5)
i.e. repayment of credits occurs proportionally to their volume with coefficient ,
which is not dependent on age. It can be used and other schemes (when credit repay-
ment begins not at once and (or) occurs in advance established equal shares.
It is easy to verify that the solution of the equation (4) looks like a travelling wave
x(t , ) u (t ) exp(x) (6)
For consistency an initial and boundary conditions at t<T it is necessary to pre-
determine u(t) on an interval t[-T,0).
From (4) - (6) follows
x(0, ) ( ) u( ) exp( ) (7)
and after replacement for -t,
u (t ) (t ) exp(t ) under -Tt<0 (8)
The total value of the considered loan (or deposit) are obtained by integration on
age
T
X ( t ) x (t , ) d
0
(9)
Substituting (6) in (9), we have
T
X ( t ) u ( t ) exp( ) d (10)
0
Integrating (4), we obtain the ordinal differential equation
dX
u (t ) X x (t , T ) u (t ) X u (t T ) exp( T ) (11)
dt
As assets with different terms of repayment are in portfolio of assets or liabilities,
so it is possible to replace scalar variable X(t) in (11) with vector. Vector`s compo-
nents are financial tools with different terms of repayment
dXk
uk (t ) k X k xk (t, Tk ) uk (t ) k X k uk (t Tk ) exp( kTk ) (12)
dt
For simplicity further we will suppose Tk=k, where k - the term, expressed in
months.
Time tools (issued loans, bonds, interbank credits, time deposits) from the mathe-
matical point of view are similar, that is why we will consider them in the context of
one and only construction, giving the general designation: Xk - to time tools in assets
and Yk – in liabilities. Then the previous model can be presented as:
dX k
uk (t ) k X k xk (t , k ) uk (t ) k X k uk (t k ) exp( k k ) (13)
dt
dSt St dt St dWt f (t )dt (14)
Mathematical Model of Banking Firm … 407
dQ dY dX dZ
k k k X k k Yk g (t ) f (t ) (15)
dt k dt k dt dt k k
dYk
vk (t ) k Yk yk (t , k ) vk (t ) k Yk vk (t k ) exp( k k ) (16)
dt
dZ Z
w(t ) (17)
dt Dz
where w(t) - inflow of on-demand deposits , vk(t) - inflow of time deposits and bor-
rowed funds; f(t) - purchase (+) or sale (-) trading securities (t/s); g(t) – operation
costs on carrying out of activities of bank; - securities portfolio return; - volatility
of securities portfolio; Wt- Wiener stochastic process; k - interest on the time deposits
and borrowed funds; k - interest on issued loans; Dz - duration (characteristic turn-
over time) on-demand deposits.
It is easily to obtain the equation of dynamics of equity by differentiation of bal-
ance equality and corresponding substitutions (13) - (17). As follows,
dC dS
k X k k Yk t f (t ) g (t ) (18)
dt k k dt
For simplicity it is supposed complete withdrawal of deposits after term in this ver-
sion of model. However it is easy to take into account possibility of prolongation of
the deposit or its transfer in category on-demand deposits. It is considered that divi-
dends are not paid.
Besides, credit risks (default risk, or a delay of payments) are not considered, that
also it is possible to take into account by entering of corresponding adjustments.
It is considered that interests on the attracted funds and the received credits are paid
according to accrual. However it is easy to set and other scheme in which interests are
accumulated on depositary accounts and are paid after term of deposit.
Let k X k X and - k Yk Y - structure of time loans and deposits.
Besides, for simplicity we will assume that there are no investments in trading se-
curities. Then dynamics of the capitals are described by the equation:
dC
X k k Y k k g (t ) , (19)
dt k k
It is giving evident representation about sensitivity of dynamics of capital to
changes of main parameters of assets and liabilities.
Main objective of shareholders and bank management is the increase in capital:
dC
max (20)
dt
subject to restrictions on financial resources and risks (credit and market, loss of li-
quidity, bankruptcy).
408 V. Selyutin and M. Rudenko
4 Conclusions
The approach to mathematical modelling of cash flow moving in asset and liability
accounts of the commercial bank based on the partial differential equations is novel
and has no analogues in the literature. At the same time, the given approach is quite
logic as reflects process of change of actives simultaneously in time and on "age".
Depending on particular theoretical or practical problems the given approach can be
realized in the various modifications, two of which are presented in the article.
As the preliminary testing has shown, the computer program created by use model
(2-3) allows provide various simulations, including stress-testing, and can be used in
the educational purposes to provide the best understanding of the dynamic processes
taking place in banking firm.
It is necessary the further development of the offered modelling approach such as
improvement of program tool and also, as required, model detailed elaboration to use
these models as part of decision support system for asset and liability management in
commercial bank. The modified model (13-18) has been proposed for these goals.
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