Finance and Banking

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Finance and Banking

• NAKE Course
• Lecture 4 Banking and Monetary Transmission

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Previous lectures have shown that..

• Loans are special as opposed to bonds private
  and public assets differ in nature
• Informational problems can lead to agency costs
  and so induce a (higher) External Finance
  Premiums
• Net worth of borrowers is an important variable
  in attracting new finance
• Monetary transmission depends not only on the
  interest rate channel but also on the bank
  lending and balance sheet channel

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This lecture

• We turn to behavioral models of banks themselves
  concern for the supply of loans
• How do banks finance themselves? Is public
  funding easy for small banks? How does the
  availability of funding affect lending?
• If there is relevance of the financing structure
  of banks, and how does this finding affect
  monetary policy? Is MM relevant to banks?
• We first follow Stein (Rand Journal, 1998), next
  we turn to Kashyap-Stein (2000) and conclude with
  the bank capital channel by Van den Heuvel (2002)

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General notions (1)

• Banks attract deposits (insurable assets) and
  non-deposits (debt and equity). Moreover they
  generate equity capital via retained profits.
  Romer-Romer (1990) argue that the financial
  structure is indifferent (Modigliani-Miller for
  banks). In that case lending is not affected by
  reductions in reservable assets
• On the other hand debt might be a problem see
  Stein (1998) and Kashyap-Stein (1995, 2000).
  Lending might be reduced after a reduction of
  deposit finance
• Equity might be a problem dividends need to be
  positive Van den Heuvel (debt has a tax advance
  over equity. Excessive equity accumulation is too
  expensive (but required in order to face capital
  restrictions)

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General notions (2)

• Deposit (or insured) finance is influenced by
  monetary policy. A drop in deposit finance will
  affect bank balance sheet decisions
• Choice between reducing private assets (loans) or
  public assets (securities) or attract more
  noninsured liabilities (bonds)
• Informational costs might affect these decisions.
  Informational costs depend e.g. on bank size
• What is the main informational problem?
  Opaqueness of the value of bank assets (including
  capital). This opaqueness can lead to adverse
  selction

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Stein (1998)

• How do banks allocate their assets among loans
  and securities and how do these allocations
  respond to shocks in the availability of insured
  deposit finance?
• How do nonfinancial firms choose between bank and
  nonbank sources of debt finance?
• How does monetary transmission work, assuming
  that the central bank can influence both the
  interest rate level and the EFP?

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Stein (1)

• Adverse selection model of bank asset and
  liability management. Bank management knows more
  about the value of the banks assets than private
  investors do
• If the bank is able to finance itself fully with
  insured deposits, there will be no problems
  (lending is undistorted), but with uninsured
  finance there are informational difficulties
  (adverse selection in Stein)
• Stein gives a model of bank portfolio choice, the
  choice between bank- and non-bank debt, and
  implications for monetary policy
• Main result banks with a higher potential to a
  better future ultimate capital position are more
  prone to a loss of insured reserves and will lend
  relatively less. The intuition is that bad
  banks have nothing to loose and simply keep on
  lending out. Banks with more informational
  problems tend to be more sensitive to shocks

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Partial equilibrium bank model (1)

• First one-period version of the model
• Bank assets reserves R, new loans L, old assets
  K. Liabilities insured deposits M, previously
  raised non-deposit finance P, and incremental
  non-deposit finance E. It is easy to assume that
  K P. R ? ?M, where ? is the fractional reserve
  requirement on insured deposits
• L R M E, or L ? M(1 - ?) E
• Banks are assumed to be monopolists in the loan
  market and face a loan demand LD a br, where
  r is the loan rate (spread between loans and
  securities). We assume the expected return on M
  and E to be fixed to 0

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Partial equilibrium bank model (2)

• Asymmetric information about the value of old
  assets K. Good banks (G) hold capital that will
  ultimately be worth KH gt KL and bad banks (B)
  KL. Define a measure of information asymmetry A
  1 KL / KH. A higher A indicates more
  informational problems, e.g. a small bank
• Suppose M falls exogenously
• There will be a so-called unique separating
  equilibrium the good and the bad are separated
  in equilibrium through the design of the
  contract. Main intuition the bad case proceeds
  as ever, the good case adjusts
• Type B maximizes interest income rL, so LB a/2
  (from the profit maximizing interest rate r
  a/2b) and must raise EB max(0, a/2 - M(1- ?))

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Partial equilibrium bank model (3)

• Type G raises less external finance and lends
  less LG LB - Z. Z is underlending. It raises
  less external finance EG EB Z. Type Gs
  equity is worth more
• Profit of G is ?G (a/2 Z)(a/(2b) Z/b)
  ?B Z2/b
• Type B can try to mimic G. Profits will then
  fall ?G ?B Z2/b. But it can sell overpriced
  equity. This gain is equal to AEG. So Z2/b AEG.
  This describes a non-mimicking equilibrium.
  Remember A 1 KL / KH

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Partial equilibrium bank model (4)

• How big is Z?
• Z2/b AEG A(EB - Z), so
  Z2 bAZ - bA EB 0, which is a quadratic
  function in Z
• So type B will use uninsured external finance
  fully and leave lending unchanged. Type G are
  reluctant to use external finance

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Partial equilibrium bank model (3)

• So if deposits are abundant, both types will lend
  at the first-best levels
• But if deposits fall short, bad banks will use
  external finance fully. Good banks will be
  reluctant to do so, and finance a fraction of the
  shortfall and so reduce lending
• So dLG/dM gt 0 and d2LG/dMdA gt 0. Good banks are
  more sensitive to a larger degree of information
  asymmetry. So small but good banks are sensitive
  to deposit shocks

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Kashyap-Stein (Carnegie-Rochester, 1995) (1)

• Banks face an adverse selection problem in the
  market for uninsured external finance
• Two period model a simple way to model
  precautionary or buffer stock motives to hold
  securities in the first period
• Loans cannot be liquidated at time 2 (and no new
  lending opportunities arise at time 2).
  Securities (no return, normalized to 0) can
  costlessly be liquidated in period 2 and so give
  a liquidity yield
• Yield on loans is the loans-securities spread

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Kashyap-Stein (2)

• Again a partial equilibrium model of bank
  portfolio behavior
• Insured deposit supply (M1 at time 1 and M2 at
  time 2) is completely out of control for banks
  (and determined by the central bank)
• Three possibilities for the bank in case of a
  contraction of M1 (1) reduce supply of loans Ls,
  (2) raise non-deposit finance E (we denote the
  incremental amounts by E1 and E2), or (3) sell
  securities S

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Simple balance sheet of banks

• Loans Ls
• Securities S

• Deposits M
• Non-deposits E

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Stochastics of deposits M

• M is stochastic a long-term unconditional mean
  M, persistence ? and uniform distribution with a
  variance ?
• Given M1, M2 is uniformly distributed on the
  interval
  ?M1 (1 - ?)m - ?/2,
  ?M1 (1 ?)m ?/2
• EM2 ?M1 (1 - ?)m
• VarM2 ?²/12, since the variance of a variable
  that is uniformly distributed on the interval
  a,b is equal to (b - a)²/12

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Non-deposits E

• At time 2 the total amount of non-deposit finance
  is E1 E2. E1 can be raised at increasing
  marginal costs ?1E1²/2, ?1 needs to be positive
  in order to have a lending channel E2 has costs
  ?2E2²/2
• r is the interest rate on loans. r - rE must be
  positive. Otherwise there is no lending channel.
  We assume rE 0
• Lemons premium makes that ?1 and ?2 are probably
  larger for smaller banks
• For ?i 0 (i 1,2) we have Modigliani-Miller
• We could use adverse selection models (like
  Stein, 1998) or costly state verification models
  (Froot, Scharfstein, Stein, 1993) to get
  quadratic cost functions as well

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Securities

• Banks use securities S as a buffer in order to
  prevent the inefficient liquidation of loans
• Loans are more profitable r gt rS (rS 0 for
  convenience)
• If E1 M2 gt L there is no problem and E2 0
• If E1 M2 lt L banks have to attract
  E2 L - E1 - M2
• So E2 max(0, L - E1 - M2)

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Net revenue for banks

• Maximize expected profits on t 1
  rL - ?1E1²/2 - E?2E2²/2 with respect to
  E1, L, S
• E?2E2²/2 ?2(L - E1 - ?M1 - (1 - ?)m ?
  /2)²/6 (see the proof on the next slide)
• E1 r / ?1 the marginal costs of obtaing
  additional funds return
• L r / ?1 3r / ?2 ?M1 (1 - ?)m - ?/2
  if the ?-parameters become small
  loan supply becomes more elastic more
  uncertainty (a higher ?) reduces supply
• S M1 E1 - L -3r / ?2 (1 - ?)(M1 - m) ?/2

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Derivation of the expected costs

• E?2E2²/2 ?2(L - E1 - ?M1 - (1-?)m ?/2)²/6
• E2 max0, L - E1 - M2
• Use M2 ?M1 (1 - ?)m - ?/2
• We are interested in the properties of
  L - E1 - ?M1 - (1 - ?)m ?/2 on a uniform
  interval
• EE2 (L - E1 - ?M1 - (1 - ?)m ?/2)/2 and
  varE2 (L - E1 - ?M1 - (1 - ?)m ?/2)²/12
• Use EX² varX EX²

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Results

• Non-deposit sources E1 depends positively on r
  but negatively on the costs ?1. Large banks
  probably attract more non-deposits
• Loan supply depends positively on the available
  amount of deposits M1 and the spread r, but
  negatively on ?1 and ?2. Large banks respond more
  to the spread.
• Securities depend negatively on the spread r and
  positively on the costs

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Equilibrium with a competitive homogeneous loan
market

• Loan demand Ld depends on general conditions Y
  and the interest rate r Ld Y kr
• We assume that a change in monetary stance has an
  identical impact on bank i and j dM1i dM1j
• Differentiate the equations we derived before wrt
  M1
• dLi / dM1 (1/?1i 3/?2i) dr / dM1 ?
• dSi / dM1 -3/?2i dr / dM1 (1-?)
• dr / dM1 a dY / dM1 b, with a and b positive.
  If dY / dM1 is small, dr / dM1 lt
  0 the lending channel
• Things become more complicated with heterogeneous
  loan demand

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Testable hypotheses

• dLi / dM1 (1/?1i 3/?2i) dr / dM1 ?
• dSi / dM1 -3/?2i dr / dM1 (1 - ?)
• Smaller banks have larger ?s. So
• Proposition 1 The lending volume of small banks
  falls more rapidly after a given contraction in
  deposits (due to higher costs of adjustment)
• Proposition 2 The securities holdings of small
  banks falls more slowly after a contraction in
  deposits (small banks value securities more at
  the margin and are less willing to reduce them)

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Is it credit supply or demand?

• Is it so that after a restriction of monetary
  policy the quantity of credit decreases through a
  decrease in demand or supply?
• Evidence on the macro level if the interest
  spread increases it is probably a fall of supply
• Evidence on the micro level if small banks react
  more strongly than large banks it would be a
  large coincidence

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Four Kashyap-Stein hypotheses

• d²L / dM dSize lt 0 smaller banks are more
  sensitive to a reduction in deposits dL / dM
• d²S / dM dSize lt 0 small banks are more
  sensitive in adjusting securities dS / dM
• d²L / dS dM lt 0 monetary policy has more effect
  on lending behavior of banks with less liquid
  balance sheets
• d³L / dS dM dSize lt 0 for large banks financial
  structure is less important

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Stein (1998, part 2) (1)

• A two-period adverse selection model
• A bank holds about 30-40 per cent of its assets
  in liquidity and securities in Stein, part 1, we
  abstracted from liquidity in securities
• The bank faces an adverse selection problem in
  the market for uninsured external finance in each
  of the two periods. It might buffer using
  liquidity
• Securities held at time 1 can costlessly be
  liquidated at time 2. It is costly to liquidate
  loans an amount J (jettison, Dutch overboord
  gooien) costs ?J2

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Stein (part 2) (2)

• Banks fund themselves with insured deposits M1
  and M2 and common equity E
• Stochastic structure of deposits is similar to
  the above one Given M1, M2 is uniformly
  distributed on the interval
• ?M1 (1 - ?)M - ?/2, ?M1 (1 - ?)m ?/2
• EM2 ?M1(1 - ?)m. m is the unconditional
  mean, ? defines the persistence of shocks
• VarM2 ?²/12, so ? is the variance measure

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Stein (part 2) (3)

• E1 and E2 are the incremental amounts of external
  finance raised.
• r is the return on loans, all other returns are
  equal to 0
• Balance sheet restrictions
• L S ? M1(1 - ?) E1 at time 1
• L J ? M2(1 - ?) E1 E2 at time 2

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Stein (part 2) (4)

• There is asymmetric information about the
  ultimate value of the old assets K. The value of
  K develops to a binomial process
• K0 is unconditional value. After time1 a public
  signal arrives. If the signal is good, with
  probability p, the value of the assets rises to
  uK0 with u gt 1. If the signal is bad (with (1 -
  p)) to dK0 with d lt 1. Outside investors observe
  the signal after period 1, but bank managers know
  in advance, when making lending and financing
  decisions at period 1. At time 2 a second public
  signal arrives (maybe to d2K0 or u2K0). Define A
  1 - d/u. A larger A again indicates more
  informational problems

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Stein (part 2) (5)

• Type G bank any bank whose private information
  (either at time 1 or 2) leads it to expect an
  increase in the value of K when the next public
  signal is released (and a type B bank the other
  way round)
• Define SF(M1) by that value of securities
  holdings that is sufficient to insulate a bank
  (not necessary to cut loans or raise new
  securities funding) SF(M1) (1- ?)(M1 - m)(1 -
  ?) ?/2

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Stein Lemma 1

• A type B bank will hold SB ? SF(M1) and so will
  lend at the first best. Type B will raise an
  amount of external finance at time 1 E1B
  max(a/2-(1- ?)?M1 (1 - ?)m - ?/2,0)
• Illustration of the proof hereafter

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Stein Lemma 2

• Suppose that E1B ? 0. Then a bank type G at time
  1 will have SG lt SF(M1). So bank G will not fully
  insulate loan supply
• Bank G will also lend less than bank B
• So bank G will cut lending if M decreases
• The link between M and LG is stringer when the
  information asymmetry A increases

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Solving the two-period model (1)

• The model can be solved backwards (start at time
  2). Remember we have good (G) and bad (B) banks
• We again look for a separating equilibrium
• Type B will raise at time 2 EB max(0,L - E1 -
  M2(1 - ?)), just enough not to liquidate loans
• Type G will raise at time 2 E2G E2B J
• The incentive constraint is, like before, ?J2
  AE2G, from which we can solve for J
• When do we liquidate (as expected at time 1)? If
  short L - E1 (1- ?)(?M1 (1 - ?)m - ?/2) lt
  0
• Expected costs of liquidation X p C(short),
  where p is the probability of an up move (and so
  being a G) and C(.) a convex cost function (with
  C(0) 0)

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Solving the two-period model (2)

• Now we turn to period 1
• Type B lends LB a/2 (remember loan demand LD
  a - br). Type B just maximizes lending profit.
  Type B will also raise enough external funds at
  time 1 so that it never has to raise further
  external funds at time 2. This amount is
• SF(M1) (1- ?)(M1 - m)(1 - ?) ?/2. A bank
  that is type B at time 1 will hold SB SF(M1)
• We have LB SB ? M1(1 - ?) E1B at time 1
• a/2 (1- ?)(M1 - m)(1 - ?) ?/2 ? M1(1 - ?)
  E1B
• So E1B max(a/2 - (1- ?)?M1 (1 - ?)m -
  ?/2,0) and XB 0 (XB the expected
  costs of liquidation for bank B)

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The consequences

• If E1B gt 0, a G-type bank in period 1 will have
  SG lt SF(M1), so hold less than full-insurance and
  lend less than the first-best a/2. Securities
  holdings are discouraged by adverse selection in
  the time 1 market
• If E1B gt 0, type G will react to a decrease in M1
  by reducing lending and d2LG/dM1dA gt 0 if loan
  demand is rather inelastic
• So, again informational asymmetries determine
  both lending and financing behavior

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The Bank Capital Channel

• Traditional monetary theories focus on the role
  of reserves in determining the volume of demand
  deposits. The bank lending channel analyzes the
  role of reserves in loan supply
• Van den Heuvel (2002) discusses the role of (an
  imperfect market for) bank equity. He assumes a
  perfect market for debt
• Lending will depend on the financial structure of
  the bank. The model also focuses on the maturity
  mismatch of assets and liabilities
• Conclusion lending by banks with low capital has
  a delayed, but amplified, reaction to interest
  rate shocks

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Heuvel (1)

• The model includes capital adequacy regulations.
  The Basle accord established minimum capital
  requirements
• The bank cannot readily issue new equity. But
  equity capital is endogenous through retained
  earnings
• There is a tax advantage on debt (earnings are
  taxed at a rate ? gt 0)
• Banks perform maturity transformation, leading to
  a serious interest rate risk. Banks are
  leveraged (compared to real firms), such that an
  interest rate increase results in a larger
  percentage change in equity (given that there is
  a profit squeeze)

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Heuvel (2)

• Assets loans L and securities S liabilities
  debt B and equity E. A L S B E
• The bank issues Nt new loans at the beginning of
  each period. Each period a fraction ?u lt 1 is
  due. 1/ ?u is the loan portfolios average
  maturity. ?t1 is the fraction of loans actually
  repaid at the end of t (and can deviate from the
  due rate)
• Loans are risky a fraction ?t1 of outstanding
  loans Lt goes bad as well as a fraction bt1 of
  the new loans Nt
• Lt1 (1 - ?t1 - ?t1) Lt (1 - ?t1 bt1)Nt

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Heuvel (3)

• Downward sloping demand curve for loans some
  market power on part of the bank (due to whatever
  reason)
• Contractual interest rate on new loans is a
  decreasing function of the amount of new loans
  made Nt ?(Nt) and ?(.) lt 0
• The average contractual interest rate on all
  outstanding loans in period t, ?at is given by
• ?at1 (1-?t1-?t1)Lt ?at(1-?t1bt1)Nt
  ?(Nt) /Lt1
• bt1 b(Nt, ?t1) and b(.,.) increases in N

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Heuvel (4)

• Banks debt B is all short. Loans have a maturity
  longer than one, so there is a maturity mismatch
• The bank can borrow liquidity freely at a rate rt
• Only B - S can be determined, so tradable
  securities S 0 is assumed
• No role for reservable or demand deposits (a bank
  can borrow/lend freely at the riskless rate r)
• Banks have perfect access to insured and
  nonreversable deposits contracts (and this is not
  the case in the basic bank lending channel models)

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Heuvel (5)

• Accounting value of equity Et Lt - Bt
• Equity evolves according to Et1 Et - Dt (1
  - ?)?t1 Dt dividends paid out and ?t1
  pre-tax accounting profits interest income on
  loans minus interest paid on banks debt net
  charge-offs on loans
• ?t1 ?at(1 - ?t1)Lt (1 - bt1)Nt ?(Nt)
  rt(Lt Et Nt Dt) - (?t1Lt bt1Nt)
  ?F
• ?F represents othe profit components

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Heuvel (6) debt

• Law of motion of debt. Note that debt has a tax
  advantage
• Bt1 (1(1-?)rt)(Bt Nt Dt) - ?t1(Lt Nt )
  (1- ?)(?at(1-?t1)Lt (1-bt1)Nt ?(Nt) ?F) -
  ?(?t1Lt bt1Nt )
• Interest costs are discounted by (1 - ?), ?t1 is
  the fraction of loans actually repaid at the end
  of t, net of tax interest receipts and the tax
  shield from charge-offs on loans

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Heuvel (7) bank balance sheet (including new
loans and dividend)

• Assets
• Loans Lt Nt
• Securities St ?St
• Total At Nt ?St

• Liabilities
• Debt Bt Nt ?St Dt
• Equity Et - Dt
• Total At Nt ?St

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Heuvel (8) capital regulation

• Basle loans are in the highest risk category,
  mortgages and residential property have a risk
  weight of 50 and riskless securities in 0.
  Basle sets ?, the regulatory minimum, equal to
  0.04 for tier 1 capital and 0.08 for tier 2
• The risk adjusted capital asset ratio RACAR at
  the beginning of the period RACARt Et / Lt
• If Et gt ?Lt then Et - Dt ? ?(Lt Nt)
• If Et lt ?Lt then Nt 0 and Dt 0 no new loans
  and no dividends

45
Heuvel (9) financial constraint

• The bank cannot issue new equity, this implies
  that dividends must be nonnegative Dt ? 0
• Shareholders do not price the risk of bank shares
  (risk neutral valuation)
• Market value of bank equity is
• Vt max Et?s0?( ?u0s-1(1 rtu)-1)Dts
• So the problem is to maximize the market value of
  the bank subject to the constraints listed above.
  The bank assumes the interest rate rt, the
  repayment rate ?t and the fraction ?t of
  outstanding loans Lt that goes bad as given

46
Heuvel (10) optimization

• State variables st (Et, Lt, ?at, rt). Control
  variables Dt and Nt
• Bellman equation V(st) maxDt 1/(1 rt) Et
  V(st1)
• Without the financial constraint we have VU. The
  lending decision is then independent of the
  financial position of the bank. The optimal
  policy functions of the unconstrained problem are
  NU(st) and DU(st)
• NU(st) is NU(rt) only the interest rate matters
  MM! Optimal financial policy is to keep the risk
  adjusted capital ratio exactly to the minimum
  DU(st) Et - ?(Lt
  NU(rt))
• Since the bank can always raise capital there is
  no need to hold a costly precautionary buffer of
  capital above the regulatory minimum

47
Heuvel (11) constrained

• A constrained bank can lend at the unconstrained
  level NU(rt) if it starts its period with
  sufficient capital Et ? ?(Lt
  NU(rt))
• Holding the additional equity is costly (debt is
  cheaper due to taxation), but will lead to a
  smaller likelihood that the financial constraint
  will bind. The bank should have Et-1 - Dt-1
  sufficiently high to ensure that Et ? ?(Lt
  NU(rt)) for all realizations of rt, ?t, and ?t
• There is a tradeoff between holding more equity
  against the tax disadvantage
• The optimal trade-off involves a positive
  probability of being financially constrained (see
  Heuvel for a proof)
• It is not the case that financially constrained
  banks will always lend less N(st) ? NU(rt)

48
Summary

• This lecture provides basic models of bank
  behavior under constraints
• Constraints lead to relevance of financial
  structure and more complicated impact of monetary
  policy
• Constraints differ in nature for instance the
  inability to raise new equity or the costs
  involved to liquidate loans