AT T R I B U T I O N 3 . 0
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Publication date: 17 June 2014 | Vol.3, Issue 1 (2014) 167-180
AUTHOR
MARCO PEDROTTI
Dept. of Banking and Finance,
University of Potsdam, Germany
[email protected]
A Model for the Interest
Margin of a Risk-neutral
Bank. The Role of the Bank
Orientation
ABSTRACT
This paper examines the impact of the bank orientation on classical banking business, distinguishing
between shareholder and stakeholder banks, and analyzes the preconditions for positive social
welfare effects from the existence of stakeholder banks. For this reason we develop a theoretical
bank decision model based on the utility approach and focus on the determination of the interest
margin. Vis-à-vis previous studies, we connect to the recent literature on the agency view considering
the lessons from recent financial developments. Moreover, we study the influence of the social
mission on the bank pricing strategy, merging these two different fields for the first time. The results
highlight the fundamental role of a strict and prudential regulation of the social mission as well as of
the existence of internal and external control mechanisms, in order to avoid a misuse of stakeholder
banks for distorting aims, to increase the system stability and to reduce market failures. Therefore,
a correct implementation of these aspects represents a precondition for a positive contribution of
stakeholder banks to the social welfare.
KEY-WORDS
SOCIAL WELFARE; STAKEHOLDER BANKS; SHAREHOLDER BANKS; COOPERATIVE BANKS; SAVINGS
BANKS; INTEREST MARGIN
JEL Classification: G21, G34, H81, P13 | DOI: http://dx.doi.org/10.5947/jeod.2014.008
167
JEOD - Vol.3, Issue 1 (2014)
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
1. Introduction
The shocks of the global financial crisis led to a reconsideration of the widely held perceptions about the
superiority of certain forms of bank ownership and gave an impulse to the research in this field (Ayadi et al.,
2010). In the past, cooperative and savings banks contributed to the economic and social progress of many
countries and nowadays represent important elements of many financial systems. It is therefore important
to investigate their business strategy with respect to the economic and social effects of their operations on
different stakeholders. The aim of this study is to model the influence of the bank orientation on the bank
pricing strategy, focusing on the credit and deposit business through the study of the interest margin.
We make a distinction between shareholder and stakeholder banks with respect to their target function.
Since shareholder banks aim at the maximization of shareholders interests through profit distribution,
their target function maximizes the expected profits under the balance sheet constraint. On the contrary
a stakeholder bank has to consider the profits and also in addition the claims of other stakeholders, which
are modeled through the introduction in the utility function of a social component. This focuses on the
stakeholder “customers” and measures the dimension of the opportunity costs for this category arising from
the transaction with the bank1. This extension of the target function influences the optimal bank spread,
represented by the interest margin.
The results from the comparative statics show that the social component reduces the interest margin
and thereby serves one of the stakeholder claims if and only if following aspects necessary for a sufficient
efficiency grade are regulated2:
- a detailed definition of the social mission;
- a clear definition of the independence criteria for the bank management;
- an internal supervisory board containing representatives of the stakeholders;
- a stringent supervision from an external supervising authority of the bank’s risk exposition.
If such requirements are not satisfied the achievement of the efficiency condition will be obstructed,
so that the social component within the target function of stakeholder banks won’t produce the expected
decrease in the interest margin.
Section 2 summarizes the literature, section 3 presents the model framework and section 4 develops the
comparative statics properties of the model and interprets the results. Section 5 summarizes and concludes.
2. Literature review and conceptual framework
This work refers to two different research fields in banking studies; we shortly describe the literature of
both sectors: (1) the interest margin and (2) the influence of bank orientation on the bank business model.
1
“Customers” represent only one of many groups of possible stakeholders of the bank. In addition to this group the literature
usually considers other groups and their various claims, like for instance the regional economy, the local public institutions, the
population and the workers (Brämer et al., 2010; Christen et al., 2007; Stiglitz et al., 1993).
2
In addition to the reduction of the interest margin the stakeholder claims usually consider also other aspects of the banking
business, like for instance the availability of a sufficient variety of financial services for all the customers or the regional
presence on the territory (Brämer et al., 2010). However, the consideration of different claims in the model framework would
exponentially increase its complexity. For this reason, we concentrate our model analysis on the dimension of the opportunity
costs as a representative claim of the stakeholder “customers” (Smith et al., 1981). At the later stage of the results’ interpretation
we abstract from the concrete model parameters and refer to fundamental aspects of the stakeholder banking connected to the
agency view (Shleifer and Vishny, 1997). Since these aspects are not specifically connected to the interest rates and concern the
whole claims of the customers, the conclusions are generally applicable to all the claims of the stakeholder “customers”.
168
JEOD - Vol.3, Issue 1 (2014)
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
The first model of the interest margin was presented by Ho and Saunders (1981) and adopted
Intermediation/Dealership
focusing on in
theorder
classical
and combining
thethehedging
and the expectedApproach,
utility approaches
to banking
analyzebusiness
the determinants
of the
bank
hedging
and
the
expected
utility
approaches
in
order
to
analyze
the
determinants
of
bank
margins.
Many of
margins. Many theoretical extensions and empirical studies followed, which led to the inclusion
and empirical
studies followed,
led to the (Maudos
inclusion of
business
feeEntrop
and
thetheoretical
business extensions
fee and different
risk sources
as marginwhich
determinants
andtheSolís
2009;
different
risk sources as margin determinants (Maudos and Solís, 2009; Entrop et al., 2012).
et al.
2012).
With
respecttotothethe
influence
ofbank
the bank
orientation
on the business
model
theretheoretical
is a broad
With respect
influence
of the
orientation
on the business
model there
is a broad
theoretical
which
from
the role
of public
to the public
mission
savings
literature, literature,
which ranges
fromranges
the role
of public
banks,
to the banks,
public mission
of savings
banksof
and
the
banks
and
the
member
value
distribution
in
cooperative
banks.
Sapienza
(2004)
and
Yeyati
member value distribution in cooperative banks. Sapienza (2004) and Yeyati et al. (2004) presentetaal.
(2004) present a detailed overview of different theoretical explanations, which comprehend the
detailed overview of different theoretical explanations, which comprehend the social, the development, the
social, the development, the macroeconomic, the political and the agency view. Focusing on the
macroeconomic, the political and the agency view. Focusing on the German savings banks Brämer et al.
German savings banks Brämer et al. (2010) present a modern interpretation of their public mission
present
modern German
interpretation
of their
public mission
describe
theother
current
German
literature
and(2010)
describe
theacurrent
literature
concerning
this and
topic.
On the
hand,
Angelini
et al.
concerning
this
topic.
On
the
other
hand,
Angelini
et
al.
(1998)
discuss
the
member
value
distribution
(1998) discuss the member value distribution by cooperative banks. Few contributions discuss the
3
3
by cooperative
banks.ofFew
contributions
discuss
theoretical
modeling
of stakeholder
banks
andof
only
theoretical
modeling
stakeholder
banks
andthe
only
recent articles
present
empirical
studies
their
recent articles
present
empirical
peculiarities
(Ferri
et al.
2012). studies of their peculiarities (Ferri et al., 2012).
To
our knowledge,
knowledge,
is the
that explicitly
twofields
different
fields of
To our
thisthis
is the
first first
studystudy
that explicitly
merges merges
these twothese
different
of literature,
literature,
using
the
interest
margin
as
an
instrument
to
study
the
peculiarities
of
the
business
model
using the interest margin as an instrument to study the peculiarities of the business model of stakeholder
of banks.
stakeholder banks.
In the following we differentiate between shareholder and stakeholder banks on the basis of
In the following we differentiate between shareholder and stakeholder banks on the basis of three different
three different criteria, namely the business orientation, the business objective and the role of
criteria, namely the business orientation, the business objective and the role of profits.
profits.
Table 1. Distinguishing criteria between shareholder and stakeholder banks
TABLE
1. DISTINGUISHING CRITERIA BETWEEN SHAREHOLDER AND STAKEHOLDER BANKS
Criteria
Shareholder banks
Stakeholder banks
Business orientation
Focus on the maximization of shareholders’
interests
Focus on the satisfaction of the claims of a
broader category of subjects (stakeholders)
than only the owners
Business objective
Management is subject to the bank value
maximization
Management follows a “double-bottom line”
approach, which describes a multistage target
system4
Role of profits
Profit realization is the main business objective
Profit realization is sought only to reach longterm cost coverage, in order to ensure the
continuity of the banking business5
Source: Own representation on the basis on Ayadi et al. (2009) and Ferri (2010)
Source: Own representation on the basis on Ayadi et al. (2009) and Ferri (2010)
45
The group of shareholder banks consists of private profit-oriented banks, whereas cooperative
and savings banks aim to satisfy the claims of their stakeholders and are therefore defined as
stakeholders-oriented.
Inetdetail,
cooperative
banksof usually
focus
the member
and savings
3
Taylor (1971) and Smith
al. (1981)
model the behavior
cooperative
banks,on
whereas
Barros and surplus
Modesto (1999)
study
public banks.
Christen et al. (2007) were the first authors to define stakeholder banks as “double-bottom line Institutions”: “In addition to
a financial objective, they also have a developmental or social objective. If their managers were asked which of the objectives is
Taylor (1971) and Smith et al. (1981) model the behavior of cooperative banks, whereas Barros and Modesto (1999)
primary, most of them would say that the non-financial objective - extending outreach to people not normally served by banks
study -public
banks.one, and that solid financial performance is a mean to that end rather than an end in and of itself ” (Christen et
is the crucial
4
Christen
et al.
(2007)
werecontributions
the first authors
to the
define
stakeholder
banks
as “double-bottom
line Institutions”:
al., 2007,
p. 2).
Following
extended
non-financial
objective
to other
fields, like for instance
sustaining local“In
addition
to a financial
objective,
also ahave
developmental
or social
their managers
were
asked which
development
(Goglio,
2009) orthey
granting
stablea and
long-term oriented
offerobjective.
of bankingIfproducts
to all classes
of population
of the (Brämer
objectives
is 2010).
primary, most of them would say that the non-financial objective - extending outreach to people not
et al.,
4
3
normally
served by banks - is the crucial one, and that solid financial performance is a mean to that end rather than an
5
Profits of stakeholder banks are usually allocated to the accumulation of reserves, which ensures the bank’s solvency in case of
end inlosses
and or
of crisis
itself”
(Christen et al. 2007, p. 2). Following contributions extended the non-financial objective to other
situations.
fields, like for instance sustaining local development (Goglio 2009) or granting a stable and long-term oriented offer of
banking products to all classes of population (Brämer et al. 2010).
5
Profits of stakeholder banks are usually allocated to the 169
accumulation of reserves, which ensures the bank’s solvency
JEOD - Vol.3, Issue 1 (2014)
in case of losses or crisis situations.
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
The group of shareholder banks consists of private profit-oriented banks, whereas cooperative and savings
banks aim to satisfy the claims of their stakeholders and are therefore defined as stakeholders-oriented. In
detail, cooperative banks usually focus on the member surplus and savings banks on the social welfare.
Since we consider both kinds of stakeholder banks, we do not focus as in Smith et al. (1981) only on the
claim of the members, but on the claim of the broader category “customers”. Therefore, the stakeholders
banks on the social welfare. Since we consider both kinds of stakeholder banks, we do not focus as
considered
in the(1981)
modelonly
include, the
withclaim
respect
banks,
with voting
rights
in
Smith
of to
thecooperative
members,
oncustomer-members
the claim
of the
banks
on et
theal.social
welfare.on
Since we
consider
both
kinds but
of6 stakeholder
banks,
webroader
do not category
focus as
in
the
assembly
as
well
as
customers
without
membership
status
.
“customers”.
considered
include,
with respect
to
in Smith et al.Therefore,
(1981) onlythe
on stakeholders
the claim of the
members,inbutthe
on model
the claim
of the broader
category
cooperative
banks,
customer-members
with
voting
rights
in
the
assembly
as
well
as
customers
“customers”. Therefore, the
stakeholders considered in the model include, with respect to
without
membership
status6.
cooperative
banks, customer-members
with voting rights in the assembly as well as customers
without membership status6.
3. Model framework
3. Model framework
3. Model framework
3.1. A basic bank model
3.1.
3.1. AA basic
basicbank
bankmodel
model
In order to first model the bank profits we use an adapted version of the basic bank model
7
fromIn Gambacorta
(2004)
For
simplicity
wean
focus
onthethe
traditional
banking
business,
In order
to model
first
model
theprofits
bankwe
profits
use
adapted
version
of the
basic
model
order
to first
the. bank
usereasons
anwe
adapted
version
of
basic
bank
model
frombank
Gambacorta
7balance sheet8. In line with Melitz and Pardue (1973) the loan and deposit
excluding
bonds
from
the
from
Gambacorta
(2004)
.
For
simplicity
reasons
we
focus
on
the
traditional
banking
business,
7
(2004) . For simplicity reasons we focus 8on the traditional banking business, excluding bonds from the
demand
functions
depend
only on sheet
the permanent
partMelitz
of the and
income,
whereas
income
excluding
bonds from
the balance
. In line with
Pardue
(1973)the
thetransitory
loan and deposit
balance
sheet8. with
In linea with
Melitz andor
Pardue
(1973) theeffect
loan and deposit
demand
functions
only
is
associated
self-financing
a consuming
does whereas
not
affect
bankdepend
business.
demand
functions depend
only on the permanent
part of theand
income,
thethe
transitory
income
on
permanent
part
ofthe
thebalance
income,sheet
whereas
the transitory
income
isthe
associated
withthe
a self-financing
or
Moreover,
we with
measure
toeffect
changes
money
market
rate through
ana
is the
associated
a self-financing
or a sensitivity
consuming
andindoes
not affect
bank
business.
altered
version
of
the
modified
duration.
consuming
effect
and
does
not
affect
the
bank
business.
Moreover,
we
measure
the
balance
sheet
sensitivity
Moreover, we measure the balance sheet sensitivity to changes in the money market rate through an
altered
version
the modified
duration.
to
changes
in theofmoney
market rate
through an altered version of the modified duration.
δ=
δ=
∑t
∑t
tCF >0
tCF >0
CF
CF
%
(
'%∑ CF( Ai ,0 → tCF ) − ∑ CF(Liabi ,0 → tCF )*(
& Ai
)
Liabi
,0 → tCF )*
' ∑ CF( Ai ,0 → tCF ) − ∑ CF(Liab
i
(tCF +1)
& Ai
)
(1+ rm (tCFLiab)i)
(tCF +1)
(1+∑r A(t ))
i
m i CF
∑A
i
(1)
(1)
i
Where CF are the cash-flows, Ai/Li the corresponding asset/liability, tCF the considered time
Where
are are
the cash-flows,
/Li the
corresponding
asset/liability,
tCF the
considered
time
and
span and
rCF
money
marketArate.
these
changes
the
profit
of span
a bank
i
m the
Where
CF
the cash-flows,
AConsidering
asset/liability,
tCFfunction
the considered
timeisrm
i/Li the corresponding
the
money
rate.
Considering
these
changes
the these
profit
of
bank
determined
by
the revedetermined
revenue
from rate.
the
credit
business
(rlL)function
net of the
(wL),ofthe
revenue
span
and rmarket
thethemoney
market
Considering
changes
theacredit
profitisrisk
function
a bank
is
mby
9
9
from
the the
risk-free
, of
thethe
cost
of
deposits
funding
(r
D),
the
cost
from
the
interest
rate
mS)
d
determined
by
thesecurities
revenue(r(r
from
the
credit
business
(r
L)
net
of
the
credit
risk
(wL),
the
revenue
nue
from
credit
business
L)
net
credit
risk
(wL),
the
revenue
from
the
risk-free
securities
(r
S)
,
l
l
m
9
risk
(TT)
and
the
operative
costs
(C):
from
the
risk-free
securities
(r
S)
,
the
cost
of
deposits
funding
(r
D),
the
cost
from
the
interest
rate
m the cost from the interest rate risk d(TT) and the operative costs (C):
the cost of deposits funding (rdD),
risk (TT) and the operative costs (C):
Π = (rl − w)L + rm S − rd D − TT − C
(2)
Π = (rl − w)L + rm S − rd D − TT − C
(2)
3.2. Utility function: shareholder vs. stakeholder banks
3.2. Utility function: shareholder vs. stakeholder banks
3.2. Utility function: shareholder vs. stakeholder banks
The utility function of the bank is defined as a linear combination of a profit (Π) and a social
The
utility
function
bank
is defined
asbyaasthe
linear
combination
aofprofit
(Π)
and
a social
costs
costs component
(θ)
andofisthe
influenced
(rcombination
deposit
rate
(rd(Π)
). and
The
utility
function
ofdirectly
the
bank
is defined
a loan
linear
a profit
a social
l) and the of
costs component (θ) and is directly influenced by the loan (rl) and the deposit rate (rd).
U (rl ,rd ) = Π(rl ,rd ) − θ (rl ,rd )
(3)
(3)
6 U (rl ,rd ) = Π(rl ,rd ) − θ (rl ,rd )
Since the latter are non-members, they cannot vote in the assembly, but can withdraw from their relationship with the bank.
A short description of the basic bank model is shown in Appendix 1.
7
Since
One the
could
argue
bonds influence
profitvote
of a in
bank
for this but
reason
have tofrom
be considered
within the
balance
latter
arethat
non-members,
theythe
cannot
theand
assembly,
canthey
withdraw
their relationship
with
the
86
6 sheet. However, the net effect of the bond business on the optimal solution would be strongly reduced by its multiplication with
bank.
Since the latter are non-members, they cannot vote in the assembly, but can withdraw from their relationship with the
7
reserve
coefficient.
Since
the reserve
coefficient
usually
very1.low values, an inclusion of the bond business would not
Athe
short
description
of the
basic
bank model
is shown
in assumes
Appendix
bank.
could
argue that
influence
the
profit
of ainbank
for1.this
reason
have see
to Gambacorta
be considered
within the
a significant
impact
the
optimal
interest
margin.
For
aand
modeling
of the
bondthey
business
(2004).
Ahave
short
description
of bonds
theon
basic
bank model
is shown
Appendix
8
balance
sheet. However, the net effect of the bond business on the optimal solution would be strongly reduced by its
9 One could argue that bonds influence the profit of a bank and for this reason they have to be considered within the
As in Gambacortathe
(2004)
we consider the Since
risk-free
securities
either as reserves
deposited
byvery
the central
bank an
or inclusion
as risk-free
multiplication
reserve
thebusiness
reserve on
coefficient
usually
assumes
values,
balance sheet. with
However,
the netcoefficient.
effect of the bond
the optimal
solution
would
be low
strongly
reduced
by its
investments. Since in both cases the interest yield corresponds to the risk-free rate, both alternatives generate the same revenue.
of
the bond business
would
notcoefficient.
have a significant
impact
oncoefficient
the optimalusually
interest
margin.very
Forlow
a modeling
of inclusion
the bond
multiplication
with the
reserve
Since the
reserve
assumes
values, an
business
see Gambacorta
(2004).
of the bond
business would
not have a significant impact on the optimal interest margin. For a modeling of the bond
9
As in Gambacorta
(2004)
we consider the risk-free securities
either as reserves deposited by the central bank or as
business
see Gambacorta
(2004).
170
9
risk-free
investments.(2004)
Since in
casesthe
the risk-free
interest
yield
corresponds
the risk-free
rate, by
boththealternatives
generate
As in Gambacorta
weboth
consider
securities
either
astoreserves
deposited
central bank
or as
JEOD - Vol.3, Issue 1 (2014)
the
same revenue.
risk-free
investments. Since in both cases the interest yield corresponds to the risk-free rate, both alternatives generate
the same revenue.
87
One
from the risk-free securities (rmS)9, the cost of deposits funding (rdD), the cost from the interest rate
risk (TT) and the operative costs (C):
D − for
TTthe−Interest
C Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Π = (rl − w)L + rm S − rAdModel
Pedrotti, M.
(2)
3.2. Utility function: shareholder vs. stakeholder banks
The utility function of the bank is defined as a linear combination of a profit (Π) and a social
component
(θ) and(θ)
is directly
influenced
by the loan
) and the deposit rate (rd). rate (rd).
costs component
and is directly
influenced
by (r
the
l loan (rl) and the deposit
U (rl ,rd ) = Π(rl ,rd ) − θ (rl ,rd )
(3)
The social costs component (θ) measures the opportunity costs of the transaction with the
The
social
costs
component
(θ)
measures
the
opportunity
costs
of
the
transaction
with
the
bank
for
the
customers
and
considers
thereby
as costs
well as
depositors.
opportunity
The
social
costs
component
(θ)
measures
the opportunity
opportunity
costs
of the
theThe
transaction
withcosts
the
The
social
costscosts
component
(θ) measures
theborrowers
opportunity
of costs
the transaction
with
the bank
for
The
social
component
(θ)
measures
the
of
transaction
with
the
bank
for
the
customers
and
considers
thereby
borrowers
as
well
as
depositors.
The
opportunity
costs
for
the
single
borrower
(OCL)
are
represented
by
the
difference
between
the
loan
rate
(r
)
and
the
l
bank
for
the
customers
and
considers
thereby
borrowers
as
well
as
depositors.
The
opportunity
costs
bank
for
the
customers
and
considers
thereby
borrowers
as
well
as
depositors.
The
opportunity
costs
the customers and considers thereby borrowers as well as depositors. The opportunity
10 costs for the single
for
the
single
borrower
(OCL)
are
represented
bybethe
the
difference
between
the
loan
and
the
cheapest
market
alternative,
which
is
assumed toby
thedifference
money market
rate the
(r
. rate
m)loan
6forthe
thethe
single
borrower
(OCL)
arecannot
represented
the
difference
between
the
loan
rate (r
and
the
for
single
(OCL)
are
represented
between
rate
(r(rlll)))market
and
Since
latterborrower
are
non-members,
they
vote in theby
assembly,
but loan
can withdraw
from
their
relationship
withthe
the
10
borrower
(OCL)
are
represented
by
the
difference
between
the
rate
(r
)
and
the
cheapest
10
l
10.
cheapest
market
alternative,
which
is
assumed
to
be
the
money
market
rate
(r
)
m
cheapestmarket
marketalternative,
alternative,which
whichisisassumed
assumedto
tobe
bethe
themoney
money
market rate
rate (r(rmm)) ..
cheapest
market
bank.
10
alternative,
isofassumed
to be the money market rate (r ) .
7
(4)
−which
rm
OCL
= rdescription
A short
the basic bank model is shown in Appendixm1.
l
8
One
could
argue
that
bonds
influence
the
profit
of
a
bank
and
for
this
reason
they
have
to
be
considered
within
the
(4)
OCL
rrl −−−rrrm
(4)
OCL===rsheet.
(4)
OCL
ll
mm
balance
However, the net effect of the bond business on the optimal solution would be strongly reduced by
its
Using with
the the
same
approach
we Since
define
opportunity
costs assumes
for thevery
single
depositor
as the
multiplication
reserve
coefficient.
the the
reserve
coefficient usually
low values,
an inclusion
Using
the
same
approach
we
define
the
opportunity
thethe
single
depositor
Using
the
same
approach
we
define
the
opportunity
costs
for
the
single
depositor
as
the
difference
between
the approach
most
profitable
market
alternative
(rmfor
) costs
and
deposit
rate
(ras
of the
bond
business
would
not have
a we
significant
impact
on thecosts
optimal
interest
margin.
For
a modeling
of the
d).the difference
Using
the
same
approach
we define
define
the
opportunity
costs
for
the
single
depositor
asbond
the
Using
the
same
the
opportunity
for
the
single
depositor
as
the
business
see
Gambacorta
(2004).
difference
between
the
most
profitable
market
alternative
(r
)
and
the
deposit
rate
(r
).
between
the
most
profitable
market
alternative
(r
)
and
the
deposit
rate
(r
).
m
d
difference
between
the
most
profitable
market
alternative
(r
)
and
the
deposit
rate
(r
).
m
d
m
d
difference
between
the
most
profitable
market
alternative
(r
)
and
the
deposit
rate
(r
).
9
As in Gambacorta (2004) we consider the risk-free securities eithermas reserves deposited by thed central bank or as
(5)
OCD
= rm − rd
risk-free investments.
Since in both cases the interest yield corresponds to the risk-free rate, both alternatives generate
(5)
−
r
OCD
=
r
(5)
OCD
rm −−rdrdd
the
same
(5)
OCD
==rrevenue.
mm
The social costs component (θ) is expressed by the sum of both opportunity costs multiplied
11
The
social
costs
component
(θ)
isisand
expressed
by
the
sum
of
both
opportunity
costs
withThe
the
respective
sensitivity
(θl is
/ (θ)
θ(θ)
the
amount
granted/collected
(Lcosts
/ D).multiplied
d) is
social
costscosts
component
(θ)
expressed
by the
sum
of
both
with the
The
social
costs
component
expressed
by the
the
sum opportunity
of both
both opportunity
opportunity
costs multiplied
multiplied
The
social
component
expressed
by
sum
of
costs
multiplied
11
11
with
the
respective
sensitivity
(θ
/
θ
)
the
amount
granted/collected
(L
/
D).
11 and
11
l
d
with
the
respective
sensitivity
(θ
/
θ
)
and
the
amount
granted/collected
(L
/
D).
respective
sensitivity sensitivity
(θl / θd) and
(L / D).
l/ θdamount
d) and granted/collected
with
the respective
(θlthe
the amount granted/collected
(L / D).
(6)
θ = θ l ⋅OCL ⋅ L + θ d ⋅OCD ⋅ D with 0 ≤ θ l ,θ d ≤ 1 and θ l + θ d ≤ 1
(6)
θ
=
θ
⋅OCL
⋅
L
+
θ
⋅OCD
⋅
D
with
0
≤
θ
,
θ
≤
1
and
θ
+
θ
≤
1
(6)
⋅OCL⋅ ⋅LL++θθddd⋅OCD
⋅OCD⋅ ⋅DDwith
with00≤≤θθll l,θ
,θddd ≤≤11and
and θθll l ++θθddd ≤≤11
(6)
θθ==θθll l⋅OCL
The bank aims to maximize the utility function through the determination of the optimal
TheThe
bank
aimsaims
to
maximize
the utility
function
through
the
of component
the optimal
credit
and
bank
to
maximize
the
utility
function
through
the
determination
of
optimal
credit
and
rate,
considering
profit
as wellthrough
as determination
the social
costs
at
the
same
The deposit
bank aims
aims
to
maximizeboth
the utility
utility(Π)
function
through
the determination
determination
of the
the
optimal
The
bank
to
maximize
the
function
the
of
the
optimal
deposit
rate,deposit
considering
both
profit (Π)
as well
as (Π)
the
social
costs
component
at the
same time at
(θ).
credit
and
rate,
considering
both
profit
as
well
as
the
social
costs
component
time
(θ).
credit
and
deposit
rate,considering
considering
both
profit
(Π)as
aswell
well
as the
the social
social costs
costs
component
at the
the same
same
credit
and
deposit
rate,
both
profit
(Π)
as
component
at
the
same
time
(θ).
time
(θ).
time (θ).
(7)
Max U = Max (Π − θ ) = Max π (rl ,rd ) + Min θ (rl ,rd )
rl ,rd
rl ,rd
(7)
Max
U
=
Max
(Π
−
θ
)
=
Max
π
(r
,r
)
+
Min
θ
(r
,r
)
(7)
Max
U = Max
(Π − θ ) = Max π (rl ,rdd))++Min
Minθθ(r
(rl l,r,rdd))
(7)
Max
rl ,rd U = Max
rl ,rd (Π − θ ) = Max π (rl ,r
rl r,rl d,rd
r ,r
l
d
l
d
d
Throughrl ,rl d this
utility function we can differentiate between shareholder and stakeholder banks.
Through this utility function we can differentiate between shareholder and stakeholder banks.
Shareholder
banks
areutility
bound
to profit
maximization,
so thatbetween
the
component
disappears
(θ = 0)banks.
and
Through
this
function
we
can
differentiate
shareholder
and
stakeholder
Shareholder
banks
are
bound
to profit
maximization,
thatsocial
theshareholder
social
component
disappears
(θ =
Through
thisutility
utility
function
wecan
can
differentiateso
between
shareholder
and stakeholder
stakeholder
banks.
Through
this
function
we
differentiate
between
and
banks.
their
utility
depends
only
on
the
generated
amount
of
profits.
On
the
contrary,
stakeholder
banks
follow
Shareholder
banks
are
bound
to
profit
maximization,
so
that
the
social
component
disappears
(θ
0)
and
their
utility
depends
only
on
the
generated
amount
of
profits.
On
the
contrary,
stakeholder
Shareholderbanks
banksare
arebound
bound toto profit
profit maximization,
maximization, so
so that
that the
the social
social component
component disappears
disappears (θ
(θ =
Shareholder
==
0)
and
their
utility
depends
only
on
the
generated
amount
of
profits.
On
the
contrary,
stakeholder
abanks
“double-bottom
line”
approach
and
must
also
consider
the
social
component
in
their
multistage
target
follow
a
“double-bottom
line”
approach
and
must
also
consider
the
social
component
in
their
0)
and
their
utility
depends
only
on
the
generated
amount
of
profits.
On
the
contrary,
stakeholder
0) and their utility depends only on the generated amount of profits. On the contrary, stakeholder
12
12
banks
follow
“double-bottom
line”
approach
and
must
also
consider
the
social
component
multistage
system
0).line”
Their
utility isand
therefore
aalso
combination
of social
profits
and social in
costs
.
system
(θ
> target
0).aaTheir
utility(θis>therefore
aapproach
combination
of
profits
and
social costs
.
banksfollow
follow
a“double-bottom
“double-bottom
line”
approach
and
must
consider
the
social
component
in their
their
banks
must
also
consider
the
component
in
their
12
12.
multistage
target
system
(θ
>
0).
Their
utility
is
therefore
a
combination
of
profits
and
social
costs
12
Substituting
(4)
and
(5)
into
(6)
and
(2)
and
(6)
into
(7)
and
maximizing
(7)
through
the
first
order
multistage
target
system
(θinto
0).
Their
utility
therefore
combination
of(7)
profits
andthe
social
costs
Substituting
(4)
and (5)
(6)
and utility
(2)
andisis(6)
into (7)aaand
maximizing
through
firstcosts
order ..
multistage
target
system
(θ
>>0).
Their
therefore
combination
of
profits
and
social
Substituting
(4)
and
(5)
into
(6)
and
(2)
and
(6)
into
(7)
and
maximizing
(7)
through
the
first
order
condition
we
obtain
the
optimal
interest
margin,
calculated
as
the
difference
between
the
optimal
Substituting
(4)
and
(5)
into
(6)
and
(2)
and
(6)
into
(7)
and
maximizing
(7)
through
the
first
order
Substituting
(4)
and
(5)
into
(6)
and
(2)
and
(6)
into
(7)
and
maximizing
(7)
through
the
first
order
condition* we obtain the optimal* 13
interest margin, calculated as the difference between the optimal credit
condition
we
obtain
the
optimal
interest
margin,
calculated
as
the
difference
between
the
optimal
credit
(r
)
and
deposit
rate
(r
)
.
l
d
condition
we
obtain
the
optimal
interest
margin,
calculated
as
the
difference
between
the
optimal
*
*
13
condition
interest margin, calculated as the difference between the optimal
** we obtain
13
(r
) and(rdeposit
rate (rdthe
)rate
. optimal
13.
l
credit
) and
deposit
(r(rdd***))13
credit
anddeposit
depositrate
rate(r
.
credit
(r(rll*l))and
d ) .
Smith et al. (1981) also discuss the dependence of the Credit Union objective function from the value of the transaction for its
Smith et They
al. (1981)
dependence
Union
objective
fromthethe
of the
members.
calculatealso
the discuss
Net Gainthe
on Loans,
definedofas the
“the Credit
difference
between
the CUfunction
loan rate and
bestvalue
alternative
10
10 market
rate
times
the level
ofThey
loan calculate
activity”
(Smith
al.,of
1981,
p.
520).
Since
aas
customer
usually
does
not have
to
transaction
its(1981)
members.
the Netet Gain
on
Loans,
defined
“the difference
between
the access
CU
loan
rate
Smith
etetfor
al.
also
discuss
the
dependence
the
Credit
Union
objective
function
from
the
value
of
the
10
Smith
al.
(1981)
also
discuss
the
dependence
of
the
Credit
Union
objective
function
from
the
value
ofthe
the
Smith
et for
al.its(1981)
also They
discuss
theanthe
dependence
of on
the
Credit
Unionetasobjective
function
from
the
value
of
the
money
market,
we
consider
the
bank
as
agent,
which
enables
such
refinancing
form.
The
bank
has
to
refinance
itself
on
the
and
the
best
alternative
market
rate
times
level
of
loan
activity”
(Smith
al.
1981,
p.
520).
Since
a
customer
usually
transaction
members.
calculate
the
Net
Gain
Loans,
defined
“the
difference
between
the
CU
loan
rate
transaction
for
its
members.
They
calculate
the
Net
Gain
on
Loans,
defined
as
“the
difference
between
the
CU
loan
rate
transaction
for
itsand
members.
They
calculate
the
NetofGain
on
Loans,
defined
asthe
“the
difference
between
the CU
loan
rate
money
market
its
surplus
from
the intermediation
activity
is measured
by
difference
between
thealoan
rateform.
and
the
and
the
alternative
rate
times
the
level
loan
activity”
etet
al.
1981,
p.
520).
Since
usually
does
notbest
have
access
to market
the
money
we
consider
the
bank
as(Smith
an agent,
which
enables
such
refinancing
The
andthe
the
best
alternative
market
ratemarket,
timesthe
the
level
ofloan
loan
activity”
(Smith
al.
1981,
p.520).
520).
Since
a customer
customer
usually
and
best
alternative
market
rate
times
level
of
activity”
(Smith
et
al.
1981,
p.
Since
a
customer
usually
money
market
rate.
For
this
reason,
the
money
market
rate
represents
in
our
opinion
a
better
choice
as
reference
rate
for
the
does
not
have
access
totoitself
the
money
consider
the
bank
as
an
agent,
which
enables
such
refinancing
form.
bank
has
to refinance
on themarket,
money we
market
and its
surplus
from
the intermediation
activity
is measured
byThe
the
does
not
have
access
the
money
market,
we
consider
the
bank
as
an
agent,
which
enables
such
refinancing
form.
The
does
not
have
access
to
the
market,
we
consider
the
bank
asthis
an
agent,
which
enablesmarket
such refinancing
form.
measurement
of thethe
opportunity
costs
for the
customers
from
theFor
bank
transaction.
bank
has
toto
refinance
itself
on
the
money
market
and
its
surplus
from
the
intermediation
activity
isis measured
by
the
difference
between
loanmoney
rate
and
the
money
market
rate.
reason,
the money
rate
represents
inThe
our
bank
has
refinance
itself
on
the
money
market
and
its
surplus
from
the
intermediation
activity
measured
by
the
bank
hasa to
refinance
itself
on
the
money
market
and itsrate.
surplus
from
the intermediation
activity
is measured
the
11
opinion
better
choice
as
reference
rate
for
the measurement
of
the
opportunity
costs
formarket
the
customers
from theby
bank
difference
between
the
loan
rate
and
the
money
market
this
reason,
the
money
rate
in
our
The sensitivity
of the
utility
function
to
the
opportunity
costs
ofFor
the
credit
(θl) and
the
deposit
market
(θd)represents
summarizes
the
difference
between
the
loan
rate
and
the
money
market
rate.
For
this
reason,
the
money
market
rate
represents
in
our
difference
between
the loan
rate andrate
the for
money
market rate. For
thisopportunity
reason, thecosts
money
market
rate represents
in our
opinion
a better
choice
reference
the
measurement
the
the
customers
from
the
transaction.
orientation
of the
bankas
respect rate
to two
different
customer of
categories.
A value ofcosts
θl = for
1 indicates
a complete
opinion
better
choice
aswith
reference
forthe
themeasurement
measurement
ofthe
the opportunity
opportunity
for
the customers
customers
fromborrower
the bank
bank
opinion
aabetter
choice
as
reference
rate
for
of
costs
for
the
from
the
bank
11
transaction.
orientation,
whereas
a
value
of
θ
=
1
implies
the
consideration
of
only
the
depositors’
claims
(Smith
et
al.,
1981).
The
sensitivity
of
the
utility
function
to
the
opportunity
costs
of
the
credit
(θ
)
and
the
deposit
market
(θ
)
summarizes
l
d
transaction.
d
transaction.
11
11The
sensitivity
the
utility
function
the
opportunity
of
the
credit
(θ
and
the
deposit
(θ
) summarizes
the
orientation
ofof
the
bank
with
respectto
to
two
different costs
customer
categories.
A
value
of θl =market
1 indicates
a complete
12
11
The
sensitivity
of
the
utility
function
the
opportunity
costs
ofthe
the
credit
(θl))l)and
and
thedeposit
deposit
market
(θ
summarizes
The
In
the
literature of
about
stakeholder
bankstoto
many
authors consider
profits
ascredit
necessary
in
order
to
cover market
the
costs(θ
ofddd))the
banking
sensitivity
the
utility
function
the
opportunity
costs
of
(θ
the
summarizes
l A
the
orientation
of
the
bank
with
respect
to
two
different
customer
categories.
value
of
θ
=
1
indicates
a
borrower
orientation,
whereas
a
value
of
θ
=
1
implies
the
consideration
of
only
the
depositors’
claims
(Smith
et al.
l
d
the
orientation
of
the
bank
with
respect
to
two
different
customer
categories.
A
value
of
θ
=
1
indicates
a complete
complete
l = 2009).
activity,
to build
up
reserves
thereby
ensuring
continuity
in the
pursuit of
the social aims
(Ayadi
etθlal.,
the
orientation
of
the
bank
with
respect
to
two
different
customer
categories.
A
value
of
1
indicates
a
complete
borrower
orientation,
whereas
aa value
of
θθdd == 11 implies
the
consideration
of
only
the
depositors’
claims
(Smith
et
1981).
borrower
orientation,
whereas
value
of
implies
the
consideration
of
only
the
depositors’
claims
(Smith
et al.
al.
13
borrower
a value
ofmargin
θdmany
= 1see
implies
the
consideration
ofnecessary
only the in
depositors’
claimsthe(Smith
et
al.
12 For the orientation,
derivationabout
of whereas
thestakeholder
optimal
interest
Appendix
2.
1981).
In
the
literature
banks
authors
consider
profits
as
order
to
cover
costs
of
the
1981).
12
1981).
12In the activity,
many
authors
consider
as
in
cover
costs
of
banking
buildstakeholder
up reservesbanks
thereby
ensuring
continuity
the pursuit
of the social
aimsto
et al.
2009).
12
In theliterature
literaturetoabout
about
stakeholder
banks
many
authors
considerinprofits
profits
as necessary
necessary
in order
order
to(Ayadi
cover the
the
costs
of the
the
13 In the literature about stakeholder banks many authors consider profits as necessary in order to cover the costs of the
banking
totobuild
reserves
thereby
ensuring
in
the
pursuit
of
the
social
aims
(Ayadi
et
al.
2009).
For theactivity,
derivation
of theup
optimal
interest
margin
see continuity
Appendix
2.
banking
activity,
build
up
reserves
thereby
ensuring
continuity
in
the
pursuit
of
the
social
aims
(Ayadi
et
al.
2009).
13
banking
activity,
to build
upoptimal
reserves
thereby
ensuring
continuity
in the pursuit of the social aims (Ayadi et al. 2009).
171
13
the
derivation
of
the
interest
margin
see
Appendix
2.
13 For
Forthe
thederivation
derivationof
ofthe
theoptimal
optimalinterest
interestmargin
margin
see- Vol.3,
Appendix
For
see
Appendix
2.2.
JEOD
Issue 1 (2014)
10
10
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
' 1$
θl
α − θd '
1
1
1 $ 1+ α − θ d − αθ l '
1$ 1
1
r
w+ &
δ t−1Δrm + &
c1 +
c2 ) + & −
+
IM * = − (l1 y p + l2 p) +
)
l0
2(1− θ l )
2 % (1− θ d )(1− θ l ) (
2 % (1− θ l )
(1− θ d ) ( 2 % (1− θ l ) (1− θ d ) )( m
(8)
The
optimal
interest
margin
depends
on factor
two factor
classes:
The
optimal
interest
margin
depends
on two
classes:
-­‐ factors independent from the ownership-type (macroeconomic factors, i.e. permanent income
- factors independent from the ownership-type (macroeconomic factors, i.e. permanent income and
(8)
and price level and reserve coefficient, as well as money market rate);
price
level
and
reserve
coefficient,
as
well
as
money
market
rate);
-­‐ factors depending on the ownership-type (bank-specific factors, i.e. credit risk, social
- factors
depending
on the
(bank-specific
component,
interest
rateownership-type
risk and operative
costs). factors, i.e. credit risk, social component, interest
The
optimal
interest
margin
depends
on
two
factor
classes: the determination of the interest
rate
risk
and
operative
costs).
As we can see from (8) the social component
influences
-­‐
factors
independent
from
the
ownership-type
(macroeconomic
factors,thei.e.
permanent
income
As we
can seea from
(8) in
thethe
social
componentofinfluences
determination
ofsocial
the
interest
margin
margin
through
change
coefficients
different the
factors.
Since
component
is
and
price
level
and
reserve
coefficient,
as
well
as
money
market
rate);
absent
in
shareholder
banks,
they
determine
their
interest
margin
in
order
to
maximize
their
profits.
through a change in the coefficients of different factors. Since the social component is absent in shareholder
-­‐ the
factors
depending
on the
ownership-type
(bank-specific
factors,
i.e. credit
On
contrary,
stakeholder
banks
areindenoted
by
a positive
social
component
thatrisk,
leadssocial
to a
banks,
they
determine
theirrate
interest
margin
ordercosts).
to maximize
their profits.
On
the contrary,
stakeholder
component,
interest
risk
and
operative
modification
of
the
optimal
interest
margin
in
comparison
to
those
of
shareholder
banks.
The
next
banks are
denoted
by a positive
social
that leads to a modification
of the optimal interest
we can
from (8)
thecomponent
social
theuse
determination
of statistics.
the margin
interest
sectionAs
focuses
on see
the influence
of
the
socialcomponent
componentinfluences
through the
of comparative
inmargin
comparison
to those
of shareholder
The nextofsection
focuses
on the influence
the social
componentis
through
a change
in thebanks.
coefficients
different
factors.
Since theof social
component
absent in
banks, they determine their interest margin in order to maximize their profits.
through
theshareholder
use of of
comparative
4. The influence
the socialstatistics.
component
On the contrary, stakeholder banks are denoted by a positive social component that leads to a
modification of the optimal interest margin in comparison to those of shareholder banks. The next
We study the influence of the social component deriving the optimal interest margin over the
section focuses on the influence of the social component through the use of comparative statistics.
sensitivity of the utility function both to the opportunity costs of the credit (θl) and of the deposit
market
(θ
also
study
the
conditions under which the social component increases the social
d). Weof
4.
the
social
component
4.The
Theinfluence
influence
of
the
social
component
welfare through the reduction of the interest margin, which represents the satisfaction of the
14
considered
claim
of low
opportunity
costs
thederiving
stakeholder
“Customers”
. interest
study
the
influence
the component
socialofcomponent
deriving
the interest
optimalmargin
margin
over the
WeWe
study
the influence
of theof
social
the optimal
over
the sensitivity
sensitivity
of
the
utility
function
both
to
the
opportunity
costs
of
the
credit
(θ
)
and
of
the
deposit
of the* utility function both to the opportunity costs of the credit (θl) and of the depositl market (θd). We
also
'
>
c
+
w
+
δ
Δr
if
θ
<
1
r
∂IM
w
+
δ
Δr
+
c
−
r
1
)
market
(θ
).
We
also
study
the
conditions
under
which
the
social
component
increases
the
social
m
1
t−1
m
l
d
(9)
t−1
m
1
m
study <the
which
0 ifconditions under
< 0 →the
( social component increases the social welfare through the reduction
2
2
∂θ l
if θ l = 1margin, which represents the satisfaction of the
welfare
through
the
the interest
)undefined
1−
θ l ) reduction of
(
*
of the interest margin, which represents the satisfaction of the considered claim14 of low opportunity costs
considered claim of low opportunity costs of the stakeholder “Customers” .
of the stakeholder “Customers”14.
From (9) we can derive the conditions for a welfare-increasing influence of the borrower')rm > c1 + w + δ t−1Δrm if θ l < 1
∂IM *
+ c1 − rm
1 w + δ t−1Δrthree
(9)
m
orientation,
different
into consideration: the operative costs for the credit
< 0 if taking
< 0 → ( elements
2
2
∂θ l
θ
=
1
undefined
if
)
1− θ l ) risk (w) and
l
* the interest
business (c1), the (credit
rate risk (δt-1Δrm), which together express the total
costs of the credit business. The social component with respect to the credit business reduces the
From
(9) we
derive
the conditions
forexceeds
a welfare-increasing
influence
of
the
borrower-orientation,
From
(9) can
canmoney
derive
the conditions
for athe
welfare-increasing
influence
thecompletely
borrowerinterest
margin
ifwethe
market
rate
total costs
and the
bank
isofnot
taking
three different
into
consideration:
the
operative
costs
for
the
credit
business
(c
),
orientation,
taking(θelements
three
different
elements
into
consideration:
the
operative
costs
for
the credit
credit
borrower-oriented
≠
1).
These
conditions
imply
the
subjection
of
an
increase
in
the
social
l
1
business
(c
),
the
credit
risk
(w)
and
the
interest
rate
risk
(δ
Δr
),
which
together
express
the
total
1
m
risk
(w) and
the
interest
rate risk
(δt-1Δrm), to
which
express
total costs of
the credit if
business.
The
welfare
through
the social
component
the together
satisfaction
oft-1the
an efficiency
condition:
the bank
is
costs
of
the
credit
business.
The
social
component
with
respect
to
the
credit
business
reduces
the
efficient
enough with
to produce
investing
funds the
at the
money
market
rate,
it can
lowerrate
the
social
component
respect atosurplus
the credit
businessitsreduces
interest
margin
if the
money
market
interest
margin
if them money
market
rate exceeds
the
total
costs and the bank is not completely
credit
rate
and
the
same
cover its
total
costs.
l towards
exceeds
the rtotal
costs rand
theatbank
is nottime
completely
borrower-oriented
(θl ¹ 1). These conditions imply
borrower-oriented
(θ
≠
1).
These
conditions
imply
the
subjection
of
in the social
l
The
first
derivative
of
(8)
over
θ
leads
to
the
following
results
for an
thetoincrease
depositor-orientation,
d
the subjection of an increase in the social welfare through the social component
the satisfaction of an
welfare
through
the
social
component
to
the
satisfaction
of
an
efficiency
condition:
if
the bank
is
which again consider the money market rate compared to the operative costs for the
deposit
efficiency
condition:
ifproduce
the banka is
efficient
enough its
to produce
athe
surplus
investing
itsrate,
funds
at the
money
efficient
enough
to
surplus
investing
funds
at
money
market
it
can
lower
the
business (c2) and the interest rate risk (δt-1Δrm) under the assumption of a not perfect depositor
market
rate, ritl towards
can lowerrmthe
rl towards
rm and
the same
credit rate
andcredit
at therate
same
time cover
itsattotal
costs.time cover its total costs.
orientation.
derivative
(8) over
θd leads
the following
the depositor-orientation,
TheThe
firstfirst
derivative
of (8)ofover
θd leads
to thetofollowing
results results
for the for
depositor-orientation,
which
which
again
consider
the
money
market
rate
compared
to
the
operative
costs
for
the
deposit
again* consider the money market rate
( compared
c2 + αδ t−1Δrmto the operative costs for the deposit business (c2) and the
(10)
if θ)d <under
1
∂IM
αδ2t−1
Δr
+ c2the
+ (α −interest
1)rm
1(c
*rm > risk (δt-1Δr
business
)
and
rate
the assumption of a not perfect depositor
m
m
(1− α )
<
0
if
<
0
→
)
interest
rate
risk
(δ
Δr
2 ) under the assumption of a not perfect depositor orientation.
2
∂θ d
*undefined if θ = 1
(1−t-1θ d ) m
orientation.
d
+
(
c2 + αδ t−1Δrm
(10)
if θ d < 1
∂IM *
1 αδ t−1Δrm + c2 + (α − 1)rm
*r >
(1− α )
< 0 if
<0→ ) m
2
2
∂θ d
(1− θ d ) margin is also*+interpreted
14
undefined ifin
θ dthe
= 1 literature as an increase in the social welfare, because of the reduction
A reduction of the interest
14
of the intermediation costs of the banking business (Saunders and Schumacher, 2000). Furthermore, a low interest margin
represents in our model the satisfaction of the interest of the stakeholders “customers”, since lower credit rates and higher
Adeposit
reduction
of the the
interest
margincosts
is also
interpreted
in the with
literature
as an increase in the social welfare, because of
rates reduce
opportunity
from
their transaction
the bank.
the reduction of the intermediation costs of the banking business (Saunders and Schumacher 2000). Furthermore, a low
interest margin represents in our model the satisfaction of the interest of the stakeholders “customers”, since lower
credit rates and higher deposit rates reduce the opportunity172
costs from their transaction with the bank.
14
JEOD - Vol.3,
1 (2014)
A reduction of the interest margin is also interpreted
in Issue
the literature
as an increase in the social welfare, because of
the reduction of the intermediation costs of the banking business (Saunders and Schumacher 2000). Furthermore, a low
interest margin represents in our model the satisfaction of the interest of the stakeholders “customers”, since lower
borrower-oriented (θl ≠ 1). These conditions imply the subjection of an increase in the social
welfare through the social component to the satisfaction of an efficiency condition: if the bank is
efficient enough to produce a surplus investing its funds at the money market rate, it can lower the
credit rate rl towards rm and at the same time cover its total costs.
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
The first derivative
of (8) over θd leads Pedrotti,
to theM.following results for the depositor-orientation,
which again consider the money market rate compared to the operative costs for the deposit
business (c2) and the interest rate risk (δt-1Δrm) under the assumption of a not perfect depositor
orientation.
(
c2 + αδ t−1Δrm
if θ d < 1
∂IM *
1 αδ t−1Δrm + c2 + (α − 1)rm
*rm >
(1− α )
< 0 if
<
0
→
)
2
2
∂θ d
*undefined if θ = 1
(1− θ d )
d
+
(10)
Also within
market
a positive
influence
of the social
is subjectedistosubjected
an efficiency
Also
withinthe
thedeposit
deposit
market
a positive
influence
of thecomponent
social component
to an
constraint.
However, in
contrast to
credit market
the conditions
(10) imply afrom
consideration
of a
efficiency
constraint.
However,
in the
contrast
to the credit
market from
the conditions
(10) imply
the
interest
rate
risk
only
with
respect
to
the
reserves/risk-free
securities,
since
their
interest
rates
cannot
consideration of the interest rate risk only with respect to the reserves/risk-free securities, since their
14
interest
ratesduring
cannot
be
aligned
the
Furthermore,
thewith
efficiency
isthe
calculated
only
be
period.
Furthermore,
the period.
efficiency
is calculated
only
respect
share
of the
Aaligned
reduction
of thethe
interest
margin
isduring
also interpreted
in the literature
as an increase
in the
social to
welfare,
because
of
the
reduction
of
the
intermediation
costs
of
the
banking
business
(Saunders
and
Schumacher
2000).
Furthermore,
a
low
15
with
respect
to
the
share
of
the
balance
sheet
not
invested
in
risk-free
securities
(1-α)
and
therefore
balance sheet not invested in risk-free securities (1-α) and therefore available for the banking business .
interest margin represents in our model 15
the satisfaction of the interest of the stakeholders “customers”, since lower
available
the banking
business
. Since
an increase
in thefunds
reserve
coefficient
will
Since
anfor
increase
in deposit
the reserve
coefficient
will reduce
thefrom
available
forwith
loans,
the bank
willreduce
have tothe
credit rates
and higher
rates reduce
the opportunity
costs
their transaction
the bank.
available
funds
for
loans,
the
bank
will
have
to
increase
the
level
of
efficiency
in
order
to
reduce
the
16
increase the level
16 of efficiency in order to reduce the interest margin .
interest margin .
Summing up the conditions for a positive welfare effect of the social component within a matrix, we
Summing up the conditions for a positive welfare effect of the social component within a
can highlight the similarities and differences between the credit and the deposit market.
matrix,
we can highlight the similarities and differences between the credit and the deposit market.
Table 2. Conditions for a positive welfare effect of the social component
TABLE
2. CONDITIONS FOR A POSITIVE WELFARE EFFECT OF THE SOCIAL COMPONENT
Object
Credit market
Deposit market
Stakeholder bank
orientation
Not complete borrower-oriented
Not complete depositor-oriented
Cost-revenue
relationship
(efficiency)
The money market rate exceeds the total costs
of the loan business, defined as the sum of
operative costs of the loan business, of
deductions for the credit risk and the cost from
the interest rate risk
The money market rate exceeds the total costs
of the deposit business (defined as the sum of
operative costs and costs from the interest rate
risk on reserves) in relation to the available
funds for the banking business17
Influence of banking
supervision /
monetary policy
Supervision of the interest rate risk based on
hard data
Supervision of the interest rate risk based on
hard data and determination of the reserve
coefficient
Art of welfare
distribution
If the efficiency condition is fulfilled a
stakeholder bank can satisfy the claim of the
“borrowers”, reducing the credit rate towards
the money market rate
If the efficiency condition is fulfilled a
stakeholder bank can satisfy the claim of the
“depositors”, increasing the deposit rate
towards the money market rate
Source: Own elaboration
Source: Own elaboration
As we can see in Table 2 the bank-orientation and the cost-revenue relationship play an
As werole
canin
seethe
in Table
the bank-orientation
the cost-revenue
relationship play
an important
role
important
credit2and
deposit market. and
Furthermore
the cost-revenue
relationship
is directly
in
the
credit
and
deposit
market.
Furthermore
the
cost-revenue
relationship
is
directly
influenced
by
influenced by the interest rate risk and for the deposit market also by the reserve coefficient. the
Both
elements
control
of external
authorities,
i.e. the Both
banking
supervision
authority
interest are
rate subjected
risk and fortothethe
deposit
market
also by the
reserve coefficient.
elements
are subjected
to
and/or
the
central
bank,
which
therefore
indirectly
affect
the
achievement
of
the
efficiency
the control of external authorities, i.e. the banking supervision authority and/or the central bank, which
condition.
therefore indirectly affect the achievement of the efficiency condition.17
In
ordertotoincrease
increase
social
welfare
the considered
stakeholders
throughofa the
reduction
In order
thethe
social
welfare
for the for
considered
stakeholders
through a reduction
interest of
the interest margin, stakeholder banks have to reach an efficient cost structure, which subjects the
margin, stakeholder banks have to reach an efficient cost structure, which subjects the effect of the social
effect of the social component to an efficiency precondition. The role of an external control and the
consideration of the efficiency condition provide a connection between our work and the literature
on the different explanations of public ownership in banking, particularly to the recent contributions
15
We
assume view
a reserve(Andrianova
coefficient α<1. et al. 2010).
to the
agency
On the other hand, a decrease of the reserve coefficient would imply an increase in the available funds for loans. As a consequence,
the same operative costs could be covered by revenues originating from a larger amount of granted loans, decreasing the cost
fraction per loan unit and enabling a reduction of the interest margin.
16
The available funds are defined as the total assets net of the reserve coefficient.
17
15
173
We assume a reserve coefficient α<1.
JEOD - Vol.3, Issue 1 (2014)
On the other hand, a decrease of the reserve coefficient would imply an increase in the available funds for loans. As a
consequence, the same operative costs could be covered by revenues originating from a larger amount of granted loans,
16
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
component to an efficiency precondition. The role of an external control and the consideration of the
efficiency condition provide a connection between our work and the literature on the different explanations
of public ownership in banking, particularly to the recent contributions to the agency view (Andrianova
et al., 2010).
Figure 1. Agency problems by public savings banks18
FIGURE 1. AGENCY PROBLEMS BY PUBLIC SAVINGS BANKS18
Source: Own elaboration
Source: Own elaboration
According to the classical approach to the agency view and to Figure 1 public savings banks
are subjected to increasing agency conflicts as a consequence of the separation between the
According
thethe
classical
approach
the agency
androle
to Figure
1 public
banks
areTax
subjected
ownershiptoand
control
of these to
banks
and theview
double
covered
by the savings
governing
party.
payers
(Principal)
own
public
saving
banks
and
aim
to
the
achievement
of
social
objectives
through
to increasing agency conflicts as a consequence of the separation between the ownership and the control
the banking activity. Due to their dispersed ownership they delegate the control of the bank to the
of these
banks and the double role covered by the governing party. Tax payers (Principal) own public
governing party (Agent), whose members usually follow political aims and have to report to the
savingowners.
banks and
aim to thethe
achievement
social
through the
Due to their
Furthermore,
governing of
party
alsoobjectives
acts as principal
withbanking
respect activity.
to the bank’s
management,
whose
objectives
differ
from
those
of
the
governing
party
as
well
as
from
those
of the
dispersed ownership they delegate the control of the bank to the governing party (Agent), whose
members
tax payers. As a result of this double separation public saving banks have two different agency
usuallyconflicts:
follow political
aims and have to report to the owners. Furthermore, the governing party also acts
on the one hand the conflict between the management of the public bank and the
as principal
withparty,
respect
to other
the bank’s
management,
whose
fromthethose
of theThis
governing
governing
on the
hand the
conflict between
theobjectives
governing differ
party and
tax payers.
leads
an those
increase
making
the control
of the
banksaving
more banks
party fact
as well
as to
from
of in
thethetaxasymmetric
payers. Asinformation,
a result of this
double
separation
public
difficult, and decreases the internal efficiency and thereby the social welfare from the banking
have two
different agency conflicts: on the one hand the conflict between the management of the public
activity19. If the internal efficiency losses exceed the welfare gains from the achievement of social
bank and
thetheother
hand
the conflict
between
the governing
party
and the tax
aims,the
thegoverning
net welfareparty,
effect on
from
activity
of public
banks will
be negative
(Shleifer and
Vishny,
1997).
Based
on
the
lessons
from
the
recent
financial
crisis
Andrianova
et
al.
(2010)
criticize
this more
payers. This fact leads to an increase in stthe asymmetric information, making the control of the bank
approach, stating that in the 21 century and in a context of weak corporate governance, 19
difficult,
and decreases the internal efficiency and thereby the social welfare from the banking activity . If
opportunistic politicians can extract more private rents using privately owned banks as a
20
the internal
efficiency
exceed in
thea welfare
gains
from
the sector
achievement
of socialand
aims,
net welfare
mechanism
. On losses
the contrary,
democracy
with
public
accountability
strictthe
control
procedures
the
distortion
of
resources
for
own
personal
advantage
would
be
more
difficult
effect from the activity of public banks will be negative (Shleifer and Vishny, 1997). Based on thetolessons
implement and the positive role of public banks would be granted (Andrianova et al. 2010). Our
from the recent financial crisis Andrianova et al. (2010) criticize this approach, stating that in the 21st
finding of a direct connection between the institutional framework, the efficiency level and the
century
and ineffect
a context
of weak banks
corporate
canAlso
extract
more private
welfare
of stakeholder
is in governance,
line with thisopportunistic
approach to thepoliticians
agency view.
according
18
The agency view usually refers to public banks in general (Shleifer and Vishny 1997). However, since we consider
only savings banks and cooperative banks as stakeholder banks, we refer in the explanation of the agency view to public
savings banks as members of the broader group of public-owned banks.
19
“[…] de facto control rights belong to the bureaucrats. These bureaucrats can be thought of as having extremely
concentrated control rights, but no significant cash flow rights because the cash flow ownership of state firms is
18
The effectively
agency view
usually refers to public banks in general (Shleifer and Vishny, 1997). However, since we consider only savings
dispersed amongst the taxpayers of the country. Moreover, the bureaucrats typically have goals that are very
banks
and cooperative
as stakeholder
banks,bywe
refer
in the interests
explanation
of theownership
agency view
to public
savingsofbanks as
different
from socialbanks
welfare,
and are dictated
their
political
[...] State
is then
an example
members
of
the
broader
group
of
public-owned
banks.
concentrated control with no cash flow rights and socially harmful objectives” (Shleifer and Vishny 1997, p. 768).
20
The
developments
in the
American
system
represent
an explanatory
case of of
extracting
private
rents concentrated
from
19
“[…] de
facto
control rights
belong
to thefinancial
bureaucrats.
These
bureaucrats
can be thought
as having
extremely
private
banks.
In significant
the last decades
regulation
was weakened
private
became
public subsidies
control
rights,
but no
cash flow
rights because
the cash and
flowmany
ownership
of banks
state firms
is effectively
dispersedtoamongst
overcome the financial crisis. At the same time the bank donations to political parties increased and many politicians
the taxpayers
of the country. Moreover, the bureaucrats typically have goals that are very different from social welfare, and are
became members of the boards of private banks, participating thereby to their increasing short term profits (Andrianova
dictated by their political interests [...] State ownership is then an example of concentrated control with no cash flow rights and
et al. 2010).
socially harmful objectives” (Shleifer and Vishny, 1997, p. 768).
174
JEOD - Vol.3, Issue 1 (2014)
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
rents using privately owned banks as a mechanism20. On the contrary, in a democracy with public sector
accountability and strict control procedures the distortion of resources for own personal advantage would
be more difficult to implement and the positive role of public banks would be granted (Andrianova et al.,
2010). Our finding of a direct connection between the institutional framework, the efficiency level and the
welfare effect of stakeholder banks is in line with this approach to the agency view. Also according to the
agency view, we subordinate a positive effect of the social component on the social welfare to an internal
efficiency precondition, which guarantees that the reduction of the interest margin is possible only if the
bank can produce a surplus from the investments on the money market21. Furthermore, in accordance
with the agency view, we highlight the influence of an external and internal regulatory environment on the
fulfillment of the social mission.
Combining the results of our model with the conclusions from the agency view, we can state that in
order to avoid a misuse of stakeholder banks for distorting aims and to reach a positive effect from the social
component in banking the following four aspects regarding the regulatory environment are fundamental22:
- a strict internal definition of the social mission must be included in the statute of the bank;
- the statute of the bank must define strict independence criteria for the management, which protect it from external distorting influence;
- within the bank’s governance structure an internal supervisory board containing representatives of the
stakeholders must be created;
- an external supervising authority must pursue a constant control of the bank’s risk exposition.
The first aspect refers to the need of a concretization of the social mission in the bank’s statute through
the determination of the considered stakeholders, of their claims and of the instruments adopted to fulfill
them23. In our model and depending on the bank orientation this would correspond to the commitment
for the bank to satisfy the claim of the borrowers/customers for mostly favorable deposits/credit rate conditions,
which should be achieved through a reduction of the interest margin if the efficiency condition is satisfied.
A concrete definition of the social aims as well as of the necessary preconditions for their fulfillment would
enable a neutral external evaluation of the bank’s performance and an increase in the transparency level.
Furthermore, the definition of independence criteria in the statute should protect the management from
political influences, which would distort their activity from the achievement of the social mission. The
third aspect regards the control of the management activity through reporting to an internal supervisory
board, which should include representatives of the different stakeholders affected by the social mission.
The direct inclusion of the affected stakeholders in the monitoring of the management activity would
enable a control of the congruity between the guidelines in the statute and their practical implementation
by the management. Moreover, this control mechanism would also increase the transparency level and the
The developments in the American financial system represent an explanatory case of extracting private rents from private banks.
In the last decades regulation was weakened and many private banks became public subsidies to overcome the financial crisis. At
the same time the bank donations to political parties increased and many politicians became members of the boards of private
banks, participating thereby to their increasing short term profits (Andrianova et al., 2010).
20
In other words, an increase of the social welfare through a reduction of the interest margin can take place only if the bank is able
to extract a net positive rent from its investments on the money market, which is distributed to the stakeholder “Customers”.
On the contrary, in case of a negative net rent, losses have to be covered by an increasing margin, what takes to increasing
intermediation costs and a decrease in the social welfare. Through this mechanism, a neutral evaluation of the bank’s performance
can take place and the transparency level of the activity of stakeholder banks is increased.
21
22
Here we abstract from the specific stakeholders “Customers” considered in the model (and their claim to most low opportunity
costs from the transaction with the bank) and derive general conclusions applicable to different claims and stakeholders.
Also Yeyati et al. (2004) underline the importance of the definition of the bank’s objective and mission for the success of a state-owned
bank.
23
175
JEOD - Vol.3, Issue 1 (2014)
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
asymmetric information arisen from the principal-agent conflict could be reduced24. In our theoretical
model this aspect would imply a reporting activity to representatives of the borrowers/customers about
the determination of the interest margin and the explanation of its changes connected to the fulfillment of
the efficiency condition. The last aspect considers the control for the risk exposition of the bank, which is
represented in the model by the credit and the interest risk components and directly influences the efficiency
condition25. Since an effective control requires specific skills and a reliable and long-term collection of
hard data, the permanent monitoring of the risk measures should be pursued by an external supervising
authority.
In conclusion, thanks to these control mechanisms a distorting use of the bank resources could be
limited and thereby a more efficient activity of the public bank would be granted. On the contrary, the
lack of one of the four aspects would favor a misuse of the bank for purposes different from the stakeholder
claims, distorting its resources from the original aims. As a result, a suboptimal contribution to the social
welfare from the intervention of the stakeholder bank would be reached26.
5. Conclusions
In this work we present a theoretical model for the study of the influence of the bank orientation on
the classical banking business, adopting the interest margin as instrument. Extending the basic bank model
of Gambacorta (2004) and considering the contributions of Barros and Modesto (1999) and Smith et al.
(1981) we develop a one-period decision model for banks based on the utility approach, which is able to
differentiate between shareholder and stakeholder banks.
To our knowledge this is the first theoretical model that analyzes the relationship between the social
mission and the bank pricing strategy, focusing on the conditions under which a stakeholder-orientation
produces an increase in the social welfare. The results highlight the fundamental role of a strict and
prudential regulation of the social mission as well as of the existence of internal and external control
mechanisms, in order to avoid a misuse of stakeholder banks for distorting aims. We therefore refer to
the literature on the agency view, particularly to its recent developments considering the lessons from the
last financial crisis. The dramatic developments of the last years showed that an extreme profit orientation
within the banking system can lead to the formation of massive agency conflicts by private banks and
thereby endanger the stability of the entire financial system, producing giant social costs for bank rescues
(Andrianova et al., 2010). The negative externalities deriving from an excessive profit orientation can be
compensated by a stakeholder-orientation, which through a long-term multistage target system can help
to increase the systems’ stability. Moreover, the consideration of the stakeholders’ claims can reduce the
intermediation costs of the banking activity, positively contributing to a reduction of the market failures
Also within the agency view a clear accounting and a constant evaluation of the mission fulfillment are defined as important
preconditions for a reduction of the principal-agent conflicts and thereby for a successful intervention of state-owned banks
(Yeyati et al., 2004; Andrianova et al., 2010).
24
An excessive increase in the risk exposition would hamper the fulfillment of the efficiency condition. Furthermore, it could also
endanger the stability of the financial system (Andrianova et al., 2010). For this reason, their measurement should be subjected
to an external supervision.
25
In our model, the fulfillment of these aspects would favour the achievement of a sufficient efficiency grade of the bank and the
reduction of the opportunity costs would be facilitated. On the contrary, the failing of one of these aspects would reduce the
bank’s transparency, favoring an increase in the cost components and taking to suboptimal level of the interest margin.
26
176
JEOD - Vol.3, Issue 1 (2014)
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
and therefore to an increase in the social welfare (Stiglitz et al., 1993).
In order to empirically prove the validity of the results, we will pursue an empirical study of the
interest margin within the German and the Italian banking markets. The particular developments of both
banking markets during the last decades offer us a useful sample for the analysis of the influence of banking
regulation on the functioning of the social component.
Appendix 1
Appendix 1
Appendix
1
We present
adapted version of the basic bank model by Gambacorta (2004). We consider ourselves
Appendix
1 anan
We present
adapted version of the basic bank model by Gambacorta (2004). We consider ourselves
to
be
in
aa one
period
model
aarisk
bank,
which
isispart
of an
market. Its
sheet
to
be
in
one
period
model of
ofversion
riskneutral
neutral
bank,bank
which
part
anoligopolistic
oligopolistic
Itsbalance
balance
sheet
We present
an adapted
of the basic
model
by of
Gambacorta
(2004).market.
We consider
ourselves
Appendix
1
We
present
an
adapted
version
of
the
basic
bank
model
by
Gambacorta
(2004).
We
consider
ourselves
is
defined
by:
is
defined
to be in a one period model of a risk neutral bank, which is part of an oligopolistic market. Its balance sheet
to be in a one period model of a risk neutral bank, which is part of an oligopolistic market. Its balance sheet
is defined
We by:
present an adapted version of the basic bank model by Gambacorta (2004). We consider ourselves
isL defined
by:
(A1.1)
K =neutral
K
=D
K period
withmodel
S =αD
to +beS in
a +one
of and
a risk
bank, which is part of an oligopolistic market. Its balance
sheet
(A1.1)
S = D +by:
K
with S = α D and K = K
isL +defined
(A1.1)
and
K securities,
= Ksecurities,
L + Swhere
=where
D + LK =
L loans,
= with
loans,
=Drisk
= Deposits,K K= =equity
equity(exogenous
(exogenousand
andbigger
biggerthan
than
the
SS ==Sαrisk
freefree
DD
= Deposits,
the
27
regulatory
capital)
,
α
=
reserve
requirements
/
risk-free
investments.
27
loans,
securities,
D = Deposits,
K = equity (exogenous and bigger than
the
(A1.1)
α=Drisk
and free
K=K
L + S =where
D + capital)
K L = with
regulatory
, aS==SSreserve
/ risk-free
investments.
where
L =demand
loans,
= riskrequirements
free securities,
D interest
= Deposits,
K loans
= equity
andonbigger
than the
27
The capital)
loan
depends
negatively
on/ the
rate on
(rl) (exogenous
and positively
the permanent
regulatory
,
α
=
reserve
requirements
risk-free
investments.
27
p
The (y
loan
demand
depends
negatively
on the
interest
rate
on loans (rl) and positively on the permanent
regulatory
, αlevel
=depends
reserveand
requirements
/ the
risk-free
investments.
income
),
the
price
thesecurities,
money
(rrate
m). on
Thepcapital)
loan
onmarket
(rl) and
positively
the permanent
where
L =demand
loans,
S =(p)
risk negatively
free
D interest
= rate
Deposits,
K loans
= equity
(exogenous
andonbigger
than the
loan
depends
negatively
onmarket
the
interest
income
(yp),
), the
thedemand
price
level
(p)and
and
themoney
money
market
rate(rrate
(r ).on loans (rl) and positively on the permanent
27 level
incomeThe
(y
price
(p)
the
rate
m).m
regulatory
,
α
=
reserve
requirements
/
risk-free
investments.
p capital)
p
income
(p)with
andlthe
m).
(A1.2)
rl (y
+ l),
ythe
+ lprice
p + l3level
rm depends
<0,money
l1 >0, on
l2 market
>0,
>0rate (rrate
Ld = l0The
negatively
thel3interest
on loans (rl) and positively on the permanent
1loan demand
2
0
p p
d
(A1.2)
income
(p)
andl the
market
+ l),y pthe
+ lprice
p + l3level
r
with
<0,money
l >0, l >0,
l3 >0 rate (rm).
L = l rl (y
(A1.2)p
+ l11deposit
y + l22 p demand
+ l3rmm depends
with l00 <0,
l11 >0, l22 >0,
>0interest rate on deposits (r ), the permanent income
Ld = l00rThe
l
3
positively
on lthe
(y )
d
d
p
pp
and
the
price
level
(p).
It
depends
negatively
on
the
money
market
rate
(r
).
(A1.2)
m
deposit
demand
depends
positively
on
the
interest
rate
on
deposits
(r
),
the
permanent
income
(y
)
l0 rThe
+
l
y
+
l
p
+
l
r
with
l
<0,
l
>0,
l
>0,
l
>0
L =The
depends0 positively
interest rate on deposits (r
l
1deposit
2 demand
3 m
1
2 on the
3
d d), the permanent income (y p)
The
deposit
demand
depends
positively
on
the
interest
rate
on
deposits
(r
d), the permanent income (y )
and the
price
(p).
ItItdepends
negatively
on
rate
(rm(r). ).
and
theprice
pricelevel
level
(p).It
dependsnegatively
negativelyon
onthe
themoney
moneymarket
market
rate
and
(p).
thed money
rate (rm).m
(A1.3)
= dThe
r + d1level
y p + ddemand
p + ddepends
rdepends
withpositively
d0 >0, d1 >0,
>0, d3 <0market
D d the
on the
rate on deposits
(rd), the permanent income
(yp)
0 d deposit
2
3 m
2 interest
d
p
(A1.3)
and
negatively
d0price
r + d level
y + d(p).
p +Itddepends
r
with
d >0, don
>0,the
d money
>0, d3 <0market rate (rm).
D =the
(A1.3)
rd + d11 y p + dto22 pthe
+ d33quantities
rmm
withgranted/collected
d00 >0, d11 >0, d22 >0,other
d3 <0 risk and costs factors affect bank profits.
D d = dIn
0 d addition
With
d
p
respect
to
the
interest
risk
we
consider
an
adapted
concept
of
the
modified
duration,
which
measures
the
(A1.3)
r
+
d
y
+
d
p
+
d
r
with
d
>0,
d
>0,
d
>0,
d
<0
D = dIn
to2 the 3quantities
granted/collected
other
risk and costs factors affect bank profits. With
0 d addition
1
m
0
1
2
3
In
addition
to
the
quantities
granted/collected
other
risk
and
costs
factors
affect
bank
profits.
With
balance
sheet
sensitivity
to
changes
in
the
money
market
rate.
respect
to the interest
we consider
an adapted concept
of and
the modified
duration,
measures
the
In addition
to the risk
quantities
granted/collected
other risk
costs factors
affect which
bank profits.
With
respect
to
the sensitivity
interest risk
we consider
an
adapted
concept
of the modified duration, which measures the
balance
sheet
to
changes
in
the
money
market
rate.risk
Into
addition
to the
quantities
granted/collected
other
andmodified
costs factors
affectwhich
bank measures
profits. With
respect
the
interest
risk
we
consider
an
adapted
concept
of
the
duration,
the
balance sheet sensitivity to changes% in the money market rate.
(
respect to the interest risk we consider
adapted
CF(
Aan
,0 →
tCF ) −market
CF(Liab
,0of
→ the
tCF )* modified duration, which measures the
∑concept
' ∑in
i money
i
balance sheet sensitivity to changes
the
rate.
%
(
&
A
Liab
balance sheet sensitivity to changes
theA ,0
money
market rate.,0 → t ) )
∑ tCF %' ∑inCF(
→ tCF ) − ∑ CF(Liab
(
(t +1)
i
i
CF *
t >0
(A1.4)
))
TT = δ t−1Δrm ( L + S ) where
& A CF( Ai ,0 →(1+
tCF r)m−(tLiab
,0 → tCF ))*
CF CF(Liab
∑
i
δ = ∑ tCF '& ∑
% A
()
A(ti Liab) )(t +1)
∑
t∑
>0 t
(A1.4)
1+
r
TT = δ t−1Δrm ( L + S ) where
(
CF
(t +1)
CF( Ai ,0 →1+
,0 → tCF )*
∑))CF(Liab
i
(A1.4)
TT = δ t−1Δrm ( L + S ) where δ = t >0 '& ∑
( tCFir)mmA−(tCFCFLiab
)
A
i
i
CF
CF
i
i
i
i
CF
CF
CF
∑
∑r A(t ))
1+
TT = δ t−1
Δris
( L + S ) where
(
δt-1
of
assets
in
the
event
of a 1% increase in the money market rate on the (A1.4)
basis of
m the loss pro unit
δ=
A composition of the balance sheet), A is the asset i, Liab
∑
the modified
duration
(derived
from
the
maturity
δt-1 is the loss pro unit of assets in the event of a 1% increase in the money marketi rate on the basis ofi
δ=
CF
∑ tCF
tCF >0
i
i
i m
i
i
i
(tCF +1)
CF
i
δt-1 isthe
the
loss pro(derived
of assets
in the
event
of
1%considered
increase
intime
the money
market
rate
the basis
of
represents
liability
i, unit
CF are
the cash-flows
andcomposition
tCFathe
span.sheet),
the
modified
duration
from
the
maturity
of the
balance
Ai is
theonasset
i, Liab
i
the
modified
duration
(derived
from
the
maturity
composition
of
the
balance
sheet),
A
is
the
asset
i,
Liab
i
i
The
production
costs
are
different
for
the
credit
and
deposit
market
and
are
described
as
a
linear
represents
the
liability
i,
CF
are
the
cash-flows
and
t
the
considered
time
span.
CF
δ
is
the
loss
pro
unit
of
assets
in
the
event
of
a
1%
increase
in
the
money
market
rate
on
the
basis
of
t-1 the loss pro unit of assets in the event of a 1% increase in the money market rate on the basis of
δt-1 is
represents
the
liability
i,
CF
are
the
cash-flows
and
t
the
considered
time
span.
CF
function
of
the
quantity
produced.
The production
costs arefrom
different
for the composition
credit and deposit
are Adescribed
as ai, linear
the modified
duration (derived
the maturity
of the market
balance and
sheet),
Liab
i is the asset
the
modified
(derived
the maturity
composition
of themarket
balanceand
sheet),
is the asset
Theof production
are from
different
for the credit
and deposit
are A
described
as ai, Liab
lineari i
i
function
theduration
quantityi,costs
produced.
represents
the
liability
CF are the cash-flows and tCF the considered time span.
function
the liability
quantity
+of
c2the
D
with costs
> 0,c
>the
0 cash-flows
C = c1 LThe
represents
i,cproduced.
CF
are
andcredit
tCF theand
considered
time span.
production
are
different
for the
deposit market
and are described as a(A1.5)
linear
1
2
(A1.5)
c2 D
withcosts
cproduced.
>
0,c
>
0
C = The
c1 L +of
function
the quantity
production
are
different
for
the
credit
and
deposit
market
and
are
described
as
a
linear
1
2
(A1.5)
+ cthe
D last component
with c1 > 0,cwe
> add
0 credit risk, modeled as the percentage of loans, which is written off
C = c1 LAs
2
2
(w).
(A1.5)
+ cthe
D
with
c
>
0,c
>
0
C = c1 LAs
last
component
we
add
credit
risk,
modeled
as
the
percentage
of
loans,
which
is
written
off
(w).
2
1
2
As the
last
component
we add credit risk, modeled as the percentage of loans, which is written off
(w).
(A1.6)
27
− w)L
with
w
0 equity partly
(r l The
exogeneity
of≥the
contradicts the current equity definition. If we consider the difference between the regulatory
the effective
we can we
assume
a bank
usually
holds as
more
than required
by the
currentisregulation
in order
(A1.6)
(rl and
− w)L
with
w equity,
≥component
0
As
the
last
addthat
credit
risk,
modeled
theequity
percentage
of loans,
which
written off
(w).
(A1.6)
(rl −tow)L
with
w
≥
0
avoid
possible
reputation
costs
and
to
be
able
to
profit
from
future
investment
chances.
The
“voluntary”
part
of
the
equity
Summing up equations (A1.1) to (A1.6) we obtain bank profits as a function of the sold quantities,
the
reduces its dependency from the sum of risk-weighted assets, so that the assumption of an exogenous equity can be classified as
refinancing
and
operative
costs
and
the
interest
rate
and
credit
risk.
(A1.6)
w)L
with
w
≥
0
(rl −
Summing
up
equations
(A1.1)
to
(A1.6)
we
obtain
bank
profits
as
a
function
of
the
sold
quantities,
the
realistic (Dewatripont and Tirole, 1994; Van den Heuvel, 2001).
Summing
up equations
to interest
(A1.6) we
bankrisk.
profits as a function of the sold quantities, the
refinancing
and operative
costs(A1.1)
and the
rateobtain
and credit
refinancing
and
operative
costs
and
the
interest
rate
and
credit
risk.
(A1.7)
− w) L + rm Sup
− rdequations
D − TT − C(A1.1) to (A1.6) we obtain bank profits as a function of the sold quantities,
Π = ( rl Summing
the
177
(A1.7)
+ rm Soperative
− rd D − TTcosts
− C and the interest rate and credit risk.
Π = ( rl − w) Land
refinancing
JEOD - Vol.3, Issue 1 (2014)
(A1.7)
Π = ( rl − w) L + rm S − rd D − TT − C
i
Π = ( rl − w) L + rm S − rd D − TT − C
(A1.7)
'& ∑ CF( A ,0 → t
∑ t '&∑ CF( A ,0 → t
&
t∑
>0ttCF
TT = δ t−1Δrm ( L + S ) where
∑
t >0 CF
where
δ
=
TT
=
δ
Δr
(
L
+
S
)
TT = δ t−1
t−1Δr m( L + S ) where δ = t >0
m
CF
CF
δ=
Ai
A
Aii
CF
CF
i
i
CF
CF
CF(Liabi ,0
,0 →
→ ttCF ))*)
∑
LiabiCF(Liab
)) −− ∑
i
CF *
Liab
))
Liab i (t +1)
(A1.4)
(A1.4)
(A1.4)
1+ rm (tCF )i )(t(tCFCF+1)
+1)
(1+
r (t ) ) CF
((1+
∑rmmAA(ti CFCF ))
∑i Ai
∑
i
i
i
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
δt-1 is the loss pro unit of assets in the event of a 1% increase in the money market rate on the basis of
is the
the loss
loss pro
pro unit
unit of
of assets
assets in
in the event
event of aa 1%
1% increase
increase in
in the money
money market
market rate
rate on the
the basis
basis of
of
t-1 is
δδt-1
the modified
duration
(derived
from thethe
maturity of
composition
of the the
balance sheet),
A is theonasset
i, Liabi
the modified
modified duration
duration (derived
(derived from
from the
the maturity
maturity composition
composition of
of the
the balance
balance sheet),
sheet), A
Aiii is
is the
the asset
asset i,
i, Liab
Liabii
the
represents the liability i, CF are the cash-flows and tCF the considered time span.
represents the
the liability
liability i,i, CF
CF are the
the cash-flows
cash-flows and
and tCF
the considered
considered time span.
span.
CF the
represents
The production
costsare
are different
for the tcredit
and deposittime
market and are described as a linear
The
production
costs
are
different
for
the
credit
and
deposit
market
and are
are described
described as
as aa linear
linear
Theofproduction
are different for the credit and deposit market and
function
the quantitycosts
produced.
function
of
the
quantity
produced.
function
of the
thequantity
quantityproduced.
produced.
function of
(A1.5)
with c > 0,c2 > 0
C = c L+c D
(A1.5)
D
with cc11 >> 0,c
0,c2 >> 00
C == cc11LL ++ cc22D
(A1.5)
with
C
1
2
1
2
As the last component we add credit risk, modeled as the percentage of loans, which is written off (w).
As
the
last
component
we
addcredit
creditrisk,
risk,modeled
modeledasas
asthe
thepercentage
percentageof
ofloans,
loans,which
which isis
iswritten
written off
off (w).
(w).
AsAs
thethe
lastlast
component
wewe
add
component
add
credit
risk,
modeled
the
percentage
of
loans,
which
written
off
(w).
(r − w)L
(rll −− w)L
w)L
(r
l
(A1.6)
(A1.6)
(A1.6)
with w ≥ 0
with w
w ≥≥ 00
with
Summing
up
equations
(A1.1)
to
(A1.6) we
obtain bankprofits
profits as a functionofofthe
the soldquantities,
quantities, the
Summing
upup
equations
(A1.1)
toto
(A1.6)
obtain
Summing
equations
(A1.1)
(A1.6)wewe
we
obtainbank
bank profits
profitsasasa afunction
function of thesold
sold quantities,the
the
Summing
up equations
(A1.1)
to interest
(A1.6)
bank
refinancing
and operative
costs
and the
rateobtain
and credit
risk. as a function of the sold quantities, the
refinancing and
operative
costs
and
the
interest
rate
and
credit
risk.
refinancing
and operative
operative costs
costs and
andthe
theinterest
interestrate
rateand
andcredit
creditrisk.
risk.
refinancing
(A1.7)
(A1.7)
(A1.7)
Π = ( r − w) L + rm S − rd D − TT − C
w))LL ++ rrmSS −− rrd D
D −− TT
TT −− C
C
Π == ((rrll −− w
Π
l
m
d
27
Appendix
2
27 The exogeneity of the equity partly contradicts the current equity definition. If we consider the difference between the
27 The exogeneity of the equity partly contradicts the current equity definition. If we consider the difference between the
The exogeneity
the equity
partlywe
contradicts
the that
current
equity
definition.
we consider
the required
difference
the
regulatory
and
theofeffective
equity,
can assume
a bank
usually
holds If
more
equity than
bybetween
the current
Appendix
2 the
regulatory
and
effective
equity,
wereputation
can assume
assume
that and
bank
usually
holds
more
equity
thaninvestment
required by
by
the current
current
regulatory
the effective
we
can
that
aa bank
usually
more
equity
than
required
the
regulation and
in order
to avoidequity,
possible
costs
to be
able holds
to
profit
from
future
chances.
The
regulation
inpart
order
to and
avoid
possible
reputation
costs
andinto
to sum
be
able
to
profitthe
from
future
investment
chances.ofThe
The
Substituting
(5)possible
into (6)
and
(2) costs
andfrom
(6)
(7)able
werisk-weighted
obtain
following
utility
function:
regulation
in
to
avoid
reputation
and
to
be
to
profit
from
future
chances.
“voluntary”
of(4)
the
equity
reduces
its
dependency
the
of
assets,
so investment
that
the assumption
an
Appendix
22order
Appendix
Substituting
(4)
and
(5)
into
(6)
and
(2)
and
(6)
into
(7)
we
obtain
the
following
utility
function:
“voluntary”
part
of
the
equity
reduces
its
dependency
from
the
sum
of
risk-weighted
assets,
so
that
the
assumption
of
an
“voluntary”
part
of
the
equity
reduces
its
dependency
from
the
sum
of
risk-weighted
assets,
so
that
the
assumption
of
an
exogenous
equity
can
be
classified
as
realistic
(Dewatripont
and
Tirole
1994;
Van
den
Heuvel
(2001).
Appendix
2 can be classified as realistic (Dewatripont and Tirole 1994; Van den Heuvel (2001).
exogenous
equity
exogenous equity can be classified as realistic (Dewatripont and Tirole 1994; Van den Heuvel (2001).
Substituting
and
(5)
into
(6)
(2)
and
(6)
into
(7)+ dwe
obtain
the
p
p
Substituting
(4)
(6)
we
following utility
utility function:
function:(A2.1)
*( r − w) ( l r(4)
+ l1 yand
+ l2 p(5)
+ l3rinto
+ rmα
rand
+ d1 y(2)
+ dand
p + d(6)
r −into
r d r(7)
y p +obtain
d2 p + d3rmthe
( d0and
) + following
0 l
m)
d
2
3 m) ( d 0 d
1
, l
Substituting
(4) and
(5)
into
(6)
and
(2)
and
(6)
into
(7)
we
obtain
the
following
utility
function:
,
p
p
p
)) ((
)) ( ( (
)) ( (
)
(
)) ((
) ) ()(
) )(
)
) )
) )
)
Max U = Max*+−δ t−1Δrm %& l0 rl + lp1 y + l2 p + l3rm + α d0 rd + d1 y p + d2 p + d3rm '( − c1l0 rl + l1 y p+ l2 p + l3rm +
rl ,rd
rl ,rd *( r − w) l r + l y p + l p + l r
+ r α d r + d y p+ d p + d r − r d r + d y p+ d p + d r +
,,,( rl l − w) l00 rl l + l11 yp p+ l22 p + l33rmm + rmmα d00 rdd + d11 y p+ d2p2 p + d33rmm − rdd d00 rdd + d11 y p + d22 p + d33rmpm +
d0drd p+ +d1dy r + +
d2 p + d3rm
,,,-*−c
rl2 −dw0 r)%d l+0 rdl 1+yl1 y+ppd+2 lp2 +
p +d3lr3mrm −+θrl m(αrl −dr0mrd) +l0drpl1py+ l1+y d2+pl2+pd+3r'ml3rm− −rdθdd0(rdrm+−dppr1dy) +
(
3 m
−
U
=
Max
δ
Δr
l
r
+
l
y
+
l
p
+
l
r
Max
+ ,−δt−1
U = Max
Δr % l r + l11 y + l22 p + l33rmm ++αα dd00rrdd ++ dd11yy ++ dd22pp++ dd33rrmm ('(−− cc11ll00rrll ++ ll11yy ++ ll22pp ++2 ll33rrmm +
+
Max
rlr,r,rd
rlr,r,rd +
t−1 mm&& 00 l l
p
p
p
l d
l d , ,
Max U = Max,+−δ t−1Δrm %& l0 rl p+p l1 y + l2 p + l3rm + α d0 rd + d1 y + dpp2 p + d3rm '( − c1l0 rl + l1 y + l2 p + l3rm pp+
rl ,rd
rl ,rd, −c
−c22 dd00rrdd ++dd11yy ++dd22pp++dd33rrmm −−θθll((rrll −−rrmm)) ll00rrll ++ll11yy ++ ll22pp++ ll33rrmm −−θθdd ( rrmm −− rrdd ) dd00rrdd ++ dd11yy ++ dd22 pp ++ dd33rrmm
The-,-,,first
p
p
−c dorder
r + d1 yconditions
+ d2 p + d3rm −for
θ l ( rlthe
− rm )credit
l0 rl + l1 yand
+ l2 pdeposit
+ l3rm − θ drate
d0 rd + d1 y p + d2 p + d3rm
( rm − rare:
d)
- 2 0d
((
( (
((
(
(
(
(
(A2.1)
(A2.1)
(A2.1)
)
)
)
The first order conditions for the credit and deposit rate are:
first
( ∂U ! The
The
first order
order conditions
conditions for
for the
the credit
credit and
and deposit
deposit rate
rate! are:
2l0 rl + l1 y p + l2 p + l3rm − l0 w − l0δ t−1Δrm − l0 c1 − θ l (2l0 rl + l1 y p + l2 p + l3rm − l0 rm )= 0
* ∂r = 0 ⇒
The
first order conditions for the credit and deposit rate are:
* l
!
!
!
(()∂U
∂U=! 0! ⇒ 2l r + l y pp + l p + l r p− l w − l δ Δr − l c − θ (2l r + l y pp + l p + l r −pl r )=! 0
∂U
⇒2l
d00α
p + d t−1
rmΔr
) −mmd−0αδ
− 0c0 r2lld+0 +
p! 0+ d3rm − d0 rm )= 0
==0!0⇒
rl l r+m l1−1 y(2d
+0l2r2dp++dl313yrmm −+ l0d0 w
l00 c11t−1−Δr
θllm(2l
l11 yθ d (2d
+ l220prd++l33drmm1 y− l0+0 rmmd2)=
***∂r
2 − l00δ3t−1
∂U
∂r
***+(∂r
p
p
l ld
)) * ∂r! = 0 ⇒ 2l0 rl + l1 y + l2 p + l3rm − l0 w − l0δ t−1Δrm − l0 c1 − θ l (2l0 rl + l1 y + l2 p + l3rm − l0 rm )= 0
!!
*∂U l!
p
p
**∂U
Δr
) ==00! ⇒
⇒dd00ααrrmm −−(2d
(2d00rrdd ++dd11yy p ++dd22pp++dd33rrmm))−− dd00αδ
αδt−1
Δr −−cc22dd00 ++θθdd(2d
(2d00rrdd ++ dd11yy p ++ dd22pp ++ dd33rrmm −− dd00rrmm)=
)=!00
t−1 mm
*+*∂r
∂U
+ *∂rdd = 0 In
⇒ dorder
α rm − (2d
+ d1 y p + dfor
p + the
d3rm ) second
− d0αδ t−1Δrorder
− c2 d0 +conditions
θ d (2d0 rd + d1 y pwe
+ d2calculate
p + d3rm − d0 rmthe
)= 0 Hessian
to0rcontrol
0
d
2
m
*+ ∂rd
(A2.2)
(A2.2)
(A2.2)
matrix:
control
for
the
second
order
conditions
2
In
order
to
control
for
Hessian matrix:
matrix:
In
order
to
control
forthe
thesecond
secondorder
orderconditions
conditions we
we calculate
calculate the Hessian
%
" ∂In
U order
U
∂ 2to
order to 'control for the second order conditions we calculate the Hessian matrix:
$ In
%
0
∂r 2
∂rl ∂rd ' " 2l0 (1-θ l )
H =" $∂ 22Ul2
'
%%' = $
∂∂222U
"$∂ ∂U U
U
2d0 (θ d -1) '&
∂U
2
'''% $#0
$$$" ∂222U
U
∂
%%
""2l
∂r
00
2d
2l0 (1∂r
(1-θθl ))
∂r
∂rl∂r
$ ∂r
∂rl ll 2∂rd ∂r
l
d '' '
H
= $ " 2l0 0 (1-θl l )
H == $$$#$ ∂r
0θ -1) ''%
∂rl d∂rd'''&=
2 l
' =$#$$00
(
2d
∂∂22U
'&'&'
(
θ
-1)
2d
U
U
H =$$ $∂∂ 2U
0
d
0
d
' #$
$ $ ∂2U
2d0 (θ d -1) '&
∂ 222U '' ' $#0
∂r
$#$#∂r
∂rdd ∂r
∂r
'
$∂rl lFrom
&
dd
'
&
the
(A1.2) and (A1.3) we know that l0<0 and d0>0, so that we can define
∂rd2 equations
'&
$# ∂rl ∂rd
(A2.3)
(A2.3)
(A2.3)
the sign of its
principal minors:
From
equations
(A1.2)
and
(A1.3)
we
know
sign
of of
its
From
thethe
equations
<0 and
anddd00>0,
>0,sosothat
thatwewecan
candefine
definethethe
sign
From
the
equations(A1.2)
(A1.2)and
and(A1.3)
(A1.3)we
weknow
know that
that ll00<0
From
the equations (A1.2) and (A1.3) we know that l00<0 and d00>0, so that we can define the sign of its
principal
minors:
principal
Hprincipal
= 2l0 (1-minors:
θ l )minors:
< 0 if θ < 1
its
1
(A2.4)
principal
minors: l
H 2 = "# 2l0 (1-θ l ) $% "# 2d0 (θ d -1) $% >0 if θ l ,θ d < 1
H
H11 == 2l
2l0 (1(1-θθl ))<<00 ifif θθl <<11
H1 = 2l0 0 (1-θl l ) < 0 if θl l < 1
H
ifif θθl ,,θθd <<11
H22 == "#"#2l
2l00(1(1-θθl ))$%$"#"#2d
2d00(Hessian
(θθdd-1)
-1)$%$%>0
>0 matrix
l
d is negatively
H 2 = #"Since
2l0 (1-θl l %)the
%$ #" 2d0 (θ d -1) %$ >0 if θ l ,θ d < 1
(A2.4)
(A2.4)
defined, we can state that the solutions derived in (A2.2)
represent a local maximum of the utility function.
Since
the
is
can
state rate
that(rthe
solutions derived in (A2.2)
Since
the Hessian
Hessian
matrix
is negatively
negatively
defined,
we
Rearranging
(A2.2)matrix
we obtain
the optimaldefined,
credit (rlwe
) and
deposit
d):
Since
themaximum
Hessian
matrix
is negatively
defined,
we
can
state
that the
solutions
in (A2.2)
Since
the
Hessian
matrix
is
negatively
defined,
we
can
state
that
the
solutions
derived inderived
(A2.2) represent
represent
a
local
of
the
utility
function.
represent a local maximum of the utility function.
represent
a
local
maximum
of
the
utility
function.
Rearranging
we
the
(r
Rearranging
(A2.2)
we
obtain
the optimal
optimal credit
credit
(rll)) and
and deposit rate (rdd):
θl
1
1 obtain
a &local
maximum
of(A2.2)
the
utility
function.
*
the
(l1 y p + l2 p + l(A2.2)
r ) + we obtain
Δrmoptimal
+ c1 ) − credit
(w + δ t−1
rm(rl) and deposit rate (rd):
(rl = − Rearranging
3 m
(A2.5)
2l0
2(1−
θl )
θ l ) (r ) and deposit rate (r ):
( Rearranging
(A2.2) we
obtain
the optimal 2(1−
credit
&&' **
θθll θ l
11
1
yypp ++p ll22ppd++pll33+rrmmd))+r+ ) − 11 1(w
Δr
d rr
Δrmm +++cc1c1))−)−− αθ−
(w++(δδδt−1
(((r&rlrl *=*==−−− 11(l(l1(d
l
1 yp +
t−1αΔr
2l
2(1−
θ
)
2(1−
) mmrm
d
1
2
3
m
t−1
m
2
r
=
−
(l
y
+
l
p
+
l
r
)
+
δ
Δr
+
c
)
−
(w
+
2l
2(1−
θll ) θ ) t−1 m 1 2(1−
(((( l
00
ll ) )rm
1
2
3 m
2d
2(1−
2(1−θθθθ
0
d
d)
2l
2(1−
θ
)
2(1−
'')(
0
l
αα −−θθddl
11
((r'** = − 11 (d y pp + d p + d r ) −
δ
α
Δr
+
c
)
−
(
rr
α
−
θ
1
1
r
=
−
(d
y
+
d
p
+
d
r
)
−
δ
α
Δr
+
c
)
−
(
11
22
t−1
mm
()((ddrd* = −2d
2(1−
θθddd)) mrmm demand:
(d
y p +following
d2 p + 3d3 3mmrm ) −assumptions
+ c222 the
) −2(1−
2d
2(1−θθdd)) (δt−1t−1αΔr
2(1−
00 meet
1
m
We
about
deposit
)(
2d
2(1−
θ
)
2(1−
θ
)
0
d
d
)
d
(A2.5)
(A2.5)
the reactivity to the deposit/money market rate is symmetric to the reactivity of the credit demand to the
We
meet following
assumptions
deposit demand:
We
assumptions
about
the
credit/money
market rate
(d0=-l0; dabout
3=-l3);the
Wemeet
meetfollowing
following
assumptions
about
the deposit
deposit demand:
demand:
-­‐-­‐-­‐ the
reactivity
to
the
deposit/money
market
rate
is178
symmetric
the reactivity
of thetocredit
demand to
the
the
reactivity
to
the
deposit/money
market
rate
in the permanent
and in theto
level is equal
the reactivity
the
-­‐ the
the reactivity
reactivity to
to changes
the deposit/money
market income
rate isis symmetric
symmetric
toprice
the reactivity
of the credit
demand of
to the
JEOD
Vol.3,
Issue
1
(2014)
credit/money
market
rate
(d
);
credit/money
market
rate
(d
=-l
d
=-l
);
00=-l
00;; d
33=-l
3
credit
demand
to
the
same
factors
(d
=l
;
d
=l
).
31
1
2
2
credit/money market rate (d0=-l0; d3=-l3);
-­‐-­‐ the
reactivity
to
changes
in
permanent
income
the pricebetween
level is the
equal
to thecredit
reactivity
of the
reactivity
totointerest
changes
ininthe
the
income
and
in
optimal
margin
then calculated
asand
the in
difference
optimal
and deposit
-­‐ the
theThe
reactivity
changes
theispermanent
permanent
income
and
in the price
level is equal
to the reactivity
of the
-­‐
principal
minors:
From
the equations (A1.2) and (A1.3) we know that l0<0 and d0>0, so that we can define the sign of its
principal minors:
H1 = 2l0 (1-θ l ) < 0 if θ l < 1
(A2.4)
"# 2l(1H
(1-θ ) $ "# 2d
H12 == 2l
if0θ(θl d<-1)
1 $% >0 if θ l ,θ d < 1
0 θ )l <%0
0
l
A Model for the Interest Margin of a Risk-neutral Bank. The Role of the Bank Orientation
Pedrotti, M.
(A2.4)
H 2 = "# 2l0 (1-θ l ) $% "# 2d0 (θ d -1) $% >0 if θ l ,θ d < 1
Since the Hessian matrix is negatively defined, we can state that the solutions derived in (A2.2)
represent
a local
of the is
utility
function.
Since
the maximum
Hessian matrix
negatively
defined, we can state that the solutions derived in (A2.2)
Rearranging (A2.2) we obtain the optimal credit (rl) and deposit rate (rd):
represent a local maximum of the utility function.
Rearranging (A2.2) we obtain the optimal credit (rl) and deposit rate (rd):
& *
θl
1
1
(l1 y p + l2 p + l3rm ) +
(w + δ t−1Δrm + c1 ) −
r
(rl = −
2l
2(1−
θ
)
2(1−
θ) m
(&
0
θl l
1 l
'r * = − 1 (l y p + l p + l r ) +
1
2
3 m
1 (w + δ t−1Δrm + c1 ) − α − θ rm
(((rl* = − 2l10 (d
y p + d2 p + d3rm )2(1−
− θ l ) (δ t−1αΔrm + c2 ) −2(1− θ l d) rm
1
()' d
2d0
2(1− θ d )
2(1− θ d )
α − θd
1
(r * = − 1 (d y p + d p + d r ) −
(δ t−1αΔrm + c2 ) −
r
d
1
2
3
m
()
2d0
2(1− θ d )
2(1− θ d ) m
(A2.5)
(A2.5)
meet
following
assumptions
aboutthe
thedeposit
depositdemand:
demand:
WeWe
meet
following
assumptions
about
theWe
reactivity
to
the
deposit/money
market
rate
is
symmetric
reactivity of the credit demand to the
meet following
assumptions about
the rate
deposit
demand:totothe
- the
reactivity
to
the deposit/money
market
is symmetric
the reactivity of the credit demand to
credit/money
market
rate
(d
0=-l0; d3=-l3);
-­‐ the credit/money
reactivity to the
deposit/money
to the reactivity of the credit demand to the
market
ratethe(dpermanent
=-lmarket
; d3=-lrate
); is symmetric
-­‐ the
the reactivity market
to changes
income
and in the price level is equal to the reactivity of the
3
credit/money
rate in
(d0=-l0;0 d30=-l3);
credit
demandto
the same
(d1=l1; d2income
=l
--­‐ the
reactivity
ininfactors
the
and
2).
the
reactivity
totochanges
changes
thepermanent
permanent
income
and in
in the
the price
price level
level isisequal
equalto
tothe
thereactivity
reactivityofofthe
the
The
optimal
interest
margin
is
then
calculated
as
the
difference
between
the
optimal
credit
and deposit
credit
to the
thesame
samefactors
factors(d(d1=l
=l
;
d
=l
).
credit demand
demand to
;
d
=l
).
11 2 2 22
rate. The optimal interest margin is then1 calculated
the difference
the optimal
credit
and deposit
The optimal interest margin is then calculated as theas
difference
betweenbetween
the optimal
credit and
deposit
rate.
-­‐
rate.
"
IM * = rl* − rd*
$
$ d0 =* −l0 * *
, 1)
θl
α − θd ,
1
1
1) 1
1
1 ) 1+ α − θ d − αθ l ,
$" IM = rl − rd
⇒ IM * = − (l1 y p + l2 p) +
w+ +
#$ d3 = −l3
. δ t−1Δrm + 2 + (1− θ ) c1 + (1− θ ) c2 . + 2 + − (1− θ ) + (1− θ ) . rm
l
2(1−
θ
)
2
(1−
θ
)(1−
θ
)
*
*
*
d
=
−l
0
l
d
l
l
d
l
d $ d0 = l 0
, 1)
θl
α − θd ,
1
1
1 ) 1+ α − θ d − αθ l ,
1) 1
1
1
$ d1 = −l
⇒ IM * = − (l1 y p + l2 p) +
r
w+ +
δ t−1Δrm + +
c1 +
c2 . + + −
+
#$ 3
3
.
l0
2(1− θ l )
2 * (1− θ d )(1− θ l ) 2 * (1− θ l )
(1− θ d ) - 2 * (1− θ l ) (1− θ d ) .- m
%$ d2 = l2
$ d1 = l1
$ d2 = l2
%
(A2.6)
(A2.6)
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