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Working Paper 9502
THE CHANGING ROLE OF BANKS AND THE
CHANGING VALUE OF DEPOSIT GUARANTEES
by Peter Ritchken, James Thomson,
and Ivilina Popova
Peter Ritchken is a professor of banking and finance
at the Weatherhead School of Management, Case
Western Reserve University; James Thomson is an
assistant vice president and economist at the Federal
Reserve Bank of Cleveland; and Ivilina Popova is a
Ph.D. candidate in the Department of Operations
Research, Case Western Reserve University.
Working papers of the Federal Reserve Bank of
Cleveland are preliminary materials circulated to
stimulate discussion and critical comment. The
views stated herein are those of the authors and
not necessarily those of the Federal Reserve Bank
of Cleveland or of the Board of Governors of the
Federal Reserve System.
May 1995
FEDERAL RESERVE BANK OF CLEVELAND
950506
clevelandfed.org/research/workpaper/1995/wp9502.pdf
ABSTRACT
This article develops a model for pricing deposit guarantees. The model treats the bank's
investments as a portfolio of default-free bonds and risky loans. The risk of the loans is
determined by individual firms' financing and investment decisions. Pushing back risk to
the level of the borrowing firms allows us to link deposit guarantees to specific
characteristics of these loans, such as their durations, and to correlations between
business risk and interest rates. Since the nature of bank loans has been changing over
time, our model should predict the accompanying change in value of the government
guarantees.
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I. Introduction
Traditional models of deposit insurance assume that a bank's assets follow an
exogenously provided stochastic process. If the process is Geometric Wiener, then, as
Merton [I9771 has shown, an isomorphic correspondence exists between loan guarantees
and common-stock put options, and a simple formula exists for deposit insurance.
Merton's analysis has since been extended in a number of ways. Merton [1978], for
example, evaluates the cost of deposit insurance taking into account surveillance costs
and random auditing times. Buser, Chen, and Kane [198:L] and Marcus [I9841 introduce
charter values into the analysis. McCulloch [1981a] and Crouhy and Galai [I9911
consider the implications of interest rate risk, and Ritchken et al. [I9931 consider the case
where the bank adapts its portfolio and capital structure decisions dynamically in order to
exploit the insurance subsidy more fully.'
The objective of this study is to develop models of deposit guarantees that capture
more realism than existing models, thereby permitting a wider range of analyses to be
performed. We extend the literature by modeling bank assets as risky debt issued by
firms.2 The value of this debt is equivalent to a portfolio consisting of a long position in
default-free bonds and a short position in put options on the assets of the bank's loan
customers. The value of the put options, in turn, depends on the investment and
financing decisions of the bank's loan customers. Pushing bank asset risk back to the
level of the borrowing firms allows us to explore several new areas more thoroughly than
have previous models. For example, the current literature has not explicitly focused on
how the types of loans made by banks affect the value of deposit guarantees. There is a
strand of literature that shows how regulatory policies increase the correlation of default
risk across bank portfolios (Penati and Protopapadakis [1988]) as well as across assets in
a given bank's portfolio (Flannery [1989]). However, little attention has been focused on
how the characteristics of bank loan customers affect the value of deposit guarantees.
What we do know is that these characteristics continue to change over time. Boyd and
Gertler [1994], for example, report that the nature of risky loans made by banks has
changed significantly over the last decade, and these changes may have important
consequences for deposit insurance. By pushing back asset risk to the level of the
The literature on deposit insurance is vast. For a review, see Flood [I9901 and Merton and Bodie 119921.
When bank assets are modeled as risky debt, the upside gains are limited to the principal and interest on
the loans. Ignoring this cap on asset-value appreciation may overstate the potential gains from moral hazard
and hence may lead to high-biased estimates of the value of deposit guarantees.
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borrowing firms, we are able to explore specific firm effects on deposit guarantees. In
particular, the value of risky loans, and hence of deposit guarantees, is influenced by the
capital structure of the borrowing firms, by correlation effects between the assets of the
firm and interest rates, by loan duration, and by other borrower-related factors.
Discerning these linkages is crucial for understanding how regulatory policies can affect
credit availability for different types of firms.
The paper proceeds as follows. In section 11, we provide some evidence that the
nature of loans issued by banks has changed over time. This motivates the need for
models of deposit insurance that explicitly capture properties of the risky loans held by
banks. Section I11 develops the basic model for an insured bank that invests in risky loans
and government bonds. Uncertainty is represented by credit and interest rate risk. In
section IV, an explicit model of deposit insurance is provided when no interest rate risk is
present. Deposit guarantees can be viewed as a put option contract on a portfolio of risky
debt and government bonds. Since risky debt itself is modeled as straight debt less a
default premium (captured by a put option), the deposit guarantee is a compound option.
Section IV analyzes this option and identifies how the quality of loans made by the bank
affects the value of the deposit guarantee. Section V generalizes the model when interest
rate risk is present. With two sources of uncertainty, the value of the deposit guarantee
depends on the correlation between credit and interest rate risks and other factors.
Section VI summarizes the paper.
II. The Changing Role of Banks
There is an ongoing debate as to the viability of banks as an industry. Gorton and Rosen
[I9921 find that banking is a declining industry fraught with overcapacity. Boyd and
Gertler [I9941 question the use of traditional measures of intermediation in assessing the
viability of the banking industry. They find that when one accounts for changes in the
types of services banks provide, the industry seems to be thriving in the new, more
competitive financial marketplace.
While the future of banks as intermediaries is far from certain, what is clear is that
the composition of bank portfolios and bank customers is changing. This is illustrated in
figure 1, which shows trends in loan composition since 1988. The percentage of the loan
portfolio invested in commercial and industrial (C&I) loans, once the mainstay of
banking, has declined over time, while the portfolio shares of other types of loans
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(particularly commercial real estate, consumer, and home mortgage loans) have risen.
Furthermore, the composition of bank C&I loan customers has shifted over time from
major corporations to smaller businesses.
Another indication of the change in bank intermediation is the changing maturity
structure of bank assets. Figure 2 shows that the average effective maturity of bank loans
and total earning assets has steadily increased since 1988. While some of this increased
maturity intermehation is a consequence of the asymmetric treatment of credit and
interest rate risks under the Bank for International Settlements7risk-based capital
guidelines (see Li et al. [1995]), the trend signals a fundamental change in the types of
loans banks are making. These changes in the composition and maturity structure of bank
loan portfolios have implications for banking regulation and federal deposit i n s ~ r a n c e . ~
In the following sections, we develop a simple model of an insured bank that contains
elements which reflect the changing nature of the outstanding loans.
III. A Model of an Insured Bank
We assume that the market for default-free bonds is competitive. Banks invest in risky
loans and government bonds. We further assume that the owners of the bank are also its
managers. At date 0, they fund the asset portfolio with a dollars of equity and
D ( O )= 1 - a dollars of deposits fully insured by a government agency. This agency
charges the bank a flat-rate premium per dollar of deposit. The value of deposit insurance
at date 0, denoted by G(O), can be viewed as government-contributed capital. The
insurance provides depositors with full protection over the period [O,T], at which time
they renew their deposits if the bank is solvent. The insurer is assumed to strictly enforce
the closure policy at date T. Specifically, if the market value of the bank's tangible assets
is below the deposit base at this date, the bank is immediately closed.
At date 0, the bank lends q dollars to a representative firm. The firm has ef dollars in
cash and invests A(0) = (q + ef) dollars into a risky project. The firm owes the bank
q*dollars, due at date T. Let B(t,T) be the value of the risky loan at date t. Clearly, at
date 0, q* is determined so that B(0,T) = q.
McCUlloch [1981b] contends that maturity intermediation is a consequence of deposit insurance and not a
natural form of intermediation for depository institutions.
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The balance sheet of the representative borrowing firm is shown below.
Balance Sheet of Representative Firm
Risky investment: A(0)
Loan from bank: B(0) = q
Firm shareholder equity: e
Total:
Total:
A(0)
ef
+q
In addition to providing loans to firms, the bank invests I(0) = 1- q dollars in
government discount bonds with maturity date s. Let P(t, s) be the date t price of a
default-free pure discount bond that pays $1 at date s. Since P(0, s) is the price of a pure
discount bond at date 0, the number of bonds purchased is (1 - q )/ P(0, s).
The bank raises D(0) = 1- a dollars in deposits, with shareholders providing a
dollars. The bank's deposits are guaranteed by the government. The value of the subsidy
arising from these guarantees is G(0). The balance sheet of the bank at date 0 is shown
below.
Balance Sheet of Bank
Risky loan: B(0, T) = q
Default-free loan: I(0) = 1- q
Deposits: D(0) = 1- a
Shareholder equity: e(0)
Government subsidy: G(0)
Total:
1 + G(0)
Total:
(1-a) + e(0)
Shareholder equity at date 0, represented by e(O), is therefore given by
e(0) = a + G(0).
(1)
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There are two sources of uncertainty, namely, the risky investment adopted by the
representative firm and the evolution of the yield curve. The dynamics of the risky
investment are given by
with A(0) = e
+ q . Here, p, (A, t) is the drift term, 0, is the instantaneous volatility,
and dz(t) is the standard Wiener increment.
Bond prices are linked to forward rates by
Here, f(t, x) is the instantaneous forward rate at time t for the time increment [x, x+dx].
Forward rates are assumed to follow a diffusion process of the form
with the forward rate function, f(O,.), initialized to the observed value. Here, p (t,s),
of(t,s), and dw(t) are the drift, the volatility structure, and the Wiener increment,
respectively. We assume that all forward rates are correlated with the asset returns. In
,
= pdt . We follow Heath, Jarrow, and Morton (hereafter
particular, ~ { d w ( t )dz(t)}
HJM) [I9921 and assume that of(t, .) is an exponentially dampened function of the form
where o,DO.
Under this model, HJM show that the price of a bond at date t is related to
its price at date 0 through
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where
They also show that the dynamics of the state variable, r(t), are given by
where, for pricing purposes, the drift can be taken as4
Now consider the value of the risky loan at date t, 0 i t I T. Following Merton [1977], a
risky loan with face value q* is equivalent to default-free debt together with a short
position in a put option on the firm's assets with exercise price q* and expiration date T.
At expiration, if the firm cannot pay q*,it surrenders its assets to the bank. Hence,
where
PE
(t,T;q*) is the value of the put option. For the volatility structure given in
equation (6), Ritchken and Sankarasubramanian [I9911 show that
pE(t,T;q*) = q * N(-d2)-A(t)N(-d
1 ),
where
and
In particular, any European interest rate claim with a cash payment at date s can be priced as
C(0) = ~ ~ [ e x ~ ( ~ ~ r ( x ) d rwhere
) C ( the
~ ) expectation
],
is taken under the process in equation (7).
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If interest rates are certain and the term structure is flat, then the above price of the risky
loan in equation (8) simplifies, with the put price computed using equation (9b) below.
where
The value of the default-free position at date t, I(t), is given by
For certain interest rates, the above equation reduces to
Let V(t) be the value of the firm's "tangible" assets at date t. Specifically,
The duration of the deposit base is assumed to be z. At that date, the level of deposits is
given by
If interest rates are certain and constant, then equation (12a) reduces to
D(z) = (1 - a)en.
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At audit date 2, the bank passes the audit test if the value of these tangible assets exceeds
the deposits.
IV. Pricing Deposit Guarantees under Interest Rate Certainty
Consider a bank with capital a , deposits 1-a, and investment consisting of q dollars in
risky loans of maturity T plus 1-q dollars in default-free bonds. The q dollars are
combined with ef dollars of firm-supplied capital and are invested in a risky project.
Since there is no interest rate uncertainty, the risk premium for the risky debt can be
computed using equations (8) and (9b). In particular, the face value of the debt, q*, is
given by the solution to
where P E (0, T;qt) is given in equation (9b). The value of the government subsidy at date
2, G(z), is given by
Substituting for D(z) and V(z), from equations (lob), (1 I), and (12b) we obtain
where K(a, q; q*) is a constant given by
Equation (15) shows that the government subsidy is a rather complex compound option.
The fair price of the deposit guarantee at date 0, G(O), is given by
where the expectation is taken under the risk-neutralized process given by equation (2),
with the drift term taken as p,(A,t) = rA(t).
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The following proposition establishes the fair value of the deposit guarantee.
Proposition
Consider a bank with capital a, deposits I-a, and investment consisting of q dollars in
risky loans of maturity T and I-q dollars in default-free bonds. The deposits earn the
riskless rate r over the period to the audit date, T, with Z
T . The
I value of the q dollars
is combined with ef dollars offirm-supplied capital and is invested in a risky project,
with volatility
0,. The value of
the government guarantee at date 0 is given by G(O),
where
where
and N , (x, y;p) is the cumulative standard bivariate normal distribution, evaluated at
(x, y ) when the correlation coejjicient is p.
Proofi See appendix.
Figure 3 shows the sensitivity of the government guarantee to the maturity of the risky
debt. Notice that as the duration of the loan increases, so too does the risk and the value
of the government subsidy. For large maturity values over the time to the audit date, the
dynamics of the bond are somewhat similar to the dynamics of an asset. However, for
shorter maturity loans, the dynamics of the bond become more predictable as its value is
drawn toward its face value. Figure 3 also shows the sensitivity of the government
subsidy to the leverage ratio of the representative borrowing firm. As this ratio expands,
the risk of default increases, as does the value of the government subsidy. In an interestrate-certainty environment, the bank will exploit the government guarantee by providing
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the most risky loans. This is accomplished by providing long-term funds to the most
highly leveraged firms.
For the degenerate case, when T=T, equation (17) reduces to
and hence
where
dl = [ln[A(O) I (q - a ) ] + o:z I 21 I o,&
d, =d, --%A
Moreover, in this case the payout to the bank's shareholders at date z is given by
Substituting equations (13b), (1 lb), and (12) into the above expression, we obtain
e(z) =
if A(z) 2 q*
(a-q)e" + q *
Max[O, ( a - q)e"
-t A(z)]
otherwise.
Figure 4 shows the payouts to the bank at date z. Note that the bank's maximum upside
potential is limited to ( a - q)en +q*. This cap stands in contrast to most models of
deposit insurance, which assume that the underlying assets have unlimited upside
potential.
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V. Deposit Guarantees under Interest Rate Uncertainty
Most studies of government guarantees have been developed in a framework where there
is only one source of uncertainty. When multiple sources of uncertainty are considered,
many of these results are overt~rned.~
We now consider a model of deposit guarantees in which there are multiple
sources of uncertainty. In particular, we consider the additional effect on guarantees
when interest rates are stochastic. Clearly, interest rate risk interacts with asset risk,
altering the overall risk exposure and affecting the value of the government subsidy. In
this section, we explore how these two uncertainties affect the value of deposit
guarantees.
Equation (14) gives the value of deposit insurance at date z, G(z), under interest
rate uncertainty. Substituting for D(z) and V(z) using equations (8), (lOa), and (12a)
yields
where
and
PE
(z, T;q*) is given in equation (9a). The fair price of the deposit guarantee for this
bank is given by
where the expectation is taken over the joint risk-neutralized process given by
As an example, in an interest-rate-certain economy, capital regulations (as embodied in the current riskbased standards) and charter regulations are substitute policies. However, when interest rates are uncertain,
Li et al. [I9951 have shown that this result does not hold.
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with A(0) = e,
+ q , and
A bivariate binomial lattice was established to numerically determine the value for G(0).6
Figure 5 shows the value of the government subsidy for loans with different maturities.
Given any correlation coefficient, the graph indicates that the value of the government
guarantee increases with maturity. It can also be seen that deposit guarantee values are
more sensitive to maturity extensions when the correlation between interest rates and
assets moves toward
Figure 6 shows the sensitivity of deposit guarantees to correlation changes for a
fixed-maturity loan. As the correlation moves toward +1, the riskless bonds and loan
portfolio tend to form a natural hedge, reducing variability and decreasing the value of the
government subsidy. If regulators are interested in the escalating costs associated with
the moral hazard issues of deposit guarantees, then their policies should consider the
nature of the loans made by banks and, in particular, the correlation effects between
interest rates and the businesses in which the bank's customers operate.
VI. Conclusion
We provide a model for deposit insurance that considers the bank's financing and
investment decisions. In particular, we assume that the bank invests in a portfolio of
default-free bonds and risky loans. Since the value of the risky loans depends on the
investment decisions of the borrowing firm, the value of the deposit guarantee is
connected to firm characteristics. By pushing back uncertainty to the level of the
borrowing firm, we are able to explore how factors like firm leverage, loan maturity, and
For a discussion of bivariate binomial lattice procedures, see Boyle, Evnine, and Gibbs [1989].
Negative correlation implies that bond prices and loan portfolios are positively correlated.
12
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correlation effects between the assets of the firm and interest rates affect the value of the
deposit guarantee. Future work will look at both the implications of alternative
regulatory systems for deposit insurance and credit availability for bank customers.
Our model has some interesting implications for deposit insurance. First, we
show that the correlation between different risks should be incorporated into any
regulatory mechanisms for deposit guarantees, whether they are explicit (risk-based
premiums) or implicit (regulatory taxes). Indeed, when banks face multiple sources of
risk, regulators need multiple regulatory toolsto minimize innovative risk-shifting
behavior by insured banks. Second,.we show that a consequence of deposit insurance is a
preference by banks to increase the mismatch between the durations of their assets and
liabilities, a phenomenon referred to by McCulloch [1981b] as misintermediation.8 Our
numerical results are consistent with the increased average maturity of bank loans and
earning assets shown in figures 2 and 3.
The fact that bank investments have limited upside potential implies that gains to
shareholders from increased risk-taking are essentially capped. This implies that moral
hazard considerations may be less important than previous analyses suggest.
In our analysis, we assume the existence of a representative firm.In practice,
banks hold a portfolio of loans. While the direction of our results will remain unchanged,
it is important to note that the beneficial role of reducing credit-related risk by
diversifying loans is not captured. It remains for future work to establish models that
assesss the impact of these additional portfolio effects.
This maturity mismatch problem will be accentuated by the current risk-based capital guidelines, which
asymmetrically tax credit and interest rate risks.
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References
Boyd, J., and M. Gertler, 1994, "Are Banks Dead, or Are the Reports Greatly
Exaggerated?" Proceedings from a Conference on Bank Structure and Competition,
Federal Reserve Bank of Chicago (May), 85- 117.
Boyle, P., J. Evnine, and S. Gibbs, 1989, "Numerical Evaluation of Multivariate
Claims," Review of Financial Studies 2, 241-250.
Buser, S. A., A. H. Chen, and E. J. Kane, 1981, "Federal Deposit Insurance, Regulatory
Policy, and Optimal Bank Capital," Journal of Finance 36,51-60.
Crouhy, M., and D. Galai, 1991, "A Contingent Claim Analysis of a Regulated
Depository Institution," Journal of Banking and Finance 15 (January), 73-90.
Flannery, M. J., 1989, "Capital Regulation and Insured Banks' Choice of Individual Loan
Default Risks," Journal of Monetary Economics 24,235-258.
Rood, M., 1990, "On the Use of Option Pricing Models to Analyze Deposit Insurance,"
Federal Reserve Bank of St. Louis, Review 72 (January/February), 19-35.
Gorton, G., and R. Rosen, 1992, "Corporate Control, Portfolio Choice, and the Decline of
Baking," Board of Governors of the Federal Reserve System, Finance and Economics
Discussion Series No. 2 15 (December).
Heath, D., R. Jarrow, and A. Morton, 1992, "Bond Pricing and the Term Structure of
Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrics 60,
77-105.
Li, A., P. Ritchken, L. Sankarasubramanian, and J. B. Thomson, 1995, "Regulatory
Taxes, Investment, and Financing Decisions for Insured Banks," Proceedings from a
k
and Competition, Federal Reserve Bank of Chicago
Conference on ~ a n Structure
(May), forthcoming.
Marcus, A., 1984, "Deregulation and Bank Financial Policy," Journal of Banking and
Finance 8 (December), 557-565.
McCulloch, J. H., 1981a, "Interest-Rate Risk and Capital Adequacy for Traditional Banks
and Financial Intermediaries," in Sherman Maisel, ed., Risk and Capital Adequacy in
Commercial Banks. University of Chicago Press and National Bureau of Economic
Research, Chicago.
McCulloch, J. H., 1981b, "Misintermediation and Macroeconomic Fluctuations,"
Journal of Monetary Economics 8 (July), 103-115.
clevelandfed.org/research/workpaper/1995/wp9502.pdf
Merton, R., 1977, "An Analytic Derivation of the Cost of Deposit Insurance and Loan
Guarantees: An Application of Modern Option Pricing Theory," Journal of Banking and
Finance 1 (June), 3- 11.
Merton, R., 1978, "On the Cost of Deposit Insurance When There are Surveillance
Costs," Journal of Business 5,439-452.
Merton, R. C., and 2. Bodie, 1992, "On the Management of Financial Guarantees,"
Financial Management 2 1 (Winter), 87- 109.
Penati, A., and A. Protopapadakis, 1988, "The Effect of Implicit Deposit Insurance on
Banks' Portfolio Choices with an Application to International 'Overexposure'," Journal of
Monetary Economics 21 (January), 107-126.
Ritchken, P., and L. Sankarasubrarpanian, 1991, "On Contingent Claim Valuation in a
Stochastic Interest Rate Economy," Case Western Reserve University, Technical
Memorandum.
Ritchken, P., J. B. Thomson, R. P. DeGennaro, and A. Li, 1993, "On Flexibility, Capital
Structure, and Investment Decisions for the Insured Bank," Journal of Banking and
Finance 17, 1133-1146.
Toft, B., 1994, "Exact Formulas for Expected Hedging Error and Transaction Costs in
Option Replication," University of California at Berkeley, Working Paper.
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Appendix
Lemma (Toft [1994])
Let N (0) and N , (o,., 0) denote the univariate and bivariate standard normal distribution
functions. Then,
Proof of Proposition
At date z, the value of the government subsidy is given by
We also have that
B(T,T) = q*e-r'T-" - pE[T,T, q*]
I(2) = (1 - q)en
~ ( 2=) (1 - q)en + q *e-r(T-7) ,
pE[T,T, q*]
D(z) = (1 - a)em.
Substituting for D(z) and V(z), the time z value of the government subsidy is given by
If V(z) < D(z), then the value of the government subsidy is
G(z) = (q - a)e" - q*e-r'T7'
+ pE(r,T;q*).
Now,
Pr{G(T) > 0) = Pr{D(r) > V(z)} = pripE (T,T, q * ) > H(T, T)),
where
H ( 2, T) = *e-r'T-7)- (q - a )e n .
The inequality for the put option will be satisfied when the underlying asset at date t ,
A(t), drops below some level A*. Hence, the sought probability is equal to the
) A*(z)), where the value A*(z) is obtained by solving
probability of the event { ~ ( z <
the equation pE(2, T, q*) = H(T,T) ,provided that H(T,T) is posihve.
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To solve this equation, we first need the value of q*, which is a solution to
where
Then, A* is a solution to
where
z =
d2 =
ln A*(~)/A(o)- p, 7
J?
0,
ln A*(s)/q* + (r - 0:/ 2 ) ( ~- 7)
0,JF7
Hence, Pr{G(z) > 0) = Pr{A(z) < A*(7)) = Pr{z < z*).
In summary, then,
Pr{G(z) > 0) =
if H(z,T) I 0
if H(z, 7')> A(z)
Pr{A(z)<A*(z)) otherwise.
The value of the government subsidy at date 0 is
where the expectation is taken under the risk-neutralized process. After substituting for
G(z), we obtain:
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~ ( O ) = e - ~ ' [ ( g - a )n
e -q *e -r(T-z)]N(Z*)+e-rz J q*e-r(T-z) N(-d2) f (s)& ~ ( r<A*
) (r)
e -rz J A(r)N(-d, If ( s ) d
A(r) <A* ( r )
Now, using the Lemma to solve the three integrals and simplifying the resulting
expressions leads to the final result.
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OT-
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Figure 2
EFFECTIVE ASSET MATURITY
Years (Assets)
1.90 1
Years (Loans)
1 1.40
Figure 2 shows the lengthening effective maturlty of bank loans and total bank assets. The slight downturn in total
asset maturity after 1992 corresponds to a shortening of the maturlty of bank security portfolios in reactlon to
changes in accounting rules that forced banks to hold a higher percentage of their security portfollos at the lesser
of book or market value. Effective maturity Is computed using the maturity/repricing breakdowns reported on the
Federal Financial Examination Councii's Reports of Condition and income. For both serles, the total dollar amount
of assets In each maturity/repricing bucket is weighted by the midpoint of the maturity range (except for the
greater-than-five-year bucket, which is given a weight of 5).
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Figure 3
Sensitivity of the Government Guarantee to the
Maturity of the Risky Debt
0.025 0.02 --
ef = 0.2
.. .. .. .... ~1
= 0.3
.,: ....
CJ = 0.4
I
1
3
5
7
9
11
13
15
17
19
Maturity, T
Figure 3 shows the sensitivity of the value of the government guarantee, G(O), to
extensions in the maturity of the risky loan. Each curve corresponds to a firm with a given
,
the risk expands, and
leverage. As the leverage increases, shareholder equity, e ~decreases,
the government guarantee becomes more valuable.
Source: Authors.
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Figure 4
Bank Shareholder Value as a Function of the value of the Representative Firm
Figure 4 shows the terminal payoff to shareholders for the case q>a. In ths case,
shareholder equity is zero if the value of the asset A(T) falls below (q-a)exp(rt). The
maximum value for shareholders occurs when the face value of the loan is paid out. For
all values of the firm larger than q*, the shareholder value is capped.
Source: Authors.
clevelandfed.org/research/workpaper/1995/wp9502.pdf
Value of the Deposit Guarantee for Different Values
of the Correlation between the Risky Asset a n d
Interest Rates
Q)
1.51 -0 a 0.5 -0 1
0
0 -,
,
I
2
3
4
5
6
7
8
9
"Q)
"o
a!!
.
a
.
10
Maturity, T
-Correlation = -0.9 -Correlation = 0.0 -Correlation = 0.9
Figure 5 plots the value of the government guarantee against loan duration for different
correlation coefficients between interest rates and the representative h ' s assets. As the
correlation increases, the default-free bond portfolio and the loan become natural hedges,
decreasing the total price variabiity and hence the value of the deposit guarantee. For any
correlation, the value of the guarantee increases with the duration of the loan. However,
the rate of increase is enhanced when the h ' s assets are most highly correlated with
bond returns. The case parameters are as follows: The leverage of the representative firm
was computed from ef = 0.10. The default-free investments were in s = two-year bonds,
with q = 0.8. The volatility structure of forward rates was given by equation (5), with
o = 0.02 and K = 0.02. The volatility of the assets was o, = 0.20.
Source: Authors.
clevelandfed.org/research/workpaper/1995/wp9502.pdf
Figure 6
Sensitivity of the Government Guarantee to the Correlation
between the Risky Project and Interest Rates
0.0035 0.003 -0.0025 -s 0.002 -8 0.0015 -0.001 -0.0005 -0
-0.9
A
I
-0.7
I
-0.5
I
-0.3
I
I
I
I
I
I
-0.1
0.1
0.3
0.5
0.7
0.9
Correlation
Figure 6 shows the effect of increasing the correlation between interest rates and the risky
asset. The case parameters are as follows: The leverage of the representative firm was
computed from ef = 0.10. The default-free investments were in s = two-year bonds, with
q = 0.8. The volatility structure of forward rates was given by equation (5), with o = 0.02
and K = 0.02. The volatility of the assets was o, = 0.20.
Source: Authors.
@FedFRASER