Pricing of a Bank’s Portfolio through a Dynamic Game
Author
Name:
Dr. Debasish Majumder
Designation:
Assistant General Manager, Reserve Bank of India
Address:
M-572, RBI Staff Quarter (Byculla),
Maratha Mandir Marg, Mumbai Central (E),
Mumbai – 400 008,
India
Tel No: +91-022-23000193; Mobile No: 9819029502
E-mail address: [email protected].
1
Pricing of a Bank’s Portfolio through a Dynamic Game
Abstract
With the process of deregulation of interest rates, the administered interest rates structure
has been dismantled and banks are getting control over their interest rate decisions.
However, the decisions of setting credit and deposit rates cannot be modeled efficiently
as a single player’s optimization exercise. It should be the outcome of a game that is
played between three interested players: the bank, the firm and the consumer. These
agents are related as funds are channeled from savings by the consumer to investment by
the firm through an intermediary which is a bank. Each player’s dynamic choices impacts
on the strategic decisions of the others. The dynamic game designed by us is a state-space
game where the equilibria that are considered are of the open-loop type. We have derived
a relationship between credit risk and the expected rate of return of the bank’s portfolio.
Consequently, the optimal credit risk and the corresponding equilibrium credit and
deposit rates are determined. This equilibrium is a rationing one where certain demands
for loans are rationed even if some borrowers are willing to pay more than the
equilibrium credit rate.
JEL Classification Code: G21, G12, E43
Key Words: Deregulation, Credit Risk, Open-Loop Strategy, Dynamic Game, Credit
Rationing
 The views expressed in this paper are of the author and not of the organization to which
he belongs.
2
1 Introduction
With the worldwide wave of liberalization, as the degree of openness increases, the first
and foremost questions that arise are: “What would institutional relations gradually tend
towards?” and “What would be the structure of financial linkages?” In our paper, we
have tried to find the answer to the above questions when three agents in an economy, a
commercial bank, firm and consumer, interact to achieve their optimal decisions. In an
economy where regulation is less, the authorities play no active role in the bilateral interlinkages of the above three economic agents. It influences indirectly by setting policies.
Agents are not only concerned about the present but also their future decisions, so the
evolution of their economic behaviour through time is of fundamental importance. Our
primary emphasis is on pricing of a bank’s portfolio via determination of the equilibrium
bank interest margin through a dynamic game.
Amidst the dramatic changes in financial intermediation worldwide, a need has been
experienced for more sophisticated technology in the pricing of the portfolio of the
intermediary. When the credit and deposit rates were exogenously determined, the
expected return on a portfolio also became predetermined. In this scenario there was
hardly any flexibility of control or need for pricing. But when interest rates are not
exogenous, but under the control of an intermediary, the flexibility in operation is
stretched. At the same time risk increases considerably. This risk may be inherited from
the choice of a flawed model for pricing or from erroneous decisions about credit and
deposit rates. On the one hand, a high credit rate may lead to excessive risk taking. On
the other, a low rate may impede profitability. Therefore, a balance in interest rate setting
and the risk-absorbing capacity of a banking firm is essential. But while setting of interest
rates or pricing of an intermediary’s portfolio is of prime attention, bilateral interlinkages
among economic agents should not be neglected. Today, many theories are available
where the problem of setting rates is thought of as bank’s problem and the consequent
solution can be achieved through maximization of a bank’s expected utility (see Klein,
1971; Sealey, 1980; Wong, 1997; Zarruk and Madura, 1992). Similarly, a firm and
consumer can solve their separate optimisation exercises to find the optimal decisions for
themselves (see Benavie, Grinols and Turnovsky, 1996; Bond and Meghir, 1994; Hay
1
and Liu, 1998; Turnovsky, 1995 Ch.14). But the impotence of the separate optimisation
problems lies in ignorance of the bilateral relations, which is a key feature of a
deregulated economy. It is misleading to neglect the inter-linkages between bank, firm
and consumer as the savings by an individual is channeled to investment by firms through
an intermediary. The consumer has the power to choose his portfolio and it is an everdiscussed issue whether he would prefer to bear risks and invest in the equity market or
he would be risk averse and save through a bank deposit. Similarly, a firm holds the
power to choose its financing mix. The intermediary always has an informational
advantage as the individual does not have the time and expertise to monitor the behaviour
of firms. It remains true, however, that both banks and savers will be at an informational
disadvantage compared with the borrower in assessing the latter’s net worth. If the three
agents in an economy act rationally to choose their portfolio depending on the acceptable
level of risk, an equilibrium must exist and it would be a rationing equilibrium as some
demands for loans would must be rationed even if some borrowers indicate a willingness
to pay more than the equilibrium interest rate.
Traditionally, the bank was thought of as an institution whose operations consisted of
granting loans and receiving deposits in virtue of lowering transactions cost of its
customers (depositors and borrowers). This traditional vision of banking has undergone a
metamorphosis during the last two decades and evolved into a new theory: The
Microeconomic Theory of Banking. On the one hand, the new theory has overturned the
traditional doctrine of “lowering transactions cost” and has elaborated a multifunctional
role of a banking firm, and, on the other, it has outlined two complementary paradigms
for developing an economic theory of banks. The incomplete information paradigm is a
consequence of different borrowers having different probabilities of repaying their loans.
The expected return to the bank obviously depends on the probability of repayments so
the bank would like to be able to identify borrowers who are more likely to repay. It is
difficult to identify “good borrowers“ and to do so requires the bank to use a variety of
screening devices. In these models, credit risk is typically endogenously determined i.e.
the loan rate charged determines credit risk faced.
The second pillar of the
microeconomic theory of banking, namely the Industrial Organization approach, is the
assumption that banks essentially offer services to their customers (depositors and
2
borrowers) and that financial transactions are only the visible counterpart to these
services. This approach provides a rich set of models (example: the Monti-Klein model)
for tackling different issues like monetary policy and some aspects of banking regulation
etc. The lender-borrower relationship, at one extreme, is the case where the borrower’s
cash flows are risky but there is no informational asymmetry. At the other extreme, is the
case of asymmetric information where the borrower’s cash flows are impossible to
observe by the lender. An intermediate case is one in which the borrower’s cash flows are
not directly observable by the lenders. However they can bear an auditing cost to know
the unobservables. The framework set up by us for modeling intermediaries’ behavior is
characterized by complete and perfect information and thus it departs from the adverse
selection aspect of interest rate-setting of banks. At the same time it supports the basics of
the latter line of work. The rationale behind the programme can be explained by the
following example: let us consider two borrowers, Borrower A and Borrower B who have
borrowed funds for their projects from Bank X. Bank X is in possession of historical data
about their loan repayment behavior. Therefore, the probability distribution of nonperforming loans of each company is data for bank X and is considered as an input in our
model. As the risk profile of each borrower is known to the bank, there is no
informational asymmetry. In the familiar manner, riskiness is proxied by the volatility of
the percentage of loan losses. Greater volatility implies greater uncertainty in future loan
repayments. Consequently, the bank would be inclined to provide loans to borrowers
whose volatility of the percentage of loan losses is significantly lower than others. Once
again in the familiar manner, this policy would not always signify a higher expected
return to the bank. Therefore, a need has been experienced for pricing of the bank’s
portfolio that would essentially depict a relationship between expected return to the bank
and the volatility of the percentage of loan losses.
The consumer’s portfolio choice theory is an ever-discussed issue in finance. Beginning
with Markowitz (1952), economists have systematically studied the portfolio choice of a
consumer. Markowitz proposed the expected mean returns–variance of returns (M-V)
rule which is basically a normative approach. The normative results of portfolio selection
were generalized into a positive theory by Sharpe (1964) and Lintner (1965). They
introduced the Capital Asset Pricing Model (CAPM) where the relationship between
3
expected return and systematic risk was explicated. Portfolio theory is described as an
analysis of diversification where the optimal choice of the proportions of different assets
is of prime importance. The concept of portfolio for an bank is different from that for a
consumer as the consumer’s sole interest is to choose optimally the proportions of
different assets and the level of consumption whereas the bank’s interest lies in the
optimal choice of deposit and credit rates. The bank performs two basic activities: it
borrows funds from individuals at a fixed rate of interest and provides credit to firms. So
the term ‘portfolio’ for a bank would be the net worth or equity which is the difference
between the value of all loans and all deposits. Consequently, the net present value of a
bank’s portfolio would be net present value of all future credit reduced by net present
value of all future deposits. Since the available funds to the bank and the credit offtake by
the firm are sensitive to deposit and lending rates, the pricing decisions of a bank’s
portfolio is a problem of setting interest rates which is different from portfolio choice
theory by the consumer. It is not always true that a high credit rate produces higher
expected return, because, in the familiar manner, it leads to a higher credit risk.
Alternatively, the bank may incline to provide loans to borrowers whose volatility of the
percentage of loan losses is significantly lower than others. Once again in the familiar
manner, this policy would not signify a higher expected return to the bank. One critical
credit rate and corresponding credit risk must exist which maximizes the expected return
on a bank’s portfolio. Our study explores the procedure of determination of the above
critical credit rate and the highest credit risk that a bank should accept.
Strategic interactions between the bank, the firm and the consumer cannot be properly
modeled as a static game since each player makes decisions at more than just one point of
time. Thus we have considered a dynamic model where all players move simultaneously
resulting in a Nash equilibrium. The game designed by us is a differential game where the
evolution of the state of the game is via differential equations. The assumption is that the
previous action history is adequately summarized in the current state vector. One can
assume, at one extreme, that the players use no information and base their strategies on
time alone. This strategy is called an Open-Loop Strategy. At the other extreme, the
players can base their strategies on the whole history of action. These strategies are called
Non-Markovian strategies. An intermediate case is one in which the players based their
strategies on the current value of the state vector, which is a Markovian strategy (see
4
Dockner, Jorgensen, Long, and Sorger, 2000) We consider an open-loop strategy which
is a normative approach and needs no dynamic information.
The rest of the paper is organised as follows: In Section 2 we have designed a game
among a bank, firm and consumer for determining the equilibrium interest rates. Pricing
decisions of the portfolio of the bank and its implications for bank’s policy is given in
Section 3. The conclusion is given in Section 4.
2 A Dynamic Game to set the Interest rate
We consider an economy that is populated by a continuum of agents. These agents can be
divided into three categories: Borrowers (or firms), bankers and depositors (savers).
Bankers intermediate in channeling the funds of depositors to investment by borrowers.
We assume that the interest rates, i.e. rates on deposits and loans, are determined through
a game played by the three economic agents. We design our game in a continuous time
framework where each player follows its utility/profit maximisation objective to
determine its decision variables.
We assume that the bank’s profit function, Π Bt , is of the following form:
Π Bt  1  θ R Lt L t  R Dt D t
(1)
where Dt and Lt are the value of deposits and loans respectively with interest rates R Dt
and R Lt , subscript t indicating time. We model credit risk by means of a random variable
, with support [0,1], to denote the proportion of non-performing loans at time t. Without
loss of generality, we assume that non-performing loans pay nothing to the bank. The
actual value of loan repayments to the bank is therefore 1  θ R Lt L t , which is less than or
equal to the value of the total contractual loan repayments, R Lt L t , depending on the
realisation of θ at time t. The change in profit in a small time interval dt is described by
the following differential equation.
5


dΠ Bt  R Lt L t  R Dt D t dt  R Lt L t dθ
(2)
The bank’s objective at date t entails maximisation of the expected present value of the
utility of profit,


Z bt  E t   e  ρt U Π Bt dt 
t

 
(3)
~
~
If we assume C t is the bank’s net worth or equity then C t would be related with Lt and Dt
through the following balance-sheet equation:
~
V( C t ) = V(Lt) - V(Dt)
(4)
~
~
where V( C t ) is the present value of C t at time t. Similarly V(Lt) and V(Dt) are the
~
present value of loans and deposits at time t. An increase in net worth, C t , in a small time
interval dt may be translated into the following differential equation:
 
~
dC t  d Π Bt  L t  D t dt
(5)
~
Equation (5) reflects the fact that a small change in net worth, C t , is due to additional
profit and an increase in net loans (i.e. loan minus deposits). The dynamic optimisation
problem for the bank involves the above constraint.
In the above optimisation exercise the control variables for the bank are interest rates, RtL
and RtD. It also involves Lt and Dt, which are the decision variables of the firm and
consumer respectively. d is the incremental credit risk having mean zero and variance
σ2. We represent below the firm’s dynamic optimisation problem to determine it’s
optimal decision about investment.
6
Without loss of generality it can be assumed that the firm’s profit is to be distributed to
its shareholders through dividends according to the following equation:
Pt YK t   R Lt L t if Pt YK t   R Lt L t
R Qt  
otherwise
0
S
t
(6)
where RtS is the stochastic rate of dividend payment, YK t  is the production function, Pt
is the output price. Let Itdt, Ltdt and Qtdt be investment, bank loan and equity,
respectively, in a small time interval dt. RtLLt is the interest cost of the loan.
There are strictly convex adjustment costs in changing capital stock of the firm, dKt,
which evolves over time according to the equation:
dKt= (It-δKt)dt
(7)
Where δ is the depreciation rate. Equation (7) implies that changes in gross investment
equals changes in capital plus its depreciation. It can be assumed that there are two
sources to change in the capital stock: through bank loans and by issuing new equity. In
the light of the above, the following equation can be written:
Itdt = Ltdt + Qtdt
(8)
Another state variable Pt is assumed to evolve according to:
dPt= Ptσpdv
(9)
where σ 2p is the variance of Pt, dv is a Wiener process with mean zero and unit variance. A
value maximising firm seeks to maximise its value to its shareholders, measured by the
expected discounted value of its future dividends:
   βt S

Z  E t   e R t Q t dt 
t

f
t
(10)
7
where R St is the stochastic rate of dividend payment.
We consider a representative consumer that chooses his portfolio of assets and rate of
consumption to maximize expected lifetime utility. The consumer cares about the
uncertainty of the future value of his total wealth, so diversification is required. In our
model we assume that the individual has two opportunities. In order to diversify his
portfolio he either invests in domestic stocks or he saves through bank deposits1.
The consumer’s objective at date t entails maximisation of the expected present value of
his utility,


Z ct  E t   e  γt UC t dt 
t

-<γ<1
(11)
subject to the stochastic wealth accumulation equation:
dWt = Wt(ηt dRtD + (1- ηt) dRtS ) + (Yt-Ct)dt
(12)
where Ct and Wt are the flows of consumption and wealth with Wt = Dt + Qt, Dt is the
consumer’s supply of deposits to the bank and Qt is the consumer’s total equity stock,
subscript t indicating the time period, η = (D/W) is the portfolio share of bank deposit, (1η) = (Q/W) is the portfolio share of equities, Y is the wage income. For computational
simplicity it can be assumed that the consumer derives all of his income from capital
gains sources i.e. Y = 0 (see Merton, 1990, Ch. 15). If RS is the stochastic real rate of
return of equity, it can be written as:
dR St  R St dt  σ S dz
(13)
where R St is the mean rate of return on equity and σ S2 is the volatility, dz is the stochastic
component of equity returns with mean zero and variance one. Conventionally, we can
ignore the stochastic part of dRtD and it can be written as dR Dt  R Dt dt , as for fully
8
insured bank deposits it can be assumed that no risks or stochastic components are
involved2.
Formally, the game played between the bank, the firm and the consumer is characterized
by a seven-tuple, F = (N, C, X, U, A, I, Z), where the constituent elements are as follows.

N is the set of players, i.e., the bank, the firm and the consumer in the present case.

C is the set of control variables. In present game, C is a vector that consists of control
variables R Lt , R Dt  for the bank, L t for the firm and C t , η t  for the consumer. That is,



C  R Lt , R Dt , L t , C t , η t  .



~
~
X is the set of state variables. That is X  C t , K t , Pt , Wt , where C t , K t , Pt 
and Wt are state variables for the bank, the firm and the consumer respectively.

U is the set of strategies. A Markovian strategy for player i is a function mi that
determines for each pair (X(t),t) the value of the control vector Ci(t). A Markovian
strategy that does not contain X(t) as an argument is called an open-loop strategy.

A is a set of actions. In the present game, the bank, the firm and the consumer move
simultaneously.

I is the information structure. The game designed in this paper is a game of complete
and perfect information.

Z is the set of payoffs. Here, Z  Z bt , Z ft , Z ct , where Z bt , Z ft and Z ct are payoffs for the
bank, the firm and the consumer.
2.1 The Solution of the Game
The dynamic game designed in the previous Section is solved using the open-loop
assumption. The open-loop assumption explicates that players leave all information
except time out of consideration or that they must choose open-loop strategies since they
cannot observe anything other than their own actions and time. Moreover, these equilibria
One can easily extend the model by considering government bonds in the individual’s diversified
portfolio (see Benavie, Grinols and Turnovsky, 1996).
2
Risk may be involved in bank deposits due to changes in the general price level. This is neglected in this
analysis.
1
9
are relatively more tractable and are a relatively good approximation where the numbers
of players is many.
In this paper we use the Lagrangian procedure as indicated by Chow (1997) to solve the
dynamic optimization problem described in the previous Section. In the simultaneous
move game the firm chooses equilibrium Lt by maximizing equation (10) subject to the
constraint given in equation (7). The result of the above optimization is characterised by
the following system of differential equations (as derived in Appendix 1):
R Lt  λ F (t)
(14)
αPt K αt -1dt  (β  δ)λ F (t)dt  E t dλ F  0
(15)




R S  R Dt
E t dλ F  L t  2t
Wt  δK t  λ FK  12 σ 2p Pt2 λ Fpp dt
σ s 1  γ 


(16)
where λ F is the corresponding Lagrange multiplier, λ Fk and λ Fpp are the first and second
derivative of λ F with respect to the state variable Kt and Pt. Akin to the firm, the consumer
chooses equilibrium Dt, where he maximises equation (11) subject to the constraint given
in equation (12). If we assume that the utility function of the consumer possesses DARA
and is of the form, U(x) = ln(x), we can derive an explicit solution of the consumer’s
optimization exercise. The result of the above optimization is characterised by the
following system of differential equations (17), (18), and (19) (as derived in the
Appendix 2):
1
 λC ( t )
Ct
(17)
1  η t Wt λCW σ S2  R Dt  R St λC t 
(18)
10
  
 

 


γλ C  λ CW Wt η t R Dt  R St  R St  C t  12 λ CWW 1  η t  σ S2 Wt2  λ CW 1  η t  σ S2 Wt  η t R Dt  R St  R St λ C
(19)
2
2
where λ C is the corresponding Lagrange multiplier and λ CW and λ CWW are the first and
second derivative of λ C with respect to the state variable Wt. The bank solves its
optimization problem, where it maximizes equation (3) subject to the constraint given in
equation (5). For the purpose of getting an explicit solution we consider, once more, that
the utility function posses DARA and is of the form, U(x) = ln(x). The solution is derived
in Appendix 3 and the results are shown below.

1
E t (1  θ t )   λ B t   R Lt L t λ CB σ θ2  0
πt 

(20)
1
E t    λ B t   0
πt 
(21)



B
ρλ B  λ CB R Lt L t  R Dt D t  (L t  D t )  12 λ CC
(R Lt L t σ θ ) 2
(22)
B
where λ B is the Lagrange multiplier, λ CB and λ CC
are the first and second derivative
~
of λ B with respect to the state variable C t . From the system of equations (20), (21) and
(22) R Lt and R Dt can be solved in terms of the unknown Lt and Dt where Lt and Dt are to
be determined from the firm’s and household’s optimization problems respectively.
The set of equations (14) to (22) is a complete system and the solution gives the Nash
equilibrium point. Solving equation set (14) to (16) and then (17) to (19) we can derive Lt
and Dt as follows (the derivation is given in Appendices 1 and 2):




R St  R Dt
R  L t  2
Wt  δK t 
σ s 1  γ 


L
t
β  δ σ 
2
p
P  αPt K
2
t
11
α-2
t





R St  R Dt

Wt  δK t α  1  (β  δ)K t 
 L t  2
σ s 1  γ 



(23)


RS  R D
D t  1  t 2 t
σS

W


(24)
t
The equation set (20, 21 & 22) gives the solution of the bank’s optimization problem as
follows (the derivation is given in Appendix 3).
2Z t
kt
ρ


σ θ2 l 2t Z t k 2t  σ θ2 l 2t
1
kt
(25)
 σ θ2 l 2t 

kC 
1  2   z1exp  2 t t2 2   0


kt 

 k t  σθ lt 
(26)
where

k  R L  R
l  R L 


Z t  R Lt L t  R Dt D t  (L t  D t )
t
t
L
t
L
t
t
D
t
Dt

t
The equation set, (23), (24), (25) and (26), is a complete system of equations where the
variables to be determined from the model are RtL, RtD, Lt, and Dt in terms of exogenous
~
variables, σ θ2 , σ s2 , σ 2p , and state variables, C t , K t , Pt , Wt . The solution of the above
system of equations gives the Nash equilibrium solution of RtL, RtD, Lt, and Dt from
which a nice relationship between risk and return may be drawn.
3 Pricing Decisions and Policy Implications
The objective in our present section is determination of the pricing formula of the
portfolio of the bank by establishing a trade-off between the expected rate of return and
the risk to be accepted. We have defined the present value of the bank’s portfolio as the
difference between the net present value of all future loans minas the net present value of
12
~
future deposits (see equation (4)) and consequently an increase of the net worth, C t in a
small time increment dt may be translated into the following differential equation:
 
~
dC t  d Π Bt  L t  D t dt
(27)
The instantaneous rate of return at time t is defined as:
αt
~

dC
 lim
dt  0
t
~
Ct

(28)
dt
Using equations (2), (27) and (28) the expected rate of return, Et(αt), can be written as:



~
E t α t   R Lt L t  R Dt D t  (L t  D t ) /C t
(29)
As RtL, RtD, Lt, Dt are not known in advance we can substitute their equilibrium values
which are the solution of the game presented in Section 2. In terms of the symbols of
Section 2, equation (29) can be rewritten as:
~
E t α t   Z t /C t
(30)
Zt can be obtained from the equilibrium solution of RtL, RtD, Lt, Dt. Using equation (30),
(25) and (26) we can establish a trade-off between Et(αt) and the -components.
THEOREM 1. The expected rate of return, Et(αt), is a concave function of the volatility
of the proportion of loan losses, σ θ2 , and Et(αt) would be a maximum when ‘the elasticity
 σ 2 dl 2 
of the return from the loan’ with respect to credit risk,  2θ t2  , equals –1.
 l t dσ θ 
Proof.
Equation (25) and (26) can be written in functional form as following:
13


(31)
φ k t , σ θ2 l 2t  0
(32)
f Z t , k t , σ θ2 l 2t  0


Differentiation of the implicit function of equation (31) and (32) yields:
dZ t
1

2 2
d σθ lt
 2
ρ
 2 2 
σ l
 θ t Zt


 φ σθ2l2t f k t

 f σ 2l2  

θ t
  φ k t




or
dZ t
σ θ2
2


l
1

t 
d σ θ2
l 2t

 



 φ σθ2l2t f k t

dl 2t 
1


 f σ 2l2   

2 
θ t
dσ θ   2
ρ   φ k t
 
 2 2 

 σθ lt

Zt 


(33)
where x and fx are the first partial derivative of the function (.) and f(.) respectively.
From Appendix 6 we can conclude that the function in the second bracket of equation
(33) cannot be zero for any value of σ θ2 . So,
dZ t
d σ θ2
 
would equal zero only when the
 σ θ2 dl 2t 
 , equals –1. It is clear that the function, Zt, or Et(αt), has the single
elasticity  2
2 
 l t dσ θ 
 σ 2 dl 2 
optimal (maximum/minimum) point when  2θ t2  is equal to –1 and the point is a
 l t dσ θ 
maximum or minimum point according as the function is concave or convex. As σ θ2 is
nonnegative and as the derivative of equation (33) is positive when σ θ2 is near zero, we
can conclude that the function, Et(αt), is a concave function of σ θ2 . The function has a
single optimum point and is increasing at the initial point.
14
 σ 2 dl 2 
THEOREM 2. There exists a critical value of σ θ2 , say σ̂ θ2 , for which  2θ t2  would
 l t dσ θ 
equal –1 and for σ θ2 = σ̂ θ2 , there always exists some equilibrium loan and deposit rate.
Proof.
As the system of equations, (23), (24), (25) and (26), is a complete system, an
equilibrium solution of RtL, RtD, Lt, and Dt must exist. Consequently, an equilibrium
solution of l t  R Lt L t  exists in terms of the exogenous variables, σ θ2 , σ s2 , σ 2p , Wt , C t , K t .
 σ 2 dl 2 
So a critical value of σ θ2 , say σ̂ θ2 , must exist for which  2θ t2  would equal –1. As the
 l t dσ θ 
equilibrium RtL and RtD would be a function of σ θ2 , σ s2 , σ 2p , Wt , C t , K t , there always exist
some equilibrium RtL, RtD when σ θ2 is equal to σ̂ θ2 . Hence our result follows.
The above results have implications for the determination of the bank’s policy. In the
famous credit rationing literature, Stiglitz and Weiss (1981) argued that credit rates
should not be determined by equalising demand and supply as the expected rate of return
depends on the probability of defaults. High interest rates lead to a high probability of
defaults. In addition to the usual direct effect of increase in returns with interest rate
increases, there is an indirect adverse selection effect acting in the opposite direction. It
can therefore be said that an increase in interest rates leads to an increase in expected
returns upto a certain stage and after that the expected return would fall. In Theorems 1
and 2 we have shown the necessary condition for a rationing equilibrium which gives the
maximum credit risk that a bank should incur and the corresponding equilibrium interest
rates.
Economic Risk Capital
15
Pricing decisions have serious policy implications for the determination of economic risk
capital. As an insurance against the uncertainty of the net worth of the portfolio, the
banking firm could well hold an amount of riskless investments. We will call this buffer
the economic risk capital of the institution. Economic risk capital eventually stands as a
cushion to absorb unexpected losses related to credit events, i.e. deterioration of credit
quality or defaults. If VaRp is the Value at Risk i.e. return at worst possible scenario at
p% confidence level and ER is the expected rate of return, then economic capital can be
calculated as below.
Economic capital = (V0/100)*[ER – VaRp]
where V0 is the current mark-to-market value of the portfolio. The diagram below shows
the economic risk capital as unexpected losses.
Several authors have described different procedures to calculate Value at Risk, but
calculation of the expected rate of return, ER, is also a critical one. The typical and
simplest solution of the problem is to work out ER by assuming a distributional form of
return. But as the exact distribution is not known and the distributional model represents
only an approximation, there is a chance that a slight perturbation in the data gives rise to
substantially different distributional outcomes and, thus, different conclusions. The
16
pricing of the bank’s portfolio through a dynamic game designed by us gives a possible
value of expected rate of return which is advantageous in two aspects: a) it does not
depend upon any distributional form b) it considers possible risk taking policies by the
bank.
As an example, if we assume that Value at Risk is calculated assuming the return
distribution is normal with mean  and variance σ θ2 and if the p-value for the VaR
calculation is five percent, then economic risk capital would be:
Economic capital = (V0/100)*[ Et(αt)+1.65 σ θ2 -]
Economic capital is the sum of two functions, one is concave and other is increasing
in σ θ2 .
4 Conclusion
Deregulation of interest rates is a major economic process which has changed the
landscape of financial intermediation globally. As a consequence, administered interest
rate structures have been dismantled and banks are gaining control over their interest rate
decisions. Against this backdrop, determination of equilibrium credit and deposit rates is
not a single agent problem, but strategic interaction exists between three protagonists: the
bank, the firm and the consumer. The spirit of this paper lies in designing a framework
for pricing of a bank’s portfolio through determination of equilibrium deposit and credit
rates. We assume that players will employ open-loop strategies which is a normative
approach and needs no information. Our results indicate that the expected rate of return
of the bank’s portfolio is a concave function of credit risk and the peak of the curve
occurs when the elasticity of returns from loans with respect to credit risk is –1. We can
infer that credit risk would boost the expected rate of return upto a certain stage and after
that the expected rate of return would fall. This critical credit risk and the corresponding
equilibrium credit and deposit rates would be the desirable rates and also would
correspond to a rationing equilibrium for the bank. In this equilibrium, certain demands
17
for credit would be rationed even if some borrowers indicate a willingness to pay more
than the equilibrium interest rates.
The idea of Economic Capital mentioned in this paper is quite different from that of the
traditional regulatory view. Economic Capital should not be reconciled with regulatory
capital requirements as one cannot be the complement of the other. The Bank’s intention
should be not only to meet the regulatory requirement, but instead, adequate capital
should be determined for future distress according to the institution’s risk taking policies.
Appendix 1
The firm’s optimisation problem can be solved by the method of Lagrange multipliers.
To apply the method, we maximize the following Lagrangian expression (with λ as
Lagrange multiplier):
 βt

£  E t   e Pt YK t   R Lt L t dt  e β(t  dt) λ F (t  dt)dK t  L t  Q t  δK t dt
t


F

(A.1)
The parameters used in equation (A.1) are described in Section 2. As the value of Qt is
unknown to the firm and to be determined by the consumer, the firm can use the reaction
function of Qt to solve its optimization exercise. The reaction of Qt would be (Wt- D t ) ,
where D t may


R St  R Dt
1


σ s2 1  γ 

be
solved
from
consumer’s
optimization
problem
as
 W (see equation (37)). So the Lagrangian expression of equation (A.1)


t
can be rewritten as:





R S  R Dt
£ F  E t  e βt Pt Yt (K t )  R Lt L t dt  e β(t  dt) λ F (t  dt) dK t  (L t  2t
Wt  δK t )dt 
σ s 1  γ 


 t


The first order conditions of maximization of equation (A.2) yield:
18
(A.2)
£ F
 0  R Lt dt  E t e βdt λ F (t  dt)dt  0
L t
(A.3)
£ F
 0  αPt K αt -1dt  e βdt E t λ F (t  dt)[ 1  δdt]  λ F (t)  0
K t
(A.4)
e-βdt can be approximated as (1- βdt) and any product of dt, dλF, and dz is of order smaller
than dt and can be neglected. Consequently, equation (A.3) and (A.4) can be simplified as
below.
R Lt  λ F (t)
(A.5)
and
αPt K αt -1dt  (β  δ)λ F (t)dt  E t dλ F  0
(A.6)
Applying Ito’s differential rule to evaluate the term E t dλ F , we get:




R St  R Dt
E t dλ  L t  2
Wt  δK t  λ FK  12 σ 2p Pt2 λ Fpp dt
σ s 1  γ 


F
(A.7)
where λ FK and λ Fpp are the first and second order derivatives of λ F (t) with respect to Kt, and
Pt respectively. Using equation (A.7), equation (A.6) can be rewritten as
1
2




R S  R Dt
σ 2p Pt2 λ Fpp dt  L t  2t
Wt  δK t  λ FK  (β  δ)λ F (t)dt  αPt K αt -1dt  0
σ s 1  γ 


(A.8)
The general solution of the differential equation, (A.8), is the sum of the complementary
function and the particular integral. To evaluate the complementary function, we consider
a trial solution,
19
λ F t   b(K t )Pt2
where b(Kt) must satisfy the following differential equation




R St  R Dt
L

Wt  δK t  b k 
 t
2
σ s 1  γ 



1
2

σ 2p Pt2  (β  δ) bK t   0
(A.9)
where bk is the first derivative of b(Kt). Solving (A.9) we get




R St  R Dt
bK t   L t  2
Wt  δK t 
σ s 1  γ 


β δ σ 
2
p
Consequently the solution of λ F (t) becomes




R S  R Dt
λ F t   L t  2t
Wt  δK t 
σ s 1  γ 


β δ σ 
2
p
Pt2
(A.10)
The above solution corresponds to the complementary function of the differential
equation (A.8). The particular integral of (A.8) is to be evaluated from:


1




R S  R Dt
λ t    12 σ 2p Pt2 D 2p   L t  2t
Wt  δK t D k  (β  δ)dt  αPt K αt -1


σ s 1  γ 




F
where
d
d2
, 2 etc. are denoted by the symbols D k , D 2p etc. Simplifications of the
dK t dPt
above equation give the particular integral as:





R S  R Dt
λ F t   αPt K αt -2  L t  2t
Wt  δK t α  1  (β  δ)K t 
σ s 1  γ 



20
(A.11)
The general solution of the differential equation, (A.8), which is sum of the
complementary function and the particular integral, is stated below.




R S  R Dt
λ F t   L t  2t
Wt  δK t 
σ s 1  γ 


β  δ σ 
2
p





R S  R Dt
Pt2  αPt K αt -2  L t  2t
Wt  δK t α  1  (β  δ)K t 
σ s 1  γ 



(A.12)
Using equation (A.12), equation (A.5) can be rewritten as below:




R St  R Dt
R  L t  2
Wt  δK t 
σ s 1  γ 


L
t
β  δ σ 
2
p
P  αPt K
2
t
α-2
t





R St  R Dt

Wt  δK t α  1  (β  δ)K t 
 L t  2
σ s 1  γ 



(A.13)
Appendix 2
The consumer’s optimization problem is solved by the method of Lagrange multipliers as
well. Let us consider the following Lagrangian expression (with λC as the Lagrange
multipliers):
  γt

£  E t   e UC t dt  e  γ ( t  dt ) λC ( t  dt ) dWt  Wt η t R Dt  R St  R St dt  C t dt  (1  η t ) Wt σ S dz 
t

(A.14)

C
 



The parameters used in equation (A.14) are described in Section 2. The first order
condition for maximization of equation (A.14) yield:
£ C
 0  U C dt  E t e  γdt λC t  dt dt  0
C t

(A.15)
 
£ C
 0  E t λC t  dt Wt σ S dz  R Dt  R St dt  0
η t


(A.16)

£ C
 0  e  γdt E t λC t  dt  1  η t R Dt  R St dt  (1  η t )σ S dz  λC t   0
Wt


21
(A.17)
where Uc is the first derivative of U(Ct) with respect to Ct. If we use Ito’s differential rule
we get
  


  λ 1  η  σ W dt  λ 1  η W σ dz
dλ C  λ CW Wt η t R Dt  R St  R St  C t dt 
1
2
C
WW
2
t
2
S
2
t
C
W
t
t
S
e-γdt can be approximated as (1- γdt) and any product of dt, dλC, and dz is of order smaller
than dt and can be neglected. Consequently, equation (A.15), (A.16) and (A.17) can be
simplified as below.
U C  λC ( t )
(A.18)
1  η t Wt λCW σ S2  R Dt  R St λC t 
(A.19)
  

 
 


γλC  λCW Wt η t R Dt  R St  R St  C t  12 λCWW 1  η t  σS2 Wt2  λCW 1  η t  σS2 Wt  ηt R Dt  R St  R St λC
(A.20)
2
2
With a view to solve explicitly the system of equations (A.18), (A.19) and (A.20) we
continue to assume that the utility function possesses DARA and is of the form U(C t) =
ln(Ct). The differential equation, (A.20), can be solved as
λC ( t ) 
1 1
Wt
γ
Using the above value of λC ( t ) we can solve equations (A.18), and (A.19) as below:
C t  γWt
1  η t  
(A.21)
R
S
t
 R Dt
σ S2

(A.22)
22
Appendix 3
The bank will determine its optimal credit and deposit rates by maximizing the following
Lagrangian expression (with λB as the Lagrange multipliers):




~
£ B  E t  e ρt U (1  θ t )R Lt L t  R Dt D t dt  e ρ(t  dt) λ B (t  dt) dC t  R Lt L t  R Dt D t  (L t  D t ) dt  R Lt L t dθ 
t

(A.23)





The variables used in expression (A.23) are defined in Section 2. First order conditions
for maximization yield:
£ B
 0  E t (1  θ t )U dt  E t e ρdt λ B (t  dt) dt  dθ   0
L
R t
(A.24)
£ B
 0  E t Udt  E t e ρdt λ B (t  dt)dt  0
D
R t
(A.25)
£ B
B
ρdt B
~  0  λ t   E t e λ (t  dt)
C t
(A.26)






where U is the first derivative of U(.). If we use Ito’s differential rule we get



B
dλ B  λ CB R Lt L t  R Dt D t  (L t  D t ) dt  λ CB R Lt L t dθ  12 λ CC
(R Lt L t σ θ ) 2 dt  

For an explicit solution of the system of equations (A.24), (A.25) and (A.26) we assume
that the utility function possesses DARA and is of the form U(x) = ln(x). e-dt can be
approximated as (1- dt) and any product of dt, dλB, and d is of order smaller than dt
and can be neglected. Consequently, equation (A.24), (A.25) and (A.26) can be
simplified as below.

23

1
E t (1  θ t )   λ B t   R Lt L t λ CB σ θ2  0
πt 













B
ρλ B  λ CB R Lt L t  R Dt D t  (L t  D t )  12 λ CC
(R Lt L t σ θ ) 2  



1
E t    λ B t   0 
πt 







The general solution of the differential equation (A.28) is
 
~
λ B t   z1 .exp αC t
(A.31)
Using the results derived in Appendices 4 and 5, the equation (A.28) and (A.29) can be
simplified as below.
λ CB 
1
k 2t
 1
λ B t   
 kt
(A.32)
 σ θ2 l 2t 
1  2 
kt 

(A.33)
where kt, lt, and Zt are defined in Section 2.1. As we can assume that kt is positive, from
equation (A.33) we get that λ B t  is negative and from equation (A.31) we can infer that
z1 is negative. If λ B t  is of the form stated in equation (A.31), using equation (A.32),
and (A.30) we can conclude that  is the negative root of the following equation.



 ρ  α R Lt L t  R Dt D t  (L t  D t )  12 α 2 (R Lt L t σ θ ) 2  0
From equation (A.34) the value of  can be approximated as below.
24
(A.34)
 2Z
ρ
α   2 t2  
 σθ lt Zt 
(A.35)
By equation (A.32) and (A.31) we can write, λ B .α 
1
and consequently using equation
k 2t
(A.33) we find
α
kt
k 2t  σ θ2 l 2t
(A.36)
Equating equation (A.35) and (A.36) yields
2Z t
k
ρ

 2 t 2 2
2 2
σθ lt Zt k t  σθ lt
(A.37)
Equating equation (A.31) and (A.33) and using the relation (A.36) we get
1
 
 kt

 σ θ2 l 2t 
k
~ 
1  2   z1 . exp   2 t 2 2 C t 
kt 

 k t  σθ lt

(A.38)
Appendix 4
1
Result 1. E θ 1 / π t   
 kt
 σ θ2 l 2t 
1  2  where all the variables are defined in Section 2.
kt 

Proof.

1
E θ 1 / π t   
L
D
 1  θ R t L t  R t D t
 1
= E θ 
 k t  θl t







25
1
 1   k t
 
=  E θ   θ  
 l t   l t
 
 1   k
  E θ  t
 l t   l t
1

k 
  θ t 

 lt 
2
k
 θ 2  t
 lt



3



(We can ignore the higher
order terms as lt>kt)
 1  l σ θ2 l 3t 
    3 
kt 
 l t  k t
 1
 
 kt
 σ θ2 l 2t 
1  2 
kt 

Appendix 5
Result 2. E θ  θ  / π t   
σ θ2 l 2t
where all the variables are defined in Section 2.
k 2t
Proof.

θ
E θ  θ  / π t   E θ 
L
D
 1  θ R t L t  R t D t




 (1  θ)R L t L t  R L t L t  R D t D t  R D t D t
 L
E θ 
1  θ R L t L t  R D t D t
R tLt

1

kt
1 
E θ 1 
lt
 k t  θl t





l
1  E θ 1  θ t

kt






1

lt
1




26




1

lt
1
 k
 kt
 
t

 1  Eθ   θ 
l
 
lt

 t
 

1
2
2
σ l2
1  k t  l σ θ l 3t  
 1    3     θ 2 t (Ignoring higher order terms of the
lt  lt  k t
kt  
k t expansion)


Appendix 6:
 φ σθ2l2t f k t
Result 3. 
φkt


 f σ2l2   cannot be zero for any value of σ θ2 , where  and f are
θ t

defined in Section 3.
Proof.
The first partial derivative of  and f yields:
φ σ2l2  
θ t

k t Ct
1
 z1 
3
2
kt
 k t  σ θ 2 l 2t

2
φ k t   
1 3σ θ l 2t

k 2t
k 4t
2Z t
f σ 2l2   
f k t
2
t
2
t
θ
θ
2
t
1
2
t
θ




 σ θ l 2t
2
2 2
t
2 2 2
θ t
2
t
kt

σ l  k
 k  σ l  




 k  σ l  
2 2 2
θ t
θ t


kC
 exp   2 t t 2 2

 k t  σθ lt



 k  σ l  

 exp  
z 
 k
 k  σ l  

2


 σ l 
k t Ct
2
t
2 2
θ t


2
2 2
t
2 2 2
t
If the equation, f σθ2l2t  
φ σ2l2 f k t
θ t
φkt
2
, holds for some value of σ θ then the following
2
equation must hold for those values of σ θ .
27

2Z t
kt

σ l  k
2 2 2
θ t
2
t

2 2 2
θ t
σ l


 k 2  σ 2 l 2 


k
C
1
t
θ t
1  2 t t 2 2 
 
2
2


2 2
7
2
k t  σ θ l t   2
 k t k t  σ θ l t 
k k 2  σ θ l 2t
 k t  3σ θ 2 l 2t  t t
2

k 2t  σ θ l 2t



















(A.39)
We have obtained equation (A.40) by specifying the values of the partial derivatives in
the relation, f σθ2l2t  
φ σ2l2 f k t
θ t
φkt
, and by using the equation (29). If the relation
2
~ 2 then σ
~ 2 can be written as a functional
(A.39) holds for some value of σ θ equal to σ
θ
θ
form
2
~
σθ  f1 C t , l t , k t 
(A.40)
2
If ~
σ θ exists then equation (A.40) must be satisfied for all realisations of C t, but if we
take Ct = 2kt and if σ θ l 2t  k 2t then the right hand side of equation (A.39) is positive
2
whereas the left hand side is negative. So for Ct = 2kt, equation (A.40) does not hold.
 φ σθ2l2t f k t

 f σ 2l2   cannot be zero for any value
Consequently we can conclude that 
θ t


 φkt

of σ θ2 .
28
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