dynamic pricing on retail term deposits of a bank
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Electronic copy available at: http://ssrn.com/abstract=2343033
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DYNAMIC PRICING ON RETAIL TERM DEPOSITS OF A BANK
Ihor Voloshyn and Mykyta Voloshyn
October 2013
Version 1
Ihor Voloshyn
PhD , PJSC "Kreditprombank"
38, Druzhby Narodiv Boulevard,
01014, Kiev, Ukraine
Tel. +38 044 496 1784
Mykyta Voloshyn
Postgraduate, National Technical
University of Ukraine
"Kiev Polytechnic Institute"
14, Polytechnic Street,
03056, Kiev, Ukraine
Tel. +38 044 454 9859
The article “An estimation of optimal interest rates on retail term deposits of a bank” by Voloshyn, І.V.,
Voloshyn, M.І. has been published in “Herald of National Bank of Ukraine”, 2009, No 12 (166). http://archive.nbuv.gov.ua/portal/soc_gum/vnbu/2009_12.pdf
Electronic copy available at: http://ssrn.com/abstract=2343033
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Abstract
A quantitative ground for decision-making in a bank on setting of optimal interest rates on retail
term deposits is proposed. The offered approach is based on the variational method of
synthesis of optimal deterministic control. To solve this problem, instead balance sheet interest
rates it is used the interest rates on deposit cash flows. An example of optimal interest rate
dynamics is enclosed. The proposed approach allows achieving the maximal net interest
income for the bank under budget constraints.
Key words: dynamic pricing, retail, term deposit, automatic control, interest expense, net
interest income, optimal interest rate, optimization, variational method
JEL Classifications: G21, C61
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1. INTRODUCTION
Term deposits of individuals are one of the most important and expensive funding
sources for Ukrainian banking system. For example, as of 1 September 2013 the share
of individuals’ term deposits in the liabilities of Ukrainian banks was 31.8% and
comprised about 41.2 billion US dollars (National Bank of Ukraine). Therefore,
competition between Ukrainian banks for individuals’ deposits become stronger. Note
that a collection of new deposits helps banks to extend their credit activity.
To manage deposit activity efficiently banks would concern about achievement of
both budgeted deposit balances and interest expense. Too high interest rates on
deposits can provide a considerable deposit inflow in a bank, but the interest expense
will also appear too high. The low interest rates result in the low interest expense, but
the deposit inflow can be below than the budgeted balances. Thus, the depositing
control is a relevant issue for a bank.
The aim of this article is to develop an approach providing quantitative ground for
decision-making in a bank on setting of optimal interest rates on retail term deposits.
The developed approach is based on the theory of automatic control.
Note that two approaches can be used to solve this issue and, for our purpose,
we will call them a balance approach and a flow one. Consider these two approaches in
more detail (Voloshyns, 2009).
2. THE BALANCE AND FLOW APPROACHES
TO ESTIMATE OPTIMAL INTEREST RATES
Consider a bank’s portfolio of retail term deposits consisting of segregated deposit
accounts of individuals. The volume of the deposit portfolio B(t) at a certain time t is
equal to the sum of all deposit balances on banking accounts.
Study the behaviour of deposit portfolio over a certain period of time T and call it
the time period of deposit control. We will split the control period into equal time
intervals Δt.
The volume of the deposit portfolio are changed as a result of attracting new
deposits (credit turnover) and repayment of existing deposits (debit turnover):
)()()(
tDttCtt
tB
, (1)
where ΔB(t) is increment of deposit balances over the time interval Δt, Ct(t) is credit
turnover, Dt(t) is debit turnover.
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A balance interest rate R(t) on the deposit portfolio at a certain time t is an
average-weighted interest rate on all deposits in the bank’s portfolio. Note that the
balance interest rate varies continuously due to cash flows through deposit accounts.
An interest rate on new deposits attracted in the bank is a flow interest rate.
To estimate optimal interest rates on deposits the most investigators use a
balance interest rate (Sealey and Lindley, 1977; Enzo Dia, 2004). According to the
approach, for example, proposed by Sealey and Lindley (1977), Enzo Dia (2004), the
optimal interest rates are obtained for the balance interest rates by optimization of
interest expense over a certain control period of time:
min)()(0
T
t
ttRtBIE , (2a)
where IE is interest expense on deposits over the time period T, B(t) is deposit
balances, R(t) is the balance interest rate of the bank .
In this approach, the supply of new deposits is assumed to be proportional to the
balance interest rate of the bank. In our view, such a supply model is not clear enough
to reflect the economic sence of this phenomenon.
Surely, there is a correlation between the deposit supply and the balance interest
rate. However, the balance interest rate on existing deposits can not directly affect on
the deposit supply. It may influence on the deposit supply only through the flow interest
rate on the new deposits. In fact, depositors see only interest rates on the new deposits,
i.e. the flow interest rates.
The balance interest rate can vary through deposit repayment while the flow
interest rate on the new deposits can remain without changes.
According to the flow approach by Tsirlin and Kazakov (2003), the deposit supply
is supposed to be proportional to the difference between the flow interest rate offered by
the bank and the market interest rate on the new deposits.
In our opinion, such a supply model much more precisely matches the economic
sence of interaction between the bank and the retail deposit market (Fig. 1). The bigger
this difference, the bigger new deposit inflow. And vice versa.
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Fig. 1. Scheme of interaction between the bank and the retail deposit market.
However, Tsirlin and Kazakov (2003) found the optimal flow interest rates
through the following way of optimization:
min)()(0
T
t
Ct ttRtCt (2b)
and not through optimization of the interest expense (2a). In the expression (2b) RCt(t) is
the bank interest rate on deposit cash flow.
The approach by Tsirlin and Kazakov (2003) is based on application of the
thermodynamic principles to description of micro-economical processes. That is why the
criterion of optimization (2b) is chosen. Note that the criterion (2b) expresses the
principle of minimal capital dissipation.
It should be observed that for the bank it is more important to optimize interest
expense, instead of capital dissipation (2b). As a result of the approach (2b), central
information for the bank about the duration of new deposits staying on accounts is lost
while this duration directly affects on the bank interest expense.
Thus, from one side, it is necessary to minimize namely the interest expense
over the control period. From other side, the interest expense is needed to express
through the flow interest rates on the new deposits, because the deposit control is
realized by them.
To express the interest expense through the flow interest rates consider Figure 2
where it is shown how to define the duration of the new deposits’ remaining on accounts
over the control period of time.
Retail market for deposits with typical interest rate u0
on deposits
The bank offered interest rate u(t) on deposits
Cash flows of deposits or credit turnover Ct(t) ~
(u(t)- u0)
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Fig. 2. The durations of the new deposits’ remaining on accounts over the control period
of time
As it is shown on Figure 2, the cash flow amount Ct(t1) of the new deposits
attracted at time moment t1 stays on accounts for T-t1 until to the end of control period
T. Correspondingly, the cash flow amount Ct(t2) of the new deposits collected at time
moment t2 remains on accounts for T-t2 until to the end of control period T, and so on.
Taking into account it write the interest expense on the new attracted deposits over the
control period:
T
Ct dttTtRtCtIE0
)()( (3)
Further use the expression (3) to define an objective function for automatic
control.
3. ASSUMPTIONS
Let there is the national currency retail deposit market. And the alternative ones do not
exist. Thus, it is no need to take into account cash transfer from one market to other.
The deposit market is assumed to be unlimited. Consequently, deposit operations of a
bank do not influence on the level of market interest rates.
Banks in this market offer only one a fixed interest rate deposit with one term to
maturity. The investigated bank is a market-taker. It has a stable credit rating and does
not extend its branch network. Thus, parameters of a supply model are not changed in
time.
The bank controls its deposits over a chosen control period of time T.
Suppose that the existing and new deposits do not mature during the control
period of time T. It means that the existing and new deposits are matured after time T.
Ct(t1) Ct(t2)
t1 t2
T-t1
T-t2
T Time
Credit turnover
0
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Besides, the new and existing deposits are not withdrawn until to time T. Thus, debit
turnovers do not exist. Over all the control period of time the bank only collects the new
deposits.
Assume that the deposit market is characterized with a certain market interest
rate u0 on deposits that is not changed during all the control period of time. The deposit
supply is proportional to a difference between the bank interest rate u(t) and the market
interest rate u0 on deposits (Fig. 1) as following by Tsirlin and Kazakov (2003). The
movement of interest rate u(t) affects on the deposit supply immediately without any
delay.
For simplicity ignore the effects of seasonality and periodicity of depositing.
Note that the assumptions made are not critical and can be weakened.
4. AN OBJECTIVE FUNCTION FOR CONTROL
Let the bank attracts deposits and immediately invests them into interest-bearing
assets, for example, loans on interest rate R. Then taking into account expression (3)
the net interest income of the bank over the control period is equal to:
T
Ct dttTtRRtCtNII0
max)()( . (4)
This expression is used as the objective control function (the criteria of
optimization) which will be maximized.
It should be observed that in the expression (4) the contribution of the existing until
to the beginning of the control period deposits to the net interest income is neglected,
because the bank can influence on the net interest income only through the interest rate
on the new collected deposits.
Such a maximizing of the net interest income serves, from one side, minimization
of interest expense and, from other side, the more quick attraction of deposits for that
they (the deposits) will work in interest-bearing assets longer.
Then, formulate the task of deposit control as follows. It is necessary to define
such optimal flow interest rate dynamics that would ensure the achievement of the
budgeted volume of deposit portfolio under criteria of maximal net interest income over
all control period of time. In other words, it is necessary to transfer the bank from the
state with B0 deposit balances to the state with BT over the fixed time period of time T
subject to maximal net interest income. This is an issue of dynamic pricing on term
deposits.
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5. A MODEL FOR AUTOMATIC CONTROL
Consider the problem of depositing control in continuous time. For convenience of
subsequent exposition, enter the following new variables:
)()( tBtx , )()( tRtu Ct , (5)
where x(t) is the deposit balance on accounts at the time t. It is the state variable. u(t),
RCt(t) are the offered flow interest rates on new deposits. It is the control variable.
Examine the simplest model of deposit supply that describes the dependence of
cash flows on change of flow interest rates in the form (Tsirlin and Kazakov, 2003):
0
)()()( ututCttx , (6)
where u0 is the market interest rate, θ is the dynamic parameter of the model, x’(t) is a
credit turnover of deposits Ct(t). Dash in equation (6) is marked the first derivative with
respect to time.
It will be recall that the existing and new deposits do not mature in the control
period T. Thus, the debit turnover Dt(t) is equal to zero.
To estimate the dynamic parameter θ, the transient response analysis is an
acceptable method. In accordance with this method, the dynamics of the deposit
balances caused a stepwise change of the offered interest rate that then remains
constant is explored. Such a situation is characteristic for the term deposits with fixed
interest rates. The bank makes immediately decision to change interest rate on deposits
and its supply curve response on it. Therefore, from the such deposit dynamics it is
possible to define the dynamic parameter θ (Voloshyn, 2008).
The equation (6) has the following boundary conditions or budget constraints:
0)0( Bx ,
TBTx )( , (7)
where B0 and BT are the volumes of deposit portfolio at the beginning and at the end of
the control period of time, respectively.
Taking into consideration the new variables (5) and the expression (6) for credit
turnover, write down the objective control function (4) in the following form:
T
dttTtuRutuNII0
0 max)()( , (8)
where R is the bank interest rate on interest-bearing assets, for example, loans. This
interest rate considers to be constant.
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6. THE SOLUTION OF PROBLEM
To solve the problem of synthesis of optimal deterministic control, use the variational
method (Kim, 2007). The Hamiltonian function of the problem (6-8) is the following:
00
)()()()( ututtTtuRutuH , (9)
where ψ(t) is Lagrange multiplier that can depend on time.
Then, write down the Euler-Lagrange equations:
0
x
H , (10)
0)()(20
ttTuRtu
u
H . (11)
The equation (11) determines the optimal control for deposits by means of the
interest rate.
From the equation (10), it is followed:
1C . (12)
Substituting the expression (12) in the equation (11), define the optimal interest
rate on deposits:
tT
CuRtu 1
02
1)( . (13)
Note that according to equation (13) the interest rate on deposits during the
control period decreases (C1>0). At t=T it tends to minus infinity, i.e. the control is
unrestricted. Therefore, it needs to lay down the following restriction: the bank interest
rate can not be less than the market one:
0)( utu .
Substituting the solution (13) in the equation (6) and integrating the got
expression, obtain:
210
)ln(2
)( CtTCtuRtx
, (14)
where the constants of integration 1C and 2C are found from the boundary conditions
(7):
Tt
tuRBC
/1ln
/2*
*
0
1
, (15)
)ln(2
)0(12
TCxC
,
)0()( xTxB ,
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where t* is the time from the beginning of the control period until to moment of time
when the bank interest rate becomes equal to the market one, i.e. u(t*) = u0. This time t*
is defined from equation (13) with taking into account the expression (15):
*
1
002
1
tT
CuRu . (16)
It should be observed, that it is needed to change the upper limit of the integral
(8). Instead T, write down t*:
*
0
0 )()(
t
dttTtuRutuNII . (17)
Thus, from a moment of time by *tt the bank stops to attract the new
deposits, because the budgeted volume of deposits x(T)= BT has been achieved.
In practice, there are possible cases when it is necessary to restrict the bank
interest rate from above:
max)( utu ,
where umax is the given maximal interest rate of the bank that the market participators
interpret as suitable.
7. AN EXAMPLE OF DEPOSITING CONTROL
To illustrate the solution (6-8), we consider the following example. The task (6-8) has six
input data. Let the control period of time T be equal to 6 months; the dynamic parameter
θ be equal to 1.69×104; the market interest rate u0 be equal to 18.0% per year; the
interest rate of investing R be equal to 25% per year; the deposit balances at the
beginning and at the end of the control period of time, accordingly, B0 be equal to $50
million and BT be equal to $150 million US dollars.
The solution for the equation (16) with respect t* is t* = 5 months. The results of
calculations are given in Table 1 and on Fig. 3 and 4.
Table 1. The main indexes of the proposed approach
# Indexes Value, million US dollars
1 Average deposit volume 114.9
2 Net interest income over 6 months from investing the attracted deposits in the working assets (formula (17))
1.43
11
0
25
50
75
100
125
150
0 1 2 3 4 5 6
Time, months
De
po
sit b
ala
nce
s, m
ln U
SD
t*
Fig. 3. Dynamics of deposit balances.
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
0 1 2 3 4 5 6
Time, months
Op
tim
al in
tere
st ra
te o
n d
ep
osits,
% p
er
ye
ar
t*
Fig. 4. Dynamics of flow interest rate on bank’s deposit.
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8. CONCLUSION
Thus, the proposed approach to estimation of optimal interest rate on retail term
deposits ensures maximal net interest income and achievement of the budgeted volume
of deposit portfolio. Although in this article, it is developed a very simple model for
depositing control, the similar more complicated models seem to be perspective for
usage in sophisticated automatic control of depositing. For this, it is necessary to
consider additionally the following:
1) portfolio of deposits with different terms to maturity;
2) cash flows from contractual repayments of matured deposits;
3) rollover and early withdrawal cash flows;
4) behaviour of market interest rates;
5) uncertainty about deposit supply, and;
6) move from continuous to discrete framework of this problem.
The following should also be noted. To take into consideration a deposit product
line (for example, deposits in foreign currencies, with different schedules of interest
payments, etc.), it is needed to build the corresponding supply models for every deposit
product.
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LITERATURE
Enzo Dia (2004). “Monopolistic Pricing in the Banking Industry: and Dynamic Model”.
Working paper series. Department of Economics University of Milan. Bicocca. No 73:
http://dipeco.economia.unimib.it/web/pdf/pubblicazioni/wp73_04.pdf retrieved on
18.10.13.
Kim, D.P. (2007) Theory of Automatic Control. V.2. Multidimensional, Nonlinear,
Optimal and Adaptive Systems. Moscow: Fizmatlit.
National bank of Ukraine (2013) “Main Indicators of Activities of Ukrainian Banks as of 1
September 2013”:
http://bank.gov.ua/control/uk/publish/article?art_id=660533&cat_id=58284 retrieved on
18.10.13.
Sealey, C.W., Lindley, J.T. (1977) “Inputs, Outputs, and Theory of Production and Cost
at Depositary Financial Institutes”. The Journal of Finance. Vol. XXXII. No 4.
Tsirlin, A.M., Kazakov, V. (2003) “Irreversibility Factor and Limiting Performance of
Financial Systems (Thermodynamic Approach)”:
http://www.qfrc.uts.edu.au/research/research_papers/rp99.pdf retrieved on 18.10.13.
Voloshyn, І.V. (2008) “Automatic Control for Term Deposits of Individuals” Financial
Risks. No. 2(51):
www.securities.com retrieved on 18.10.13.
Voloshyn, І.V., Voloshyn, M.І. (2009) “An estimation of optimal interest rates on retail
term deposits of a bank”. Herald of National Bank of Ukraine. No 12 (166).
http://archive.nbuv.gov.ua/portal/soc_gum/vnbu/2009_12.pdf retrieved on 18.10.13.