40
Preliminary Draft. Do not cite. Risk-based capital, lending rate and credit rationing Kanak Patel and Wentao He
Preliminary Draft.
Do not cite.
Risk-based capital, lending rate andcredit rationing
Kanak Patel and Wentao He
RISK-BASED CAPITAL, LENDING RATE AND CREDIT RATIONING
Kanak Patela and Wentao Heb
This paper is concerned with the impact of higher regulatory Tier I and Tier
II capital ratio requirement on the magnitude of credit rationing and the prof-
itability of bank. Based on Stiglitz and Weiss (1981), we analyse a model in
which a bank may adjust the asset or liability side of its balance sheet. The
model uses the rate of return required by shareholders of the bank to capture
possible reactions under binding or non-binding capital constraint. It has been
well documented that the introduction of Basel III capital requirement resulted
in most banks adjusting the asset side of their balance sheets because the cost
of issuing equity was too high. We show that both a poorly capitalised bank and
a bank holding capital buffer react in the same manner to change in regulatory
requirement by raising the lending rate it charges on risky loans and by shrinking
its risky loan portfolio. A number of possible scenarios are shown to explain how
the higher lending rate can affect the magnitude of credit rationing depending
on the specific forms of the demand and supply functions. We deduce that both
the higher lending rate and the change in magnitude of credit rationing have an
adverse effect on the profitability of bank.
Keywords: Basel Capital regulation, Bank capital structure, Bank Portfolio
Allocation Decision, Default Rate, Bank Costs Function.
1. INTRODUCTION
In the aftermath of 2007 financial crisis, a raft of unprecedented measures
including liquidity support, extended deposit insurance, asset purchase pro-
grammes, and recapitalisation of banks were launched by central banks to
rescue the global financial system from contagion of bank failures. A con-
1April 2012 versionaMagdalene College, University of Cambridge, Cambridge CB3 0AG. Email:
[email protected] College,University of Cambridge, Cambridge CB2 1TL. Email:
1
fluence of rising interest rate 1, falling house prices, and uncertainty sur-
rounding the values of complex structured mortgage and credit products,
had fuelled credit and liquidity risk to spiral upwards. In the run-up to the
crisis, the composition of bank balance sheets for large banks had changed
significantly. The implementation of the Basel Capital Accord in the late
1980s and early 1990s changed the economic landscape of banking activities.
Incentivized by domestic and international competition under the Basel I
regulatory capital requirement, the US banks devised asset backed commer-
cial paper (ABCP) vehicles to reduce the exposure of their assets from a
credit perspective and incur lower regulatory capital charges associated with
related liquidity exposures of such vehicles. The most striking development
of the regulation was widening of the gap between risk-weighted assets and
total assets. Many banks had significantly expanded their off balance sheet
activities, largely by increasing their holdings of highly rated securities that
carried low risk weightings for regulatory capital purposes. IMF (2008) re-
port pointed out that ’This trend is evident in the 10 largest publicly listed
banks from Europe and the United States, which doubled in aggregate as-
sets in the last five years to 15 trillion euros, while risk-weighted assets,
which drive the capital requirement, grew more moderately to reach about
5 trillion euros’. The growth in banks’ total assets was engineered by inno-
vative off-balance-sheet bank conduits and structured investment vehicles
(SIVs) that allowed commercial banks to offer their corporate customers
low-cost, off-balance-sheet funding. In addition to the significant manage-
ment fees and trading income, SIVs had the added advantage of being able
to their sell the investment in the capital notes to their client base. More
importantly, banks were able to record a lower or no risk weight under Basel
I for the associated assets and for backup credit lines extended to SPV.
1Between 2004 and 2006, the Federal Reserve Board raised interest rates 17 times,
increasing them from 1 percent to 5.25 percent.
2
The loophole in regulatory capital spurred banks to reconfigure their as-
sets using credit risk transfer instruments such as Collateralized Debt Obli-
gations (CDOs) and credit default swaps (CDS). This was done either by
purchasing insurance against credit losses using CDSs (reducing the gross
risk of a loan portfolio) or by removing the riskiest (first loss) portions of
a loan portfolio using CDOs. Apart from regulatory arbitrage, the growth
in off-balance sheet activity was driven by competition from (unregulated)
money market mutual funds (MMMFs), which had diminished banks cost
advantage in acquiring funds, and had eroded their profitability from tra-
ditional loan markets. Banks that had adopted aggressive off-balance sheet
trading and investment activities became vulnerable to illiquidity in the
wholesale money markets, earnings volatility from marked to-market as-
sets, and illiquidity in structured finance markets. These concealed risks of
their exposures to off-balance sheet vehicles, which had not been captured
by disclosures or regulations, came under the spotlight when dislocations in
credit and funding markets surfaced. Poignantly, as more and more informa-
tion about the multi-layered complex structures became publicly available,
banks that had adopted off-balance sheet strategy most aggressively were
heavily penalised equity markets. Between 2006 and 2009, the overall loss in
market capitalization of the top-30 banks was a 52 percent, which includes a
significant stock market recovery during 2009 (Laeven and Valencia (2010)).
The downward spiral of falling asset prices and deleveraging precipi-
tated contraction of the supply of secured financing, particularly, to highly-
leveraged market users. In IMF (2008) IMF emphasised ’It is now clear
that the current turmoil is more than simply a liquidity event, reflecting
deep-seated balance sheet fragilities and weak capital bases, which means
its effects are likely to be broader, deeper, and more protracted.’ Because
leveraged institutions suffer mark-to-market losses of x dollar have to reduce
their position by x dollar times their leverage ratio, the ultimate impact on
3
new lending to businesses and households was enormous. During August and
December of 2008, Ivashina and Scharfstein (2010) estimate 36% reduction
in monthly loan origination by a bank with the median deposits-to-assets
ratio relative to the previous year while the same ratio one standard devia-
tion below the mean the reduction goes as high as 49%. Greenlaw, Hatzius,
Kashyap and Shin (2008) estimate shows ’$2.3 trillion contraction in in-
termediary balance sheets, of which roughly $1 trillion would represent a
decline in lending to households, businesses, and other non-levered entities.’
Bernanke (2006) in his speech on implementation of Basel II remarked
’Much more so than in the past, banks today are able to manage and con-
trol obligor and portfolio concentrations, maturities, and loan sizes, and to
address and even eliminate problem assets before they create losses Basel
II will make it easier for supervisors to identify banks whose capital is not
commensurate with their risk levels and to evaluate emerging risks in the
banking system as a whole. From the perspective of bank management and
stockholders, the availability of advanced methods for managing interest
rate risk leads to a more favorable risk-return trade-off. For supervisors, the
benefit is a greater resilience of the banking system in the face of a risk
that figured prominently in some past episodes of banking problems’. With
the hindsight of the credit crunch, it is evident that financial institutions
applied risk models in ways that significantly underestimated certain risk
exposures, and consequently, their capital was not commensurate with ex-
posures. Given the public and private sector costs of the credit crunch, the
focus has shifted towards assessing the implications of Basel II and Basel
III risk-management requirements and bank supervision.
When the regulator raises the capital requirement, one option for the
banks is to issue more equity to fund their current level of lending. If the
bank is able to raise new required equity, the action of the regulator (raising
minimum capital requirement) will have no effect on credit supply. However,
4
as have been edvidence in recent down market condition that the proposal
of Basel III to raise both Tier 1 and Tier 2 capital requirement and intro-
duce the capital conservation buffer, the banks have resorted to adjust the
asset side of their balance sheets in order to satisfy the minimum capital
requirement instead of issuing equity. The underlying reason is that banks
have found the cost of issuing equity too high, especially when their share
prices have been badly hit after the recent financial crisis. Under the cir-
cumstance, the supply of credit contracts, which is one of the major factors
that determines the magnitude of credit rationing. The second factor is the
demand for credit, which depends on the banks’ lending rates. A natural
question to ask is: how does higher capital requirement affect the lending
rate? One can argue that given the higher capital requirement, banks may
want to reduce the volatility of their asset values as well. Hence, they re-
duce their exposure to risky borrowers and shift to the safe borrowers and
give up some returns. This implies that banks reduce the lending rate to
attract safe borrowers as argued by SW. Assuming normal demand curve,
the magnitude of credit rationing increases as the gap between the supply
and demand widens. However, one can argue that banks would not lower
their lending rate because they will not be able to cover the cost of capital.
When the regulator raises capital requirement, banks’ shareholders required
higher rate of return. Even if the shareholders’ required rate of return re-
mains the same, higher equity implies that banks need to generate extra
profit to compensate for the additional equity. One option for banks is to
lend more to risky borrowers in order to charge a higher rate of return,
however, the demand woule be lower and this will offset the effect of lower
credit supply. The magnitude of credit rationing depends on which of the
two effects dominate, supply or demand.
The combined effect of regulatory and economic forces has a number
of implications for banks’ profitability and credit supply. First and fore-
5
most, the waxing issue in the current global economic climate is holding
an adequate amount of capital to ensure financial stability and to support
a recovery. To the extent that any of the major banks are still seriously
undercapitalized, the presence of the assets creates an incentive to gamble
for reclamation. For a clearly solvent bank, the decision to hang on to or
dispose of the assets would be based on a profit-maximizing motive. For
a bank that is close to insolvent, the incentive to remove the risk is much
lower. If the assets lose value and drive the bank into insolvency, then the
inability to resolve such an institution could create a zombie bank.
Second, in order to reduce funding needs and meet capital and liquidity
requirements many banks have forced many banks to shrink operations. For
example, Barclays sold Barclays Global Investor to Blackrock to raise $6.6
billion cash in 2009, Lloyds sold A$1.7 billion (November 2011) worth of
distressed property loans to Morgan Stanley and Goldman Sachs Group Inc
and, according to a recent report on Reuters June(2012) a unit of UK-based
Lloyds Banking Group plc (LYG) plans to sell A$1.9bn portfolio of troubled
property loans to Australian private equity property funds managed by
Blackstone Group LP (BX) and Morgan Stanley (MS).
Third, lower-yields on liquid assets, by repricing in funding and credit
markets, can adversely affect bank returns. As a result, banks may have to
constrain the supply of credit or raise lending rates to bolster returns. But,
even with such actions, returns may not be sufficient to cover cost of capital.
Moreover, if banks try to pass on the higher cost of capital to clients and
increase lending rates, it can inadvertently increase default risk. While this
may lay bare that much bank activity was made profitable by levering up, it
will have implications for the global economy. In the G3 (US, Euro Area and
Japan), the Institute of International Finance estimated that the combined
impact of new banking regulations may be to cut gross domestic product by
0.5− 0.6% per year over five years and could cost some 7.5 million jobs in
6
the process. According to IMF (2008)), ’The repricing has been triggered by
tighter lending conditions across the major economies, making credit more
difficult to access for corporates and households. Faced with the increasing
probability of unintended balance sheet expansion and losses, banks have
become increasingly reluctant to extend credit while securitization markets
may remain impaired. Combined with widening spreads, this increases the
risks to the economy of a credit crunch.’
The aim of this paper is to provide a structural framework to study
the effects on the borrowing interest rate, the magnitude of the credit ra-
tioning and the bank’s profitability when the regulatory capital requirement
changes. It integrates the concept of minimum capital requirement into the
credit rationing model suggested by Stiglitz and Weiss (1981)(henceforth
SW). By considering the capital structure of a typical commercial bank, we
propose a simple mechanism for the regulator to set the minimum capital
requirement ratio to achieve its goal in different scenarios. Based on SW
model, Agur (2011) develops a static model, which focuses on maximising
bank’s total value, to study the trade-off between financial stability and
credit rationing when capital requirement is raised. The study attempts
to link capital requirement with credit rationing. His is result is based on
the premise that ’smaller balance sheet reduces the amount of credit that
bank can supply’. However, Agur only focuses on the liability side of bank’s
balance sheet and fails to consider the significance of bank’s asset alloca-
tion between safe and risky assets. Basel Accord sets zero-risk weighting
for safe assets (government securities), which implies holding an extra unit
of government security does not require bank to hold more capital. In this
sense, bank can either change its asset composition (i.e. shift from risky
loan to government securities) or raise equity to response to increase in
capital requirement. Moreover, the option approach in Agur (2011), which
is used to show higher capital requirement, implies more credit rationing
7
is problematic due to the fact that higher strike price does not necessary
imply higher volatility of the underlying asset. This is a major shortcoming
of this model and it leads to a too strong conclusion that higher capital
requirement implies more credit rationing.
Our model considers both side of bank’s balance sheet and incorporates
bank’s asset allocation decision between sake and risky assets. We specify
bank’s rate of return to show that risk weighted capital requirement only
affects the lending rate within a certain region, and higher risk weighted
capital requirement may either increase or decrease the magnitude of credit
rationing. One may ask why higher capital requirement can reduce the mag-
nitude of credit rationing. Well, this is because bank will change its lend-
ing rate in response to a change in risk weighted capital requirement. The
change in lending rate will have a direct impact on the demand side of the
credit market. According to Agur (2011), the magnitude of credit rationing
is defined as the different between supply and demand in the credit market.
Thus, if the change in demand for credit exceeds the supply, the magnitude
of credit rationing will be smaller. By incorporating bank’s asset allocation
decision, we demonstrate that the lending rate is an increasing function of
capital requirement.
The rest of the paper is organised as follows. The section 2 presents a
review of events that have resulted in shrinkage of bank’s capital and con-
traction of credit supply. Section 3 discusses the relevant literature. Section
4 presents the model of the relationship between bank’s capital require-
ment and credit supply. Section 5 analyses the impact of minimum capital
requirement on bank’s lending rate. Section 6 investigates some wider impli-
cations of capital requirement on the magnitude of credit rationing. Section
7 discusses the effect of capital requirement on bank’s profitability. Section
8 concludes.
8
2. A HISTORIC SYNOPSIS OF BANKS’ ON AND OFF BALANCE SHEETLEVERAGE
By necessity, banks are highly geared. Banks had taken on excessive lever-
age in the period leading up to subprime crisis and, consequently, engaged
in excessive risk-taking. Inderst and Mueller (2008), model banks’ optimal
capital structure and show how competition for borrowers leads to an ’un-
derinvestment problem,’ unless banks are levered up sufficiently. Based on
’functional approach’, the authors argue that an important function of banks
is to make risky loans in a competitive environment. To illustrate how more
leverage was fostered under Basel I, Merton (1995) offered the following
example (pp. 468-469): If a bank were managing and holding mortgages on
houses, it would have to maintain a capital requirement of 4%. If, instead, it
were to continue to operate in the mortgage market in terms of origination
and servicing, but sells the mortgages and uses the proceeds to buy US gov-
ernment bonds, then under the BIS rules, US government bonds produce no
capital requirement and the bank would thus have no capital maintenance.
SIVs and Conduits were set up mainly as off-balance-sheet entities that
allowed banks to extend their lending without the pressure of regulatory
capital requirements. Most SIVs and conduits had back-up liquidity facili-
ties with banks. Along with SIVs and conduits, finance companies such as
Countrywide and Thornburg Mortgage, Northern Rock borrowed short in
ABCP markets to underwrite loans that they then sold to broker-dealers
for securitization. ABCP conduits and SIVs changed the way credit was
intermediated and risk was transformed in the financial system. In a period
of low interest rates, repackaging low-grade assets into investment-grade as-
sets by using complex financial instruments such as CDOs, cash flow CDOs,
was highly lucrative. The SIVs involved five groups of players: money mar-
ket mutual funds (MMMF) institutional investors (pension funds, insurance
companies, hedge fund), credit rating agencies, underwriters and traders (of-
9
ten both within the same banks). By pooling and tranching the cash flows
from (seemingly) imperfectly correlated assets, CDOs allowed institutional
investors to gain exposure to assets that, on their own, had been too risky,
while banks looking to take more risk receive potentially higher returns by
holding the most junior or ’equity’ CDO tranches. For banks (Merril Lynch,
Citigroup, Credit Suises, Goldman Sachs, Bear Stern, Deutsche Bank, etc.)
and rating agencies, CDOs generated underwriting and rating fees, respec-
tively. According to JPMorgan estimates, $6 trillion worth of credit was
intermediated through the shadow banking system as of the second quar-
ter of 2007 compared with the $10 trillion intermediated through regulated
banks funded primarily by deposits. Astonishingly, some investment banks
continued to market new CDOs (and synthetic CDOs) in 2007, even after
RMBS securities lost value and mortgage delinquencies intensified.
The process of transformation of risk involved funding illiquid long term
assets (CDOs, cash flow CDOs, CMO) with staggered, off balance sheet,
short term ABCPs in the unregulated wholesale market rather than from
the traditional retail deposits. The operational structure of conduits and
SIVs was highly risky; it lacked the solid foundation of adequate capital and
transparency. Moreover, SIVs often had multi layers of leverage because they
owned leveraged vehicles (CDOs), particularly those backed by subprime
and so-called Alt-A mortgages. Whilst most ABCP conduits had liquidity
support to cover at least 100% of the value of ABCP issued, SIVs relied
on capital and liquidity models, approved by ratings agencies, to manage
liquidity risk. ABCPs ratings were contingent on liquidity support and the
ratings of the credit so that a downgrade in the short-term or long-term
debt ratings of any of the parties may result in a reevaluation and possible
downgrade of the ABCPs. As Dodd and Mills (2008) succinctly pointed
out, ’The principal risk management strategy was to plan to trade rapidly
out of a loss-making position. But such a strategy, which relies on markets
10
remaining liquid, failed when markets rapidly became illiquid.’ Indeed, the
first signs of failure of this strategy surfaced on July 31, 2007, when two Bear
Stearns’s hedge funds filed for bankruptcy and a week later BNP Paribas
halted withdrawals from its three investment funds.
The crisis in the US subprime mortgage market picked up pace in August
2007. In mid July, Standard & Poor’s (S & P) and Moody’s each down-
graded over 400 or more residential mortgage backed security (RMBS). On
August 29, 2007, Standard & Poor’s downgraded the ratings of the short-
term notes issued by Cheyne Finance by six notches, which just two weeks
before it had declared those same notes to be the highest investment grade.
Cheyne Finance become the first SIV to default on its ABCP debt after the
administrator of the troubled fund won court backing to declare it in breach
of insolvency tests. The first sale of the assets of SIV Cheyne Finance was
described as ’fire sale’ by Moody’s Investors Service.
Mass downgrades by Moody’s and S & P sent shockwaves through the
financial markets and led investors to speculate about the next investment
vehicle to fall. The speed at which the downgrades occurred was an in-
dication of how quickly RMBS prices and values of assets in CDOs had
deteriorated. In January 2008, S & P again shocked the markets by its ac-
tions on over 6,300 RMBS and 1,900 CDOs (including downgrading and
placing securities on credit watch with negative implications) and triggered
sales of assets that had lost investment grade status. Investors like pension
funds, insurance companies, and banks were suddenly forced to reduce their
exposures to RMBS and CDO holdings because they had lost their invest-
ment grade status. New securitizations were unable to find investors since
RMBS and CDO securities held by financial firms lost much of their value.
Plunging asset prices meant market-value thresholds embedded in the SIVs
started to be hit. By early 2009 the total value assets SIVs’ was virtually
close to zero from the peak of $400bn in July 2007; of the total 29 SIVs,
11
7 defaulted, 18 were restructured or were consolidated onto the sponsoring
banks’ balance sheets. The subprime RMBS market initially froze and then
collapsed and CDO investors and underwriters.
The frantic exodus by money market funds from ABCP market proved
to be catastrophic for the off balance sheet credit supply. Rapidly shrinking
ABCP market on one side and plummeting values of illiquid assets on the
other side meant that the whole process of issuing, underwriting and mar-
keting high risk, low quality assets suddenly sent the banking system into a
tailspin. Banks that had sponsored SIVs came under heavy pressure to take
their assets onto their own balance sheets and/or fire sale those assets.
A full blown liquidity and credit crunch hit banks balance sheets that
precipitated deleveraging and wiped significant chunks of their equity. CRS
Report for Congress described ’By September, not a single ’bulge bracket’
investment bank remained standing: they had either failed (Lehman Broth-
ers), merged (Merrill Lynch and Bear Stearns), or converted themselves into
commercial bank holding companies (Goldman Sachs and Morgan Stanley)’
2. FDIC insured banks fell by 568 from June 2007 to April 2010 3.
Coordinated central bank actions were taken to support troubled banks
aimed at both asset side and liability side of banks’ balance sheets. Liquid-
ity support and extended deposit insurance were the first to be instigated
to contain the panic in the ABCP market. In the wake of the demise of
Northern Rock, HSBOS, Bear Stern, Countrywide, Lehman Brothers, Mer-
rill Lynch, and many other banks, money market fund and conduits, liq-
uidity needs rose sharply across markets. On the asset side, liquidity was
provided through purchase of illiquid assets outright or by accepting for the
purposes of collateralized lending. In August 2007, a series of emergency ac-
tions by the European Central Bank (ECB) injected a further US$85 billion
2CRS Report for Congress, Containing Financial Crisis, Updated November 24, 2008.3http://www.fdic.gov/bank/individual/failed/banklist.html
12
in liquidity through various mechanisms, highlighting the seriousness of the
crisis. The Federal Reserve introduced three programs with varying degrees
of success. The Commercial Paper Funding Facility (CPFF) and the Asset-
Backed Commercial Paper Money Market Fund Liquidity Facility (AMLF)
lending programs were created to enhance liquidity by reducing extension
risk and by reducing the risk of suspension of redemptions at money market
mutual funds that hold CP. The Treasury, in an effort to assure investors
during a run on money market funds, then a $3 trillion industry, that fu-
ture suspension of redemptions would not occur, also offered insurance for
the value of MMMF shares held to funds. Central banks managed to avert
run on deposits by guarantees of deposit and non-deposit liabilities in a
number of different forms. Guarantees in respect of non-deposit liabilities
in the UK were restricted to ’new’ borrowing and granted only under cer-
tain conditions, such as a defined quantum of recapitalization. A blanket
guarantee of liabilities was put in place in Ireland while in Italy guarantee
or support was offered to a particular class of non-deposit liabilities, such
as inter-bank claims. Central banks, both inside and outside the Euroarea
offered emergency liquidity assistance to individual banks under such terms
as they choose, with the credit risk remaining at national level.
The asset purchase programmes 4 of troubled assets (and high-quality
assets) eased the pressure of deleveraging, fire sale, haircuts by distressed
banks inflicting losses on other institutions. The first signs of distress emerged
in 2006 when HSBC, the world’s third-largest bank, disclosed its bad-debt
provisions soar to $10.8 billion as a result of defaults in its subprime port-
folio. As asset valuation uncertainties increased, troubled banks began to
offload their assets at distressed prices. In deteriorating market conditions,
fire sales intensified and capital losses of leveraged institutions went up,
4The Emergency Economic Stabilization Act of 2008 authorized Troubled Asset Relief
Program to restore liquidity and stability to the US financial.
13
credit terms became tighter with higher haircuts/initial margins on assets.
Problems intensified with the bailout of Bear Stearns, and later in the year
with the collapse of investment bank Lehman Brothers, and the government
bailouts of insurer AIG and mortgage lenders Freddie Mac and Fannie Mae.
The Troubled Asset Relief Program (TARP), ”the bailout legislation” as it
has come to known, was established to buy troubled assets from ailing banks
and other financial institutions and then dispose of them. Some 707 financial
institutions received $204.9 billion as part of the Capital Purchase Program,
and as of March 31, 351 regional and community banks were still in the
program. Another 83 financial institutions were in the TARP Community
Development Capital Initiative, bringing the total number of institutions
still in TARP to 434. The Capital Purchase Program (CPP) was set up to
Infuse capital into troubled financial institutions, had the obvious positive
effect of increasing banks’ available capital.
As part of the co-ordinated Action plan, various EU governments on a
national level pledged a total of EUR 1,873 billion for guarantees of their
banking sectors. In February 2008, the UK Government nationalised North-
ern Rock Bank plc, which was the first UK bank failure of the 2007-2009
financial crisis. The government also took controlling interest in Royal Bank
of Scotland Group Plc and Lloyds Banking Group plc and injected ?500 bil-
lion ($750 billion) in the eight largest banks and building societies. Barclays
also raised £ 5.8 billion of new capital in 2008 from the state investment
funds and royal families of Qatar and Abu Dhabi. In July 2007, the Ger-
man government and financial regulators were granted approval by the EU
Commission to bailout 9 billion EUR ($11.7 billion) of IKB. The Financial
Market Stabilization Supplementary Act was passed April 2009 that paved
the way for the nationalisation of Hypo Real Estate Holding AG. The law
extends the financial market stabilization law agreed in 2008, giving the gov-
ernment powers to seize control of banks whose failure would pose a risk to
14
the stability of the financial system. The Bad Bank Act, passed July 2009,
provided private banks relief on holdings of illiquid assets by allowing them
to transfer assets to a special entity and receive government-guaranteed
bonds issued by this special entity in exchange.
The actions of central banks and governments in coping with the global
financial crisis have raised many issues. There is no doubt that the ex-
ceptional rescue measures and monetary policy reaction to the crisis have
helped to stabilise the banking system. Raising the level of banks capital can
be valuable in turbulent times; however, it only affords partial protection
to the banking system. Faced with a combination of deteriorating economic
conditions and unfavourable profit outlook, increased credit risk, and the
steep cost of raising new capital, banks did not have any other choice than
curtail their portfolio and drastically reduce new lending. Indeed, between
2008 and 2010, commercial bank lending was reduced by 25% and M1 money
multiplier was reduced almost half. The most recent rules allow a 10 times
gearing ratio, which implies that a 10% write off of its loan portfolio could
wipe out its capital and no bank geared at such level could withstand a run
on its deposits whatever its level of capital.
3. MODEL FRAMEWORK
We consider 3 participants in the credit market: Regulator, Borrowers
and Banks.
Regulator:
The regulator sets the minimum capital requirement for bank. Here, we
just consider the Tier 1 capital. This is the originally amount of paid up
capital stock (shares) of the bank, net retained profit and other qualified
Tier 1 capital. We only consider a single period in our model, so we only
need to consider the amount of equity at the beginning of the period as
15
Tier 1 capital. Let ϕ be the minimum capital requirement (capital ratio)5
set by the regulator. For a traditional bank (or building society) that holds
a portfolio of risky loan and safe government security, we can define the
capital ratio as
ϕ =E
wLL+ wGG
where E is the equity of the bank, L is the amount of risky loan the bank
has issued, G is the amount of government security the bank is holding over
the period, wL and wG are the risk weighting for risky loan and government
security, respectively.
To simplify our analysis, we further assume that the risk weighting for
the government security is zero and the risk weighting for risky loan is 1.
We have
ϕL = E
Borrowers:
There are N borrowers, each has one and only one investment project.
Without loss of generality, we can number these borrowers/projects in nu-
merical order, which represents their riskiness. i.e. θ ∈ 1, 2, 3, . . . , N , and
the bigger the θ, the greater the volatility of project returns, which implies
greater risk of the project. These projects are assumed to be observably iden-
tical in the sense of mean preserving spreads. This means that all projects
have the same expected return, but different variances. If we let R to be
the return on a project, f(R, θ) be the density function of R, and F (R, θ)
be the distribution function of returns, the mean preserving spread can be
interpreted mathematically as follows.
For any θ1 > θ2, if∫ ∞0
Rf(r, θ1)dR =
∫ ∞0
Rf(r, θ2)dR
5We will use these two terms interchangablely in the remaining analysis.
16
then for any y ≥ 0, ∫ y
0
F (r, θ1)dR ≥∫ y
0
F (r, θ2)dR
We assume that borrowers have identical initial wealth and need to bor-
row among B from the bank to start their projects. The rate of interest
borrowers have to pay is rL, which is determined by the bank. Based on
these set up, SW showed that for a given level of interest rate on the loan,
there is a critical value θ∗ such that a borrwer borrows from the bank if and
only if θ > θ∗. This critical value will increase if the level of interest rate
increase. (i.e. ∂θ∗
∂rL> 0)
Banks:
There are M identical banks in the credit market. We assume this M is
small and therefore the banks are operating in an oligopoly market. Hence,
we just need to analyse the action of a representative bank. To simplify our
model, we assume there are only two types of assets banks hold in their
portfolios. One of the assets is the risky asset (loan), and the other one
is the government security, which provides risk free return rG. We further
assume that the risky assets are observably identical initially by the way
we model the borrowers, which will ensure that all the risky asset in banks’
balance sheets have the same risk weighting.
Let L be the amount of loan on the bank’s balance sheet, G be the amount
of government securities held by the bank, and E be the original amount
of bank’s capital. We will assume that the bank cannot adjust without cost
the amount of capital when the capital requirement changes. For example,
when the regulator proposes to increase the capital requirement, bank tries
to sell risky loan before adjusting the equity portion of its balance sheet.
This implies that the market for bank capital is not frictionless and we
assume the amount of equity is constant.
The bank funds the portfolio of its assets by issuing deposits, which has
17
an interest rate rD. We assume the amount of deposits is influenced only
by the monetary policy set by the central bank and therefore is out of the
control of individual banks.6 Hence, the liability side of the bank’s balance
sheet is fixed, which implies:
L+G = constant
Furthermore, we can normalise the rate of interest on deposits rD to zero,
and then rL and rG just represent the spreads between the two.
Let δ to be the expected loss given default per unit of L. This is different
from the rate of default of the loan in the sense that δ excludes the amount
of money bank can recover from the collateral in the case of default and
just represent how much money the bank will lose. In this sense, we can say
that (rL − δ)L is the one period return from the risky loan.
Objective Function:
Next we can introduce the bank’s objective function. In this paper, we
focus on the case that the bank’s objective is to maximise the rate of return
on equity, which is assumed to be constant. The return on equity is one of
the main parameters the shareholders are concerned with. As a result, we
set the bank’s objective function as:
(3.1)max v(rL) = (rL−δ)L+rGG
E
= rL−δϕ
+ rGGE
This is the rate of return per unit of equity of the bank, which is the key
measure of the bank’s profitability. This is the main distinguishing feature of
our model compared to the existing literature that focuses on the absolute
value of the bank. Note that L is not equal to the amount of loan the bank
wishes to supply, instead it is the amount the bank actually can lend out
6Relaxing this assumption could be a potential future research area.
18
under some given economic conditions. So L is equal to the smaller of the
two demand or supply.
We assume that the government securities are risk free, and therefore
have a zero weighting when we calculate the risk weighted assets. After the
recent European sovereign debt crisis, we should bear in mind that investing
in the government debt is not necessarily risk free and the government may
default. In this sense, we should assign a default rate to the return on
government securities as well. But this will make the model more complex
and introduce more scenarios in our analysis. For simplicity, we do not
introduce the government default here, but this is a possible area of research
in the future.
4. CAPITAL REQUIREMENT AND BORROWING INTEREST RATE
We consider the effect of a change in capital requirement on the rate of
interest on the loan under two scenarios. First, we consider the case that
when the bank’s capital constraint is binding. This represents the poorly
capitalised banks that have issued the maximum amount of loan. As a com-
parison, we also study the well capitalised banks whose capital constraint
is not binding.
4.1. Capital constraint is binding
In this part, we assume the bank’s capital constraint is binding, which is
a result of the optimal capital structure under the Modigliani-Miller (MM)
propositions. Under MM proposition (with taxes) the valued of the firm is
maximised at 100% of debt and the required rate of return on equity is an
increasing function of the leverage ratio. This implies that an increase in
the equity will actually reduce the required rate of return, but due to the
cost of equity is much higher than the cost of debt, we will assume that
the banks will choose the maximum amount of leverage. Hence, the capital
19
constraint for this type of banks is always binding.
Under this framework, we can show the following proposition.
Proposition 1A: If the capital constraint is binding, the lending interest
rate rL will be an increasing function of ϕ.
Here, we provide two different ways to prove this proposition. The first
one is straight forward, and the second is more robust.
Proof 1:
By definition, a bank’s risk weighted capital ratio is calculated as:
ϕ =E
wLL+ wGG
where wL and wG are the risk weightings for risky loan and government
security, respectively.
To simplify the analysis, we assume the risk weightings for risky loan is 1
and for government security is 0. Then, the risk weighted capital ratio can
be expressed as
ϕ =E
L
Differentiate this with respect to the interest rate rL, we can have
(4.1)∂ϕ
∂rL=∂ϕ
∂L
L
∂rL= − E
L2
∂L
∂rL
We know that both E and L2 are positive. Hence, we just need to deter-
mine the sign of ∂L∂rL
in order to examine the relationship between ϕ and
rL.
We deduce that total amount of lending is a decreasing function of rL
(i.e. ∂L∂rL≤ 0). We know that L is the amount of risky loan, and it is equal
to the smaller of the demand or supply of loan. Also, assuming the normal
demand curve is downward sloping any increase in interest rate will cause
the demand for credit to fall. On the other hand, a decrease in interest rate
does not imply L would increase. This is because the capital constrain is
binding given the assumption and the equity is assumed to be fixed in the
20
short run, so the bank cannot increase the supply of credit. Moreover, by
considering the rate of return (3.1), we know that the bank will not reduce
the supply of credit when the interest rate goes down. If it does reduce
the supply of credit, it needs to reallocate the reduced amount of capital
to government security, which has a lower rate of return. In this case, the
bank would simply supply either the quantity of demanded for credit or the
maximum amount it can lend under the capital constraint depending on
which quantity is smaller. As a result, the total amount of lending L will
remain the same. So we have ∂L∂rL≤ 0.
We can illustrate the above arguement in a standard demand and supply
diagram (Figure 1 on page 35). We have the normal downward sloping
demand curve and the supply curve is perfectly inelastic. The quantity of
supply is zero when the interest rate is less than or equal to δ + rG7, and
the bank cannot change the quantity supplied for any interest rate above
δ + rG. Suppose the demand is equal to supply initially at point A and
L = LA. If the interest rate falls to r1L, the demand will increase to point B,
so L = min(QD, QS) = LA, where QD, QS are the quantity of demand and
supply, respectively. If the interest rate raises to r2L, the demand will fall to
point C, so L = min(QD, QS) = LC . Hence, if the bank charges an interest
rate rL > δ+ rG, then ∂L∂rL≤ 0. If rL < δ+ rG, then the bank will hold only
the government security. But on the other hand, we know the interest rate
rL is set by the bank, which implies that rL will never be set at such a level.
Hence, (4.1) implies that ∂ϕ∂rL≥ 0 for any ϕ > 0. Q.E.D.
An alternative way of obtaining the same result given below.
Proof 2:
7Note that rL < δ + rG could happen either at a very low interest rate or at a very
high interest rate since the loss given default δ will change as interest rate changes. We
refer to the two critical levels of interest rates that equal to δ+rG as ruL and rdL in Figure
1 on page 35.
21
By differentiating (3.1) with respect to rL, and applying the first order
condition, we have
(4.2)∂ϕ
∂rL=
ϕ
rL − δ(1− ∂δ
∂rL− rGE
∂L
∂rLϕ)
Given that rL − δ has to be greater than 0. If this is not the case, the
bank will only hold the government security. We then have
rL > δ
Hence
(4.3) 1 >∂δ
∂rL
This implies that the loss given default rate is changing at a smaller rate
than the interest rate.
Also, ϕ > 0 by definition. Following the same logic in the first proof, we
have ∂L∂rL≤ 0. We consider the case ∂L
∂rL= 0 and ∂L
∂rL< 0 separately.
Case I: ∂L∂rL
= 0
∂ϕ
∂rL=
ϕ
rL − δ(1− ∂δ
∂rL) > 0
Since ∂ϕ∂rL
always has the same sign as ∂rL∂ϕ
, we know ∂rL∂ϕ
for any ϕ > 0.
Hence, the conclusion is any increase in capital requirement will increase
the interest rate on the loan.
Case II: ∂L∂rL
< 0
By reaggranging (4.1), we have
(4.4)∂ϕ
∂rL
> 0 if ϕ >
1− ∂δ∂rL∂L∂rL
ErG
= ϕ∗
< 0 if ϕ <1− ∂δ
∂rL∂L∂rL
ErG
= ϕ∗
22
Since ∂L∂rL≤ 0, and we know that E and rG are both positive number.
Also, from (4.2), we know ∂δ∂rL
< 1, then1− ∂δ
∂rL∂L∂rL
ErG
< 0. We know that the
capital requirement ϕ has to be greater than zero, and therefore we have
ϕ > 0 >1− ∂δ
∂rL∂L∂rL
E
rG
Hence, we can deduce that ∂ϕ∂rL
> 0 for any ϕ > 0, which is the same
result as shown above. Q.E.D.
The second proof is more robust ( ∂ϕ∂rL
> 0 compared to ∂ϕ∂rL≥ 0 the first
one). The result indicates that an increase in capital requirement will cause
interest rate on the bank loan to rise and the bank’s loan portfolio shrinks.
On the other hand, a decrease in capital requirement can has the opposite
effect - the bank to will reduce interest rate on the loan and increase the
size of its loan portfolio.
Proposition 1A implies that when the regulator increases capital require-
ment, poorly capitalised banks 8 will raise the interest rate they charge on
the loan and shrink the risky loan to meet the capital requirement. A conse-
quence of this is that the total lending volume in the market will fall, which
can cause the investment activity and even the whole economy to slow down.
This suggests that if the most of the banks in banking system are poorly
capitalised and the economy is in a recession, the regulator should not in-
crease the capital requirement in such a distressed environment. It should
only raise capital requirement when the economy is in a growth phase. In
summary, a countercyclical capital requirement policy is preferable in the
case of a poorly capitalised banking system.
4.2. Capital constraint is not binding
In this part, we assume the bank’s capital constrain is NOT binding. This
is a more realistic case since the majority of banks (and building societies)
8By poorly capitalised banks, we mean the banks whose capital constraint is binding.
23
do hold capital buffers in practice. Though this type of practice contradict
the Modigliani-Miller’s proposition (with taxes) for capital structure, we
can argue that it is in line with the static trade-off theory, which seeks
to balance the costs of financial distress with the tax shield benefits from
using debt. Under the static trade-off theory, banks will choose the capital
structure that minimise their weighted average cost of capital, which can
be different from one bank to another since the distress costs are different.
In this sense, the banks will not choose the maximum level of leverage and
therefore capital buffer exits. We refer to this as a well capitalised banking
system.
However, if a bank does hold capital buffer, the return on equity will
decrease since the capital buffer doesn’t generate any additional profit. We
can see this from (3.1):
v(rL) ==(rL − δ)L+ rGG
E
Capital buffer implies E will become grater but L and G doesn’t change.
Hence, a lower rate of return. These banks are choosing a suboptimal rate
of return since they a different form of objective function that minimises
the weighted average cost of capital. Nonetheless, our analysis for poorly
capitalised banks is still valid in this case if we assume these banks keep
the proportion of the capital buffer fixed. The constraint for these well
capitalised banks becomes the capital requirement plus their target capital
buffer ratio, and this new constraint is binding initially, so Proposition 1A
is still valid here.
We can obtain the same conclusion here as we had for the case of poorly
capitalised banks. An increase in capital requirement will cause the well
capitalised banks’ lending volume to decrease and therefore have an adverse
effect on the economic activity. As a result, the regulator should not try to
increase the capital requirement when the economy is in a recession. Again,
24
a countercyclical capital requirement policy is preferable.
4.3. Discussion
We should bear in mind that the effect of change in capital requirement
on the interest rate is limited if rL = δ + rG. The above analysis assumed
that the bank will only set the interest rate rL such that rL > δ + rG.
The case rL < δ + rG could happen when either rL is too low or rL is too
high. It is easy to understand when rL is too low the bank will not want to
lend, but not why the bank does not want to lend when the interest rate is
too high? This is because the loss given default δ will raise as rL increases,
which can cause the risk adjusted return to be lower than the return on the
government security. This is in line with SW result that if the bank raises
the interest rate, safe borrowers will withdraw from the credit market earlier
than the risky borrowers. As a result, the bank will set the interest rate in
a reasonable range to make sure the risk adjusted return rL− δ > rG. Let’s
say for any rL > ru and rL < rd, the risk adjusted return from loan will
be lower than the return from the government security (i.e. rL < δ + rG).
Proposition 1A suggests rL is positively correlated with ϕ, so an increase in
capital requirement ϕ will cause the bank to raise interest rate rL. So there
will be a corresponding value of ϕ, say ϕu, that makes rL = ru. The bank
will not adjust its interest rate if the capital requirement is raised above
this upper bound.
In summary, for any
ϕ > ϕu
∂rL∂ϕ
= 0
Hence, we know that an increase in capital requirement will not have any
effect on the interest rate rL.
Similar conclusion can be reached for the lower bound of rL. There is
25
a lower bound ϕd for changing capital requirement can effectively change
interest rate. For any
ϕ < ϕd
∂rL∂ϕ
= 0
Putting this together, we have:
(4.5)∂rL∂ϕ
> 0 if ϕ ∈ [ϕd, ϕu]
= 0 Otherwise
Combining the above analysis with our discussion on well capitalised
banks, Proposition 1A should be restated as
Proposition 1B: If the capital requirement is within certain range [ϕd, ϕu],
the lending interest rate rL will be positively correlated with ϕ. If the cap-
ital requirement is outside this range, changes in capital requirement will
not have any effect on the interest rate rL. This result holds for both poorly
capitalised bank and well capitalised bank.
5. CREDIT RATIONING MAGNITUDE
Recall that there are N borrowers, and each has one and only one in-
vestment project. The numerical order of these N projects represents their
riskiness. i.e. θ ∈ 1, 2, 3, . . . , N , and the bigger the θ, the greater the volatil-
ity of the project return implies greater risk of the project. These borrowers
have identical initial wealth and need to borrow among B from the banks
to start their projects. SW showed that for a given level of interest rate on
the loan, there is a critical value θ∗ such that a borrower borrows from the
bank if and only if θ > θ∗. This critical value will increase if the level of
interest rate increase. (i.e. ∂θ∗
∂rL> 0). As a result, N − θ∗ is the number of
borrowers who apply for loan given an interest rate rL, and B(N − θ∗) is
the aggregate demand for loan in the credit market.
26
Also note that there are M identical banks in the credit market and M is
small. As a result, the banks are operating in an oligopolistic market, and
we just need to analyse the action of a representative bank. Let S be the
amount of loan a typical bank would like to supply given an interest rate rL.
Note that this S is different from the L in the previous section. S represents
the quantity of credit the bank is willing to supply and L is the smaller of
the quantity of supplied (or demanded). Hence, MS is the aggregate supply
of loan in the credit market.
Now, let us define the magnitude of credit rationing as the difference
between demand and supply:
(5.1) Ω = B(N − θ∗)−MS
Recall that SW’s definition of credit rationing is:
Among a group of observationally identical borrowers some will receive
loan and others not. For those borrowers who have been denied loans would
not be able to borrow even if they indicate a willingness to pay more than
the market interest rate or to put up more collateral than that demanded by
the bank.
Our definition of the magnitude of credit rationing is in line with SW’s
definition. Once Ω > 0 the bank needs to randomly reject borrowers from
an observationally identical group of borrowers.
We will investigate how the magnitude of credit rationing changes with
the capital requirement. We only consider the the banks that have capital
constraint binding in this part of the analysis. This is because we assume
that well capitalised banks have capital buffer target and therefore are not
subject to a tighter binding constraint. Hence, these two type of banks will
take the same action as they did before the capital requirement changed.
Based on our definition of the magnitude of credit rationing and Propo-
sition 1B, we can prove another proposition as follows:
27
Proposition 2: Depending on the value of capital requirement, a rise in
the minimum capital requirement can either increase or decrease the mag-
nitude of credit rationing.
Proof: Differentiate (4.3) with respect to ϕ, we can get
(5.2)∂Ω∂ϕ
= ∂(B(N−θ∗))∂θ∗
∂θ∗
∂rL
∂rL∂ϕ−M ∂S
∂ϕ
= −B ∂θ∗
∂rL
∂rL∂ϕ−M ∂S
∂ϕ
SW’s result shows that the critical value of the riskiness of loan applicant
θ∗ is an increasing function of rL. So we have ∂θ∗
∂rL> 0.
From the proof in Proposition 1, we know that the supply curve for credit
is perfectly inelastic, and increase in capital requirement will shift the whole
supply curve to the left. Hence, we have ∂S∂ϕ
< 0. This also can be deduced
from the fact that the bank’s equity is fixed in the short run and the only
way it can increase the risk weighted capital ratio is to reduce the risk
weighted asset, which is the loan portfolio. Hence, an increase in capital
requirement will the supply of reduce credit.
From Proposition 1B, we know ∂rL∂ϕ
> 0 for any ϕ ∈ [ϕd, ϕu], and ∂rL∂ϕ
= 0
otherwise. We consider these two cases separately.
Case I: ∂rL∂ϕ
= 0 If ϕ /∈ [ϕd, ϕu], then ∂rL∂ϕ
= 0. (5.2) becomes:
∂Ω
∂ϕ= −M∂S
∂ϕ
Since ∂S∂ϕ< 0, we have ∂Ω
∂ϕ> 0 in this case.
Case II: ∂rL∂ϕ
> 0
In this case, rearranging the terms in (5.2) to give:
(5.3)∂Ω
∂ϕ
> 0 if ∂rL∂ϕ
<−M ∂S
∂ϕ
B ∂θ∗∂rL
= 0 if ∂rL∂ϕ
=−M ∂S
∂ϕ
B ∂θ∗∂rL
< 0 if ∂rL∂ϕ
>−M ∂S
∂ϕ
B ∂θ∗∂rL
28
Since ∂S∂ϕ< 0, the right hand side of ∂rL
∂ϕ<−M ∂S
∂ϕ
B ∂θ∗∂rL
is positive.
From (4.2), we know that ∂ϕ∂rL
= ϕrL−δ
(1 − ∂δ∂rL− rG
E∂L∂rL
ϕ). Hence, we can
find out the critical value for the minimum capital requirement ratio to
determine the sign of ∂Ω∂ϕ
by solving ϕ from
(5.4)rL − δ
ϕ(1− ∂δ∂rL− rG
E∂L∂rL
ϕ)<−M ∂S
∂ϕ
B ∂θ∗
∂rL
Since ∂rL∂ϕ
> 0, we know 1− ∂δ∂rL− rG
E∂L∂rL
ϕ > 0, the above inequality become
(5.5) f(ϕ) = M∂S
∂ϕ
rGE
∂S
∂rLϕ2 −M∂L
∂ϕ(1− ∂δ
∂rL)ϕ− (rL − δ)B
∂θ∗
∂rL> 0
This is a quadratic function. Since M ∂S∂ϕ
rGE
∂L∂rL
> 0, and f(0) = (rL −δ)(−B) ∂θ
∗
∂rL, which is less than 0 by assumption, we can conclude that the
positive root of f(ϕ) = 0 is the critical value for ϕ in this case. Let’s call this
value ϕ+, but due to the complexity, we do not give an explicitly expression
here. Hence, we have
(5.6)∂Ω
∂ϕ
> 0 if ϕ > ϕ+
= 0 if ϕ = ϕ+
< 0 if 0 < ϕ < ϕ+
We can summarise the two cases together
Scenario I: ϕ+ < ϕd
(5.7)∂Ω
∂ϕ> 0 if ϕ > 0
Scenario II: ϕ+ ∈ [ϕd, ϕu]
(5.8)∂Ω
∂ϕ
> 0 if 0 < ϕ < ϕd
< 0 if ϕd < ϕ < ϕ+
> 0 if ϕ+ < ϕ
29
Scenario III: ϕ+ > ϕu
(5.9)∂Ω
∂ϕ
> 0 if 0 < ϕ < ϕd
< 0 if ϕd < ϕ < ϕu
> 0 if ϕu < ϕ
We ignore the case that ∂Ω∂ϕ
= 0 when ϕ = ϕ+ in the above summary since
this only happens if the change in ϕ is infinitely small around the point ϕ+,
but we know that the change in capital requirement cannot be infinitely
small, so this case has been ruled out. Q.E.D.
We also can present the above result in a graph. See Figure 2 on page
36 for the Scenario I. As ? suggests, we assume the initial interest rate is
below the market clearing level and credit rationing exist. In Figure 2, r0L
is the initial interest rate and L0 is the initial lending volume. The red line
represent the initial magnitude of credit rationing. If the regulator raises
capital requirement, we know that the supply curve of credit will shift to
left and the demand for credit will move along the demand curve due to
the change in interest rate. Suppose the interest rate is increases to r1L and
the supply curve shifts to L1S. This implies that the effect of higher capital
requirement has a greater on the supply than increasing interest rate. We
can see from Figure 2 that the green line represents the new magnitude of
credit rationing, and which is longer than the red line. Hence, the magnitude
of credit rationing increases as the capital requirement increases in this case.
When the capital requirement rises above ϕu (the corresponding interest
rate is ru), we know from the previous discussion that the bank will not
raise the interest rate any further. Instead, the bank will charge an interest
rate marginally below ru, and decrease the supply of credit to a level that
marginally satisfies the capital requirement. We call this L2S in 2. So if the
current capital requirement is above ϕu and is raised by the regulator, the
interest rate will not change, but L2S will shift further to the left. Hence, the
30
magnitude of credit rationing (Orange line) will increase. 9
See Figure 3 on page 37 for the Scenario II. The only difference between
this case and the scenario I is that when the capital requirement rises above
a certain level within the range of [ϕd, ϕu], there is a possibility that the
reduction in the quantity of demand is greater than the decrease in quantity
of supply. We can see from 3 that when the regulator raises the capital
requirement for the first time, interest rate increases to r1L and supply curve
shifts to L1S, so the magnitude of credit rationing (green line) increases. But
if the capital requirement is raised again, the interest rate can increase to
r′L and the supply curve will shift to L
′S in this scenario, and the magnitude
of credit rationing is reduced (black line). When ϕ > ϕu, the effect will be
the same as Scenario I.
Scenario III is presented in Figure 4 on page 38. In this case, any increase
in capital requirement within the range [ϕd, ϕu] will reduce the magnitude of
credit rationing. One explanation for this could be the quantity of demand is
more sensitive to the change in capital requirement than the supply curve.
But if the capital requirement is raised outside the range of [ϕd, ϕu], the
magnitude of credit rationing will be increased.
6. CAPITAL REQUIREMENT AND BANK’S PROFITABILITY
Apart from the effect of change in capital requirement on the magnitude
of credit rationing, the regulator may be also interested in how the bank’s
profitability will change. One could imagine that an increase in capital re-
quirement can reduce the bank’s profitability. If the bank can costlessly
adjust its equity without changing its asset side, then the return on equity
will be reduced since the numerator of the measure stays the same, but
9Note that we only consider the change in magnitude of credit rationing within the
range ϕ ∈ [ϕd, ϕu] and ϕ > ϕu separately. The case that δ increases from the range
[ϕd, ϕu] to some ϕ > ϕu will need to know the exact value of each parameters to evaluate.
31
the denominator increases. However, this will not be the case in our model
since we assume the bank cannot change its equity in the short run, and a
consequence of this is that the bank needs to shrink its risky asset in order
to satisfy the increase in capital requirement. For the measure of the return
on equity, the denominator will not change in this case, and we need to find
how the numerator changes when the bank changes its asset allocation.
We differentiate (3.1) with respect to ϕ to investigate how the bank’s
return would change if the capital requirement change. Some algebra can
lead us to:
(6.1)∂v
∂ϕ=ϕ∂rL∂ϕ− (rL − δ − rG)
ϕ2
From Proposition 1B, we know that ∂rL∂ϕ
> 0 for any ϕ ∈ [ϕd, ϕu], and
∂rL∂ϕ
= 0 otherwise. We consider these two cases separately.
Case I: ∂rL∂ϕ
= 0
This case only happens if ϕ /∈ [ϕd, ϕu], which indicates that the capital
requirement policy is ineffective. Alternatively, we can also say that this
case happens only happens if rL − δ ≤ rG. Under such a scenario, the bank
has two choices. It can either not allocate any portion of its assets into the
risky loan or only charge the interest rate rdL and ruL that corresponds to ϕd
and ϕu. The former case implies the bank does not function as a financial
intermediary, and therefore, its profitability does not affect by the capital
requirement. The later case means rL − δ − rG = 0, so ∂v∂ϕ
= 0.
Case II: ∂rL∂ϕ
> 0
In this case, capital requirement policy is effective.
The denominator of (6.1) is ϕ2 > 0, and therefore we just need to consider
the numerator
(6.2) ϕ∂rL∂ϕ
> rL − δ − rG
From (4.2), we know that ∂ϕ∂rL
= ϕrL−δ
(1− ∂δ∂rL− rG
E∂L∂rL
ϕ). Hence, we have
32
∂v∂ϕ> 0 if and only if
(6.3)rL − δ
1− ∂δ∂rL− rG
E∂LrLϕ> rL − δ − rG
Since ∂rL∂ϕ
> 0, we have 1 − ∂δ∂rL− rG
E∂LrLϕ > 0. Rearranging the terms in
(6.3) can give us that ∂v∂ϕ> 0 if and only if
(6.4) ϕ <1− ∂δ
∂rL− rL−δ
rL−δ−rGrGE
∂L∂rL
= ϕ
Note that ∂L∂rL≤ 0, so the denominator is less than zero. So if 1 − ∂δ
∂rL−
rL−δrL−δ−rG
> 0, then ϕ < 0.
Since the capital requirement has to be greater than zero, we can conclude
that ∂v∂ϕ
< 0 for any ϕ ∈ [ϕd, ϕu]. This implies that an increase in capital
requirement will reduce the bank’s profitability.
If 1 − ∂δ∂rL− rL−δ
rL−δ−rG< 0, there will be a chance that ϕ ∈ [ϕd, ϕu]. As a
result, an increase in capital requirement up to ϕ can increase the bank’s
profitability.
We can summarise the above analysis as follow:
For ϕ /∈ [ϕd, ϕu]
(6.5)∂v
∂ϕ< 0 for anyϕ ∈ [ϕd, ϕu]
For ϕ ∈ [ϕd, ϕu]
(6.6)∂v
∂ϕ
> 0 if ϕd < ϕ < ϕ
< 0 if ϕ < ϕ < ϕu
Hence, we have shown the following proposition
Proposition 3: Risk weighted capital requirement policy only has effect
on bank’s profitability in the region of [ϕd, ϕu]. Inside this region, there is
an optimal risk weighted capital ratio that maximises bank’s profitability.
33
7. CONCLUSION
This paper considers the maximisation problem of market return on
bank’s equity in order to investigate the effect of increase in capital re-
quirement on profitability, lending rate and credit supply. We specify bank’s
asset allocation between safe and risky assets; when the capital requirement
changes, bank can change the composition of its asset allocation. Under
the assumption of constant equity in the short run we derive three results.
Our solution depends on specification of upper and lower bounds of capital
requirement. First, within these bounds, the lending rate is shown to be an
increasing function of capital requirement. We build on Stiglitz and Weiss
(1981) model to study the effect of capital requirement on credit rationing.
Our main contribution is to show quantitatively the magnitude of credit
rationing. Second, following from the first result the magnitude of credit
rationing is shown to depend crucially on the specification of supply and
demand function. This result suggests that the regulator should take the
magnitude of credit rationing into account in setting capital requirement.
In practice the regulator has multiple objectives, of which stability of bank-
ing system has been an overriding consideration in the aftermath of 2007
crisis. However, there is a ample evidence that suggests forced deleveraging,
fired sales of assets and portfolio restructuring, aimed at the sole purpose of
recapitalisation, have resulted in contraction of credit supply to corporates
and households. The timing of setting higher capital requirement is crucial
as banks may not have the option to raise equity in deteriorating economic
conditions. As we have shown, a flexible counter cyclical regulatory cap-
ital requirement is more desirable for accommodating economic activities.
Third, there is an optimal risk weighted capital requirement that maximises
bank’s profitability within the bounds specified. Outside of these specified
bounds, risk weighted capital requirement is shown to be ineffective.
34
APPENDIX A
Figure 1.— Credit Demand and Supply diagram
35
Figure 2.— Credit rationing magnitude Scenario I
36
Figure 3.— Credit rationing magnitude Scenario II
37
Figure 4.— Credit rationing magnitude Scenario III
38
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39
