# Bank Valuation and Its Connections with the Subprime Mortgage

Authors: C. H. Fouche; J. Mukuddem-Petersen; M. A. Petersen; M. C. Senosi
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

The ongoing subprime mortgage crisis (SMC) and implementation of Basel II
Capital Accord regulation have resulted in issues related to bank
valuation and profitability becoming more topical. Profit is a major
indicator of financial crises for households, companies, and financial institutions. An SMC-related example of this is the U.S. bank,
Wachovia Corp., which reported major losses in the first quarter of 2007
and eventually was bought by Citigroup in September 2008. A first
objective of this paper is to value a bank subject to Basel II based
on premiums for market, credit, and operational risk. In this case, we
investigate the discrete-time dynamics of banking assets, capital, and
profit when loan losses and macroeconomic conditions are explicitly
considered. These models enable us to formulate an optimal bank
valuation problem subject to cash flow, loan demand, financing, and
balance sheet constraints. The main achievement of this paper is bank
value maximization via optimal choices of loan rate and supply which
leads to maximal deposits, provisions for deposit withdrawals, and bank
profitability. The aforementioned loan rates and capital provide
connections with the SMC. Finally, OECD data confirms that loan loss
provisioning and profitability are strongly correlated with the
business cycle.


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## Body

## 1. Introduction

In this paper, we mainly consider bank valuation (Bank
value is commonly defined in terms of the market value of the investors equity (stock
market capitalization if a company is quoted) plus the market value of the nett
financial debt) and profitability when loan losses and macroeconomic conditions
are explicitly considered. We note that in the acquisition of bank equity, a
valuation gives the stock analyst (possibly acting on behalf of a potential
shareholder) an independent estimate of a fair price of the bank's shares. As
far as profitability is concerned, we are motivated by the fact that it is a
major indicator of financial crises for households, companies, and financial
institutions. An example of the latter from the subprime mortgage crisis (SMC)
that became more apparent in 2007 and 2008 is that both the failure of the
Lehmann Brothers investment bank and the acquisition in September 2008 of
Merrill Lynch and Bear Stearns by Bank of America and JP Morgan Chase,
respectively, were preceded by a decrease in profitability and an increase in
the price of loans and loan losses (see Subsection5.1 for a
diagrammatic overview of the SMC). In this paper, we discuss the relationship
between our banking models and the SMC as well as the subsequent credit crunch
that has had a profound impact on the global banking industry from 2007
onwards. These connections are forged via the bank's risk premium, sensitivity
of changes in capital to loan extension, Central Bank base rate, own loan rate,
loan demand, loan losses and default rate, loan loss provisions, choice between
raising deposits and interbank borrowing, liquidity, profit, as well as bank
valuation. In addition, we establish connections between our models and the
Basel II Capital Accord. These associations are mainly determined via total
bank capital, the bank capital constraint, and the procyclicality of approaches
to Basel II Credit Risk.Loan pricing models usually have components related to
the financial funding cost, a risk premium to compensate for the risk of
default by the borrower, a premium reflecting market power exercised by the
bank and the sensitivity of the cost of capital raised to changes in loans
extended. On the other hand, loan losses can be associated with an offsetting
expense called theloan loss
provision (LLP) which is charged against nett profit. This offset will
reduce reported income but has no impact on taxes, although when the assets are
finally written off, a tax-deductible expense is created. An important factor
influencing loan loss provisioning is regulation and supervision. Measures of
capital adequacy are generally calculated using the book values of assets and
equity. The provisioning of loans and their associated write offs will cause a
decline in these capital adequacy measures, and may precipitate increased
regulation by bank authorities. Greater levels of regulation generally entail
additional costs for the bank. Currently, this regulation mainly takes the form
of the Basel II Capital Accord (see Subsection 5.2.1 for a
diagrammatic overview of Basel II; also [1, 2]) that has been implemented on a worldwide basis since 2008.The impact of a risk-sensitive framework such as Basel
II on macroeconomic stability of banks is an important issue. In this regard,
we note that the 1996 Amendment's Internal Models Approach (IMA) determines the
capital requirements on the basis of the institutions' internal risk
measurement systems. The minimum capital requirement is then the sum of a
premium to covercredit risk,
general market risk and operational risk. The credit risk premium
is made up of risk-weighted loans and the market risk premium is equal to a
multiple of the average reported two-week VaRs in the preceding 60 trading
days. Banks are required to report daily their value-at-risk (VaR) at a 99%
confidence level over both a one day and two weeks (10 trading days) horizon.
In order for a bank to determine their minimum capital requirements they will
first decide on a planning horizon. This planning horizon is then partitioned
into non-overlapping backtesting-periods, which is in turn divided into
non-overlapping reporting periods. At the start of each reporting period the
bank has to report its VaR for the current period and the actual loss from the
previous period. The market risk premium for the current reporting period is
then equal to the multiple m of the reported VaR. At the end of each
backtesting period, the number of reporting periods in which actual loss
exceeded VaR is counted and this determines the multiple m for the next backtesting period according to a
given increasing scale. Usually the premium to cover operational risk equals
the sum of the premiums for each of eight business lines. The operational risk
premium is discussed further in Subsection 2.4.A popular approach to the study of banking valuation
and profitability involves a financial system that is assumed to be imperfectly
competitive. As a consequence, profits (see, for instance, [3, 4]) are ensured by virtue of
the fact that the nett loan interest margin is greater than the marginal
resource cost of deposits and loans. Besides competition policy, the decisions
related to capital structure play a significant role in bank behavior. Here,
the relationship between bank capital, credit and macroeconomic activity is of
crucial importance. In this regard, it is a widely accepted fact that certain
financial variables such as capital, credit, asset prices, profitability and
provisioning (also bond spreads, ratings from credit rating agencies, leverage
and risk-weighted capital adequacy ratios, other ratios such as write-off/loan
ratios and perceived risk) exhibit cyclical tendencies. The cyclicality of a
financial variable is related to its relationship with the business cycle or a
proxy of the business cycle such as the output gap. Here the output gap is defined as the amount by
which a country's output, or GDP, falls short of what it could be given its
available resources. In particular, “procyclicality” has become a
buzzword in discussions about banking regulation. In essence, the movement in a
financial variable is said to be “procyclical” if it tends to amplify
business cycle fluctuations. As such, procyclicality is an inherent property of
any financial system. A consequence of procyclicality is that banks tend to
restrict their lending activity during economic downturns because of their
concern about loan quality and the probability of loan defaults. This
exacerbates the recession since credit constrained businesses and individuals
cut back on their investment activity. On the other hand, banks expand their
lending activity during boom periods, thereby contributing to a possible
overextension of the economy that may transform an economic expansion into an
inflationary spiral. Our contribution emphasizes the cyclicality of bank
profitability and provisioning for loan losses.By way of addressing the issues raised above, we
present a two-period discrete-time banking model involving on-balance sheet
variables such as assets (cash, bonds, shares, loans, Treasuries and reserves),
liabilities (deposits and interbank borrowing), bank capital (shareholder
equity, subordinate debt and loan loss reserves) and off-balance sheet items
such as intangible assets (see, for instance, [5, 6]). In turn, the aforementioned models enable us to
formulate an optimization problem that seeks to establish a maximal value of
the bank by a stock analyst by choosing an appropriate loan rate and supply.
Under cash flow, loan demand, financing and balance sheet constraints, the
solution to this problem also yields a procedure for profit maximization in
terms of the loan rate and deposits. Here profits are not only expressed as a
function of loan losses but also depend heavily on provisions for loan losses.


### 1.1. Relation to Previous Literature

In this subsection, we consider the association
between our contribution and previously published literature. The issues that
we highlight include loan pricing, bank valuation and profitability, the role
of bank capital, credit models for monetary policy, macroeconomic activity,
cyclicality concerns and discrete-time modeling and optimization as well as the
SMC and Basel II.A number of recent papers on loan pricing are related
to our contribution. For instance, [7, 8] analyzes and estimates the possible effects of Basel
II on the pricing of bank loans. In this regard, the authors discuss two
approaches for credit risk capital requirements, viz., the Internal Ratings
Based (IRB) and Standardized approaches, and distinguish between retail and
corporate customers. As is the case in our contribution, their loan pricing
equation is based on a model of a bank facing uncertainty operating in an
imperfectly competitive loan market. The main results in [8] indicate that high quality
corporate and retail customers will enjoy a reduction in loan rates in banks
that adopt the IRB approach while high risk customers will benefit by shifting
to banks that adopt the Standardized approach. In a perfectly competitive
market, the work in [9] considers
corporate loans where, as in the model underlying the Basel II IRB approach, a
single factor explains the correlation in defaults across firms. The results
from [8] also hold
true for corporate customers when comparing the IRB and Standardized
approaches. In addition, [9] shows that only a very high social cost of bank failure
might justifyy the proposed IRB capital charges. A partial reason for this is
that nett interest income from performing loans is not considered to be a
buffer against loan losses.The most common method to value a bank is to calculate
the present value of the bank`s future cash flows. For instance, in [10] a regression model is
derived to address the problem of valuing a bank. Similar to this is [11] where a regression model is
derived for the change in market value for a specific bank. These papers, and
others not mentioned explicitly, discuss activities that add value to the bank
making it attractive for potential shareholders. Also, the extent of exposure
to emerging markets plays a role in the valuation of the bank. Most of the
studies considered, has a statistical background. By contrast, the novelty of
our contribution is that we use control laws to find the optimal bank value.
The work in [12] claims
that profitability by bank function is determined by subtracting all direct and
allocable indirect expenses from total gross revenue generated by that
function. This computation results in the nett revenue (yield) that excludes
cost of funds. From the nett yield the cost of funds is subtracted to determine
the nett profit of the bank by function. Coyne represents four major leading
functions, viz., investments, real estate mortgage loans, installment loans as
well as commercial and agricultural loans. The work in [13] has a discussion on the determinants of commercial
bank profitability in common with our paper. The contribution [14] demonstrates by means of
technical arguments that banks' profits will not decrease if the growth rate of
sales is higher than the absolute growth rate of the bank's own loan rate. This
rate will decrease when it is necessary to stimulate growth and provide
liquidity.The most important role of capital is to mitigate the
moral hazard problem that results from asymmetric information between banks,
depositors and borrowers. In the presence of asymmetric information about the
LLP, bank managers may be aware of asset quality problems unknown to outside
analysts. Provisioning the assets may convey a clearer picture regarding the
worth of these assets and precipitate a (negative) market adjustment. The
Modigliani-Miller theorem forms the basis for modern thinking on capital
structure (see [15]).
In an efficient market, their basic result states that, in the absence of
taxes, insolvency costs and asymmetric information, the bank value is
unaffected by how it is financed. In this framework, it does not matter if bank
capital is raised by issuing equity or selling debt or what the dividend policy
is. By contrast, in our contribution, in the presence of loan market frictions,
the bank value is dependent on its financial structure (see, for instance,
[16–18]). In this case, it is
well-known that the bank's decisions about lending and other issues may be
driven by the capital adequacy ratio (CAR) (see, for instance, [19–23]). Further evidence of the impact
of capital requirements on bank lending activities are provided by [24, 25]. A new line of research
into credit models for monetary policy has considered the association between
bank capital and loan demand and supply (see, for instance, [26–31]). This credit channel is commonly known as the bank capital channel and propagates that a
change in interest rates can affect lending via bank capital.We also discuss the effect of macroeconomic activity
on a bank's capital structure and lending activities (see, for instance,
[32]). With regard to
the latter, for instance, there is considerable evidence to suggest that
macroeconomic conditions impact the probability of default and loss given
default on loans (see, for instance, [32, 33]). Our contribution has a close connection with
[29] via our interest
in how cyclicality relates to profitability and provisioning. In particular,
the fact that provisioning (profitability) behaves procyclically by falling
(rising) during economic booms and rising (falling) during recessions (see, for
instance, [27–29, 34–36]) is incorporated in our models. The working paper
[37] provides us with
a direct connection between the present contribution and the SMC. In the said
paper, it is claimed that the rise and fall of the subprime mortgage market
follows a classic credit boom-bust scenario in which unsustainable growth leads
to the collapse of the market. In other words, this means that procyclicality
of bank credit has led to the crisis in credit markets—a situation that we
allow for in our model.Several discussions related to discrete-time
optimization problems for banks have recently surfaced in the literature (see,
for instance, [18, 22, 32, 38]). Also, some recent papers
using dynamic optimization methods in analyzing bank regulatory capital
policies include [39]
for Basel II and [40–42] for Basel market risk
capital requirements. In [22], a discrete-time dynamic banking model of imperfect competition
is presented, where the bank can invest in a prudent or a gambling asset. For
both these options, a maximization problem that involves bank value is
formulated. On the other hand, [38] examines a problem related to the optimal risk
management of banks in a continuous-time stochastic dynamic setting. In
particular, the authors minimize market and capital adequacy risk that involves
the safety of the assets held and the stability of sources of capital,
respectively (see, also, [43]).The working paper [37] explains the fundamentals of the SMC in some detail.
A model that has become important during this crisis is the Diamond-Dybvig
model (see, for instance, [44, 45]). Despite the fact that these contributions consider
a simpler model than ours, they are able to explain important features of bank
liquidity that reflect reality. The quarterly reports [46, 47] of the Federal Deposit
Insurance Corporation (FDIC) intimate that profits decreased from $35.6
billion to $19.3 billion during the first quarter of 2008 versus the previous
year, a decline of 46%.


### 1.2. Outline of The Paper

We extend aspects of the literature mentioned in
Subsection1.1 in several directions. Firstly, taking our lead from Basel II,
by contrast to [29],
the risk-weight for the assets appearing on and off the balance sheet may vary
with time. In the second place, in the spirit of the Basel II, we incorporate
market, credit and operational risk at several levels in our discrete-time
models. Here we recognize that most contributions (see, for instance, [41]) only consider market and
credit risk as in the previous regulatory paradigm (Basel I Capital Accord).
Furthermore, we incorporate both Treasuries and reserves as part of the
provisions for deposit withdrawals whereas [29] only discusses the role of Treasuries. Fourthly, we
include loan losses and its provisioning as an integral part of our analysis
(compare with [29, 36]). Also, we
provide substantive evidence of the procyclicality of credit, profitability and
provisioning for OECD countries (compare with [27, 28, 34, 35]). In the
sixth place, we recognize the important role that intangible assets play in
determining bank profit and valuation (compare with [5, 6]). Also, we determine the
value of a bank subject to capital requirements based on reported Value-at-Risk
(VaR) and operational measures, as in the Basel Committee's Internal Models
Approach (see, for instance, [48]). Finally, we forge connections between our banking
models and the SMC as well as Basel II.The main problems to emerge from the previous
paragraph can be formulated as follows.Problem 1 (modeling bank valuation and loan losses).
Can we model the value of a bank and quantify losses from its lending
activities in discrete-time? (Sections2 and 3). Can we confirm that these
models are realistic in some respects? (Section 4).Problem 2 (optimal bank valuation problem).
Which
decisions about loan rates, deposits and Treasuries must be made in order to
attain an optimal bank value for a shareholder? (Theorem3.1 in Section 3).Problem 3 (connections with the SMC and Basel II).
How do the banking models developed in our paper relate to the SMC and Basel II
Capital Accord?  (Section5).The paper is structured as follows. In Subsection2.1
of Section 2, we describe general bank assets (shares, bonds, cash, intangible
assets, Treasuries and reserves). Also, in this section, we construct models
for bank loan supply, demand and losses as well as for provisions for loan
losses (see Subsections  2.2). In Section 3, we present models for capital (with
a risk-based capital requirement) and profit (Subsections 3.1 and 3.2). A
description of how a bank may be valued by a stock analyst for a shareholder is
given in Subsection 3.3, while an optimal valuation problem is formulated and
solved in Subsection 3.4. By way of corroborating our choice of models, in
Section 4, historical evidence (see Subsection 4.1) and illustrative examples
(see Subsection 4.2) reflecting the cyclicality of provisions and profitability
and the correlation between these financial variables, respectively, are
presented. Aspects of the relationships between bank valuation and the SMC as
well as Basel II are analyzed in Section 5. Next, Section 6 offers a few
concluding remarks and topics for possible future research. Finally, relevant
appendices are provided in the appendices.


## 1.1. Relation to Previous Literature

In this subsection, we consider the association
between our contribution and previously published literature. The issues that
we highlight include loan pricing, bank valuation and profitability, the role
of bank capital, credit models for monetary policy, macroeconomic activity,
cyclicality concerns and discrete-time modeling and optimization as well as the
SMC and Basel II.A number of recent papers on loan pricing are related
to our contribution. For instance, [7, 8] analyzes and estimates the possible effects of Basel
II on the pricing of bank loans. In this regard, the authors discuss two
approaches for credit risk capital requirements, viz., the Internal Ratings
Based (IRB) and Standardized approaches, and distinguish between retail and
corporate customers. As is the case in our contribution, their loan pricing
equation is based on a model of a bank facing uncertainty operating in an
imperfectly competitive loan market. The main results in [8] indicate that high quality
corporate and retail customers will enjoy a reduction in loan rates in banks
that adopt the IRB approach while high risk customers will benefit by shifting
to banks that adopt the Standardized approach. In a perfectly competitive
market, the work in [9] considers
corporate loans where, as in the model underlying the Basel II IRB approach, a
single factor explains the correlation in defaults across firms. The results
from [8] also hold
true for corporate customers when comparing the IRB and Standardized
approaches. In addition, [9] shows that only a very high social cost of bank failure
might justifyy the proposed IRB capital charges. A partial reason for this is
that nett interest income from performing loans is not considered to be a
buffer against loan losses.The most common method to value a bank is to calculate
the present value of the bank`s future cash flows. For instance, in [10] a regression model is
derived to address the problem of valuing a bank. Similar to this is [11] where a regression model is
derived for the change in market value for a specific bank. These papers, and
others not mentioned explicitly, discuss activities that add value to the bank
making it attractive for potential shareholders. Also, the extent of exposure
to emerging markets plays a role in the valuation of the bank. Most of the
studies considered, has a statistical background. By contrast, the novelty of
our contribution is that we use control laws to find the optimal bank value.
The work in [12] claims
that profitability by bank function is determined by subtracting all direct and
allocable indirect expenses from total gross revenue generated by that
function. This computation results in the nett revenue (yield) that excludes
cost of funds. From the nett yield the cost of funds is subtracted to determine
the nett profit of the bank by function. Coyne represents four major leading
functions, viz., investments, real estate mortgage loans, installment loans as
well as commercial and agricultural loans. The work in [13] has a discussion on the determinants of commercial
bank profitability in common with our paper. The contribution [14] demonstrates by means of
technical arguments that banks' profits will not decrease if the growth rate of
sales is higher than the absolute growth rate of the bank's own loan rate. This
rate will decrease when it is necessary to stimulate growth and provide
liquidity.The most important role of capital is to mitigate the
moral hazard problem that results from asymmetric information between banks,
depositors and borrowers. In the presence of asymmetric information about the
LLP, bank managers may be aware of asset quality problems unknown to outside
analysts. Provisioning the assets may convey a clearer picture regarding the
worth of these assets and precipitate a (negative) market adjustment. The
Modigliani-Miller theorem forms the basis for modern thinking on capital
structure (see [15]).
In an efficient market, their basic result states that, in the absence of
taxes, insolvency costs and asymmetric information, the bank value is
unaffected by how it is financed. In this framework, it does not matter if bank
capital is raised by issuing equity or selling debt or what the dividend policy
is. By contrast, in our contribution, in the presence of loan market frictions,
the bank value is dependent on its financial structure (see, for instance,
[16–18]). In this case, it is
well-known that the bank's decisions about lending and other issues may be
driven by the capital adequacy ratio (CAR) (see, for instance, [19–23]). Further evidence of the impact
of capital requirements on bank lending activities are provided by [24, 25]. A new line of research
into credit models for monetary policy has considered the association between
bank capital and loan demand and supply (see, for instance, [26–31]). This credit channel is commonly known as the bank capital channel and propagates that a
change in interest rates can affect lending via bank capital.We also discuss the effect of macroeconomic activity
on a bank's capital structure and lending activities (see, for instance,
[32]). With regard to
the latter, for instance, there is considerable evidence to suggest that
macroeconomic conditions impact the probability of default and loss given
default on loans (see, for instance, [32, 33]). Our contribution has a close connection with
[29] via our interest
in how cyclicality relates to profitability and provisioning. In particular,
the fact that provisioning (profitability) behaves procyclically by falling
(rising) during economic booms and rising (falling) during recessions (see, for
instance, [27–29, 34–36]) is incorporated in our models. The working paper
[37] provides us with
a direct connection between the present contribution and the SMC. In the said
paper, it is claimed that the rise and fall of the subprime mortgage market
follows a classic credit boom-bust scenario in which unsustainable growth leads
to the collapse of the market. In other words, this means that procyclicality
of bank credit has led to the crisis in credit markets—a situation that we
allow for in our model.Several discussions related to discrete-time
optimization problems for banks have recently surfaced in the literature (see,
for instance, [18, 22, 32, 38]). Also, some recent papers
using dynamic optimization methods in analyzing bank regulatory capital
policies include [39]
for Basel II and [40–42] for Basel market risk
capital requirements. In [22], a discrete-time dynamic banking model of imperfect competition
is presented, where the bank can invest in a prudent or a gambling asset. For
both these options, a maximization problem that involves bank value is
formulated. On the other hand, [38] examines a problem related to the optimal risk
management of banks in a continuous-time stochastic dynamic setting. In
particular, the authors minimize market and capital adequacy risk that involves
the safety of the assets held and the stability of sources of capital,
respectively (see, also, [43]).The working paper [37] explains the fundamentals of the SMC in some detail.
A model that has become important during this crisis is the Diamond-Dybvig
model (see, for instance, [44, 45]). Despite the fact that these contributions consider
a simpler model than ours, they are able to explain important features of bank
liquidity that reflect reality. The quarterly reports [46, 47] of the Federal Deposit
Insurance Corporation (FDIC) intimate that profits decreased from $35.6
billion to $19.3 billion during the first quarter of 2008 versus the previous
year, a decline of 46%.


## 1.2. Outline of The Paper

We extend aspects of the literature mentioned in
Subsection1.1 in several directions. Firstly, taking our lead from Basel II,
by contrast to [29],
the risk-weight for the assets appearing on and off the balance sheet may vary
with time. In the second place, in the spirit of the Basel II, we incorporate
market, credit and operational risk at several levels in our discrete-time
models. Here we recognize that most contributions (see, for instance, [41]) only consider market and
credit risk as in the previous regulatory paradigm (Basel I Capital Accord).
Furthermore, we incorporate both Treasuries and reserves as part of the
provisions for deposit withdrawals whereas [29] only discusses the role of Treasuries. Fourthly, we
include loan losses and its provisioning as an integral part of our analysis
(compare with [29, 36]). Also, we
provide substantive evidence of the procyclicality of credit, profitability and
provisioning for OECD countries (compare with [27, 28, 34, 35]). In the
sixth place, we recognize the important role that intangible assets play in
determining bank profit and valuation (compare with [5, 6]). Also, we determine the
value of a bank subject to capital requirements based on reported Value-at-Risk
(VaR) and operational measures, as in the Basel Committee's Internal Models
Approach (see, for instance, [48]). Finally, we forge connections between our banking
models and the SMC as well as Basel II.The main problems to emerge from the previous
paragraph can be formulated as follows.Problem 1 (modeling bank valuation and loan losses).
Can we model the value of a bank and quantify losses from its lending
activities in discrete-time? (Sections2 and 3). Can we confirm that these
models are realistic in some respects? (Section 4).Problem 2 (optimal bank valuation problem).
Which
decisions about loan rates, deposits and Treasuries must be made in order to
attain an optimal bank value for a shareholder? (Theorem3.1 in Section 3).Problem 3 (connections with the SMC and Basel II).
How do the banking models developed in our paper relate to the SMC and Basel II
Capital Accord?  (Section5).The paper is structured as follows. In Subsection2.1
of Section 2, we describe general bank assets (shares, bonds, cash, intangible
assets, Treasuries and reserves). Also, in this section, we construct models
for bank loan supply, demand and losses as well as for provisions for loan
losses (see Subsections  2.2). In Section 3, we present models for capital (with
a risk-based capital requirement) and profit (Subsections 3.1 and 3.2). A
description of how a bank may be valued by a stock analyst for a shareholder is
given in Subsection 3.3, while an optimal valuation problem is formulated and
solved in Subsection 3.4. By way of corroborating our choice of models, in
Section 4, historical evidence (see Subsection 4.1) and illustrative examples
(see Subsection 4.2) reflecting the cyclicality of provisions and profitability
and the correlation between these financial variables, respectively, are
presented. Aspects of the relationships between bank valuation and the SMC as
well as Basel II are analyzed in Section 5. Next, Section 6 offers a few
concluding remarks and topics for possible future research. Finally, relevant
appendices are provided in the appendices.


## 2. Discrete-time banking model

Throughout, we suppose that(Ω,F,(ℱt)t≥0,P) is a filtered probability space. Also, we deal
with an individual bank that precommits to a loan quantity via its dividends
policy in the tth period, which is subsequently followed by
the loan rate competition in the (t+1)th period. As is well-known, the bank balance
sheet consists of assets (uses of funds) and liabilities (sources of funds)
that are balanced by bank capital (see, for instance, [17]) according to the
well-known relation (2.1)Totalassets(A)=Totalliabilities(Γ)+Totalbankcapital(K).In period t,  the main on-balance sheet items in (2.1) can
specifically be identified as(2.2)At=Λtm+Wt+Ct+St+Bt,Wt=Tt+Rt;Γt=Dt+Bt;Kt=ntEt−1+Ot+Rtl,where Λm,C,S,B,T,R,D,B,n,E,O, and  Rl are the market value of short- and long-term
loans, cash, short- and long-term securities, bonds, Treasuries, reserves,
deposits, interbank borrowing including borrowing from the Central Bank, number
of shares, market price of the bank's common equity, subordinate debt and loan
loss reserves, respectively.The balance sheet reflects the fact that banks are
active in the primary market by raising deposits,D, from and extending credit, Λ,  to the public. Also, banks operate in the
secondary market in order to bridge the gap between surpluses and deficits in
its reserves, R and Rl.  This involves transactions with other
commercial banks (interbank lending), with the Central Bank (monetary loans or
deposits with the Central Bank) and Treasury (buying and selling Treasury
securities) as well as in the financial markets (buying and selling
securities). Also the bank holds capital, K, as required by the regulator, which serves as
a cushion against unexpected losses (primarily from its loan portfolio).


### 2.1. General Bank Assets

In this subsection, we discuss on- and off-balance
sheet bank assets such as shares, bonds and cash, Treasuries, reserves and
intangible assets.


#### 2.1.1. Shares

Of the first three general bank asset classes, shares,S, have historically been the most prominent
performers over the long term. Since the returns from shares usually exceed the
returns from both bonds and cash and have significantly outpaced inflation,
they are important to a portfolio for growth of capital over time. Over the
short term, however, shares can be volatile and as a result there is regulation
related to banks holding shares. In the sequel, the rate of return on shares in
the tth period, St, is denoted by rtS.


#### 2.1.2. Bonds

Whereas shares represent equity, or part ownership of
the companies that issue them, bonds,B, represent debt. Municipalities and governments
all use bonds as a way to raise cash. When banks buy bonds, they are lending
money to the issuer in exchange for fixed interest payments over a set number
of years and a promise to pay the original amount back in the future. Bonds are
valuable to banks more for the income they provide than for growth potential.
Since the income they pay is fixed it is generally reliable and steady. The
primary risk in bond market investing comes from interest rate changes. When
interest rates rise, a bond's market value decreases. Another potential risk of
owning bonds is default, which can occur when the bond issuer is no longer able
either to pay the interest or repay the principal. The latter is negated by the
fact that banks mainly buy government and municipal bonds with a very small
likelihood of default. Below, the rate of return on bonds in the tth period, Bt, is denoted by rtB.


#### 2.1.3. Cash

Cash,C, is a term assigned to very short-term savings
instruments such as money market securities. These investments can be used to
meet near-term financial needs or to protect a portion of an investment
portfolio from price fluctuation. The downside of cash securities is that they
offer no real opportunities for long-term growth. Though economic conditions
and factors such as changing interest rates can impact both stocks and bonds,
these markets perform independently of each other and can therefore serve as a
balance within the portfolio of a bank. In the sequel, the rate of return on
cash in the tth period, Ct, is denoted by rtC.


#### 2.1.4. Intangible Assets

In the contemporary banking industry, shareholder
value is often created byintangible
assets which consist of patents, trademarks, brand names, franchises and
economic goodwill. Such goodwill consists of the intangible advantages a bank
has over its competitors such as an excellent reputation, strategic location,
business connections, and so forth. In addition, such assets can comprise a large part
of the bank's total assets and provide a sustainable source of wealth creation.
Intangible assets are used to compute Tier 1 bank capital and have a
risk-weight of 100% according to Basel II regulation (see Table 1). In
practice, valuing these off-balance sheet items constitutes one of the
principal difficulties with the process of bank valuation by a
stock analyst.
The reason for this is that intangibles may be considered to be
“risky” assets for which the future service potential is hard to
measure. Despite this, our model assumes that the measurement of these intangibles
is possible (see, for instance, [5, 6]).Table 1
Risk categories, risk-weights and representative items.
Risk categoryRisk-weightBanking items10%Cash, bonds, treasuries, reserves220%Shares350%Home loans4100%Intangible assets5100%Loans to private agentsIn reality, valuing this off-balance sheet item
constitutes one of the principal difficulties with the process of bank
valuation (see, for instance, [5, 6]). Nevertheless, we denote the value of intangible
assets, in the tth period, by It and the return on these assets by rtIIt, where(2.3)rtI=It+1−ItIt.


#### 2.1.5. Treasuries

Treasuries in the tth period, Tt, coincide with securities that are issued by
national Treasuries at a rate denoted by rT. In essence, they are the debt financing
instruments of the federal government. There are four types of Treasuries,
viz., Treasury bills, Treasury notes, Treasury bonds and savings bonds. All of
the Treasury securities besides savings bonds are very liquid and are heavily
traded on the secondary market.


#### 2.1.6. Reserves

Bank reserves are the deposits held in accounts with a national
institution (for instance, the Federal Reserve) plus money that is physically
held by banks (vault cash). Such reserves are constituted by money that is not
lent out but is earmarked to cater for withdrawals by depositors. Since it is
uncommon for depositors to withdraw all of their funds simultaneously, only a
portion of total deposits may be needed as reserves. As a result of this
description, we may introduce a reserve-deposit ratio, γ, for which(2.4)Rt=γDt.The bank uses the remaining
deposits to earn profit, either by issuing loans or by investing in assets such
as Treasuries and stocks.


### 2.2. Loans

In this subsection, we consider loan and their supply
and demand, loan losses and the provisioning for such losses.


#### 2.2.1. Loans and Their Demand and Supply

We suppose that, after providing liquidity, the bank
lends in the form oftth period loans, Λt, at the bank's own loan rate, rtΛ. This loan rate, for profit maximizing banks,
is determined by the risk premium (or yield differential), given
by(2.5)ϱt=rtΛ−rt,the industry's market power as
determined by its concentration, N, the market elasticity of demand for loans, η, base rate, rt, the marginal cost of raising funds in the
secondary market, crw, and the product of the cost of elasticity
(equity) raised, cE, and the sensitivity of the required capital to
changes in the amount of loans extended,(2.6)∂K∂Λ.In this situation, we may
express the bank's own loan rate, rΛ, as(2.7)rtΛ=ϱ+(1+rt)Nη+crw+cE∂K∂Λ+E[l],where(2.8)N=∑i=1nSi2is the Herfindahl-Hirschman
index of the concentration in the loan market,(2.9)Si=ΛiΛis the market share of bank i in the loan market, but in our contribution we
only use one bank, therefore N=1 and(2.10)η=−∂Λ∂rtΛrtΛΛis the elasticity of demand for
loans. Also, in our model, besides the risk premium, we include E[l] which constitutes the amount of provisioning
that is needed to match the average expected losses faced by the loans.In this paragraph, we provide a brief discussion of
loan demand and supply. Taking our lead from the equilibrium arguments in
[30], we denote both
these credit price processes by Λ={Λt}t≥0. In this case, the bank faces a Hicksian demand for loans given
by(2.11)Λt=l0−l1rtΛ+l2Mt+σtΛ.We note that the loan demand in
(2.11) is an increasing function of M and a decreasing function of rtΛ. Also, we assume that σtΛ is the random shock to the loan demand with
support [Λ¯,Λ¯] that is independent of an exogenous stochastic
variable, xt, to be characterized below. In addition, we
suppose that the loan supply process, Λ, follows the first-order autoregressive
stochastic process(2.12)Λt+1=μtΛΛt+σt+1Λ,where μtΛ=rtΛ−cΛ−rd(Mt) and σt+1Λ denotes zero-mean stochastic shocks to loan
supply.Remark 2.1 (loan demand and supply).
Banks respond
differently to shocks that affect loan demand,Λ, when the minimum capital requirements are
calculated by using risk-weighted assets. In the Hicksian case, these responses
are usually sensitive to macroeconomic conditions that are related to the term l2Mt in (2.11). Here we may broaden the analysis
quite considerably by supposing that M={Mt}t≥0 follows the first-order autoregressive
stochastic process Mt+1=μMMt+σt+1M,where σt+1M denotes zero-mean stochastic shocks to
macroeconomic activity.


#### 2.2.2. Loan Losses and Provisioning

The bank's investment in loans may yield substantial
returns but may also result in loan losses. In line with reality, our dynamic
bank model allows for loan losses for which provision can be made.Total loan loss provisions, P, mainly affects the bank in the following ways.
Reported nett profit will be less for the period in which the provision is
taken. If the bank eventually writes off the asset, the write off will reduce
taxes and thus increase the banks cash flows.
Empirical evidence suggests that P is affected by macroeconomic activity, M, so that the notation P(Mt) for period t loan loss provisioning is in order (see, for
instance, [34, 35]).For thevalue of the aggregate loan losses, L, and the default rate, rd, we have that(2.13)L(Mt)=rd(Mt)Λt,where rd∈[0,1] increases when macroeconomic conditions
deteriorate according to(2.14)0≤rd(Mt)≤1,∂rd(Mt)∂Mt<0.We note that the above
description of the loan loss rate is consistent with empirical evidence that
suggests that bank losses on loan portfolios are correlated with the business
cycle under any capital adequacy regime (see, for instance, [34–36, 49]).As was mentioned before, the contribution [34] (see, also, [36, 49]) highlights the fact that
normally provisions for expected loan losses, (αϱ+E[l])Λt, where 0≤α≤1 and ϱ is the risk premium from (2.5), and loan loss
reserves, Rl, act as buffers against expected and unexpected
loan losses, respectively. Firstly, we have to distinguish between total
provisioning for loan losses, P, and loan loss reserves, Rl. Provisioning is a decision made by bank
management about the size of the buffer that must be set aside in a particular time
period in order to cover loan losses, L. However, not all of P may be used in a time period with the amount
left over constituting loan loss
reserves, Rl, so that for period t we have(2.15)Rtl=P(Mt)−L(Mt),P>L.Our model for provisioning in
period t+1 can be taken to be(2.16)P(Mt+1)={(αϱ+E[l])Λt,forP>L⁢Expectedlosses(αϱ+E[l])Λt+Rt+1l,forP≤L⁢Expectedlosses+Unexpectedlosses,We note that our model
determines the provisions for period t+1 in the tth period which is a reasonable assumption.
Our suspicion is that provisioning, P, is a decreasing function of current
macroeconomic conditions, M, so that(2.17)∂P(Mt)∂Mt<0.This claim has resonance with
the idea of procyclicality where we expect the provisioning to decrease during
booms, when macroeconomic activity increases. By contrast, provisioning may
increase during recessions because of an elevated probability of default and/or
loss given default on loans. This suspicion is confirmed in Section 4 where
empirical data from OECD countries comparing macroeconomic activity (via the
output gap) and provisioning (via the provisions-to-total assets ratio) is
examined.


### 2.3. Liabilities

In this subsection, we consider deposits and
provisioning for deposit withdrawals as well as interbank borrowing.


#### 2.3.1. Deposits

The bank takes deposits,Dt, at a constant marginal cost, cD, that may be associated with cheque clearing
and bookkeeping. It is assumed that deposit taking is not interrupted even in
times when the interest rate on
deposits or deposit rate, rtD, is less than the interest rate on Treasuries or bond rate, rtT, We suppose that the dynamics of the deposit
rate process, rD={rtD}t≥0, is determined by the first-order
autoregressive stochastic process(2.18)rt+1D=μrDrtD+σt+1rD,where σt+1rD is zero-mean stochastic shocks to the deposit
rate.Remark 2.2 (deposit rate and monetary policy).
In
some quarters, the deposit rate,rD, is considered to be a strong approximation of
bank monetary policy. Since such policy is usually affected by macroeconomic
activity, M, we expect the aforementioned items to share an
intimate connection. However, in our analysis, we assume that the shocks σt+1D and σt+1M  to rD  and M,  respectively, are uncorrelated. Essentially,
this means that a precise monetary policy is lacking in our bank model. This
interesting relationship is the subject of further investigation.


#### 2.3.2. Provisioning for Deposit Withdrawals

We have to consider the possibility that unanticipated
deposit withdrawals will occur. By way of making provision for these
withdrawals, the bank is inclined to hold Treasuries and reserves that are both
very liquid. In our contribution, we assume that theunanticipated deposit withdrawals, u, originates from the probability density function, f(u), that is independent of time. For sake of
argument, we suppose that the unanticipated deposit withdrawals have a uniform
distribution with support [0,D¯] so that the cost of liquidation, cl, or additional external funding is a quadratic
function of the sum of Treasuries and reserves, W. In addition, for any t, if we have that(2.19)u>Wt,where Wt=Tt+Rt,
then bank assets are liquidated at some penalty rate, rtp. In this case, the cost of deposit withdrawals is(2.20)cw(Wt)=rtp∫Wt∞[u−Wt]f(u)du=rtp2D¯[D¯−Wt]2.Remark 2.3 (deposit withdrawals and bank liquidity).
A vital component of the process of deposit withdrawal is liquidity. The level
of liquidity in the banking sector affects the ability of banks to meet
commitments as they become due (such as deposit withdrawals) without incurring
substantial losses from liquidating less liquid assets. Liquidity, therefore,
provides the defensive cash or near-cash resources to cover banks' liabilities.


#### 2.3.3. Borrowing from Other Banks

Interbank borrowing including borrowing from the
Central Bank provides a further source of funds. In the sequel, the amount
borrowed from other banks is denoted byB, while the interbank borrowing rate (for
instance, known as the Libor rate in the United Kingdom) and marginal borrowing
costs are denoted by rB and cB, respectively. Of course, when our bank borrows
from the Central Bank, we have rB=r, where r is the base rate appearing in (2.5). Another
important issue here is the comparison between the cost of raising and holding
deposits, (rD+cD)D, and the cost of interbank borrowing, (rB+cB)B. In this regard, a bank in need of capital
would have to choose between raising deposits and borrowing from other banks on
the basis of overall cost. In other words, the expression(2.21)min{(rD+cD)D,(rB+cB)B}is of some consequence. For sake
of argument, in the sequel, we assume that(2.22)(rD+cD)D=min{(rD+cD)D,(rB+cB)B}.


### 2.4. Operational Risk

The Basel II framework outlines three quantitative
approaches for determining an operational risk capital premium: the Basic
Indicator approach, the Standardized approach, and the Advanced Measurement
approach. The Basic Indicator and the Standardized approaches are simple and
generate results on the basis of predetermined multipliers. More specifically,
the capital premium for operational risk, under the Standardized approach
outlined in the Basel II, may be expressed as(2.23)O=max[∑k=18βkgk,0],where,  g1−8 is three-year average of gross income for each of eight business lines, and β1−8 is fixed percentage relating level of required capital to level of gross income
for each of eight business lines.Theβ-values for operational risk are provided in
the document [1].


## 2.1. General Bank Assets

In this subsection, we discuss on- and off-balance
sheet bank assets such as shares, bonds and cash, Treasuries, reserves and
intangible assets.


### 2.1.1. Shares

Of the first three general bank asset classes, shares,S, have historically been the most prominent
performers over the long term. Since the returns from shares usually exceed the
returns from both bonds and cash and have significantly outpaced inflation,
they are important to a portfolio for growth of capital over time. Over the
short term, however, shares can be volatile and as a result there is regulation
related to banks holding shares. In the sequel, the rate of return on shares in
the tth period, St, is denoted by rtS.


### 2.1.2. Bonds

Whereas shares represent equity, or part ownership of
the companies that issue them, bonds,B, represent debt. Municipalities and governments
all use bonds as a way to raise cash. When banks buy bonds, they are lending
money to the issuer in exchange for fixed interest payments over a set number
of years and a promise to pay the original amount back in the future. Bonds are
valuable to banks more for the income they provide than for growth potential.
Since the income they pay is fixed it is generally reliable and steady. The
primary risk in bond market investing comes from interest rate changes. When
interest rates rise, a bond's market value decreases. Another potential risk of
owning bonds is default, which can occur when the bond issuer is no longer able
either to pay the interest or repay the principal. The latter is negated by the
fact that banks mainly buy government and municipal bonds with a very small
likelihood of default. Below, the rate of return on bonds in the tth period, Bt, is denoted by rtB.


### 2.1.3. Cash

Cash,C, is a term assigned to very short-term savings
instruments such as money market securities. These investments can be used to
meet near-term financial needs or to protect a portion of an investment
portfolio from price fluctuation. The downside of cash securities is that they
offer no real opportunities for long-term growth. Though economic conditions
and factors such as changing interest rates can impact both stocks and bonds,
these markets perform independently of each other and can therefore serve as a
balance within the portfolio of a bank. In the sequel, the rate of return on
cash in the tth period, Ct, is denoted by rtC.


### 2.1.4. Intangible Assets

In the contemporary banking industry, shareholder
value is often created byintangible
assets which consist of patents, trademarks, brand names, franchises and
economic goodwill. Such goodwill consists of the intangible advantages a bank
has over its competitors such as an excellent reputation, strategic location,
business connections, and so forth. In addition, such assets can comprise a large part
of the bank's total assets and provide a sustainable source of wealth creation.
Intangible assets are used to compute Tier 1 bank capital and have a
risk-weight of 100% according to Basel II regulation (see Table 1). In
practice, valuing these off-balance sheet items constitutes one of the
principal difficulties with the process of bank valuation by a
stock analyst.
The reason for this is that intangibles may be considered to be
“risky” assets for which the future service potential is hard to
measure. Despite this, our model assumes that the measurement of these intangibles
is possible (see, for instance, [5, 6]).Table 1
Risk categories, risk-weights and representative items.
Risk categoryRisk-weightBanking items10%Cash, bonds, treasuries, reserves220%Shares350%Home loans4100%Intangible assets5100%Loans to private agentsIn reality, valuing this off-balance sheet item
constitutes one of the principal difficulties with the process of bank
valuation (see, for instance, [5, 6]). Nevertheless, we denote the value of intangible
assets, in the tth period, by It and the return on these assets by rtIIt, where(2.3)rtI=It+1−ItIt.


### 2.1.5. Treasuries

Treasuries in the tth period, Tt, coincide with securities that are issued by
national Treasuries at a rate denoted by rT. In essence, they are the debt financing
instruments of the federal government. There are four types of Treasuries,
viz., Treasury bills, Treasury notes, Treasury bonds and savings bonds. All of
the Treasury securities besides savings bonds are very liquid and are heavily
traded on the secondary market.


### 2.1.6. Reserves

Bank reserves are the deposits held in accounts with a national
institution (for instance, the Federal Reserve) plus money that is physically
held by banks (vault cash). Such reserves are constituted by money that is not
lent out but is earmarked to cater for withdrawals by depositors. Since it is
uncommon for depositors to withdraw all of their funds simultaneously, only a
portion of total deposits may be needed as reserves. As a result of this
description, we may introduce a reserve-deposit ratio, γ, for which(2.4)Rt=γDt.The bank uses the remaining
deposits to earn profit, either by issuing loans or by investing in assets such
as Treasuries and stocks.


## 2.1.1. Shares

Of the first three general bank asset classes, shares,S, have historically been the most prominent
performers over the long term. Since the returns from shares usually exceed the
returns from both bonds and cash and have significantly outpaced inflation,
they are important to a portfolio for growth of capital over time. Over the
short term, however, shares can be volatile and as a result there is regulation
related to banks holding shares. In the sequel, the rate of return on shares in
the tth period, St, is denoted by rtS.


## 2.1.2. Bonds

Whereas shares represent equity, or part ownership of
the companies that issue them, bonds,B, represent debt. Municipalities and governments
all use bonds as a way to raise cash. When banks buy bonds, they are lending
money to the issuer in exchange for fixed interest payments over a set number
of years and a promise to pay the original amount back in the future. Bonds are
valuable to banks more for the income they provide than for growth potential.
Since the income they pay is fixed it is generally reliable and steady. The
primary risk in bond market investing comes from interest rate changes. When
interest rates rise, a bond's market value decreases. Another potential risk of
owning bonds is default, which can occur when the bond issuer is no longer able
either to pay the interest or repay the principal. The latter is negated by the
fact that banks mainly buy government and municipal bonds with a very small
likelihood of default. Below, the rate of return on bonds in the tth period, Bt, is denoted by rtB.


## 2.1.3. Cash

Cash,C, is a term assigned to very short-term savings
instruments such as money market securities. These investments can be used to
meet near-term financial needs or to protect a portion of an investment
portfolio from price fluctuation. The downside of cash securities is that they
offer no real opportunities for long-term growth. Though economic conditions
and factors such as changing interest rates can impact both stocks and bonds,
these markets perform independently of each other and can therefore serve as a
balance within the portfolio of a bank. In the sequel, the rate of return on
cash in the tth period, Ct, is denoted by rtC.


## 2.1.4. Intangible Assets

In the contemporary banking industry, shareholder
value is often created byintangible
assets which consist of patents, trademarks, brand names, franchises and
economic goodwill. Such goodwill consists of the intangible advantages a bank
has over its competitors such as an excellent reputation, strategic location,
business connections, and so forth. In addition, such assets can comprise a large part
of the bank's total assets and provide a sustainable source of wealth creation.
Intangible assets are used to compute Tier 1 bank capital and have a
risk-weight of 100% according to Basel II regulation (see Table 1). In
practice, valuing these off-balance sheet items constitutes one of the
principal difficulties with the process of bank valuation by a
stock analyst.
The reason for this is that intangibles may be considered to be
“risky” assets for which the future service potential is hard to
measure. Despite this, our model assumes that the measurement of these intangibles
is possible (see, for instance, [5, 6]).Table 1
Risk categories, risk-weights and representative items.
Risk categoryRisk-weightBanking items10%Cash, bonds, treasuries, reserves220%Shares350%Home loans4100%Intangible assets5100%Loans to private agentsIn reality, valuing this off-balance sheet item
constitutes one of the principal difficulties with the process of bank
valuation (see, for instance, [5, 6]). Nevertheless, we denote the value of intangible
assets, in the tth period, by It and the return on these assets by rtIIt, where(2.3)rtI=It+1−ItIt.


## 2.1.5. Treasuries

Treasuries in the tth period, Tt, coincide with securities that are issued by
national Treasuries at a rate denoted by rT. In essence, they are the debt financing
instruments of the federal government. There are four types of Treasuries,
viz., Treasury bills, Treasury notes, Treasury bonds and savings bonds. All of
the Treasury securities besides savings bonds are very liquid and are heavily
traded on the secondary market.


## 2.1.6. Reserves

Bank reserves are the deposits held in accounts with a national
institution (for instance, the Federal Reserve) plus money that is physically
held by banks (vault cash). Such reserves are constituted by money that is not
lent out but is earmarked to cater for withdrawals by depositors. Since it is
uncommon for depositors to withdraw all of their funds simultaneously, only a
portion of total deposits may be needed as reserves. As a result of this
description, we may introduce a reserve-deposit ratio, γ, for which(2.4)Rt=γDt.The bank uses the remaining
deposits to earn profit, either by issuing loans or by investing in assets such
as Treasuries and stocks.


## 2.2. Loans

In this subsection, we consider loan and their supply
and demand, loan losses and the provisioning for such losses.


### 2.2.1. Loans and Their Demand and Supply

We suppose that, after providing liquidity, the bank
lends in the form oftth period loans, Λt, at the bank's own loan rate, rtΛ. This loan rate, for profit maximizing banks,
is determined by the risk premium (or yield differential), given
by(2.5)ϱt=rtΛ−rt,the industry's market power as
determined by its concentration, N, the market elasticity of demand for loans, η, base rate, rt, the marginal cost of raising funds in the
secondary market, crw, and the product of the cost of elasticity
(equity) raised, cE, and the sensitivity of the required capital to
changes in the amount of loans extended,(2.6)∂K∂Λ.In this situation, we may
express the bank's own loan rate, rΛ, as(2.7)rtΛ=ϱ+(1+rt)Nη+crw+cE∂K∂Λ+E[l],where(2.8)N=∑i=1nSi2is the Herfindahl-Hirschman
index of the concentration in the loan market,(2.9)Si=ΛiΛis the market share of bank i in the loan market, but in our contribution we
only use one bank, therefore N=1 and(2.10)η=−∂Λ∂rtΛrtΛΛis the elasticity of demand for
loans. Also, in our model, besides the risk premium, we include E[l] which constitutes the amount of provisioning
that is needed to match the average expected losses faced by the loans.In this paragraph, we provide a brief discussion of
loan demand and supply. Taking our lead from the equilibrium arguments in
[30], we denote both
these credit price processes by Λ={Λt}t≥0. In this case, the bank faces a Hicksian demand for loans given
by(2.11)Λt=l0−l1rtΛ+l2Mt+σtΛ.We note that the loan demand in
(2.11) is an increasing function of M and a decreasing function of rtΛ. Also, we assume that σtΛ is the random shock to the loan demand with
support [Λ¯,Λ¯] that is independent of an exogenous stochastic
variable, xt, to be characterized below. In addition, we
suppose that the loan supply process, Λ, follows the first-order autoregressive
stochastic process(2.12)Λt+1=μtΛΛt+σt+1Λ,where μtΛ=rtΛ−cΛ−rd(Mt) and σt+1Λ denotes zero-mean stochastic shocks to loan
supply.Remark 2.1 (loan demand and supply).
Banks respond
differently to shocks that affect loan demand,Λ, when the minimum capital requirements are
calculated by using risk-weighted assets. In the Hicksian case, these responses
are usually sensitive to macroeconomic conditions that are related to the term l2Mt in (2.11). Here we may broaden the analysis
quite considerably by supposing that M={Mt}t≥0 follows the first-order autoregressive
stochastic process Mt+1=μMMt+σt+1M,where σt+1M denotes zero-mean stochastic shocks to
macroeconomic activity.


### 2.2.2. Loan Losses and Provisioning

The bank's investment in loans may yield substantial
returns but may also result in loan losses. In line with reality, our dynamic
bank model allows for loan losses for which provision can be made.Total loan loss provisions, P, mainly affects the bank in the following ways.
Reported nett profit will be less for the period in which the provision is
taken. If the bank eventually writes off the asset, the write off will reduce
taxes and thus increase the banks cash flows.
Empirical evidence suggests that P is affected by macroeconomic activity, M, so that the notation P(Mt) for period t loan loss provisioning is in order (see, for
instance, [34, 35]).For thevalue of the aggregate loan losses, L, and the default rate, rd, we have that(2.13)L(Mt)=rd(Mt)Λt,where rd∈[0,1] increases when macroeconomic conditions
deteriorate according to(2.14)0≤rd(Mt)≤1,∂rd(Mt)∂Mt<0.We note that the above
description of the loan loss rate is consistent with empirical evidence that
suggests that bank losses on loan portfolios are correlated with the business
cycle under any capital adequacy regime (see, for instance, [34–36, 49]).As was mentioned before, the contribution [34] (see, also, [36, 49]) highlights the fact that
normally provisions for expected loan losses, (αϱ+E[l])Λt, where 0≤α≤1 and ϱ is the risk premium from (2.5), and loan loss
reserves, Rl, act as buffers against expected and unexpected
loan losses, respectively. Firstly, we have to distinguish between total
provisioning for loan losses, P, and loan loss reserves, Rl. Provisioning is a decision made by bank
management about the size of the buffer that must be set aside in a particular time
period in order to cover loan losses, L. However, not all of P may be used in a time period with the amount
left over constituting loan loss
reserves, Rl, so that for period t we have(2.15)Rtl=P(Mt)−L(Mt),P>L.Our model for provisioning in
period t+1 can be taken to be(2.16)P(Mt+1)={(αϱ+E[l])Λt,forP>L⁢Expectedlosses(αϱ+E[l])Λt+Rt+1l,forP≤L⁢Expectedlosses+Unexpectedlosses,We note that our model
determines the provisions for period t+1 in the tth period which is a reasonable assumption.
Our suspicion is that provisioning, P, is a decreasing function of current
macroeconomic conditions, M, so that(2.17)∂P(Mt)∂Mt<0.This claim has resonance with
the idea of procyclicality where we expect the provisioning to decrease during
booms, when macroeconomic activity increases. By contrast, provisioning may
increase during recessions because of an elevated probability of default and/or
loss given default on loans. This suspicion is confirmed in Section 4 where
empirical data from OECD countries comparing macroeconomic activity (via the
output gap) and provisioning (via the provisions-to-total assets ratio) is
examined.


## 2.2.1. Loans and Their Demand and Supply

We suppose that, after providing liquidity, the bank
lends in the form oftth period loans, Λt, at the bank's own loan rate, rtΛ. This loan rate, for profit maximizing banks,
is determined by the risk premium (or yield differential), given
by(2.5)ϱt=rtΛ−rt,the industry's market power as
determined by its concentration, N, the market elasticity of demand for loans, η, base rate, rt, the marginal cost of raising funds in the
secondary market, crw, and the product of the cost of elasticity
(equity) raised, cE, and the sensitivity of the required capital to
changes in the amount of loans extended,(2.6)∂K∂Λ.In this situation, we may
express the bank's own loan rate, rΛ, as(2.7)rtΛ=ϱ+(1+rt)Nη+crw+cE∂K∂Λ+E[l],where(2.8)N=∑i=1nSi2is the Herfindahl-Hirschman
index of the concentration in the loan market,(2.9)Si=ΛiΛis the market share of bank i in the loan market, but in our contribution we
only use one bank, therefore N=1 and(2.10)η=−∂Λ∂rtΛrtΛΛis the elasticity of demand for
loans. Also, in our model, besides the risk premium, we include E[l] which constitutes the amount of provisioning
that is needed to match the average expected losses faced by the loans.In this paragraph, we provide a brief discussion of
loan demand and supply. Taking our lead from the equilibrium arguments in
[30], we denote both
these credit price processes by Λ={Λt}t≥0. In this case, the bank faces a Hicksian demand for loans given
by(2.11)Λt=l0−l1rtΛ+l2Mt+σtΛ.We note that the loan demand in
(2.11) is an increasing function of M and a decreasing function of rtΛ. Also, we assume that σtΛ is the random shock to the loan demand with
support [Λ¯,Λ¯] that is independent of an exogenous stochastic
variable, xt, to be characterized below. In addition, we
suppose that the loan supply process, Λ, follows the first-order autoregressive
stochastic process(2.12)Λt+1=μtΛΛt+σt+1Λ,where μtΛ=rtΛ−cΛ−rd(Mt) and σt+1Λ denotes zero-mean stochastic shocks to loan
supply.Remark 2.1 (loan demand and supply).
Banks respond
differently to shocks that affect loan demand,Λ, when the minimum capital requirements are
calculated by using risk-weighted assets. In the Hicksian case, these responses
are usually sensitive to macroeconomic conditions that are related to the term l2Mt in (2.11). Here we may broaden the analysis
quite considerably by supposing that M={Mt}t≥0 follows the first-order autoregressive
stochastic process Mt+1=μMMt+σt+1M,where σt+1M denotes zero-mean stochastic shocks to
macroeconomic activity.


## 2.2.2. Loan Losses and Provisioning

The bank's investment in loans may yield substantial
returns but may also result in loan losses. In line with reality, our dynamic
bank model allows for loan losses for which provision can be made.Total loan loss provisions, P, mainly affects the bank in the following ways.
Reported nett profit will be less for the period in which the provision is
taken. If the bank eventually writes off the asset, the write off will reduce
taxes and thus increase the banks cash flows.
Empirical evidence suggests that P is affected by macroeconomic activity, M, so that the notation P(Mt) for period t loan loss provisioning is in order (see, for
instance, [34, 35]).For thevalue of the aggregate loan losses, L, and the default rate, rd, we have that(2.13)L(Mt)=rd(Mt)Λt,where rd∈[0,1] increases when macroeconomic conditions
deteriorate according to(2.14)0≤rd(Mt)≤1,∂rd(Mt)∂Mt<0.We note that the above
description of the loan loss rate is consistent with empirical evidence that
suggests that bank losses on loan portfolios are correlated with the business
cycle under any capital adequacy regime (see, for instance, [34–36, 49]).As was mentioned before, the contribution [34] (see, also, [36, 49]) highlights the fact that
normally provisions for expected loan losses, (αϱ+E[l])Λt, where 0≤α≤1 and ϱ is the risk premium from (2.5), and loan loss
reserves, Rl, act as buffers against expected and unexpected
loan losses, respectively. Firstly, we have to distinguish between total
provisioning for loan losses, P, and loan loss reserves, Rl. Provisioning is a decision made by bank
management about the size of the buffer that must be set aside in a particular time
period in order to cover loan losses, L. However, not all of P may be used in a time period with the amount
left over constituting loan loss
reserves, Rl, so that for period t we have(2.15)Rtl=P(Mt)−L(Mt),P>L.Our model for provisioning in
period t+1 can be taken to be(2.16)P(Mt+1)={(αϱ+E[l])Λt,forP>L⁢Expectedlosses(αϱ+E[l])Λt+Rt+1l,forP≤L⁢Expectedlosses+Unexpectedlosses,We note that our model
determines the provisions for period t+1 in the tth period which is a reasonable assumption.
Our suspicion is that provisioning, P, is a decreasing function of current
macroeconomic conditions, M, so that(2.17)∂P(Mt)∂Mt<0.This claim has resonance with
the idea of procyclicality where we expect the provisioning to decrease during
booms, when macroeconomic activity increases. By contrast, provisioning may
increase during recessions because of an elevated probability of default and/or
loss given default on loans. This suspicion is confirmed in Section 4 where
empirical data from OECD countries comparing macroeconomic activity (via the
output gap) and provisioning (via the provisions-to-total assets ratio) is
examined.


## 2.3. Liabilities

In this subsection, we consider deposits and
provisioning for deposit withdrawals as well as interbank borrowing.


### 2.3.1. Deposits

The bank takes deposits,Dt, at a constant marginal cost, cD, that may be associated with cheque clearing
and bookkeeping. It is assumed that deposit taking is not interrupted even in
times when the interest rate on
deposits or deposit rate, rtD, is less than the interest rate on Treasuries or bond rate, rtT, We suppose that the dynamics of the deposit
rate process, rD={rtD}t≥0, is determined by the first-order
autoregressive stochastic process(2.18)rt+1D=μrDrtD+σt+1rD,where σt+1rD is zero-mean stochastic shocks to the deposit
rate.Remark 2.2 (deposit rate and monetary policy).
In
some quarters, the deposit rate,rD, is considered to be a strong approximation of
bank monetary policy. Since such policy is usually affected by macroeconomic
activity, M, we expect the aforementioned items to share an
intimate connection. However, in our analysis, we assume that the shocks σt+1D and σt+1M  to rD  and M,  respectively, are uncorrelated. Essentially,
this means that a precise monetary policy is lacking in our bank model. This
interesting relationship is the subject of further investigation.


### 2.3.2. Provisioning for Deposit Withdrawals

We have to consider the possibility that unanticipated
deposit withdrawals will occur. By way of making provision for these
withdrawals, the bank is inclined to hold Treasuries and reserves that are both
very liquid. In our contribution, we assume that theunanticipated deposit withdrawals, u, originates from the probability density function, f(u), that is independent of time. For sake of
argument, we suppose that the unanticipated deposit withdrawals have a uniform
distribution with support [0,D¯] so that the cost of liquidation, cl, or additional external funding is a quadratic
function of the sum of Treasuries and reserves, W. In addition, for any t, if we have that(2.19)u>Wt,where Wt=Tt+Rt,
then bank assets are liquidated at some penalty rate, rtp. In this case, the cost of deposit withdrawals is(2.20)cw(Wt)=rtp∫Wt∞[u−Wt]f(u)du=rtp2D¯[D¯−Wt]2.Remark 2.3 (deposit withdrawals and bank liquidity).
A vital component of the process of deposit withdrawal is liquidity. The level
of liquidity in the banking sector affects the ability of banks to meet
commitments as they become due (such as deposit withdrawals) without incurring
substantial losses from liquidating less liquid assets. Liquidity, therefore,
provides the defensive cash or near-cash resources to cover banks' liabilities.


### 2.3.3. Borrowing from Other Banks

Interbank borrowing including borrowing from the
Central Bank provides a further source of funds. In the sequel, the amount
borrowed from other banks is denoted byB, while the interbank borrowing rate (for
instance, known as the Libor rate in the United Kingdom) and marginal borrowing
costs are denoted by rB and cB, respectively. Of course, when our bank borrows
from the Central Bank, we have rB=r, where r is the base rate appearing in (2.5). Another
important issue here is the comparison between the cost of raising and holding
deposits, (rD+cD)D, and the cost of interbank borrowing, (rB+cB)B. In this regard, a bank in need of capital
would have to choose between raising deposits and borrowing from other banks on
the basis of overall cost. In other words, the expression(2.21)min{(rD+cD)D,(rB+cB)B}is of some consequence. For sake
of argument, in the sequel, we assume that(2.22)(rD+cD)D=min{(rD+cD)D,(rB+cB)B}.


## 2.3.1. Deposits

The bank takes deposits,Dt, at a constant marginal cost, cD, that may be associated with cheque clearing
and bookkeeping. It is assumed that deposit taking is not interrupted even in
times when the interest rate on
deposits or deposit rate, rtD, is less than the interest rate on Treasuries or bond rate, rtT, We suppose that the dynamics of the deposit
rate process, rD={rtD}t≥0, is determined by the first-order
autoregressive stochastic process(2.18)rt+1D=μrDrtD+σt+1rD,where σt+1rD is zero-mean stochastic shocks to the deposit
rate.Remark 2.2 (deposit rate and monetary policy).
In
some quarters, the deposit rate,rD, is considered to be a strong approximation of
bank monetary policy. Since such policy is usually affected by macroeconomic
activity, M, we expect the aforementioned items to share an
intimate connection. However, in our analysis, we assume that the shocks σt+1D and σt+1M  to rD  and M,  respectively, are uncorrelated. Essentially,
this means that a precise monetary policy is lacking in our bank model. This
interesting relationship is the subject of further investigation.


## 2.3.2. Provisioning for Deposit Withdrawals

We have to consider the possibility that unanticipated
deposit withdrawals will occur. By way of making provision for these
withdrawals, the bank is inclined to hold Treasuries and reserves that are both
very liquid. In our contribution, we assume that theunanticipated deposit withdrawals, u, originates from the probability density function, f(u), that is independent of time. For sake of
argument, we suppose that the unanticipated deposit withdrawals have a uniform
distribution with support [0,D¯] so that the cost of liquidation, cl, or additional external funding is a quadratic
function of the sum of Treasuries and reserves, W. In addition, for any t, if we have that(2.19)u>Wt,where Wt=Tt+Rt,
then bank assets are liquidated at some penalty rate, rtp. In this case, the cost of deposit withdrawals is(2.20)cw(Wt)=rtp∫Wt∞[u−Wt]f(u)du=rtp2D¯[D¯−Wt]2.Remark 2.3 (deposit withdrawals and bank liquidity).
A vital component of the process of deposit withdrawal is liquidity. The level
of liquidity in the banking sector affects the ability of banks to meet
commitments as they become due (such as deposit withdrawals) without incurring
substantial losses from liquidating less liquid assets. Liquidity, therefore,
provides the defensive cash or near-cash resources to cover banks' liabilities.


## 2.3.3. Borrowing from Other Banks

Interbank borrowing including borrowing from the
Central Bank provides a further source of funds. In the sequel, the amount
borrowed from other banks is denoted byB, while the interbank borrowing rate (for
instance, known as the Libor rate in the United Kingdom) and marginal borrowing
costs are denoted by rB and cB, respectively. Of course, when our bank borrows
from the Central Bank, we have rB=r, where r is the base rate appearing in (2.5). Another
important issue here is the comparison between the cost of raising and holding
deposits, (rD+cD)D, and the cost of interbank borrowing, (rB+cB)B. In this regard, a bank in need of capital
would have to choose between raising deposits and borrowing from other banks on
the basis of overall cost. In other words, the expression(2.21)min{(rD+cD)D,(rB+cB)B}is of some consequence. For sake
of argument, in the sequel, we assume that(2.22)(rD+cD)D=min{(rD+cD)D,(rB+cB)B}.


## 2.4. Operational Risk

The Basel II framework outlines three quantitative
approaches for determining an operational risk capital premium: the Basic
Indicator approach, the Standardized approach, and the Advanced Measurement
approach. The Basic Indicator and the Standardized approaches are simple and
generate results on the basis of predetermined multipliers. More specifically,
the capital premium for operational risk, under the Standardized approach
outlined in the Basel II, may be expressed as(2.23)O=max[∑k=18βkgk,0],where,  g1−8 is three-year average of gross income for each of eight business lines, and β1−8 is fixed percentage relating level of required capital to level of gross income
for each of eight business lines.Theβ-values for operational risk are provided in
the document [1].


## 3. Bank valuation

In this section, we discuss bank regulatory capital,
binding capital constraints, retained earnings and the valuation of a bank by a
stock analyst.


### 3.1. Bank Regulatory Capital

In this subsection, we provide a general description
of bank capital and then specify the components of total bank capital that we
use in our study.


#### 3.1.1. General Description of Bank Capital

According to Basel II, three types of capital can be
identified, viz., Tier 1, 2 and 3 capital, which we describe in more detail
below.Tier 1 capital comprises
ordinary share capital (or equity) of the bank and audited revenue reserves,
for example, retained earnings less current year's losses, future tax benefits and
intangible assets (for more information see, for instance, [5, 6]). Tier 1 capital or core
capital acts as a buffer against losses without a bank being required to cease
trading. Tier 2 capital includes
unaudited retained earnings; revaluation reserves; general provisions for bad
debts (e.g., loan loss reserves); perpetual cumulative preference shares (i.e.,
preference shares with no maturity date whose dividends accrue for future
payment even if the bank's financial condition does not support immediate
payment) and perpetual subordinated debt (i.e., debt with no maturity date
which ranks in priority behind all creditors except shareholders). Tier 2
capital or supplementary capital can absorb losses in the event of a wind-up
and so provides a lesser degree of protection to depositors. Tier 3 capital consists of subordinated
debt with a term of at least 5 years and redeemable preference shares which may
not be redeemed for at least 5 years. Tier 3 capital can be used to provide a
hedge against losses caused by market risks if Tier 1 and Tier 2 capital are
insufficient for this.


#### 3.1.2. Specific Components of Total Bank Capital

For the purposes of our study,regulatory capital, K, is the book value of bank capital defined as
the difference between the accounting value of the assets and liabilities. More
specifically, Tier 1 capital is represented by period t−1's market value of the bank equity, ntEt−1, where nt is the number of shares and Et is the period t market price of the bank's common equity. Tier
2 capital mainly consists of subordinate debt, Ot, that is subordinate to deposits and hence
faces greater credit risk and loan loss reserves, Rtl. Subordinate debt issued in period t−1 are represented by a one-period bond that pays
an interest rate, rO. Also, we assume that loan loss reserves held
in period t−1 changes at the rate, rRl. Tier 3 capital is not considered at all. In
the sequel, we take the bank's total regulatory capital, K, in period t to be (3.1)Kt=ntEt−1+Ot+Rtl.For Kt given by (3.1), we obtain the balance sheet constraint(3.2)Wt=Dt+Bt−Λt−Ct−Bt−St+Kt.


#### 3.1.3. Binding Capital Constraints

In order to describe the binding capital constraint,
we consider risk-weighted assets (RWAs) that are defined by placing each on- and off-balance sheet item into a risk category. The more risky assets are
assigned a larger weight. Table1 provides a few illustrative risk
categories, their risk-weights and representative items.As a result, RWAs are a weighted sum of the various
assets of the banks. In the sequel, we denote the risk-weight on intangible
assets, cash, bonds, shares, loans, Treasuries and reserves byωI,ωC,ωB,ωS,ωΛ,ωT, and,  ωR, respectively. In particular, we can identify a
special risk-weight on loans, ωΛ=ω(Mt), that is a decreasing function of current
macroeconomic conditions so that(3.3)∂ω(Mt)∂Mt<0.This is in line with the
procyclical notion that during booms, when macroeconomic activity increases,
the risk-weights will decrease. On the other hand, during recessions,
risk-weights may increase because of an elevated probability of default and/or
loss given default on loans.Thebank
capital constraint is defined by the inequality(3.4)Kt≥ρ(at+12.5(m⁢VaR+O))where(3.5)at=ωIIt+ωCCt+ωBBt+ωSSt+ωΛΛt+ωTTt+ωRRt,and ρ≈0.08. The formulation of (3.4) and the choice of
this particular value for ρ is informed by page 12 of  “Part 2: The
First Pillar-Minimum Capital Requirements” of [2]. This excerpt from the
document outlining Basel II states that“Part 2
presents the calculation of the total minimum capital requirements for credit,
market and operational risk. The capital ratio is calculated using the
definition of regulatory capital and risk-weighted assets. The total capital
ratio must be no lower that 8%.…    Total risk-weighted assets are determined
by multiplying the capital requirements for market risk and operational risk by
12.5 (i.e., the reciprocal of the minimum capital ratio of 8%) and adding the
resulting figures to the sum of risk-weighted assets for credit risk.”Also,mVaR and O in (3.4) are as described in Sections 1 and 2
of this paper, respectively.In accordance with Table1, if we assume that the
risk-weights associated with intangible assets, shares, cash, bonds,
Treasuries, reserves and loans may be taken to be ωI≠0,ωS≠0,ωC=ωB=ωT=ωR=0 and ωΛ=ω(Mt), respectively, then equation (3.4) becomes the
capital constraint(3.6)Kt≥ρ[ω(Mt)Λt+ωIIt+ωSSt+12.5(m⁢VaR+O)].


### 3.2. Profits and Retained Earnings

In this subsection, we discuss profits and its
relation to retained earnings.


#### 3.2.1. Profits

We assume that (2.4) holds. As far as profit, Π, is concerned, we use the basic fact that
profits can be characterized as the difference between income and expenses that
are reported in the bank's income statement. In our contribution, income is solely constituted by the
returns on intangible assets, rtIIt, cash, rtCCt, bonds, rtBBt, shares, rtSSt, loans, rtΛΛt, and Treasuries, rtTTt. Furthermore, we assume that the level of
macroeconomic activity is denoted by Mt. In our case we consider the cost of monitoring
and screening of loans and capital, cΛΛt, interest paid to depositors, rtDDt, the cost of taking deposits, cDDt, the cost of deposit withdrawals, cw(Wt), the value of loan losses, L(Mt), and total loan loss provisions, P(Mt) as expenses, in period t. Here rD and cD are the deposit rate and marginal cost of
deposits, respectively. Summing all the costs mentioned to operating costs and
supposing that (2.13) holds and that Wt=Tt+γDt, then the bank's profits are given by the
expression(3.7)Πt=(rtΛ−cΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+cD)Dt−cw(Wt)−P(Mt)−rtTγDt,where rI,rC,rB and rS are the rates of return of the intangible
assets, cash, bonds and shares, respectively. Furthermore, by considering (2.17)
and (3.7), we suspect that profit, Π, is an increasing function of current
macroeconomic conditions, M, so that(3.8)∂Πt∂Mt>0.This is connected with
procyclicality where we expect profitability to increase during booms, when
macroeconomic activity increases. By contrast, profitability may decrease
during recessions because of, among many other factors, an increase in
provisioning (see (3.7)). Importantly, examples of this phenomenon is
provided in Subsection 4.2 of Section 4 where the correlation between
macroeconomic activity, provisioning and profitability is established.


#### 3.2.2. Profits and Its Relationship with Retained Earnings

To establish the relationship between bank
profitability and the Basel Accord a model of bank financing is introduced that
is based on [26]. We
know that bank profits, Πt, are used to meet the bank's commitments that
include dividend payments on equity, ntdt,interest and principal payments on subordinate
debt, (1+rtO)Ot. The retained earnings, Etr, subsequent to these payments may be computed
by using(3.9)Πt=Etr+ntdt+(1+rtO)Ot.In standard usage, retained earnings
refer to earnings that are not paid out in dividends, interest or taxes. They
represent wealth accumulating in the bank and should be capitalized in the
value of the bank's equity. Retained earnings also are defined to include bank
charter value income. Normally, charter value refers to the present value
of anticipated profits from future lending.In each period, banks invest in fixed assets
(including buildings and equipment) which we denote byFt. The bank is assumed to maintain these assets
throughout its existence so that the bank must only cover the costs related to
the depreciation of fixed assets, ΔFt. These activities are financed through
retaining earnings and the eliciting of additional debt and equity, so
that(3.10)ΔFt=Etr+(nt+1−nt)Et+Ot+1+Rt+1l.We can use (3.9) and (3.10) to
obtain an expression for bank capital of the form(3.11)Kt+1=nt(dt+Et)+(1+rtO)Ot−Πt+ΔFt,where Kt is defined by (3.1).


### 3.3. Bank Valuation for A Shareholder

If the expression for retained earnings given by (3.9)
is substituted into (3.10), the nett
cash flow generated by the bank for a shareholder is given
by(3.12)Nt=Πt−ΔFt=ntdt+(1+rtO)Ot−Kt+1+ntEt.In addition, we have the
relationship(3.13)Bankvalueforashareholder=Nettcashflow+Ex‐dividendbankvalue.This translates to the
expression(3.14)Nt+Kt+1,where Kt is defined by (3.1). Furthermore, the stock
analyst evaluates the expected future cash flows in j periods based on a stochastic discount factor, δt,j such that the value of the bank
is(3.15)Nt+Et[∑j=1∞δt,jNt+j].


### 3.4. Optimal Bank Value for A Shareholder

In this subsection, we make use of the modeling of
assets, liabilities and capital of the preceding section to solve an optimal
bank valuation problem.


#### 3.4.1. Statement of The Optimal Bank Valuation Problem

Suppose that the bank valuation performance criterion,J, at t is given by(3.16)Jt=Πt+lt[Kt−ρ(ω(Mt)Λt+ωIIt+ωSSt+12.5(m⁢VaR+O))]−ctdw[Kt+1]+Et[δt,1V(Kt+1,xt+1)],where lt is the Lagrangian multiplier for the total
capital constraint, Kt is defined by (3.1), Et[⋅] is the expectation conditional on the bank's
information at time t and xt is the deposit withdrawals in period t with probability distribution f(xt). Also, ctdw is the deadweight cost of total capital that
consists of equity, subordinate debt and loan loss reserves. The optimal bank
valuation problem is to maximize the bank value given by (3.15). We can now
state the optimal valuation problem as follows.Problem 4 (statement of the optimal bank valuation problem).
Suppose that the total capital constraint and the performance
criterion,J, are given by
(3.6) and (3.16), respectively.
The optimal bank valuation problem is to maximize the value of the bank given
by (3.15) by choosing the loan rate,
deposits and regulatory capital for (3.17)V(Kt,xt)=maxrtΛ,Dt,Πt⁢Jt,subject to the cash flow,
balance sheet, financing constraint and loan demand given by (3.7), (3.2), (3.11)  and (2.11), respectively.


#### 3.4.2. Solution to The Optimal Bank Valuation Problem for Expected Losses

In this subsection, we find a solution to Problem4
when the capital constraint (3.6) holds as well as when it does not. In this
regard, the main result can be stated and proved as follows. Theorem 3.1 (solution to the optimal bank valuation problem (holding)).
Suppose thatJ and V are given by (3.16) and (3.17), respectively, and P(Mt)=αϱ+E[l]Λt−1. When the capital constraint given by (3.6) holds (i.e., lt>0), a solution to the optimal bank valuation
problem yields an optimal bank loan supply and loan rate of the
form(3.18)(3.19)respectively. In this case, the
corresponding optimal deposits, provisions for deposit withdrawals and profits
are given by(3.20)Dt*=D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ]+Ktρω(Mt)−ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt)+Ct+St+Bt−Kt−Bt,Wt*=D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ],Πt*=(Ktρω(Mt)−ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt))×{1l1(l0−Ktρω(Mt)+ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt)+l2Mt+σtΛ)−(cΛ+(rtD+cD+rtTγ)+rd(Mt))}−(rtD+cD+rtTγ)(Ct+Bt+St−Kt−Bt)+(D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ])((1−γ)rtT−(rtD+cD))−cw(Wt)−P(Mt)+rtIIt+rtCCt+rtBBt+rtSSt, respectively.Proof.
An immediate consequence of the prerequisite
that the capital constraint (3.6) holds, is that loan supply is closely related
to the capital adequacy constraint and is given by (3.18). Also, the dependence
of changes in the loan rate on macroeconomic activity may be fixed
as(3.21)∂rtΛ*∂Mt=l2l1. Equation (3.18) follows from (3.6) and the fact that
the capital constraint holds. This also leads to equality in (3.6). In (3.19)
we substituted the optimal value for  Λt into control law (2.11) to get the optimal
default rate. We obtain the optimal Wt using the following steps. Firstly, we rewrite
(3.2) to make deposits the dependent variable so that(3.22)Dt=Wt+Λt+Ct+Bt+St−Kt−Bt. Next, we note that the first-order conditions (for
verification of these conditions see Appendix A in the appendices) are given
by(3.23)(3.24)(3.25)(3.26) Here
F(⋅) is the cumulative distribution of the shock to
the loans. Using (3.26) we can see that (3.24) becomes
(3.27)∂Πt∂Dt=0.Looking at the form of Πt given in (3.7) and the
equation(3.28)cw(Wt)=rtp2D¯[D¯−Wt]2it follows that(3.29)Πt=(rtΛ−cΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+cD)Dt−rtp2D¯[D¯−Wt]2−P(Mt)−rtTγDt.Finding the partial derivatives
of profit, Πt,  with respect to deposit, Dt, we have that(3.30)∂Πt∂Dt=rtT−(rtD+cD)+rtpD¯(D¯−Wt)−rtTγ=(1−γ)rtT−(rtD+cD)+rtpD¯(D¯−Wt)=0.This would then give us the
optimal value for Dt.
Using (3.2) and all the optimal values calculated to date, we can find optimal
deposits, and the same goes for optimal profits.Remark 3.2 (solution to the optimal bank valuation problem).
Theorem3.1 addresses a very important issue in bank
operations that is related to the optimal supply of loans under regulatory
constraint. In order for the bank to fulfill its primary role as a credit
provider, (3.18) should satisfy the condition Ktρω(Mt)>ωIIt+ωSSt+12.5(mVaR+O)ω(Mt).In other words, the optimal loan
supply, Λ*, should have a positive value. A similar
comment can be made about the optimal loan rate, rΛ* in (3.19). If we substitute (3.26) in Subsection 3.4 into the optimal decisions for the loan rate and deposits
represented by (3.23) and (3.24), respectively, we can obtain a
time-independent solution for the optimal bank valuation problem. This leads to
a significant reduction in the technical difficulty of the procedure.In the case where the constraint (3.6) does not hold,
the following corollary follows directly.Corollary 3.3 (solution to the optimal bank valuation problem (not holding)).
Suppose thatJ and V are given by (3.16) and (3.17), respectively, and P(Mt)=αϱ+E[l]Λt−1. When the capital constraint (3.6) does not hold  (i.e.,  lt=0), a solution to the optimal bank valuation
problem stated in Problem 4 yields the optimal bank loan supply and loan rate(3.31)(3.32)respectively. In this case, the
corresponding Wt,
deposits and profits are given by(3.33)Wtn*=D¯+D¯(1−γ)rtp(rtT−(rtD+cD)1−γ),Dtn*=D¯+D¯(1−γ)rtp(rtT−(rtD+cD)1−γ)+Λtn*+Ct+Bt+St−Kt−Bt,Πtn*=12(l0+l2Mt+σtΛ)−l12(cΛ+(rtD+cD)+rd(Mt)+rtT(γ))×{12l1(l0+l2Mt+σtΛ)−12(cΛ+(rtD+cD)+rd(Mt)+rtTγ)}−(rtD+cD+rtTγ)(Ct+Bt+St−Kt−Bt)+(D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ])×((1−γ)rtT−(rtD+cD))−cw(Wt)−P(Mt)+rtIIt+rtCCt+rtBBt+rtSSt,respectively.Proof.
For the situation where capital constraint
(3.6) does not hold (i.e., lt=0),  using equation (3.26) and the fact that lt=0, we can see that (3.23) becomes(3.34)∂Πt∂rtΛ=0.
As in the proof of Theorem3.1, looking at the form of Πt given in (3.7) and (3.28), we have equation
(3.29). Therefore(3.35)∂Πt∂rtΛ=Λt−l1(rtΛ−cΛ−rd(Mt))+l1rtT+rtpD¯(D¯−Wt)l1=0.By substitute (3.30) into (3.35)
and using (2.11) would give us optimal loans and loan rate given by (3.31) and (3.32), respectively. Furthermore we can find the optimal deposit, deposit
withdrawals and profits.


## 3.1. Bank Regulatory Capital

In this subsection, we provide a general description
of bank capital and then specify the components of total bank capital that we
use in our study.


### 3.1.1. General Description of Bank Capital

According to Basel II, three types of capital can be
identified, viz., Tier 1, 2 and 3 capital, which we describe in more detail
below.Tier 1 capital comprises
ordinary share capital (or equity) of the bank and audited revenue reserves,
for example, retained earnings less current year's losses, future tax benefits and
intangible assets (for more information see, for instance, [5, 6]). Tier 1 capital or core
capital acts as a buffer against losses without a bank being required to cease
trading. Tier 2 capital includes
unaudited retained earnings; revaluation reserves; general provisions for bad
debts (e.g., loan loss reserves); perpetual cumulative preference shares (i.e.,
preference shares with no maturity date whose dividends accrue for future
payment even if the bank's financial condition does not support immediate
payment) and perpetual subordinated debt (i.e., debt with no maturity date
which ranks in priority behind all creditors except shareholders). Tier 2
capital or supplementary capital can absorb losses in the event of a wind-up
and so provides a lesser degree of protection to depositors. Tier 3 capital consists of subordinated
debt with a term of at least 5 years and redeemable preference shares which may
not be redeemed for at least 5 years. Tier 3 capital can be used to provide a
hedge against losses caused by market risks if Tier 1 and Tier 2 capital are
insufficient for this.


### 3.1.2. Specific Components of Total Bank Capital

For the purposes of our study,regulatory capital, K, is the book value of bank capital defined as
the difference between the accounting value of the assets and liabilities. More
specifically, Tier 1 capital is represented by period t−1's market value of the bank equity, ntEt−1, where nt is the number of shares and Et is the period t market price of the bank's common equity. Tier
2 capital mainly consists of subordinate debt, Ot, that is subordinate to deposits and hence
faces greater credit risk and loan loss reserves, Rtl. Subordinate debt issued in period t−1 are represented by a one-period bond that pays
an interest rate, rO. Also, we assume that loan loss reserves held
in period t−1 changes at the rate, rRl. Tier 3 capital is not considered at all. In
the sequel, we take the bank's total regulatory capital, K, in period t to be (3.1)Kt=ntEt−1+Ot+Rtl.For Kt given by (3.1), we obtain the balance sheet constraint(3.2)Wt=Dt+Bt−Λt−Ct−Bt−St+Kt.


### 3.1.3. Binding Capital Constraints

In order to describe the binding capital constraint,
we consider risk-weighted assets (RWAs) that are defined by placing each on- and off-balance sheet item into a risk category. The more risky assets are
assigned a larger weight. Table1 provides a few illustrative risk
categories, their risk-weights and representative items.As a result, RWAs are a weighted sum of the various
assets of the banks. In the sequel, we denote the risk-weight on intangible
assets, cash, bonds, shares, loans, Treasuries and reserves byωI,ωC,ωB,ωS,ωΛ,ωT, and,  ωR, respectively. In particular, we can identify a
special risk-weight on loans, ωΛ=ω(Mt), that is a decreasing function of current
macroeconomic conditions so that(3.3)∂ω(Mt)∂Mt<0.This is in line with the
procyclical notion that during booms, when macroeconomic activity increases,
the risk-weights will decrease. On the other hand, during recessions,
risk-weights may increase because of an elevated probability of default and/or
loss given default on loans.Thebank
capital constraint is defined by the inequality(3.4)Kt≥ρ(at+12.5(m⁢VaR+O))where(3.5)at=ωIIt+ωCCt+ωBBt+ωSSt+ωΛΛt+ωTTt+ωRRt,and ρ≈0.08. The formulation of (3.4) and the choice of
this particular value for ρ is informed by page 12 of  “Part 2: The
First Pillar-Minimum Capital Requirements” of [2]. This excerpt from the
document outlining Basel II states that“Part 2
presents the calculation of the total minimum capital requirements for credit,
market and operational risk. The capital ratio is calculated using the
definition of regulatory capital and risk-weighted assets. The total capital
ratio must be no lower that 8%.…    Total risk-weighted assets are determined
by multiplying the capital requirements for market risk and operational risk by
12.5 (i.e., the reciprocal of the minimum capital ratio of 8%) and adding the
resulting figures to the sum of risk-weighted assets for credit risk.”Also,mVaR and O in (3.4) are as described in Sections 1 and 2
of this paper, respectively.In accordance with Table1, if we assume that the
risk-weights associated with intangible assets, shares, cash, bonds,
Treasuries, reserves and loans may be taken to be ωI≠0,ωS≠0,ωC=ωB=ωT=ωR=0 and ωΛ=ω(Mt), respectively, then equation (3.4) becomes the
capital constraint(3.6)Kt≥ρ[ω(Mt)Λt+ωIIt+ωSSt+12.5(m⁢VaR+O)].


## 3.1.1. General Description of Bank Capital

According to Basel II, three types of capital can be
identified, viz., Tier 1, 2 and 3 capital, which we describe in more detail
below.Tier 1 capital comprises
ordinary share capital (or equity) of the bank and audited revenue reserves,
for example, retained earnings less current year's losses, future tax benefits and
intangible assets (for more information see, for instance, [5, 6]). Tier 1 capital or core
capital acts as a buffer against losses without a bank being required to cease
trading. Tier 2 capital includes
unaudited retained earnings; revaluation reserves; general provisions for bad
debts (e.g., loan loss reserves); perpetual cumulative preference shares (i.e.,
preference shares with no maturity date whose dividends accrue for future
payment even if the bank's financial condition does not support immediate
payment) and perpetual subordinated debt (i.e., debt with no maturity date
which ranks in priority behind all creditors except shareholders). Tier 2
capital or supplementary capital can absorb losses in the event of a wind-up
and so provides a lesser degree of protection to depositors. Tier 3 capital consists of subordinated
debt with a term of at least 5 years and redeemable preference shares which may
not be redeemed for at least 5 years. Tier 3 capital can be used to provide a
hedge against losses caused by market risks if Tier 1 and Tier 2 capital are
insufficient for this.


## 3.1.2. Specific Components of Total Bank Capital

For the purposes of our study,regulatory capital, K, is the book value of bank capital defined as
the difference between the accounting value of the assets and liabilities. More
specifically, Tier 1 capital is represented by period t−1's market value of the bank equity, ntEt−1, where nt is the number of shares and Et is the period t market price of the bank's common equity. Tier
2 capital mainly consists of subordinate debt, Ot, that is subordinate to deposits and hence
faces greater credit risk and loan loss reserves, Rtl. Subordinate debt issued in period t−1 are represented by a one-period bond that pays
an interest rate, rO. Also, we assume that loan loss reserves held
in period t−1 changes at the rate, rRl. Tier 3 capital is not considered at all. In
the sequel, we take the bank's total regulatory capital, K, in period t to be (3.1)Kt=ntEt−1+Ot+Rtl.For Kt given by (3.1), we obtain the balance sheet constraint(3.2)Wt=Dt+Bt−Λt−Ct−Bt−St+Kt.


## 3.1.3. Binding Capital Constraints

In order to describe the binding capital constraint,
we consider risk-weighted assets (RWAs) that are defined by placing each on- and off-balance sheet item into a risk category. The more risky assets are
assigned a larger weight. Table1 provides a few illustrative risk
categories, their risk-weights and representative items.As a result, RWAs are a weighted sum of the various
assets of the banks. In the sequel, we denote the risk-weight on intangible
assets, cash, bonds, shares, loans, Treasuries and reserves byωI,ωC,ωB,ωS,ωΛ,ωT, and,  ωR, respectively. In particular, we can identify a
special risk-weight on loans, ωΛ=ω(Mt), that is a decreasing function of current
macroeconomic conditions so that(3.3)∂ω(Mt)∂Mt<0.This is in line with the
procyclical notion that during booms, when macroeconomic activity increases,
the risk-weights will decrease. On the other hand, during recessions,
risk-weights may increase because of an elevated probability of default and/or
loss given default on loans.Thebank
capital constraint is defined by the inequality(3.4)Kt≥ρ(at+12.5(m⁢VaR+O))where(3.5)at=ωIIt+ωCCt+ωBBt+ωSSt+ωΛΛt+ωTTt+ωRRt,and ρ≈0.08. The formulation of (3.4) and the choice of
this particular value for ρ is informed by page 12 of  “Part 2: The
First Pillar-Minimum Capital Requirements” of [2]. This excerpt from the
document outlining Basel II states that“Part 2
presents the calculation of the total minimum capital requirements for credit,
market and operational risk. The capital ratio is calculated using the
definition of regulatory capital and risk-weighted assets. The total capital
ratio must be no lower that 8%.…    Total risk-weighted assets are determined
by multiplying the capital requirements for market risk and operational risk by
12.5 (i.e., the reciprocal of the minimum capital ratio of 8%) and adding the
resulting figures to the sum of risk-weighted assets for credit risk.”Also,mVaR and O in (3.4) are as described in Sections 1 and 2
of this paper, respectively.In accordance with Table1, if we assume that the
risk-weights associated with intangible assets, shares, cash, bonds,
Treasuries, reserves and loans may be taken to be ωI≠0,ωS≠0,ωC=ωB=ωT=ωR=0 and ωΛ=ω(Mt), respectively, then equation (3.4) becomes the
capital constraint(3.6)Kt≥ρ[ω(Mt)Λt+ωIIt+ωSSt+12.5(m⁢VaR+O)].


## 3.2. Profits and Retained Earnings

In this subsection, we discuss profits and its
relation to retained earnings.


### 3.2.1. Profits

We assume that (2.4) holds. As far as profit, Π, is concerned, we use the basic fact that
profits can be characterized as the difference between income and expenses that
are reported in the bank's income statement. In our contribution, income is solely constituted by the
returns on intangible assets, rtIIt, cash, rtCCt, bonds, rtBBt, shares, rtSSt, loans, rtΛΛt, and Treasuries, rtTTt. Furthermore, we assume that the level of
macroeconomic activity is denoted by Mt. In our case we consider the cost of monitoring
and screening of loans and capital, cΛΛt, interest paid to depositors, rtDDt, the cost of taking deposits, cDDt, the cost of deposit withdrawals, cw(Wt), the value of loan losses, L(Mt), and total loan loss provisions, P(Mt) as expenses, in period t. Here rD and cD are the deposit rate and marginal cost of
deposits, respectively. Summing all the costs mentioned to operating costs and
supposing that (2.13) holds and that Wt=Tt+γDt, then the bank's profits are given by the
expression(3.7)Πt=(rtΛ−cΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+cD)Dt−cw(Wt)−P(Mt)−rtTγDt,where rI,rC,rB and rS are the rates of return of the intangible
assets, cash, bonds and shares, respectively. Furthermore, by considering (2.17)
and (3.7), we suspect that profit, Π, is an increasing function of current
macroeconomic conditions, M, so that(3.8)∂Πt∂Mt>0.This is connected with
procyclicality where we expect profitability to increase during booms, when
macroeconomic activity increases. By contrast, profitability may decrease
during recessions because of, among many other factors, an increase in
provisioning (see (3.7)). Importantly, examples of this phenomenon is
provided in Subsection 4.2 of Section 4 where the correlation between
macroeconomic activity, provisioning and profitability is established.


### 3.2.2. Profits and Its Relationship with Retained Earnings

To establish the relationship between bank
profitability and the Basel Accord a model of bank financing is introduced that
is based on [26]. We
know that bank profits, Πt, are used to meet the bank's commitments that
include dividend payments on equity, ntdt,interest and principal payments on subordinate
debt, (1+rtO)Ot. The retained earnings, Etr, subsequent to these payments may be computed
by using(3.9)Πt=Etr+ntdt+(1+rtO)Ot.In standard usage, retained earnings
refer to earnings that are not paid out in dividends, interest or taxes. They
represent wealth accumulating in the bank and should be capitalized in the
value of the bank's equity. Retained earnings also are defined to include bank
charter value income. Normally, charter value refers to the present value
of anticipated profits from future lending.In each period, banks invest in fixed assets
(including buildings and equipment) which we denote byFt. The bank is assumed to maintain these assets
throughout its existence so that the bank must only cover the costs related to
the depreciation of fixed assets, ΔFt. These activities are financed through
retaining earnings and the eliciting of additional debt and equity, so
that(3.10)ΔFt=Etr+(nt+1−nt)Et+Ot+1+Rt+1l.We can use (3.9) and (3.10) to
obtain an expression for bank capital of the form(3.11)Kt+1=nt(dt+Et)+(1+rtO)Ot−Πt+ΔFt,where Kt is defined by (3.1).


## 3.2.1. Profits

We assume that (2.4) holds. As far as profit, Π, is concerned, we use the basic fact that
profits can be characterized as the difference between income and expenses that
are reported in the bank's income statement. In our contribution, income is solely constituted by the
returns on intangible assets, rtIIt, cash, rtCCt, bonds, rtBBt, shares, rtSSt, loans, rtΛΛt, and Treasuries, rtTTt. Furthermore, we assume that the level of
macroeconomic activity is denoted by Mt. In our case we consider the cost of monitoring
and screening of loans and capital, cΛΛt, interest paid to depositors, rtDDt, the cost of taking deposits, cDDt, the cost of deposit withdrawals, cw(Wt), the value of loan losses, L(Mt), and total loan loss provisions, P(Mt) as expenses, in period t. Here rD and cD are the deposit rate and marginal cost of
deposits, respectively. Summing all the costs mentioned to operating costs and
supposing that (2.13) holds and that Wt=Tt+γDt, then the bank's profits are given by the
expression(3.7)Πt=(rtΛ−cΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+cD)Dt−cw(Wt)−P(Mt)−rtTγDt,where rI,rC,rB and rS are the rates of return of the intangible
assets, cash, bonds and shares, respectively. Furthermore, by considering (2.17)
and (3.7), we suspect that profit, Π, is an increasing function of current
macroeconomic conditions, M, so that(3.8)∂Πt∂Mt>0.This is connected with
procyclicality where we expect profitability to increase during booms, when
macroeconomic activity increases. By contrast, profitability may decrease
during recessions because of, among many other factors, an increase in
provisioning (see (3.7)). Importantly, examples of this phenomenon is
provided in Subsection 4.2 of Section 4 where the correlation between
macroeconomic activity, provisioning and profitability is established.


## 3.2.2. Profits and Its Relationship with Retained Earnings

To establish the relationship between bank
profitability and the Basel Accord a model of bank financing is introduced that
is based on [26]. We
know that bank profits, Πt, are used to meet the bank's commitments that
include dividend payments on equity, ntdt,interest and principal payments on subordinate
debt, (1+rtO)Ot. The retained earnings, Etr, subsequent to these payments may be computed
by using(3.9)Πt=Etr+ntdt+(1+rtO)Ot.In standard usage, retained earnings
refer to earnings that are not paid out in dividends, interest or taxes. They
represent wealth accumulating in the bank and should be capitalized in the
value of the bank's equity. Retained earnings also are defined to include bank
charter value income. Normally, charter value refers to the present value
of anticipated profits from future lending.In each period, banks invest in fixed assets
(including buildings and equipment) which we denote byFt. The bank is assumed to maintain these assets
throughout its existence so that the bank must only cover the costs related to
the depreciation of fixed assets, ΔFt. These activities are financed through
retaining earnings and the eliciting of additional debt and equity, so
that(3.10)ΔFt=Etr+(nt+1−nt)Et+Ot+1+Rt+1l.We can use (3.9) and (3.10) to
obtain an expression for bank capital of the form(3.11)Kt+1=nt(dt+Et)+(1+rtO)Ot−Πt+ΔFt,where Kt is defined by (3.1).


## 3.3. Bank Valuation for A Shareholder

If the expression for retained earnings given by (3.9)
is substituted into (3.10), the nett
cash flow generated by the bank for a shareholder is given
by(3.12)Nt=Πt−ΔFt=ntdt+(1+rtO)Ot−Kt+1+ntEt.In addition, we have the
relationship(3.13)Bankvalueforashareholder=Nettcashflow+Ex‐dividendbankvalue.This translates to the
expression(3.14)Nt+Kt+1,where Kt is defined by (3.1). Furthermore, the stock
analyst evaluates the expected future cash flows in j periods based on a stochastic discount factor, δt,j such that the value of the bank
is(3.15)Nt+Et[∑j=1∞δt,jNt+j].


## 3.4. Optimal Bank Value for A Shareholder

In this subsection, we make use of the modeling of
assets, liabilities and capital of the preceding section to solve an optimal
bank valuation problem.


### 3.4.1. Statement of The Optimal Bank Valuation Problem

Suppose that the bank valuation performance criterion,J, at t is given by(3.16)Jt=Πt+lt[Kt−ρ(ω(Mt)Λt+ωIIt+ωSSt+12.5(m⁢VaR+O))]−ctdw[Kt+1]+Et[δt,1V(Kt+1,xt+1)],where lt is the Lagrangian multiplier for the total
capital constraint, Kt is defined by (3.1), Et[⋅] is the expectation conditional on the bank's
information at time t and xt is the deposit withdrawals in period t with probability distribution f(xt). Also, ctdw is the deadweight cost of total capital that
consists of equity, subordinate debt and loan loss reserves. The optimal bank
valuation problem is to maximize the bank value given by (3.15). We can now
state the optimal valuation problem as follows.Problem 4 (statement of the optimal bank valuation problem).
Suppose that the total capital constraint and the performance
criterion,J, are given by
(3.6) and (3.16), respectively.
The optimal bank valuation problem is to maximize the value of the bank given
by (3.15) by choosing the loan rate,
deposits and regulatory capital for (3.17)V(Kt,xt)=maxrtΛ,Dt,Πt⁢Jt,subject to the cash flow,
balance sheet, financing constraint and loan demand given by (3.7), (3.2), (3.11)  and (2.11), respectively.


### 3.4.2. Solution to The Optimal Bank Valuation Problem for Expected Losses

In this subsection, we find a solution to Problem4
when the capital constraint (3.6) holds as well as when it does not. In this
regard, the main result can be stated and proved as follows. Theorem 3.1 (solution to the optimal bank valuation problem (holding)).
Suppose thatJ and V are given by (3.16) and (3.17), respectively, and P(Mt)=αϱ+E[l]Λt−1. When the capital constraint given by (3.6) holds (i.e., lt>0), a solution to the optimal bank valuation
problem yields an optimal bank loan supply and loan rate of the
form(3.18)(3.19)respectively. In this case, the
corresponding optimal deposits, provisions for deposit withdrawals and profits
are given by(3.20)Dt*=D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ]+Ktρω(Mt)−ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt)+Ct+St+Bt−Kt−Bt,Wt*=D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ],Πt*=(Ktρω(Mt)−ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt))×{1l1(l0−Ktρω(Mt)+ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt)+l2Mt+σtΛ)−(cΛ+(rtD+cD+rtTγ)+rd(Mt))}−(rtD+cD+rtTγ)(Ct+Bt+St−Kt−Bt)+(D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ])((1−γ)rtT−(rtD+cD))−cw(Wt)−P(Mt)+rtIIt+rtCCt+rtBBt+rtSSt, respectively.Proof.
An immediate consequence of the prerequisite
that the capital constraint (3.6) holds, is that loan supply is closely related
to the capital adequacy constraint and is given by (3.18). Also, the dependence
of changes in the loan rate on macroeconomic activity may be fixed
as(3.21)∂rtΛ*∂Mt=l2l1. Equation (3.18) follows from (3.6) and the fact that
the capital constraint holds. This also leads to equality in (3.6). In (3.19)
we substituted the optimal value for  Λt into control law (2.11) to get the optimal
default rate. We obtain the optimal Wt using the following steps. Firstly, we rewrite
(3.2) to make deposits the dependent variable so that(3.22)Dt=Wt+Λt+Ct+Bt+St−Kt−Bt. Next, we note that the first-order conditions (for
verification of these conditions see Appendix A in the appendices) are given
by(3.23)(3.24)(3.25)(3.26) Here
F(⋅) is the cumulative distribution of the shock to
the loans. Using (3.26) we can see that (3.24) becomes
(3.27)∂Πt∂Dt=0.Looking at the form of Πt given in (3.7) and the
equation(3.28)cw(Wt)=rtp2D¯[D¯−Wt]2it follows that(3.29)Πt=(rtΛ−cΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+cD)Dt−rtp2D¯[D¯−Wt]2−P(Mt)−rtTγDt.Finding the partial derivatives
of profit, Πt,  with respect to deposit, Dt, we have that(3.30)∂Πt∂Dt=rtT−(rtD+cD)+rtpD¯(D¯−Wt)−rtTγ=(1−γ)rtT−(rtD+cD)+rtpD¯(D¯−Wt)=0.This would then give us the
optimal value for Dt.
Using (3.2) and all the optimal values calculated to date, we can find optimal
deposits, and the same goes for optimal profits.Remark 3.2 (solution to the optimal bank valuation problem).
Theorem3.1 addresses a very important issue in bank
operations that is related to the optimal supply of loans under regulatory
constraint. In order for the bank to fulfill its primary role as a credit
provider, (3.18) should satisfy the condition Ktρω(Mt)>ωIIt+ωSSt+12.5(mVaR+O)ω(Mt).In other words, the optimal loan
supply, Λ*, should have a positive value. A similar
comment can be made about the optimal loan rate, rΛ* in (3.19). If we substitute (3.26) in Subsection 3.4 into the optimal decisions for the loan rate and deposits
represented by (3.23) and (3.24), respectively, we can obtain a
time-independent solution for the optimal bank valuation problem. This leads to
a significant reduction in the technical difficulty of the procedure.In the case where the constraint (3.6) does not hold,
the following corollary follows directly.Corollary 3.3 (solution to the optimal bank valuation problem (not holding)).
Suppose thatJ and V are given by (3.16) and (3.17), respectively, and P(Mt)=αϱ+E[l]Λt−1. When the capital constraint (3.6) does not hold  (i.e.,  lt=0), a solution to the optimal bank valuation
problem stated in Problem 4 yields the optimal bank loan supply and loan rate(3.31)(3.32)respectively. In this case, the
corresponding Wt,
deposits and profits are given by(3.33)Wtn*=D¯+D¯(1−γ)rtp(rtT−(rtD+cD)1−γ),Dtn*=D¯+D¯(1−γ)rtp(rtT−(rtD+cD)1−γ)+Λtn*+Ct+Bt+St−Kt−Bt,Πtn*=12(l0+l2Mt+σtΛ)−l12(cΛ+(rtD+cD)+rd(Mt)+rtT(γ))×{12l1(l0+l2Mt+σtΛ)−12(cΛ+(rtD+cD)+rd(Mt)+rtTγ)}−(rtD+cD+rtTγ)(Ct+Bt+St−Kt−Bt)+(D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ])×((1−γ)rtT−(rtD+cD))−cw(Wt)−P(Mt)+rtIIt+rtCCt+rtBBt+rtSSt,respectively.Proof.
For the situation where capital constraint
(3.6) does not hold (i.e., lt=0),  using equation (3.26) and the fact that lt=0, we can see that (3.23) becomes(3.34)∂Πt∂rtΛ=0.
As in the proof of Theorem3.1, looking at the form of Πt given in (3.7) and (3.28), we have equation
(3.29). Therefore(3.35)∂Πt∂rtΛ=Λt−l1(rtΛ−cΛ−rd(Mt))+l1rtT+rtpD¯(D¯−Wt)l1=0.By substitute (3.30) into (3.35)
and using (2.11) would give us optimal loans and loan rate given by (3.31) and (3.32), respectively. Furthermore we can find the optimal deposit, deposit
withdrawals and profits.


## 3.4.1. Statement of The Optimal Bank Valuation Problem

Suppose that the bank valuation performance criterion,J, at t is given by(3.16)Jt=Πt+lt[Kt−ρ(ω(Mt)Λt+ωIIt+ωSSt+12.5(m⁢VaR+O))]−ctdw[Kt+1]+Et[δt,1V(Kt+1,xt+1)],where lt is the Lagrangian multiplier for the total
capital constraint, Kt is defined by (3.1), Et[⋅] is the expectation conditional on the bank's
information at time t and xt is the deposit withdrawals in period t with probability distribution f(xt). Also, ctdw is the deadweight cost of total capital that
consists of equity, subordinate debt and loan loss reserves. The optimal bank
valuation problem is to maximize the bank value given by (3.15). We can now
state the optimal valuation problem as follows.Problem 4 (statement of the optimal bank valuation problem).
Suppose that the total capital constraint and the performance
criterion,J, are given by
(3.6) and (3.16), respectively.
The optimal bank valuation problem is to maximize the value of the bank given
by (3.15) by choosing the loan rate,
deposits and regulatory capital for (3.17)V(Kt,xt)=maxrtΛ,Dt,Πt⁢Jt,subject to the cash flow,
balance sheet, financing constraint and loan demand given by (3.7), (3.2), (3.11)  and (2.11), respectively.


## 3.4.2. Solution to The Optimal Bank Valuation Problem for Expected Losses

In this subsection, we find a solution to Problem4
when the capital constraint (3.6) holds as well as when it does not. In this
regard, the main result can be stated and proved as follows. Theorem 3.1 (solution to the optimal bank valuation problem (holding)).
Suppose thatJ and V are given by (3.16) and (3.17), respectively, and P(Mt)=αϱ+E[l]Λt−1. When the capital constraint given by (3.6) holds (i.e., lt>0), a solution to the optimal bank valuation
problem yields an optimal bank loan supply and loan rate of the
form(3.18)(3.19)respectively. In this case, the
corresponding optimal deposits, provisions for deposit withdrawals and profits
are given by(3.20)Dt*=D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ]+Ktρω(Mt)−ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt)+Ct+St+Bt−Kt−Bt,Wt*=D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ],Πt*=(Ktρω(Mt)−ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt))×{1l1(l0−Ktρω(Mt)+ωIIt+ωSSt+12.5(m⁢VaR+O)ω(Mt)+l2Mt+σtΛ)−(cΛ+(rtD+cD+rtTγ)+rd(Mt))}−(rtD+cD+rtTγ)(Ct+Bt+St−Kt−Bt)+(D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ])((1−γ)rtT−(rtD+cD))−cw(Wt)−P(Mt)+rtIIt+rtCCt+rtBBt+rtSSt, respectively.Proof.
An immediate consequence of the prerequisite
that the capital constraint (3.6) holds, is that loan supply is closely related
to the capital adequacy constraint and is given by (3.18). Also, the dependence
of changes in the loan rate on macroeconomic activity may be fixed
as(3.21)∂rtΛ*∂Mt=l2l1. Equation (3.18) follows from (3.6) and the fact that
the capital constraint holds. This also leads to equality in (3.6). In (3.19)
we substituted the optimal value for  Λt into control law (2.11) to get the optimal
default rate. We obtain the optimal Wt using the following steps. Firstly, we rewrite
(3.2) to make deposits the dependent variable so that(3.22)Dt=Wt+Λt+Ct+Bt+St−Kt−Bt. Next, we note that the first-order conditions (for
verification of these conditions see Appendix A in the appendices) are given
by(3.23)(3.24)(3.25)(3.26) Here
F(⋅) is the cumulative distribution of the shock to
the loans. Using (3.26) we can see that (3.24) becomes
(3.27)∂Πt∂Dt=0.Looking at the form of Πt given in (3.7) and the
equation(3.28)cw(Wt)=rtp2D¯[D¯−Wt]2it follows that(3.29)Πt=(rtΛ−cΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+cD)Dt−rtp2D¯[D¯−Wt]2−P(Mt)−rtTγDt.Finding the partial derivatives
of profit, Πt,  with respect to deposit, Dt, we have that(3.30)∂Πt∂Dt=rtT−(rtD+cD)+rtpD¯(D¯−Wt)−rtTγ=(1−γ)rtT−(rtD+cD)+rtpD¯(D¯−Wt)=0.This would then give us the
optimal value for Dt.
Using (3.2) and all the optimal values calculated to date, we can find optimal
deposits, and the same goes for optimal profits.Remark 3.2 (solution to the optimal bank valuation problem).
Theorem3.1 addresses a very important issue in bank
operations that is related to the optimal supply of loans under regulatory
constraint. In order for the bank to fulfill its primary role as a credit
provider, (3.18) should satisfy the condition Ktρω(Mt)>ωIIt+ωSSt+12.5(mVaR+O)ω(Mt).In other words, the optimal loan
supply, Λ*, should have a positive value. A similar
comment can be made about the optimal loan rate, rΛ* in (3.19). If we substitute (3.26) in Subsection 3.4 into the optimal decisions for the loan rate and deposits
represented by (3.23) and (3.24), respectively, we can obtain a
time-independent solution for the optimal bank valuation problem. This leads to
a significant reduction in the technical difficulty of the procedure.In the case where the constraint (3.6) does not hold,
the following corollary follows directly.Corollary 3.3 (solution to the optimal bank valuation problem (not holding)).
Suppose thatJ and V are given by (3.16) and (3.17), respectively, and P(Mt)=αϱ+E[l]Λt−1. When the capital constraint (3.6) does not hold  (i.e.,  lt=0), a solution to the optimal bank valuation
problem stated in Problem 4 yields the optimal bank loan supply and loan rate(3.31)(3.32)respectively. In this case, the
corresponding Wt,
deposits and profits are given by(3.33)Wtn*=D¯+D¯(1−γ)rtp(rtT−(rtD+cD)1−γ),Dtn*=D¯+D¯(1−γ)rtp(rtT−(rtD+cD)1−γ)+Λtn*+Ct+Bt+St−Kt−Bt,Πtn*=12(l0+l2Mt+σtΛ)−l12(cΛ+(rtD+cD)+rd(Mt)+rtT(γ))×{12l1(l0+l2Mt+σtΛ)−12(cΛ+(rtD+cD)+rd(Mt)+rtTγ)}−(rtD+cD+rtTγ)(Ct+Bt+St−Kt−Bt)+(D¯+D¯(1−γ)rtp[rtT−(rtD+cD)1−γ])×((1−γ)rtT−(rtD+cD))−cw(Wt)−P(Mt)+rtIIt+rtCCt+rtBBt+rtSSt,respectively.Proof.
For the situation where capital constraint
(3.6) does not hold (i.e., lt=0),  using equation (3.26) and the fact that lt=0, we can see that (3.23) becomes(3.34)∂Πt∂rtΛ=0.
As in the proof of Theorem3.1, looking at the form of Πt given in (3.7) and (3.28), we have equation
(3.29). Therefore(3.35)∂Πt∂rtΛ=Λt−l1(rtΛ−cΛ−rd(Mt))+l1rtT+rtpD¯(D¯−Wt)l1=0.By substitute (3.30) into (3.35)
and using (2.11) would give us optimal loans and loan rate given by (3.31) and (3.32), respectively. Furthermore we can find the optimal deposit, deposit
withdrawals and profits.


## 4. Historical evidence and examples

In this section, we firstly provide evidence to
support the fact that the output gap (proxy for the business cycle) and the
provisions for loan losses-to-total assets ratio are negatively correlated. In
essence this means that (2.17) holds and provisions for loan losses are
procyclical. Secondly, in Subsection 4.2, we investigate the correlation
between output gap and provisions in relation to profitability. Also, the
historical data provides support for our modeling choices for provisions and
profitability in Sections  2 and  3.Throughout this section, we rely on historical data
from member countries of the Organization for Economic Corporation and
Development (OECD) as supplied on the website [50]. The specific countries or
regions for which data was accessed are Australia, Finland, Italy, Japan,
Norway, Spain, Sweden, the United Kingdom and the United States.


### 4.1. Procyclicality of Provisions for Loan Losses

In this subsection, we look at empirical evidence that
provisions for loan losses is procyclical. In other words, we would like to
verify that (2.17) holds.


#### 4.1.1. Provisioning for Australia, Norway, Spain and Sweden

This subsubsection provides empirical evidence that
provisions for loan losses for the period 1986 to 2000 were procyclical in
Australia, Norway, Spain and Sweden (see Figures1 and 2).Output gap versus provisions for loan
losses-to-total assets ratio for Australia and Norway (1986–2000).
(a)(b)Output gap versus provisions for loan
losses-to-total assets ratio for Spain and Sweden (1986–2000).
(a)(b)


#### 4.1.2. Provisioning for Finland, Italy, Japan and United Kingdom

This subsubsection provides empirical evidence that
provisions for loan losses in Finland, Italy, Japan and United Kingdom were
procyclical for the period 1986 to 2000 (see Figures3 and 4).Output gap versus provisions for loan losses-to-total
assets ratio for Finland and Italy (1986–2000).
(a)(b)Output gap versus provisions for loan losses-to-total
assets ratio for Japan and United Kingdom (1986–2000).
(a)(b)


#### 4.1.3. Provisioning for The United States

This subsubsection provides empirical evidence that in
the U.S. provisions for loan losses were procyclical for the period 1986 to
2000 (see Figure5).Figure 5
Output gap versus provisions for loan losses-to-total
assets ratio for the United States (1986–2000).


#### 4.1.4. Discussion of Provisioning for The 9 Oecd Countries

In this sububsection, we provide a brief discussion of
some of the outstanding features of the data for provisioning for loan losses
provided in Subsubsections4.1.1, 4.1.2 and 4.1.3.The data for Australia from Figure1 shows that
provisions failed to increase substantially in the late 1980's, when credit and
asset prices were growing rapidly and the financial imbalances were developing.
Moreover, the peak in provisions did not occur until at least one year after
the economy had clearly slowed down.The data for Norway from Figure1 exhibits a similar
behavior as the data for Finland from Figure 3 and the data for Spain from
Figure 2. In these cases, provisions failed to increase substantially in the
late 1980's, when credit and asset prices were growing rapidly and the
financial imbalances were developing. In each of these figures the peak for
provisions did not occur until the recession. However, one of the differences
between these figures is the amount by which provisions increased when the
economy had clearly slowed down.In the data for Finland from Figure3 we see a similar
situation as in the data for Italy from Figure 3 where provisions failed to
overlap the output gap during the recession. Again this can be link with
Japanese banking problems. Although in countries like the United States (see
Figure 5) and Australia and Norway (see Figure 1), the provisions overlapped
the output gap during the recessions, the situation in Italy (see Figure 3) is
total different. It seems that even after the banking problems that Japan
experienced had been resolved, in Italy the situation changed slightly.From the data for Japan from Figure4, we can conclude
that the level of provisioning only increased substantially during the second
half of the 1990's, long after the problems in the Japanese banking system had
been widely recognized.The (low) positive correlation between provisions and
the business cycle in the United States (see Figure5) appears to be driven by
the surge in provisions in the second half of the 1980's. This phenomenon seems
to reflect the delayed cleaning of the balance sheets following the developing
countries' debt crisis of the early 1980's.


### 4.2. Correlations between Profitability and Provisions for Loan Losses

As has been suggested in Subsection4.1, bank
provisions are strongly procyclical and are negatively correlated with the
business cycle. For instance, Figure 5 shows that provisions typically do not
increase until after economic growth has slowed down considerably and often not
until the economy is in complete recession. In the main, the behavior of
provisions translates into a clear procyclical pattern in bank profitability,
which further encourages procyclical lending practices. As is shown in Table 2, this pattern appears to be strongest in those countries that experienced
banking system problems in the 1990's.Table 2
Correlations
between provisions and profitability.
CountriesProvisionsProfitabilityAustralia−0,880,71Finland—0,81Germany−0,21−0,42Italy−0,210,25Japan−0,430,22Norway−0,350,54South Africa−0,850,74Spain−0,410,84Sweden−0,830,60United Kingdom−0,380,12United States0,140,24In the main, the behavior of provisions translates
into a procyclical pattern in bank profitability, which further encourages
procyclical lending practices. Our claim is thus that profit and provisions are
negatively correlated. However, from Table2, we also conclude that the
profitability of German banks is not procyclical. This may be due to their
ability to smooth profits through hidden reserves. The procyclical nature of
bank profits has arguably also contributed to the bank equity prices being
positively correlated with the business cycle, although the correlation is
typically weaker than that for profitability, reflecting the forward-looking
nature of the equity market.


## 4.1. Procyclicality of Provisions for Loan Losses

In this subsection, we look at empirical evidence that
provisions for loan losses is procyclical. In other words, we would like to
verify that (2.17) holds.


### 4.1.1. Provisioning for Australia, Norway, Spain and Sweden

This subsubsection provides empirical evidence that
provisions for loan losses for the period 1986 to 2000 were procyclical in
Australia, Norway, Spain and Sweden (see Figures1 and 2).Output gap versus provisions for loan
losses-to-total assets ratio for Australia and Norway (1986–2000).
(a)(b)Output gap versus provisions for loan
losses-to-total assets ratio for Spain and Sweden (1986–2000).
(a)(b)


### 4.1.2. Provisioning for Finland, Italy, Japan and United Kingdom

This subsubsection provides empirical evidence that
provisions for loan losses in Finland, Italy, Japan and United Kingdom were
procyclical for the period 1986 to 2000 (see Figures3 and 4).Output gap versus provisions for loan losses-to-total
assets ratio for Finland and Italy (1986–2000).
(a)(b)Output gap versus provisions for loan losses-to-total
assets ratio for Japan and United Kingdom (1986–2000).
(a)(b)


### 4.1.3. Provisioning for The United States

This subsubsection provides empirical evidence that in
the U.S. provisions for loan losses were procyclical for the period 1986 to
2000 (see Figure5).Figure 5
Output gap versus provisions for loan losses-to-total
assets ratio for the United States (1986–2000).


### 4.1.4. Discussion of Provisioning for The 9 Oecd Countries

In this sububsection, we provide a brief discussion of
some of the outstanding features of the data for provisioning for loan losses
provided in Subsubsections4.1.1, 4.1.2 and 4.1.3.The data for Australia from Figure1 shows that
provisions failed to increase substantially in the late 1980's, when credit and
asset prices were growing rapidly and the financial imbalances were developing.
Moreover, the peak in provisions did not occur until at least one year after
the economy had clearly slowed down.The data for Norway from Figure1 exhibits a similar
behavior as the data for Finland from Figure 3 and the data for Spain from
Figure 2. In these cases, provisions failed to increase substantially in the
late 1980's, when credit and asset prices were growing rapidly and the
financial imbalances were developing. In each of these figures the peak for
provisions did not occur until the recession. However, one of the differences
between these figures is the amount by which provisions increased when the
economy had clearly slowed down.In the data for Finland from Figure3 we see a similar
situation as in the data for Italy from Figure 3 where provisions failed to
overlap the output gap during the recession. Again this can be link with
Japanese banking problems. Although in countries like the United States (see
Figure 5) and Australia and Norway (see Figure 1), the provisions overlapped
the output gap during the recessions, the situation in Italy (see Figure 3) is
total different. It seems that even after the banking problems that Japan
experienced had been resolved, in Italy the situation changed slightly.From the data for Japan from Figure4, we can conclude
that the level of provisioning only increased substantially during the second
half of the 1990's, long after the problems in the Japanese banking system had
been widely recognized.The (low) positive correlation between provisions and
the business cycle in the United States (see Figure5) appears to be driven by
the surge in provisions in the second half of the 1980's. This phenomenon seems
to reflect the delayed cleaning of the balance sheets following the developing
countries' debt crisis of the early 1980's.


## 4.1.1. Provisioning for Australia, Norway, Spain and Sweden

This subsubsection provides empirical evidence that
provisions for loan losses for the period 1986 to 2000 were procyclical in
Australia, Norway, Spain and Sweden (see Figures1 and 2).Output gap versus provisions for loan
losses-to-total assets ratio for Australia and Norway (1986–2000).
(a)(b)Output gap versus provisions for loan
losses-to-total assets ratio for Spain and Sweden (1986–2000).
(a)(b)


## 4.1.2. Provisioning for Finland, Italy, Japan and United Kingdom

This subsubsection provides empirical evidence that
provisions for loan losses in Finland, Italy, Japan and United Kingdom were
procyclical for the period 1986 to 2000 (see Figures3 and 4).Output gap versus provisions for loan losses-to-total
assets ratio for Finland and Italy (1986–2000).
(a)(b)Output gap versus provisions for loan losses-to-total
assets ratio for Japan and United Kingdom (1986–2000).
(a)(b)


## 4.1.3. Provisioning for The United States

This subsubsection provides empirical evidence that in
the U.S. provisions for loan losses were procyclical for the period 1986 to
2000 (see Figure5).Figure 5
Output gap versus provisions for loan losses-to-total
assets ratio for the United States (1986–2000).


## 4.1.4. Discussion of Provisioning for The 9 Oecd Countries

In this sububsection, we provide a brief discussion of
some of the outstanding features of the data for provisioning for loan losses
provided in Subsubsections4.1.1, 4.1.2 and 4.1.3.The data for Australia from Figure1 shows that
provisions failed to increase substantially in the late 1980's, when credit and
asset prices were growing rapidly and the financial imbalances were developing.
Moreover, the peak in provisions did not occur until at least one year after
the economy had clearly slowed down.The data for Norway from Figure1 exhibits a similar
behavior as the data for Finland from Figure 3 and the data for Spain from
Figure 2. In these cases, provisions failed to increase substantially in the
late 1980's, when credit and asset prices were growing rapidly and the
financial imbalances were developing. In each of these figures the peak for
provisions did not occur until the recession. However, one of the differences
between these figures is the amount by which provisions increased when the
economy had clearly slowed down.In the data for Finland from Figure3 we see a similar
situation as in the data for Italy from Figure 3 where provisions failed to
overlap the output gap during the recession. Again this can be link with
Japanese banking problems. Although in countries like the United States (see
Figure 5) and Australia and Norway (see Figure 1), the provisions overlapped
the output gap during the recessions, the situation in Italy (see Figure 3) is
total different. It seems that even after the banking problems that Japan
experienced had been resolved, in Italy the situation changed slightly.From the data for Japan from Figure4, we can conclude
that the level of provisioning only increased substantially during the second
half of the 1990's, long after the problems in the Japanese banking system had
been widely recognized.The (low) positive correlation between provisions and
the business cycle in the United States (see Figure5) appears to be driven by
the surge in provisions in the second half of the 1980's. This phenomenon seems
to reflect the delayed cleaning of the balance sheets following the developing
countries' debt crisis of the early 1980's.


## 4.2. Correlations between Profitability and Provisions for Loan Losses

As has been suggested in Subsection4.1, bank
provisions are strongly procyclical and are negatively correlated with the
business cycle. For instance, Figure 5 shows that provisions typically do not
increase until after economic growth has slowed down considerably and often not
until the economy is in complete recession. In the main, the behavior of
provisions translates into a clear procyclical pattern in bank profitability,
which further encourages procyclical lending practices. As is shown in Table 2, this pattern appears to be strongest in those countries that experienced
banking system problems in the 1990's.Table 2
Correlations
between provisions and profitability.
CountriesProvisionsProfitabilityAustralia−0,880,71Finland—0,81Germany−0,21−0,42Italy−0,210,25Japan−0,430,22Norway−0,350,54South Africa−0,850,74Spain−0,410,84Sweden−0,830,60United Kingdom−0,380,12United States0,140,24In the main, the behavior of provisions translates
into a procyclical pattern in bank profitability, which further encourages
procyclical lending practices. Our claim is thus that profit and provisions are
negatively correlated. However, from Table2, we also conclude that the
profitability of German banks is not procyclical. This may be due to their
ability to smooth profits through hidden reserves. The procyclical nature of
bank profits has arguably also contributed to the bank equity prices being
positively correlated with the business cycle, although the correlation is
typically weaker than that for profitability, reflecting the forward-looking
nature of the equity market.


## 5. Bank valuation and its connections with the subprime mortgage crisis
and Basel II

In this section, we consider connections between the
discrete-time stochastic models derived in the preceding discussions and the
SMC as well as the Basel II Capital Accord.


### 5.1. Bank Valuation and The Subprime Mortgage Crisis

The SMC is an ongoing crisis characterized by
shrinking liquidity in global credit markets and banking systems. A downturn in
the U.S. housing market, risky practices by lenders and borrowers and excessive
individual and corporate debt levels have affected the world economy adversely
on a number of levels. The SMC has exposed pervasive weaknesses in the global
financial system and regulatory framework. The connections between this crisis
and our banking models are mainly forged via the bank's(i)
risk premium,ϱ,   from   (2.5) in Subsection 2.2.1 (see HM1 and
HM2 in Figure 6);(ii)
required capital sensitivity to changes in the
amount of loans extended as given by (2.6) in Subsection 2.2.1 (see FM2 in
Figure 6);(iii)
base rate,r, from (2.7) in Subsection 2.2.1 decided upon by
Central Banks as well as interbank loan rates (see HM1, HM3, FM4 and GIR1 in
Figure 6);(iv)
own loan rate,rΛ, from (2.7) and (2.7) and (3.32) in
Subsections 2.2.1 and 3.4.2, respectively (see HM1, HM3, FM4 and GIR1 in Figure 6);(v)
loan demand represented by (2.11) in Subsection
2.2.1 (see HM3, FM4, GIR1 and GIR5 in Figure 6);(vi)
loan losses and default rate given by (2.13) in
Subsection 2.2.2 (see HM4, FM1, FM3, GIR3 and GIR5 in Figure 6);(vii)
loan loss provisions from (2.16) and (2.17) in
Subsection 2.2.2 and illustrated in Subsection 4.1 (see HM4, FM1, FM3, GIR3 and
GIR5 in Figure 6);(viii)
choice between raising deposits and interbank
borrowing (including borrowing from the Central Bank) as reflected by (2.21) in
Subsection 2.3.3 (see GIR1 in Figure 6);(ix)
liquidity as described by Remark2.3 (see FM4
in Figure 6);(x)
profit,Π, given by (3.7) in Subsection 3.2.1 and
illustrated in Subsection 4.2 (see HM2, HM4, FM1, FM3, GIR1 and GIR5 in Figure 6);(xi)
bank valuation performance criterion,J, given by (3.16) in Subsection 3.4.1 (see HM4,
HM6, FM1, FM2 and FM3 in Figure 6).
In this subsection, we provide a diagrammatic overview
of and sketch a background to the SMC. Furthermore, we briefly consider the
connections between our banking models and the SMC.Figure 6
Diagrammatic overview of the subprime mortgage crisis
(compare [51]).


#### 5.1.1. Diagrammatic Overview of The Subprime Mortgage Crisis

A diagrammatic overview of the SMC (see, for instance,
[51]) may be
represented as in Figure 6.


#### 5.1.2. Background to The Subprime Mortgage Crisis

Most of the information contained in this subsection
was sourced from [51].
The SMC was initiated by the deflation of the United States housing bubble
(see, for instance, [52, 53]) and high default rates on subprime and adjustable
rate mortgages (ARM). Loan incentives, such as easy initial terms and low loan
rates, in combination with escalating housing prices encouraged borrowers to
assume difficult mortgages on the belief they would be able to quickly
refinance at more favorable terms (see HM1 in Figure 6). Great concern was also
expressed about the rapid growth in business loans at commercial banks with
excessively easy credit standards. Some analysts claim that competition for
lenders had greatly increased, causing banks to reduce loan rates and ease
credit standards in order to issue new credit. Others are of the opinion that
as the economic expansion continued and past loan losses were forgotten, banks
exhibited a greater propensity for risk. However, once U.S. housing prices
started to fall moderately in 2006-2007,
refinancing became more difficult (see HM2 and HM3 in Figure 6). Defaults and
foreclosure activity increased dramatically, as easy initial terms expired,
home prices failed to go up as anticipated, and ARM interest rates reset
higher. Foreclosures accelerated in the U.S. in late 2006 and triggered a
global financial crisis through 2007 and 2008. During 2007, nearly 1.3 million
U.S. housing properties were subjected to foreclosure activity; up 79% from
2006 (see [54] for
more details; also HM4 in Figure 6). The mortgage lenders that retained credit
risk were the first to be affected, as borrowers became unable or unwilling to
make payments (see HM5 and HM6 in Figure 6).Major banks and other financial institutions globally
had reported losses of approximately $435 billion from SMC-related activities
by Thursday, 17 July 2008 (see [55, 56]; also FM1 in Figure 6). By using securitization
strategies, many mortgage lenders passed the rights to the mortgage payments
and related credit risk to third-party investors via mortgage-backed securities
(MBSs) and collateralized debt obligations (CDOs). Corporate, individual and
institutional investors holding MBS or CDO suffered significant losses, as the
underlying mortgage asset value decreased. Stock markets in many countries
declined significantly (see FM2 in Figure 6). The broader international
financial sector first began to experience the fallout from the SMC in February
2007 with the $10.5 billion writedown of HSBC, which was the first major CDO
or Mortgage Bankers Association (MBA) related loss to be reported. During 2007,
at least 100 mortgage companies had either failed, suspended operations or been
sold. Top management did not escape without blemish, as the CEOs of Merrill
Lynch and Citigroup were forced to resign within a week of each other.
Subsequently, merger deals were struck by many institutions. In addition,
Northern Rock and Bear Stearns required emergency assistance from central
banks. IndyMac was shut down by the FDIC on Sunday, 11 July 2008. Moreover, on
Sunday, 14 September 2008, after performing banking duties for more than 150
years, Lehman Brothers filed for bankruptcy as a consequence of losses stemming
from the SMC (see FM3 in Figure 6). Subsequent to this many U.S. and other
banks throughout the world also failed. The widespread dispersion of credit
risk and the unclear effect on financial institutions caused reduced lending
activity and increased spreads on higher interest rates. Similarly, the ability
of corporations to obtain funds through the issuance of commercial paper was
affected. This aspect of the crisis is consistent with a credit crunch. There
are a number of reasons why banks may suddenly make obtaining a loan more
difficult or increase the costs of obtaining a loan. This may be due to an
anticipated decline in the value of the collateral used by the banks when
issuing loans; an increased perception of risk regarding the solvency of other
banks within the banking system; a change in monetary conditions (e.g.,
where the central bank suddenly and unexpectedly raises interest rates or
capital requirements); the central government imposing direct credit controls
and instructing the banks not to engage in further lending activity. The
subprime crisis has adversely affected several inputs in the economy, resulting
in downward pressure on economic growth. Fewer and more expensive loans tend to
result in decreased business investment and consumer spending (see FM4 and FM5
in Figure 6).Liquidity concerns drove central banks around the
world to take action to provide funds to member banks to encourage lending to
worthy borrowers and to restore faith in the commercial paper markets (see GIR1
in Figure6). With interest rates on a large number of subprime and other ARM
due to adjust upward during the 2008 period, U.S. legislators, the U.S.
Treasury Department, and financial institutions took action. A systematic
program to limit or defer interest rate adjustments was implemented to reduce
the effect. In addition, lenders and borrowers facing defaults have been
encouraged to cooperate to enable borrowers to stay in their homes. Banks have
sought and received over $250 billion in additional funds from investors to
offset losses (see [57]
for more information). The risks to the broader economy created by the
financial market crisis and housing market downturn were primary factors in
several decisions by the U.S. Federal Reserve to cut interest rates and the
Economic Stimulus Package (ESP) passed by Congress and signed by President Bush
on Wednesday, 13 February 2008 (see, for instance, [58–60]; also GIR2 in Figure 6). Bush also announced a plan
voluntarily and temporarily to freeze the mortgages of a limited number of
mortgage debtors holding ARMs. A refinancing facility called FHA-Secure was
also created. This action is part of an ongoing collaborative effort between
the U.S. government and private industry to help some subprime borrowers called
the Hope Now Alliance (see GIR3 in Figure 6). The U.S. government also
bailed-out key financial institutions, assuming significant additional
financial commitments (see GIR4 in Figure 6). Following a series of ad-hoc
market interventions to bailout particular firms, a $700 billion systemic
rescue plan was accepted by the U.S. House of Representatives on Friday, 3
October 2008. These actions are designed to stimulate economic growth and
inspire confidence in the financial markets (see GIR5 in Figure 6). By November
2008, banks in Europe, Asia, Australia and South America had followed the
example of the U.S. government by putting rescue plans in place.


#### 5.1.3. Connections between Our Models and The Subprime Mortgage Crisis

The connections between our banking models and the SMC
are complicated. However, a first step towards understanding this relationship
entails identifying the problematic loan subportfolios. In this regard, it is
possible to decompose the (total) loans,Λ, discussed above as(5.1)Λ=∑i=1mΛi,where m is the number of loan subportfolios with i being the index of each loan subportfolio so that i=1,2,…,m. We note that the Basel II IRB approach to
credit risk (see [2])
dictates that m≤15. In particular, this approach identifies 15
credit risk exposure types (loan subportfolios) that may be listed
as
i=1: Project finance (PF);i=2: Object finance (OF);i=3: Commodities finance (CF);i=4: Income producing real estate (IPRE);i=5: Specialized lending high volatility commercial real estate
(SLHVCRE)i=6: Specialized lending not including high volatility commercial real
estate
(SLNIHVCRE)i=7: Bank exposure
(BE);i=8: Sovereign exposure (SE);i=9: Retail residential mortgage (RRM);i=10: Home equity line of credit (HELOC);i=11: Other retail exposure (ORE);i=12: Qualifying revolving retail exposure (QRRE);i=13: Small to medium size enterprises with corporate treatment (SMECT)i=14: Small to medium size enterprises with retail treatment
(SMERT)i=15: Equity exposure not held in the trading book
(EENHTB)
with i = 1–6 and i =  9–12 constituting corporate and retail
exposures, respectively. As a result of the SMC, market trade in almost all of
these loan subportfolios became sluggish. In particular, a problematic loan
subportfolio corresponds to i=4 which encompasses income producing real
estate. Furthermore, in commercial real estate (i.e., i=5 and i=6) there is a slowing due to the tightening
credit and slowing growth, the former a direct result of the SMC. Although
capital remains available for residential loans, the credit crunch is
pronounced in commercial lending. Moreover, i=7 which corresponds to interbank lending had
been identified as a problematic subportfolio. This is because banks were not
lending to each other during the early stages of the crisis (see FM4 in Figure 6).The risk premium,ϱ, from (2.5) has had a part to play in the SMC
(see HM1 in Figure 6). This premium is part of the expression for the bank's
own loan rate, rΛ, and its size is an indication of perceived
credit risk. A study by the U.S. Federal Reserve indicated that the average
difference in mortgage interest rates between subprime (“subprime
markup”) and prime mortgages declined from ϱ=2.8 percentage points (280 basis points) in 2001,
to ϱ=1.3 percentage points in 2007. In other words, the
risk premium required by lenders to offer a subprime loan declined dramatically
during the aforementioned period. This occurred even though subprime borrower
and loan quality declined overall during the 2001–2006 period. In
fact, this state of affairs should have had the opposite effect. However, it is
clear that such conditions are consistent with classic boom and recession
credit cycles (see [37]).The sensitivity of the required capital to changes in
the amount of loans extended from (2.6) is an important issue in the SMC. Bank
capital levels were depleted with banks experiencing high debt levels (see FM2
in Figure 6). As a consequence, for many banks (2.6) begun to take on negative
values during the SMC.The Central Bank base rate,r, from (2.7) in Subsection 2.2.1 has also had a
role to play in trying to alleviate global economic pressures during the SMC.
For instance, on Wednesday, 8 October 2008, central banks in the U.S. (U.S.
Federal Reserve), England, China, Canada, Sweden, Switzerland and the European
Central Bank cut rates in a coordinated effort to assist the world economy.
This rate will be lowered when it is necessary to stimulate growth and provide
liquidity.Despite the discussion above, the most significant
relationships between our models and the SMC are established via the bank's own
loan rate,rΛ, given by (2.7) (see also (3.19) and (3.32) as
well as HM1, HM3, FM4 and GIR1 in Figure 6). As we have noted in (2.7), this
loan rate may be expressed as(5.2)rtΛ=ρ+(1+rt)Nη+crw+cE∂K∂Λ+E[l].In general terms, rΛ can be written as a function of credit risk,
market structure (power), marginal cost of debt, marginal cost of equity and
the sensitivity of capital to loans extended. The representation of the bank's
interest setting intimates that banks will experience positive returns in good
times when the actual rate of default, rd, is lower than the provisioning for expected
losses, αϱ+E[l], and may not be able to cover their expected
losses when rd>αϱ+E[l] (see equation (2.13) for loan losses and
default rate). In the latter case, bank capital may be needed to cover these
excess (and unexpected) losses. If this capital is not enough then the bank
will face insolvency. During the latter half of 2008, we have seen a decline in
such capital.A combination of factors resulting from the SMC have
led to problems in the commercial real estate market as regards loan demand
(see (2.11); also HM3, FM4, GIR1 and GIR5 in Figure 6). According to the
National Association of Realtors (NAR) there is a slowing in commercial real
estate due to the tightening credit and slowing growth, the former a direct
result of the SMC. Although capital remains available for residential loans,
the credit crunch is pronounced in commercial lending. Patricia Nooney,
chairperson of the Realtors Commercial Alliance Committee (RCAC), portrayed the
decline as unusual since“transactions are being curtailed not for
lack of demand, but for serious challenges in obtaining financing.” (see
[61] for more
information).Next, we discuss loan losses and default rate given by
(2.13) in Subsection 2.2.2 (see HM4, FM1, FM3, GIR3 and GIR5 in Figure 6). An
acceleration in loan growth, as was experienced prior to the SMC, eventually
leads to a surge in loan losses (see, for instance, (2.13) for loan losses and
default rate) resulting in reduced bank profits (see equation (3.7) for bank
profit) and precipitating a new round of bank failures. As experience during
the mortgage crisis has shown, such a slump in banking could not only threaten
the deposit insurance fund but also slow the economy by entrenching credit
crunches. The view that faster loan growth leads to higher loan losses should
not be taken lightly; nor should it be accepted without question. If loan
growth increases because banks become more willing to lend, credit standards
should fall and loan losses should eventually rise. For many distressed banks
suffering because of the SMC, increased loan loss provisions from (2.16) and
(2.17) translate to a decrease in earnings.Usually, when capital is needed, banks have to choose
between raising and holding deposits and borrowing from the Central Bank as
reflected by (2.21) in Subsection 2.3.3.Contracted liquidity in the global credit markets and
banking system (as described by Remark2.3; see also FM4 in Figure 6) is an
ongoing economic problem. These liquidity concerns drove central banks around
the world to take action to provide funds to member banks to encourage lending
to worthy borrowers and to restore faith in the commercial paper markets (see
GIR1 in Figure 6).We recall that the bank's profits are given by (3.7)
in the form(5.3)Πt=(rtΛ−ctΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+ctD)Dt−cw(Wt)−P(Mt)−rtTγDt.It is important to note that
this expression represents the bank's profits from unsecuritized loans.
However, in the SMC, the attention has been on borrowing under securitization
strategies (see Figure 9 in Subsection C.1). In principle, we can provide an
analogue of (3.7) in a securitization context by introducing additional
variables. In this regard, suppose that rr is the average residual rate in securitized
transactions, prc the average expected fraction of loan losses
on securitized debt, cli the marginal cost of mortgage loan insurance
under securitization, cmw the marginal cost to monitor a loan that was
securitized, ctw the marginal transaction costs to securitize a
loan amortized over a loan term, pw the fraction of the loan amount securitized
and p˜w the fraction of the mortgage loan principal, Λ0, realized as a new loan in securitization. Then
the bank's profits under securitization, Πtw, may be presented in the form (5.4)Πtw=(rtr−ptrc−ctli−ctmw−cttw)ptwp˜twΛt+(rtΛ−ctlc−cttw)ptw(1−p˜tw)Λt+(1−ptw)[(rtΛ−ctΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt⁢−(rtD+ctD)Dt−cw(Wt)−P(Mt)−rtTγDt]−Vta−Et−Ft.Here we note that, in terms of
the organization of the special purposes vehicle (SPV), there are different
states that can be associated with the different possible corporate forms of
the SPV. The SPV can be an LLC (E1), LLP (E2), C-corporation (E3) or a trust (E4). Each of these SPVs has its own unique tax
benefits and problems as well as degree of asset protection and limited
liability under legal prescripts. Furthermore, we denote the optimal state
during the lifetime of the SPV by E*, with deviations from E* being given by |E*−Eti|, where i=1,2,3,…,4. These deviations measure the loss and
opportunity costs arising from the use of suboptimal corporate structures as
SPVs. Usually, such loss manifests as elevated legal fees, increased losses due
to low limited-liability and/or the value of additional time spent in dispute
resolution. In this case, the formula for E in (5.4) may be given by(5.5)Et=max{|E*−Et1|,|E*−Et2|,|E*−Et3|,|E*−Et4|,0}.Furthermore, in the
securitization context of (5.4), we assume that financing can take the form of Ff, where the transaction is an assignment and Fs, where the transaction is a true sale. In the
former case, the SPV is not subsidized by the sponsor. Alternatively, the
impaired collateral is only substituted by the sponsor. In this situation,
where Ff is the optimal state but Fs is being used instead, the transaction incurs
losses and vice versa. On the other hand, where the correct transaction format
is used, there is neither a gain nor a loss. Hence the value of the transaction
format choice (compare with (5.4) where F is used) is(5.6)Ft=max{|Ftf−Fts|,0}.Furthermore, we denote the value
of the adverse selection problem by Va. In this case, the transaction is an assignment
and the sponsor is the servicer. Here, the sponsor will typically substitute
impaired collateral, and hence is incentivized not to prefer the best
collateral as a replacement. In this regard, we denote impaired collateral,
value of replacement collateral and the value of average-quality collateral
available for replacement by C0,Ci…Cn and Ca, respectively. We assume that collateral
consists of units of the same or similar sizes where Pi is the probability of there being impaired
collateral and Pa the probability that the servicer/sponsor will
offer medium-or low-quality collateral as replacement for the impaired
collateral. In this situation, it follows that Va in (5.4) may be represented by(5.7)Va=∫0t{∑Ca∗Pi(1−Pa)}. Another remark about profitability is that during the
SMC, profits at the 8 533 U.S. banks insured by the FDIC fell from $35.2
billion to $646 million (effectively by 89%) during Quarter 4 of 2007 when
compared with the previous year. This was largely due to escalating loan losses
and provisions for loan losses. This decline in profits contributed to the
worst bank and thrift quarterly performance since 1990. In 2007, these banks earned
approximately $100 billion, which represented a decline of 31% from the
record profit of $145 billion in 2006. Profits decreased from $35.6 billion
to $19.3 billion during the first quarter of 2008 versus the previous year, a
decline of 46% (see [46, 47] for more detail). The contribution [34] considers the following
strategy to be optimal for banks to shield their profits from loan losses. The
loan loss reserves, Rl, is built up in every period that P>L. On the other hand, when P≤L the bank is allowed to draw on Rl from the current period. When Rl or other capital becomes inadequate, at some
point, the bank will face insolvency.Bank valuations as realized by (3.16) in Subsection
3.4.1 have declined dramatically during the SMC as a direct result of sharp
decreases in profitability and bank capital.


### 5.2. Bank Valuation and Basel Ii

In this subsection, we analyze the connections between
our banking model and the Basel II Capital Accord (see SubsectionB on pages
43 to 44 for a diagrammatic overview of Basel II; also [1, 2]). These relationships are
forged via the(i)
description of total bank capital given by
(3.1) in Subsection 3.1.2;(ii)
description of the bank capital constraint
given by (3.4) and (3.6) in Subsection 3.1.3;(iii)
procyclicality described in Subsections3.1.3
and 4.1.


#### 5.2.1. Diagrammatic Overview of The Basel Ii Capital Accord

Next, we provide a diagrammatic overview of the Basel
II Capital Accord  as shown in Figure7.Figure 7
Diagrammatic overview of the Basel II Capital Accord.


#### 5.2.2. Bank Regulatory Capital

Despite the analysis in Subsubsections3.1.1 and 3.1.2
of Section 3, bank capital is notoriously difficult to define, monitor and
measure. For instance, the measurement of equity depends on how all of a bank's
financial instruments and other assets are valued. The description of the
shareholder equity component of bank capital, E, given by (3.1), is largely motivated by the
following two observations. Firstly, it is meant to reflect the nature of the
book value of equity. Our intention is also to recognize that the book and
market value of equity is highly correlated.Under Basel II, bank capital requirements have
replaced reserve requirements (see Subsection2.1) as the main constraint on
the behavior of banks (see, for instance, [62]). A first motivation for this is that bank capital
has a major role to play in overcoming the moral hazard problem arising from
asymmetric information between banks, creditors and debtors. Also, bank regulators
require capital to be held to protect themselves against the costs of financial
distress, agency problems and the reduction of market discipline caused by the
safety net. Subsection 3.1.3 suggests that a close relationship exists between
bank capital holding and macroeconomic activity in the loan market (see
equation (3.6) describing the bank capital constraint).


#### 5.2.3. Procyclicality of Basel Ii Regulation

Our paper has discussed the procylicality induced by
the Basel II Capital Accord's approach to credit risk in Subsections3.1.3 and
4.1. In this regard, the Standardized approach to credit risk differentiates
assets not only according to the borrower but according to riskiness proxied by
credit rating agencies' assessment of the borrower. This approach represents an
improvement in the sense that corporations are rated. However, it is deficient
with respect to the viewpoint that ratings properly reflect risk and the
inducing of procyclical capital charges which will lead to overlending during
booms and underlending during recessions. Also, the capital formula for the IRB
approach to credit risk is problematic. In short, the IRB capital formula is
deduced by considering the large-portfolio asymptotic dynamics of a Merton
model with a single common risk-factor. Many empirical studies have confirmed
that this formula gives rise to procyclicality.In addition, Basel II dictates that a macroeconomic
shock will affect the loan risk-weights in the CAR. In general, a negative
(positive) shock results in the tightening (loosening) of the capital
constraint (see equation (3.6) describing the bank capital constraint). As a
consequence, in terms of a possible binding capital constraint, banks are free
to increase (decrease) the loan supply when macroeconomic conditions Mt improve (deteriorate). On the other hand, if
the risk-weights are constant, a shock does not affect the loan supply but
rather results in a change in the loan rate when the capital constraint binds.
It is not always true that Basel II risk-sensitive weights lead to an increase
(decrease) in bank capital when macroeconomic activity in the loan market
increases (decreases). A simple explanation for this is that macroeconomic
conditions do not necessarily only affect loan demand but also influences the
total capital constraint (see equation (3.6) describing the bank capital
constraint). Furthermore, banks do not necessarily need to raise new capital to
expand their loan supply, since a positive macroeconomic shock may result in a
decrease in the RWAs with a commensurate increase in CARs (compare the minimum
capital constraint as expressed in (3.4) and (3.6)). Similarly, banks are not
compelled to decrease their capital when the loan demand decreases since the
capital constraint usually tightens in response to a negative macroeconomic
shock. A further complication is that an improvement in the latter conditions
may result in an increase in the loan demand and, as a consequence, an increase
in the probability that the capital constraint may be binding (see equations
(2.11) and (3.6) for the loan demand and bank capital constraint, resp.).
Banks may react to this situation by increasing capital to maximize profits
(compare the definition of the return on equity (ROE) measure of profitability).
Our main conclusion is that bank capital is procyclical because it is dependent
on fluctuations in loan demand which, in turn, is reliant on macroeconomic
activity (see equation (3.6) describing the bank capital constraint).


## 5.1. Bank Valuation and The Subprime Mortgage Crisis

The SMC is an ongoing crisis characterized by
shrinking liquidity in global credit markets and banking systems. A downturn in
the U.S. housing market, risky practices by lenders and borrowers and excessive
individual and corporate debt levels have affected the world economy adversely
on a number of levels. The SMC has exposed pervasive weaknesses in the global
financial system and regulatory framework. The connections between this crisis
and our banking models are mainly forged via the bank's(i)
risk premium,ϱ,   from   (2.5) in Subsection 2.2.1 (see HM1 and
HM2 in Figure 6);(ii)
required capital sensitivity to changes in the
amount of loans extended as given by (2.6) in Subsection 2.2.1 (see FM2 in
Figure 6);(iii)
base rate,r, from (2.7) in Subsection 2.2.1 decided upon by
Central Banks as well as interbank loan rates (see HM1, HM3, FM4 and GIR1 in
Figure 6);(iv)
own loan rate,rΛ, from (2.7) and (2.7) and (3.32) in
Subsections 2.2.1 and 3.4.2, respectively (see HM1, HM3, FM4 and GIR1 in Figure 6);(v)
loan demand represented by (2.11) in Subsection
2.2.1 (see HM3, FM4, GIR1 and GIR5 in Figure 6);(vi)
loan losses and default rate given by (2.13) in
Subsection 2.2.2 (see HM4, FM1, FM3, GIR3 and GIR5 in Figure 6);(vii)
loan loss provisions from (2.16) and (2.17) in
Subsection 2.2.2 and illustrated in Subsection 4.1 (see HM4, FM1, FM3, GIR3 and
GIR5 in Figure 6);(viii)
choice between raising deposits and interbank
borrowing (including borrowing from the Central Bank) as reflected by (2.21) in
Subsection 2.3.3 (see GIR1 in Figure 6);(ix)
liquidity as described by Remark2.3 (see FM4
in Figure 6);(x)
profit,Π, given by (3.7) in Subsection 3.2.1 and
illustrated in Subsection 4.2 (see HM2, HM4, FM1, FM3, GIR1 and GIR5 in Figure 6);(xi)
bank valuation performance criterion,J, given by (3.16) in Subsection 3.4.1 (see HM4,
HM6, FM1, FM2 and FM3 in Figure 6).
In this subsection, we provide a diagrammatic overview
of and sketch a background to the SMC. Furthermore, we briefly consider the
connections between our banking models and the SMC.Figure 6
Diagrammatic overview of the subprime mortgage crisis
(compare [51]).


### 5.1.1. Diagrammatic Overview of The Subprime Mortgage Crisis

A diagrammatic overview of the SMC (see, for instance,
[51]) may be
represented as in Figure 6.


### 5.1.2. Background to The Subprime Mortgage Crisis

Most of the information contained in this subsection
was sourced from [51].
The SMC was initiated by the deflation of the United States housing bubble
(see, for instance, [52, 53]) and high default rates on subprime and adjustable
rate mortgages (ARM). Loan incentives, such as easy initial terms and low loan
rates, in combination with escalating housing prices encouraged borrowers to
assume difficult mortgages on the belief they would be able to quickly
refinance at more favorable terms (see HM1 in Figure 6). Great concern was also
expressed about the rapid growth in business loans at commercial banks with
excessively easy credit standards. Some analysts claim that competition for
lenders had greatly increased, causing banks to reduce loan rates and ease
credit standards in order to issue new credit. Others are of the opinion that
as the economic expansion continued and past loan losses were forgotten, banks
exhibited a greater propensity for risk. However, once U.S. housing prices
started to fall moderately in 2006-2007,
refinancing became more difficult (see HM2 and HM3 in Figure 6). Defaults and
foreclosure activity increased dramatically, as easy initial terms expired,
home prices failed to go up as anticipated, and ARM interest rates reset
higher. Foreclosures accelerated in the U.S. in late 2006 and triggered a
global financial crisis through 2007 and 2008. During 2007, nearly 1.3 million
U.S. housing properties were subjected to foreclosure activity; up 79% from
2006 (see [54] for
more details; also HM4 in Figure 6). The mortgage lenders that retained credit
risk were the first to be affected, as borrowers became unable or unwilling to
make payments (see HM5 and HM6 in Figure 6).Major banks and other financial institutions globally
had reported losses of approximately $435 billion from SMC-related activities
by Thursday, 17 July 2008 (see [55, 56]; also FM1 in Figure 6). By using securitization
strategies, many mortgage lenders passed the rights to the mortgage payments
and related credit risk to third-party investors via mortgage-backed securities
(MBSs) and collateralized debt obligations (CDOs). Corporate, individual and
institutional investors holding MBS or CDO suffered significant losses, as the
underlying mortgage asset value decreased. Stock markets in many countries
declined significantly (see FM2 in Figure 6). The broader international
financial sector first began to experience the fallout from the SMC in February
2007 with the $10.5 billion writedown of HSBC, which was the first major CDO
or Mortgage Bankers Association (MBA) related loss to be reported. During 2007,
at least 100 mortgage companies had either failed, suspended operations or been
sold. Top management did not escape without blemish, as the CEOs of Merrill
Lynch and Citigroup were forced to resign within a week of each other.
Subsequently, merger deals were struck by many institutions. In addition,
Northern Rock and Bear Stearns required emergency assistance from central
banks. IndyMac was shut down by the FDIC on Sunday, 11 July 2008. Moreover, on
Sunday, 14 September 2008, after performing banking duties for more than 150
years, Lehman Brothers filed for bankruptcy as a consequence of losses stemming
from the SMC (see FM3 in Figure 6). Subsequent to this many U.S. and other
banks throughout the world also failed. The widespread dispersion of credit
risk and the unclear effect on financial institutions caused reduced lending
activity and increased spreads on higher interest rates. Similarly, the ability
of corporations to obtain funds through the issuance of commercial paper was
affected. This aspect of the crisis is consistent with a credit crunch. There
are a number of reasons why banks may suddenly make obtaining a loan more
difficult or increase the costs of obtaining a loan. This may be due to an
anticipated decline in the value of the collateral used by the banks when
issuing loans; an increased perception of risk regarding the solvency of other
banks within the banking system; a change in monetary conditions (e.g.,
where the central bank suddenly and unexpectedly raises interest rates or
capital requirements); the central government imposing direct credit controls
and instructing the banks not to engage in further lending activity. The
subprime crisis has adversely affected several inputs in the economy, resulting
in downward pressure on economic growth. Fewer and more expensive loans tend to
result in decreased business investment and consumer spending (see FM4 and FM5
in Figure 6).Liquidity concerns drove central banks around the
world to take action to provide funds to member banks to encourage lending to
worthy borrowers and to restore faith in the commercial paper markets (see GIR1
in Figure6). With interest rates on a large number of subprime and other ARM
due to adjust upward during the 2008 period, U.S. legislators, the U.S.
Treasury Department, and financial institutions took action. A systematic
program to limit or defer interest rate adjustments was implemented to reduce
the effect. In addition, lenders and borrowers facing defaults have been
encouraged to cooperate to enable borrowers to stay in their homes. Banks have
sought and received over $250 billion in additional funds from investors to
offset losses (see [57]
for more information). The risks to the broader economy created by the
financial market crisis and housing market downturn were primary factors in
several decisions by the U.S. Federal Reserve to cut interest rates and the
Economic Stimulus Package (ESP) passed by Congress and signed by President Bush
on Wednesday, 13 February 2008 (see, for instance, [58–60]; also GIR2 in Figure 6). Bush also announced a plan
voluntarily and temporarily to freeze the mortgages of a limited number of
mortgage debtors holding ARMs. A refinancing facility called FHA-Secure was
also created. This action is part of an ongoing collaborative effort between
the U.S. government and private industry to help some subprime borrowers called
the Hope Now Alliance (see GIR3 in Figure 6). The U.S. government also
bailed-out key financial institutions, assuming significant additional
financial commitments (see GIR4 in Figure 6). Following a series of ad-hoc
market interventions to bailout particular firms, a $700 billion systemic
rescue plan was accepted by the U.S. House of Representatives on Friday, 3
October 2008. These actions are designed to stimulate economic growth and
inspire confidence in the financial markets (see GIR5 in Figure 6). By November
2008, banks in Europe, Asia, Australia and South America had followed the
example of the U.S. government by putting rescue plans in place.


### 5.1.3. Connections between Our Models and The Subprime Mortgage Crisis

The connections between our banking models and the SMC
are complicated. However, a first step towards understanding this relationship
entails identifying the problematic loan subportfolios. In this regard, it is
possible to decompose the (total) loans,Λ, discussed above as(5.1)Λ=∑i=1mΛi,where m is the number of loan subportfolios with i being the index of each loan subportfolio so that i=1,2,…,m. We note that the Basel II IRB approach to
credit risk (see [2])
dictates that m≤15. In particular, this approach identifies 15
credit risk exposure types (loan subportfolios) that may be listed
as
i=1: Project finance (PF);i=2: Object finance (OF);i=3: Commodities finance (CF);i=4: Income producing real estate (IPRE);i=5: Specialized lending high volatility commercial real estate
(SLHVCRE)i=6: Specialized lending not including high volatility commercial real
estate
(SLNIHVCRE)i=7: Bank exposure
(BE);i=8: Sovereign exposure (SE);i=9: Retail residential mortgage (RRM);i=10: Home equity line of credit (HELOC);i=11: Other retail exposure (ORE);i=12: Qualifying revolving retail exposure (QRRE);i=13: Small to medium size enterprises with corporate treatment (SMECT)i=14: Small to medium size enterprises with retail treatment
(SMERT)i=15: Equity exposure not held in the trading book
(EENHTB)
with i = 1–6 and i =  9–12 constituting corporate and retail
exposures, respectively. As a result of the SMC, market trade in almost all of
these loan subportfolios became sluggish. In particular, a problematic loan
subportfolio corresponds to i=4 which encompasses income producing real
estate. Furthermore, in commercial real estate (i.e., i=5 and i=6) there is a slowing due to the tightening
credit and slowing growth, the former a direct result of the SMC. Although
capital remains available for residential loans, the credit crunch is
pronounced in commercial lending. Moreover, i=7 which corresponds to interbank lending had
been identified as a problematic subportfolio. This is because banks were not
lending to each other during the early stages of the crisis (see FM4 in Figure 6).The risk premium,ϱ, from (2.5) has had a part to play in the SMC
(see HM1 in Figure 6). This premium is part of the expression for the bank's
own loan rate, rΛ, and its size is an indication of perceived
credit risk. A study by the U.S. Federal Reserve indicated that the average
difference in mortgage interest rates between subprime (“subprime
markup”) and prime mortgages declined from ϱ=2.8 percentage points (280 basis points) in 2001,
to ϱ=1.3 percentage points in 2007. In other words, the
risk premium required by lenders to offer a subprime loan declined dramatically
during the aforementioned period. This occurred even though subprime borrower
and loan quality declined overall during the 2001–2006 period. In
fact, this state of affairs should have had the opposite effect. However, it is
clear that such conditions are consistent with classic boom and recession
credit cycles (see [37]).The sensitivity of the required capital to changes in
the amount of loans extended from (2.6) is an important issue in the SMC. Bank
capital levels were depleted with banks experiencing high debt levels (see FM2
in Figure 6). As a consequence, for many banks (2.6) begun to take on negative
values during the SMC.The Central Bank base rate,r, from (2.7) in Subsection 2.2.1 has also had a
role to play in trying to alleviate global economic pressures during the SMC.
For instance, on Wednesday, 8 October 2008, central banks in the U.S. (U.S.
Federal Reserve), England, China, Canada, Sweden, Switzerland and the European
Central Bank cut rates in a coordinated effort to assist the world economy.
This rate will be lowered when it is necessary to stimulate growth and provide
liquidity.Despite the discussion above, the most significant
relationships between our models and the SMC are established via the bank's own
loan rate,rΛ, given by (2.7) (see also (3.19) and (3.32) as
well as HM1, HM3, FM4 and GIR1 in Figure 6). As we have noted in (2.7), this
loan rate may be expressed as(5.2)rtΛ=ρ+(1+rt)Nη+crw+cE∂K∂Λ+E[l].In general terms, rΛ can be written as a function of credit risk,
market structure (power), marginal cost of debt, marginal cost of equity and
the sensitivity of capital to loans extended. The representation of the bank's
interest setting intimates that banks will experience positive returns in good
times when the actual rate of default, rd, is lower than the provisioning for expected
losses, αϱ+E[l], and may not be able to cover their expected
losses when rd>αϱ+E[l] (see equation (2.13) for loan losses and
default rate). In the latter case, bank capital may be needed to cover these
excess (and unexpected) losses. If this capital is not enough then the bank
will face insolvency. During the latter half of 2008, we have seen a decline in
such capital.A combination of factors resulting from the SMC have
led to problems in the commercial real estate market as regards loan demand
(see (2.11); also HM3, FM4, GIR1 and GIR5 in Figure 6). According to the
National Association of Realtors (NAR) there is a slowing in commercial real
estate due to the tightening credit and slowing growth, the former a direct
result of the SMC. Although capital remains available for residential loans,
the credit crunch is pronounced in commercial lending. Patricia Nooney,
chairperson of the Realtors Commercial Alliance Committee (RCAC), portrayed the
decline as unusual since“transactions are being curtailed not for
lack of demand, but for serious challenges in obtaining financing.” (see
[61] for more
information).Next, we discuss loan losses and default rate given by
(2.13) in Subsection 2.2.2 (see HM4, FM1, FM3, GIR3 and GIR5 in Figure 6). An
acceleration in loan growth, as was experienced prior to the SMC, eventually
leads to a surge in loan losses (see, for instance, (2.13) for loan losses and
default rate) resulting in reduced bank profits (see equation (3.7) for bank
profit) and precipitating a new round of bank failures. As experience during
the mortgage crisis has shown, such a slump in banking could not only threaten
the deposit insurance fund but also slow the economy by entrenching credit
crunches. The view that faster loan growth leads to higher loan losses should
not be taken lightly; nor should it be accepted without question. If loan
growth increases because banks become more willing to lend, credit standards
should fall and loan losses should eventually rise. For many distressed banks
suffering because of the SMC, increased loan loss provisions from (2.16) and
(2.17) translate to a decrease in earnings.Usually, when capital is needed, banks have to choose
between raising and holding deposits and borrowing from the Central Bank as
reflected by (2.21) in Subsection 2.3.3.Contracted liquidity in the global credit markets and
banking system (as described by Remark2.3; see also FM4 in Figure 6) is an
ongoing economic problem. These liquidity concerns drove central banks around
the world to take action to provide funds to member banks to encourage lending
to worthy borrowers and to restore faith in the commercial paper markets (see
GIR1 in Figure 6).We recall that the bank's profits are given by (3.7)
in the form(5.3)Πt=(rtΛ−ctΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+ctD)Dt−cw(Wt)−P(Mt)−rtTγDt.It is important to note that
this expression represents the bank's profits from unsecuritized loans.
However, in the SMC, the attention has been on borrowing under securitization
strategies (see Figure 9 in Subsection C.1). In principle, we can provide an
analogue of (3.7) in a securitization context by introducing additional
variables. In this regard, suppose that rr is the average residual rate in securitized
transactions, prc the average expected fraction of loan losses
on securitized debt, cli the marginal cost of mortgage loan insurance
under securitization, cmw the marginal cost to monitor a loan that was
securitized, ctw the marginal transaction costs to securitize a
loan amortized over a loan term, pw the fraction of the loan amount securitized
and p˜w the fraction of the mortgage loan principal, Λ0, realized as a new loan in securitization. Then
the bank's profits under securitization, Πtw, may be presented in the form (5.4)Πtw=(rtr−ptrc−ctli−ctmw−cttw)ptwp˜twΛt+(rtΛ−ctlc−cttw)ptw(1−p˜tw)Λt+(1−ptw)[(rtΛ−ctΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt⁢−(rtD+ctD)Dt−cw(Wt)−P(Mt)−rtTγDt]−Vta−Et−Ft.Here we note that, in terms of
the organization of the special purposes vehicle (SPV), there are different
states that can be associated with the different possible corporate forms of
the SPV. The SPV can be an LLC (E1), LLP (E2), C-corporation (E3) or a trust (E4). Each of these SPVs has its own unique tax
benefits and problems as well as degree of asset protection and limited
liability under legal prescripts. Furthermore, we denote the optimal state
during the lifetime of the SPV by E*, with deviations from E* being given by |E*−Eti|, where i=1,2,3,…,4. These deviations measure the loss and
opportunity costs arising from the use of suboptimal corporate structures as
SPVs. Usually, such loss manifests as elevated legal fees, increased losses due
to low limited-liability and/or the value of additional time spent in dispute
resolution. In this case, the formula for E in (5.4) may be given by(5.5)Et=max{|E*−Et1|,|E*−Et2|,|E*−Et3|,|E*−Et4|,0}.Furthermore, in the
securitization context of (5.4), we assume that financing can take the form of Ff, where the transaction is an assignment and Fs, where the transaction is a true sale. In the
former case, the SPV is not subsidized by the sponsor. Alternatively, the
impaired collateral is only substituted by the sponsor. In this situation,
where Ff is the optimal state but Fs is being used instead, the transaction incurs
losses and vice versa. On the other hand, where the correct transaction format
is used, there is neither a gain nor a loss. Hence the value of the transaction
format choice (compare with (5.4) where F is used) is(5.6)Ft=max{|Ftf−Fts|,0}.Furthermore, we denote the value
of the adverse selection problem by Va. In this case, the transaction is an assignment
and the sponsor is the servicer. Here, the sponsor will typically substitute
impaired collateral, and hence is incentivized not to prefer the best
collateral as a replacement. In this regard, we denote impaired collateral,
value of replacement collateral and the value of average-quality collateral
available for replacement by C0,Ci…Cn and Ca, respectively. We assume that collateral
consists of units of the same or similar sizes where Pi is the probability of there being impaired
collateral and Pa the probability that the servicer/sponsor will
offer medium-or low-quality collateral as replacement for the impaired
collateral. In this situation, it follows that Va in (5.4) may be represented by(5.7)Va=∫0t{∑Ca∗Pi(1−Pa)}. Another remark about profitability is that during the
SMC, profits at the 8 533 U.S. banks insured by the FDIC fell from $35.2
billion to $646 million (effectively by 89%) during Quarter 4 of 2007 when
compared with the previous year. This was largely due to escalating loan losses
and provisions for loan losses. This decline in profits contributed to the
worst bank and thrift quarterly performance since 1990. In 2007, these banks earned
approximately $100 billion, which represented a decline of 31% from the
record profit of $145 billion in 2006. Profits decreased from $35.6 billion
to $19.3 billion during the first quarter of 2008 versus the previous year, a
decline of 46% (see [46, 47] for more detail). The contribution [34] considers the following
strategy to be optimal for banks to shield their profits from loan losses. The
loan loss reserves, Rl, is built up in every period that P>L. On the other hand, when P≤L the bank is allowed to draw on Rl from the current period. When Rl or other capital becomes inadequate, at some
point, the bank will face insolvency.Bank valuations as realized by (3.16) in Subsection
3.4.1 have declined dramatically during the SMC as a direct result of sharp
decreases in profitability and bank capital.


## 5.1.1. Diagrammatic Overview of The Subprime Mortgage Crisis

A diagrammatic overview of the SMC (see, for instance,
[51]) may be
represented as in Figure 6.


## 5.1.2. Background to The Subprime Mortgage Crisis

Most of the information contained in this subsection
was sourced from [51].
The SMC was initiated by the deflation of the United States housing bubble
(see, for instance, [52, 53]) and high default rates on subprime and adjustable
rate mortgages (ARM). Loan incentives, such as easy initial terms and low loan
rates, in combination with escalating housing prices encouraged borrowers to
assume difficult mortgages on the belief they would be able to quickly
refinance at more favorable terms (see HM1 in Figure 6). Great concern was also
expressed about the rapid growth in business loans at commercial banks with
excessively easy credit standards. Some analysts claim that competition for
lenders had greatly increased, causing banks to reduce loan rates and ease
credit standards in order to issue new credit. Others are of the opinion that
as the economic expansion continued and past loan losses were forgotten, banks
exhibited a greater propensity for risk. However, once U.S. housing prices
started to fall moderately in 2006-2007,
refinancing became more difficult (see HM2 and HM3 in Figure 6). Defaults and
foreclosure activity increased dramatically, as easy initial terms expired,
home prices failed to go up as anticipated, and ARM interest rates reset
higher. Foreclosures accelerated in the U.S. in late 2006 and triggered a
global financial crisis through 2007 and 2008. During 2007, nearly 1.3 million
U.S. housing properties were subjected to foreclosure activity; up 79% from
2006 (see [54] for
more details; also HM4 in Figure 6). The mortgage lenders that retained credit
risk were the first to be affected, as borrowers became unable or unwilling to
make payments (see HM5 and HM6 in Figure 6).Major banks and other financial institutions globally
had reported losses of approximately $435 billion from SMC-related activities
by Thursday, 17 July 2008 (see [55, 56]; also FM1 in Figure 6). By using securitization
strategies, many mortgage lenders passed the rights to the mortgage payments
and related credit risk to third-party investors via mortgage-backed securities
(MBSs) and collateralized debt obligations (CDOs). Corporate, individual and
institutional investors holding MBS or CDO suffered significant losses, as the
underlying mortgage asset value decreased. Stock markets in many countries
declined significantly (see FM2 in Figure 6). The broader international
financial sector first began to experience the fallout from the SMC in February
2007 with the $10.5 billion writedown of HSBC, which was the first major CDO
or Mortgage Bankers Association (MBA) related loss to be reported. During 2007,
at least 100 mortgage companies had either failed, suspended operations or been
sold. Top management did not escape without blemish, as the CEOs of Merrill
Lynch and Citigroup were forced to resign within a week of each other.
Subsequently, merger deals were struck by many institutions. In addition,
Northern Rock and Bear Stearns required emergency assistance from central
banks. IndyMac was shut down by the FDIC on Sunday, 11 July 2008. Moreover, on
Sunday, 14 September 2008, after performing banking duties for more than 150
years, Lehman Brothers filed for bankruptcy as a consequence of losses stemming
from the SMC (see FM3 in Figure 6). Subsequent to this many U.S. and other
banks throughout the world also failed. The widespread dispersion of credit
risk and the unclear effect on financial institutions caused reduced lending
activity and increased spreads on higher interest rates. Similarly, the ability
of corporations to obtain funds through the issuance of commercial paper was
affected. This aspect of the crisis is consistent with a credit crunch. There
are a number of reasons why banks may suddenly make obtaining a loan more
difficult or increase the costs of obtaining a loan. This may be due to an
anticipated decline in the value of the collateral used by the banks when
issuing loans; an increased perception of risk regarding the solvency of other
banks within the banking system; a change in monetary conditions (e.g.,
where the central bank suddenly and unexpectedly raises interest rates or
capital requirements); the central government imposing direct credit controls
and instructing the banks not to engage in further lending activity. The
subprime crisis has adversely affected several inputs in the economy, resulting
in downward pressure on economic growth. Fewer and more expensive loans tend to
result in decreased business investment and consumer spending (see FM4 and FM5
in Figure 6).Liquidity concerns drove central banks around the
world to take action to provide funds to member banks to encourage lending to
worthy borrowers and to restore faith in the commercial paper markets (see GIR1
in Figure6). With interest rates on a large number of subprime and other ARM
due to adjust upward during the 2008 period, U.S. legislators, the U.S.
Treasury Department, and financial institutions took action. A systematic
program to limit or defer interest rate adjustments was implemented to reduce
the effect. In addition, lenders and borrowers facing defaults have been
encouraged to cooperate to enable borrowers to stay in their homes. Banks have
sought and received over $250 billion in additional funds from investors to
offset losses (see [57]
for more information). The risks to the broader economy created by the
financial market crisis and housing market downturn were primary factors in
several decisions by the U.S. Federal Reserve to cut interest rates and the
Economic Stimulus Package (ESP) passed by Congress and signed by President Bush
on Wednesday, 13 February 2008 (see, for instance, [58–60]; also GIR2 in Figure 6). Bush also announced a plan
voluntarily and temporarily to freeze the mortgages of a limited number of
mortgage debtors holding ARMs. A refinancing facility called FHA-Secure was
also created. This action is part of an ongoing collaborative effort between
the U.S. government and private industry to help some subprime borrowers called
the Hope Now Alliance (see GIR3 in Figure 6). The U.S. government also
bailed-out key financial institutions, assuming significant additional
financial commitments (see GIR4 in Figure 6). Following a series of ad-hoc
market interventions to bailout particular firms, a $700 billion systemic
rescue plan was accepted by the U.S. House of Representatives on Friday, 3
October 2008. These actions are designed to stimulate economic growth and
inspire confidence in the financial markets (see GIR5 in Figure 6). By November
2008, banks in Europe, Asia, Australia and South America had followed the
example of the U.S. government by putting rescue plans in place.


## 5.1.3. Connections between Our Models and The Subprime Mortgage Crisis

The connections between our banking models and the SMC
are complicated. However, a first step towards understanding this relationship
entails identifying the problematic loan subportfolios. In this regard, it is
possible to decompose the (total) loans,Λ, discussed above as(5.1)Λ=∑i=1mΛi,where m is the number of loan subportfolios with i being the index of each loan subportfolio so that i=1,2,…,m. We note that the Basel II IRB approach to
credit risk (see [2])
dictates that m≤15. In particular, this approach identifies 15
credit risk exposure types (loan subportfolios) that may be listed
as
i=1: Project finance (PF);i=2: Object finance (OF);i=3: Commodities finance (CF);i=4: Income producing real estate (IPRE);i=5: Specialized lending high volatility commercial real estate
(SLHVCRE)i=6: Specialized lending not including high volatility commercial real
estate
(SLNIHVCRE)i=7: Bank exposure
(BE);i=8: Sovereign exposure (SE);i=9: Retail residential mortgage (RRM);i=10: Home equity line of credit (HELOC);i=11: Other retail exposure (ORE);i=12: Qualifying revolving retail exposure (QRRE);i=13: Small to medium size enterprises with corporate treatment (SMECT)i=14: Small to medium size enterprises with retail treatment
(SMERT)i=15: Equity exposure not held in the trading book
(EENHTB)
with i = 1–6 and i =  9–12 constituting corporate and retail
exposures, respectively. As a result of the SMC, market trade in almost all of
these loan subportfolios became sluggish. In particular, a problematic loan
subportfolio corresponds to i=4 which encompasses income producing real
estate. Furthermore, in commercial real estate (i.e., i=5 and i=6) there is a slowing due to the tightening
credit and slowing growth, the former a direct result of the SMC. Although
capital remains available for residential loans, the credit crunch is
pronounced in commercial lending. Moreover, i=7 which corresponds to interbank lending had
been identified as a problematic subportfolio. This is because banks were not
lending to each other during the early stages of the crisis (see FM4 in Figure 6).The risk premium,ϱ, from (2.5) has had a part to play in the SMC
(see HM1 in Figure 6). This premium is part of the expression for the bank's
own loan rate, rΛ, and its size is an indication of perceived
credit risk. A study by the U.S. Federal Reserve indicated that the average
difference in mortgage interest rates between subprime (“subprime
markup”) and prime mortgages declined from ϱ=2.8 percentage points (280 basis points) in 2001,
to ϱ=1.3 percentage points in 2007. In other words, the
risk premium required by lenders to offer a subprime loan declined dramatically
during the aforementioned period. This occurred even though subprime borrower
and loan quality declined overall during the 2001–2006 period. In
fact, this state of affairs should have had the opposite effect. However, it is
clear that such conditions are consistent with classic boom and recession
credit cycles (see [37]).The sensitivity of the required capital to changes in
the amount of loans extended from (2.6) is an important issue in the SMC. Bank
capital levels were depleted with banks experiencing high debt levels (see FM2
in Figure 6). As a consequence, for many banks (2.6) begun to take on negative
values during the SMC.The Central Bank base rate,r, from (2.7) in Subsection 2.2.1 has also had a
role to play in trying to alleviate global economic pressures during the SMC.
For instance, on Wednesday, 8 October 2008, central banks in the U.S. (U.S.
Federal Reserve), England, China, Canada, Sweden, Switzerland and the European
Central Bank cut rates in a coordinated effort to assist the world economy.
This rate will be lowered when it is necessary to stimulate growth and provide
liquidity.Despite the discussion above, the most significant
relationships between our models and the SMC are established via the bank's own
loan rate,rΛ, given by (2.7) (see also (3.19) and (3.32) as
well as HM1, HM3, FM4 and GIR1 in Figure 6). As we have noted in (2.7), this
loan rate may be expressed as(5.2)rtΛ=ρ+(1+rt)Nη+crw+cE∂K∂Λ+E[l].In general terms, rΛ can be written as a function of credit risk,
market structure (power), marginal cost of debt, marginal cost of equity and
the sensitivity of capital to loans extended. The representation of the bank's
interest setting intimates that banks will experience positive returns in good
times when the actual rate of default, rd, is lower than the provisioning for expected
losses, αϱ+E[l], and may not be able to cover their expected
losses when rd>αϱ+E[l] (see equation (2.13) for loan losses and
default rate). In the latter case, bank capital may be needed to cover these
excess (and unexpected) losses. If this capital is not enough then the bank
will face insolvency. During the latter half of 2008, we have seen a decline in
such capital.A combination of factors resulting from the SMC have
led to problems in the commercial real estate market as regards loan demand
(see (2.11); also HM3, FM4, GIR1 and GIR5 in Figure 6). According to the
National Association of Realtors (NAR) there is a slowing in commercial real
estate due to the tightening credit and slowing growth, the former a direct
result of the SMC. Although capital remains available for residential loans,
the credit crunch is pronounced in commercial lending. Patricia Nooney,
chairperson of the Realtors Commercial Alliance Committee (RCAC), portrayed the
decline as unusual since“transactions are being curtailed not for
lack of demand, but for serious challenges in obtaining financing.” (see
[61] for more
information).Next, we discuss loan losses and default rate given by
(2.13) in Subsection 2.2.2 (see HM4, FM1, FM3, GIR3 and GIR5 in Figure 6). An
acceleration in loan growth, as was experienced prior to the SMC, eventually
leads to a surge in loan losses (see, for instance, (2.13) for loan losses and
default rate) resulting in reduced bank profits (see equation (3.7) for bank
profit) and precipitating a new round of bank failures. As experience during
the mortgage crisis has shown, such a slump in banking could not only threaten
the deposit insurance fund but also slow the economy by entrenching credit
crunches. The view that faster loan growth leads to higher loan losses should
not be taken lightly; nor should it be accepted without question. If loan
growth increases because banks become more willing to lend, credit standards
should fall and loan losses should eventually rise. For many distressed banks
suffering because of the SMC, increased loan loss provisions from (2.16) and
(2.17) translate to a decrease in earnings.Usually, when capital is needed, banks have to choose
between raising and holding deposits and borrowing from the Central Bank as
reflected by (2.21) in Subsection 2.3.3.Contracted liquidity in the global credit markets and
banking system (as described by Remark2.3; see also FM4 in Figure 6) is an
ongoing economic problem. These liquidity concerns drove central banks around
the world to take action to provide funds to member banks to encourage lending
to worthy borrowers and to restore faith in the commercial paper markets (see
GIR1 in Figure 6).We recall that the bank's profits are given by (3.7)
in the form(5.3)Πt=(rtΛ−ctΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt−(rtD+ctD)Dt−cw(Wt)−P(Mt)−rtTγDt.It is important to note that
this expression represents the bank's profits from unsecuritized loans.
However, in the SMC, the attention has been on borrowing under securitization
strategies (see Figure 9 in Subsection C.1). In principle, we can provide an
analogue of (3.7) in a securitization context by introducing additional
variables. In this regard, suppose that rr is the average residual rate in securitized
transactions, prc the average expected fraction of loan losses
on securitized debt, cli the marginal cost of mortgage loan insurance
under securitization, cmw the marginal cost to monitor a loan that was
securitized, ctw the marginal transaction costs to securitize a
loan amortized over a loan term, pw the fraction of the loan amount securitized
and p˜w the fraction of the mortgage loan principal, Λ0, realized as a new loan in securitization. Then
the bank's profits under securitization, Πtw, may be presented in the form (5.4)Πtw=(rtr−ptrc−ctli−ctmw−cttw)ptwp˜twΛt+(rtΛ−ctlc−cttw)ptw(1−p˜tw)Λt+(1−ptw)[(rtΛ−ctΛ−rd(Mt))Λt+rtTWt+rtIIt+rtCCt+rtBBt+rtSSt⁢−(rtD+ctD)Dt−cw(Wt)−P(Mt)−rtTγDt]−Vta−Et−Ft.Here we note that, in terms of
the organization of the special purposes vehicle (SPV), there are different
states that can be associated with the different possible corporate forms of
the SPV. The SPV can be an LLC (E1), LLP (E2), C-corporation (E3) or a trust (E4). Each of these SPVs has its own unique tax
benefits and problems as well as degree of asset protection and limited
liability under legal prescripts. Furthermore, we denote the optimal state
during the lifetime of the SPV by E*, with deviations from E* being given by |E*−Eti|, where i=1,2,3,…,4. These deviations measure the loss and
opportunity costs arising from the use of suboptimal corporate structures as
SPVs. Usually, such loss manifests as elevated legal fees, increased losses due
to low limited-liability and/or the value of additional time spent in dispute
resolution. In this case, the formula for E in (5.4) may be given by(5.5)Et=max{|E*−Et1|,|E*−Et2|,|E*−Et3|,|E*−Et4|,0}.Furthermore, in the
securitization context of (5.4), we assume that financing can take the form of Ff, where the transaction is an assignment and Fs, where the transaction is a true sale. In the
former case, the SPV is not subsidized by the sponsor. Alternatively, the
impaired collateral is only substituted by the sponsor. In this situation,
where Ff is the optimal state but Fs is being used instead, the transaction incurs
losses and vice versa. On the other hand, where the correct transaction format
is used, there is neither a gain nor a loss. Hence the value of the transaction
format choice (compare with (5.4) where F is used) is(5.6)Ft=max{|Ftf−Fts|,0}.Furthermore, we denote the value
of the adverse selection problem by Va. In this case, the transaction is an assignment
and the sponsor is the servicer. Here, the sponsor will typically substitute
impaired collateral, and hence is incentivized not to prefer the best
collateral as a replacement. In this regard, we denote impaired collateral,
value of replacement collateral and the value of average-quality collateral
available for replacement by C0,Ci…Cn and Ca, respectively. We assume that collateral
consists of units of the same or similar sizes where Pi is the probability of there being impaired
collateral and Pa the probability that the servicer/sponsor will
offer medium-or low-quality collateral as replacement for the impaired
collateral. In this situation, it follows that Va in (5.4) may be represented by(5.7)Va=∫0t{∑Ca∗Pi(1−Pa)}. Another remark about profitability is that during the
SMC, profits at the 8 533 U.S. banks insured by the FDIC fell from $35.2
billion to $646 million (effectively by 89%) during Quarter 4 of 2007 when
compared with the previous year. This was largely due to escalating loan losses
and provisions for loan losses. This decline in profits contributed to the
worst bank and thrift quarterly performance since 1990. In 2007, these banks earned
approximately $100 billion, which represented a decline of 31% from the
record profit of $145 billion in 2006. Profits decreased from $35.6 billion
to $19.3 billion during the first quarter of 2008 versus the previous year, a
decline of 46% (see [46, 47] for more detail). The contribution [34] considers the following
strategy to be optimal for banks to shield their profits from loan losses. The
loan loss reserves, Rl, is built up in every period that P>L. On the other hand, when P≤L the bank is allowed to draw on Rl from the current period. When Rl or other capital becomes inadequate, at some
point, the bank will face insolvency.Bank valuations as realized by (3.16) in Subsection
3.4.1 have declined dramatically during the SMC as a direct result of sharp
decreases in profitability and bank capital.


## 5.2. Bank Valuation and Basel Ii

In this subsection, we analyze the connections between
our banking model and the Basel II Capital Accord (see SubsectionB on pages
43 to 44 for a diagrammatic overview of Basel II; also [1, 2]). These relationships are
forged via the(i)
description of total bank capital given by
(3.1) in Subsection 3.1.2;(ii)
description of the bank capital constraint
given by (3.4) and (3.6) in Subsection 3.1.3;(iii)
procyclicality described in Subsections3.1.3
and 4.1.


### 5.2.1. Diagrammatic Overview of The Basel Ii Capital Accord

Next, we provide a diagrammatic overview of the Basel
II Capital Accord  as shown in Figure7.Figure 7
Diagrammatic overview of the Basel II Capital Accord.


### 5.2.2. Bank Regulatory Capital

Despite the analysis in Subsubsections3.1.1 and 3.1.2
of Section 3, bank capital is notoriously difficult to define, monitor and
measure. For instance, the measurement of equity depends on how all of a bank's
financial instruments and other assets are valued. The description of the
shareholder equity component of bank capital, E, given by (3.1), is largely motivated by the
following two observations. Firstly, it is meant to reflect the nature of the
book value of equity. Our intention is also to recognize that the book and
market value of equity is highly correlated.Under Basel II, bank capital requirements have
replaced reserve requirements (see Subsection2.1) as the main constraint on
the behavior of banks (see, for instance, [62]). A first motivation for this is that bank capital
has a major role to play in overcoming the moral hazard problem arising from
asymmetric information between banks, creditors and debtors. Also, bank regulators
require capital to be held to protect themselves against the costs of financial
distress, agency problems and the reduction of market discipline caused by the
safety net. Subsection 3.1.3 suggests that a close relationship exists between
bank capital holding and macroeconomic activity in the loan market (see
equation (3.6) describing the bank capital constraint).


### 5.2.3. Procyclicality of Basel Ii Regulation

Our paper has discussed the procylicality induced by
the Basel II Capital Accord's approach to credit risk in Subsections3.1.3 and
4.1. In this regard, the Standardized approach to credit risk differentiates
assets not only according to the borrower but according to riskiness proxied by
credit rating agencies' assessment of the borrower. This approach represents an
improvement in the sense that corporations are rated. However, it is deficient
with respect to the viewpoint that ratings properly reflect risk and the
inducing of procyclical capital charges which will lead to overlending during
booms and underlending during recessions. Also, the capital formula for the IRB
approach to credit risk is problematic. In short, the IRB capital formula is
deduced by considering the large-portfolio asymptotic dynamics of a Merton
model with a single common risk-factor. Many empirical studies have confirmed
that this formula gives rise to procyclicality.In addition, Basel II dictates that a macroeconomic
shock will affect the loan risk-weights in the CAR. In general, a negative
(positive) shock results in the tightening (loosening) of the capital
constraint (see equation (3.6) describing the bank capital constraint). As a
consequence, in terms of a possible binding capital constraint, banks are free
to increase (decrease) the loan supply when macroeconomic conditions Mt improve (deteriorate). On the other hand, if
the risk-weights are constant, a shock does not affect the loan supply but
rather results in a change in the loan rate when the capital constraint binds.
It is not always true that Basel II risk-sensitive weights lead to an increase
(decrease) in bank capital when macroeconomic activity in the loan market
increases (decreases). A simple explanation for this is that macroeconomic
conditions do not necessarily only affect loan demand but also influences the
total capital constraint (see equation (3.6) describing the bank capital
constraint). Furthermore, banks do not necessarily need to raise new capital to
expand their loan supply, since a positive macroeconomic shock may result in a
decrease in the RWAs with a commensurate increase in CARs (compare the minimum
capital constraint as expressed in (3.4) and (3.6)). Similarly, banks are not
compelled to decrease their capital when the loan demand decreases since the
capital constraint usually tightens in response to a negative macroeconomic
shock. A further complication is that an improvement in the latter conditions
may result in an increase in the loan demand and, as a consequence, an increase
in the probability that the capital constraint may be binding (see equations
(2.11) and (3.6) for the loan demand and bank capital constraint, resp.).
Banks may react to this situation by increasing capital to maximize profits
(compare the definition of the return on equity (ROE) measure of profitability).
Our main conclusion is that bank capital is procyclical because it is dependent
on fluctuations in loan demand which, in turn, is reliant on macroeconomic
activity (see equation (3.6) describing the bank capital constraint).


## 5.2.1. Diagrammatic Overview of The Basel Ii Capital Accord

Next, we provide a diagrammatic overview of the Basel
II Capital Accord  as shown in Figure7.Figure 7
Diagrammatic overview of the Basel II Capital Accord.


## 5.2.2. Bank Regulatory Capital

Despite the analysis in Subsubsections3.1.1 and 3.1.2
of Section 3, bank capital is notoriously difficult to define, monitor and
measure. For instance, the measurement of equity depends on how all of a bank's
financial instruments and other assets are valued. The description of the
shareholder equity component of bank capital, E, given by (3.1), is largely motivated by the
following two observations. Firstly, it is meant to reflect the nature of the
book value of equity. Our intention is also to recognize that the book and
market value of equity is highly correlated.Under Basel II, bank capital requirements have
replaced reserve requirements (see Subsection2.1) as the main constraint on
the behavior of banks (see, for instance, [62]). A first motivation for this is that bank capital
has a major role to play in overcoming the moral hazard problem arising from
asymmetric information between banks, creditors and debtors. Also, bank regulators
require capital to be held to protect themselves against the costs of financial
distress, agency problems and the reduction of market discipline caused by the
safety net. Subsection 3.1.3 suggests that a close relationship exists between
bank capital holding and macroeconomic activity in the loan market (see
equation (3.6) describing the bank capital constraint).


## 5.2.3. Procyclicality of Basel Ii Regulation

Our paper has discussed the procylicality induced by
the Basel II Capital Accord's approach to credit risk in Subsections3.1.3 and
4.1. In this regard, the Standardized approach to credit risk differentiates
assets not only according to the borrower but according to riskiness proxied by
credit rating agencies' assessment of the borrower. This approach represents an
improvement in the sense that corporations are rated. However, it is deficient
with respect to the viewpoint that ratings properly reflect risk and the
inducing of procyclical capital charges which will lead to overlending during
booms and underlending during recessions. Also, the capital formula for the IRB
approach to credit risk is problematic. In short, the IRB capital formula is
deduced by considering the large-portfolio asymptotic dynamics of a Merton
model with a single common risk-factor. Many empirical studies have confirmed
that this formula gives rise to procyclicality.In addition, Basel II dictates that a macroeconomic
shock will affect the loan risk-weights in the CAR. In general, a negative
(positive) shock results in the tightening (loosening) of the capital
constraint (see equation (3.6) describing the bank capital constraint). As a
consequence, in terms of a possible binding capital constraint, banks are free
to increase (decrease) the loan supply when macroeconomic conditions Mt improve (deteriorate). On the other hand, if
the risk-weights are constant, a shock does not affect the loan supply but
rather results in a change in the loan rate when the capital constraint binds.
It is not always true that Basel II risk-sensitive weights lead to an increase
(decrease) in bank capital when macroeconomic activity in the loan market
increases (decreases). A simple explanation for this is that macroeconomic
conditions do not necessarily only affect loan demand but also influences the
total capital constraint (see equation (3.6) describing the bank capital
constraint). Furthermore, banks do not necessarily need to raise new capital to
expand their loan supply, since a positive macroeconomic shock may result in a
decrease in the RWAs with a commensurate increase in CARs (compare the minimum
capital constraint as expressed in (3.4) and (3.6)). Similarly, banks are not
compelled to decrease their capital when the loan demand decreases since the
capital constraint usually tightens in response to a negative macroeconomic
shock. A further complication is that an improvement in the latter conditions
may result in an increase in the loan demand and, as a consequence, an increase
in the probability that the capital constraint may be binding (see equations
(2.11) and (3.6) for the loan demand and bank capital constraint, resp.).
Banks may react to this situation by increasing capital to maximize profits
(compare the definition of the return on equity (ROE) measure of profitability).
Our main conclusion is that bank capital is procyclical because it is dependent
on fluctuations in loan demand which, in turn, is reliant on macroeconomic
activity (see equation (3.6) describing the bank capital constraint).


## 6. Concluding remarks and future research

In this paper, we have forged connections between
discrete-time stochastic banking models and macroeconomic activity as well as
the SMC and Basel II Capital Accord. Furthermore, we solved an optimal bank
valuation problem that, amongst other things, maximized profit under several
realistic banking constraints. A comprehensive illustration of some of the
concepts discussed in the main body of the paper was provided. The discussion
of procyclicality confirms that loan loss provisions increase as world
economies move further into recession. Also, we demonstrate that banks increase
loan loss provisions as loan problems become more apparent. In short, this
contribution represents a starting point for studying modeling issues related
to the SMC and, to a lesser extent, Basel II. We envisage that our study on the
modeling of the SMC will be extended in several directions in the near future.Future modeling research should consider the bank's
lending responses to business cycles as it pertains to the nexus between the
proximity to its capital constraints and its forthrightness in recognizing loan
losses. This relationship may not be constant over time. Further research
should also establish a solid basis for comparison of extant valuation practice
and demonstrate the superiority of our model to another model. In particular,
we need to learn more about the inadequacies of current practice as a basis for
substantiating the need for our modeling paradigm. Moreover, future reserach
should provide empirical support to demonstrate the superiority of our model
over other models or the accuracy of its performance relative to any market
benchmark. Hence, even if one could utilize the model in some valuation project
to estimate the value of the bank's common shareholder's equity, we have no
idea of how close we might be to the truth with respect to market values. Since
fair market values are of great interest to valuators, a test of reasonable
congruence with such values would seem to be prerequisite to the use of the
valuation model for purposes of business valuation. Although large sample
empirical data would be preferred, smaller sample analysis using recent actual
acquisitions might also be helpful. All these facts have to be incorporated in
future modeling programmes.Another research topic will involve complex models of
bank items driven by Lévy processes (see, for instance, Protter in [63, Chapter I, Section 4]).
Such processes have an advantage over the more traditional modelling tools such
as Brownian motion because the behavior of bank loans, wealth, capital and CARs
are characterized by jumps. As a result of this, recent research has strived to
replace the existing Brownian motion-based bank models (see, for instance, [3, 18, 20, 23, 64]) by systems driven by more
general processes. Also, a study of the optimal capital structure should
ideally involve the consideration of taxes and costs of financial distress,
transformation costs, asymmetric bank information and the regulatory safety
net. Another research area that is of ongoing interest is the (credit, market,
operational, liquidity, securitization) risk minimization of bank operations
within a regulatory framework (see, for instance, [38, 43]). Another risk that becomes
important is interest rate risk at the point of loan issuing. For instance, an alternative optimization problem
would be to maximize the risk-free rate of interest in order to provide a
shareholder with an incentive to invest money. An example of a over-simplified
model that has become more important during the current SMC and global credit
crunch is the Diamond-Dybvig model (see, for instance, [44, 45]). Their contributions
proposes a much simpler model than ours but yet explains key aspects of what
bank liquidity means in reality ceteris parabis. In future, it will be
worthwhile to explore connections with the work of Diamond and Dybvig on
liquidity.



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*Source: 740845-2009-03-10.xml*