Comparing Insurance Costs for
Time and Demand Deposits
Gongmeng Chen
Department of Accountancy, The Hong, Kong Polytechnic University
Mandy Li
School of Business and Administration, Open University of Hong Kong
This study analyzes the differential risks imposed by time deposits and demand deposits
on the deposit insurer. By modeling deposit insurance as an exchange option, a general
model is developed that is applicable to the pricing of insurance for both types of
deposits.
In this model, time deposits are viewed as a special case of demand deposits with the
withdrawal known rather than random. The analytical model shows that demand deposits
should cost less to insure than time deposits. This and other results are illustrated via a
numericaI example.
Keywords: deposit insurance, option pricing, risk premium
Introduction
This paper applies option pricing theory to compare the insurance costs for
time and demand deposits 1 .Deposit insurance has been a controversial public
policy issue since inception (Keeton 1990). The strongest arguments in support
of it are the maintenance of confidence in the banking system and the
protection of small depositors who are less well informed and who have little
market power (Ho 1991). On the other hand, deposit insurance is known to
induce excessive risk-taking by banks, lax monitoring by regulators, and
imprudent behavior on the part of depositors (Carr et al. 1995; Shiers 1994).
At present, some form of deposit insurance exists in the following Asian
and Pacific-Rim countries: the United States, Canada, Japan, Taiwan, India,
and the Philippines. Countries in the same region with no deposit insurance
include China, Australia, New Zealand, Malaysia, Singapore, Indonesia, and
Hong Kong. On balance, deposit insurance seems to be gaining acceptance
around the globe. Examples include the recently mandated deposit insurance
in Oman (E1-Nafie 1995), the current proposal for deposit insurance in Korea, as
well as many activities to reform existing deposit insurance systems in the
1
Gongmen Chen is Assistant Professor of the Department of Accountancy, The Hong
Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. Mandy Li is Assistant
Professor of the School of Business and Administration, Open University of Hong Kong,
30 Good Shepherd Street, Homantin, Hong Kong.
light of the globalization of financial markets in recent years. Examples of systems
being reformed include deposit insurance systems in the United States (Craine
1995), Taiwan (Engbarth 1995), Japan (Daimon 1995), and Europe (Bruyneel
1995; Roth, 1994).
The fair pricing of insurance premium is one of the most hotly debated issues in
deposit insurance reform, which should be relevant for new systems yet to be
designed. As with other forms of insurance, deposit insurance should be priced
according to the risk. By insuring a financial institution's deposits, the deposit
insuring agency assumes a potential liability because the financial institution
might default. The greater the risk of the default, the greater is the agency's
potential liability. So deposit insurance should be priced according to the potential
liability imposed by the financial institutions on the insuring agency.
Surprisingly, this is not always the case. In the United States, for instance,
risk-based premium was not taken seriously until a decade ago. This led to the
debate whether U.S. banks were fairly charged by the U.S. Federal Deposit
Insurance Corporation. On the one hand, some studies, such as Marcus and
Shaked (1984), suggest that most banks were overcharged for deposit insurance. In
contrast, others such as Buser et al. (1981) suggest just the opposite.
Like other forms of mispricing, unfair pricing of insurance premium is
economically inefficient and would result in unjust transfers of wealth between
taxpayers at large and bank depositors, and between healthy and unhealthy banks.
In addition, unfair pricing of premiums would lead to dead-weight losses due to
undesirable opportunistic behavior on the part of bank
management. In the
literature on deposit insurance pricing, a good deal of effort has been devoted to
the issue of fair pricing. In Merton (1977), deposit insurance is shown as a put
option in which context risk-based insurance pricing is discussed. In Allen and
Sanders (1993), Merton's pricing approach is modified by modeling deposit
insurance as a callable put option. That is, the deposit insuring agency issues a
put option to the bank with no expiration date but has the right to terminate the put
option at any time. In Duan and Yu (1994), an effort is made to cast the Merton
(1977) analysis in a multiperiod framework, thus leading to more accurate
determination of the insurance premium. In Duan et al. (1995), the bank's
interest-rate risk exposure is taken into consideration in pricing the insurance. In
Pennacchi (1987), an extension is made to incorporate the effectiveness of control
by government regulators in the pricing of deposit insurance. In Nagarajan and
Sealey (1995), it is shown that risk-based insurance premium can be replaced by
other means, including an optimal policy of allowing insolvent banks to operate
under regulatory supervision. In the last two studies, the question is how
effectively can the bank regulators make a capital-deficient bank take remedial
action such as adding more capital or reducing liability. If the regulatory rules can
be enforced effectively and without delay, then deposit insurance can be set at a
lower level. Otherwise, the insurer must be set at a higher level. In short, a large
track of the literature has been devoted to addressing the issue of fair pricing of
deposit insurance premium.
The focus of this study is on the relative pricing of time and demand deposits.
Its intended contribution to the literature of deposit insurance is to differentiate the
risk associated with, and thus the insurance cost of, the two types of deposits. The
question is which of the two kinds of deposit cost more to insure against default?
Merton (1977) derives a valuation formula for time deposits by applying the Black
and Scholes (1973) option pricing model. In his analysis, the insurance for a time
deposit is viewed as a put option issued by the deposit insuring agency. So the
insurance premium is equal to the value of the put option. Merton's analysis does
not differentiate between time and demand deposits. Rather, he assumes that the
time-to-maturity of a demand deposit is the length of time until the next regulatory
audit of the bank. This approach ignores the probability that the demand deposit
could be withdrawn at any time between the current moment and the next
regulatory audit. Ronn and Verma (1986) make a similar assumption in their
analysis of deposit insurance cost. In Crouchy and Galai (1991), a demand deposit
is viewed as if it were a series of overnight time deposits. For instance, a 100-day
demand deposit is treated as equivalent to a 100 overnight time deposits. However,
the default risk associated with a 100-day time deposit is different from that
associated with 100 overnight time deposits. In short, prior research has not given
much attention to the fact that time and demand deposits impose different levels of
risk on the deposit insuring agency. Nor has this difference been noted in practice.
In the United States, for instance, deposit insurance premium is calculated as a
fixed proportion of total domestic deposits without differentiating between time
and demand deposits. An investigation of how the two forms of liability impose
differential risk on the insuring agency would lead to a more accurate
determination of deposit insurance premium, thus furthering our understanding of
deposit insurance pricing.
This paper uses Margrabe's (1978) exchange option approach to model
deposit insurance costs, but treats time and demand deposits differently. The
central thesis is that time deposits and demand deposits impose different levels
of risk on the insuring agency. More specifically, it will be shown that demand
deposits, which are subject to random withdrawals by depositors, impose a lower
level of risk on the insurer and thus should cost less to insure than time deposits.
To accomplish this task, suppose there are two banks, A and B, that are identical in
every aspect except for their liabilities. Bank A's liabilities consist of only time
deposits with a time-to-maturity of T, while bank B's liabilities consist of only
demand deposits which can be withdrawn at any time t where 0 < t ≤ T. For
analytical convenience, it is assumed that the two banks will be dissolved at the
end of T. We will first set up the insurance for time deposits as an exchange option
and derive a pricing model accordingly. Then, the insurance for demand deposits
is viewed as a compound option. A general pricing model for deposit insurance
cost is derived. We find that the pricing model for time deposit insurance is a
special case of a general pricing model.
Comparison of the costs between time deposits and demand deposits is analyzed
numerically via an example.
The remainder of the paper is organized as follows. The next section models
the insurance cost for time deposits. The third section establishes a general pricing
model for the insurance cost of demand deposits. This is followed by a numerical
example in the fourth section. The final section concludes the paper.
Insurance Cost for a Time Deposit
Consider an option that gives the holder a right to exchange one asset, say asset 1,
for another, say asset 2. The two assets have the same present value to the investor.
Let S1 and S2 denote the prices of the two assets, respectively, at the expiration
date. Let T be the time horizon from the present to the expiration date. Then, let
U(S1, S2, T) be the value of the option to exchange asset 2 for asset 1 with the
time-to-maturity of T. When T = 0, that is, at the expiration date, the holder of the
option could exercise the option by exchanging asset 2 for asset 1. The holder will
do so if the price of asset 1 is greater than that of asset 2. Conversely, if the price
of asset 1 is less than that of asset 2, the holder will let the option expire. Thus, the
value of the exchange option at the expiration date is given by
U(S1, S2, 0) = max(S1-S2, 0)
(1)
With the condition
0≤U(S1, S2, T) ≤S1
(2)
The assumptions underlying Margrabe's model are:
1. capital market is perfect; that is, there are no transactions costs or taxes, and
perfect competition prevails; information is costless and is received
simultaneously by all individuals;
2. there are no restrictions on short sales;
3. no dividends are paid on either asset 1 or asset 2;
4. the option can be exercised only at the expiration date; and
5. the prices of both asset 1 and asset 2 follow the geometric Brownian motion;
that is,
dSi = Si μi dt + Siσ i dzi , i = 1, 2
(3)
where μi is the instantaneous expected rate of return on asset i; a constant, is the
instantaneous standard deviation per unit of time of the rate of return on asset i;
and dz is a standard Weiner process.
Based on the above assumptions, it can be shown that
U(S1, S2, T) - S1N(d1) – S2N(d2),
(4)
where
ln( S1 S2 ) + T σ 2 2
d1 =
,
σ T
d 2 = d1 − σ T ,
σ 2 = σ 12 + σ 22 + 2 ρ12σ 1σ 2
and ρ12 is the correlation between two assets and N( ) is the cumulative normal
distribution function.
Suppose a bank's liabilities consist of only time deposits all with the
same maturity date. Let T be the time to maturity, at which time the bank is
assumed to be dissolved. Let D(T) be the present value of time deposits. Consider
the case where a deposit insurance scheme does not exist. At maturity, if the
bank's value, V(0), is greater than the terminal value of the deposits, D(0), the
depositors will receive the full amount, D(0). On the other hand, if,V(0) is less
than D(0), then the depositors wilI retain the full amount of the bank's assets and
suffer a loss of D(0) - V(0). In sum, the depositors will receive the amount,
min[D(0),V(0)], at the maturity date. Now consider the other case where deposit
insurance exists such that the depositors are guaranteed D(0) regardless of the
value of V(0). The payoff at maturity from the insurer must then become
G(0) = D(0) - min[D(0), V(0)]
= max[D(0) -V(0), 0].
(5)
The payoff structure in (5) is the same as the payoff structure in (1). Thus, the
deposit insurance can be viewed as an option to exchange the bank's assets, V, for
the time deposits, D. Applying Eq. (4) yields the following formula for the cost of
time deposit insurance:
(6)
G(T) = DN(x1) - VN(x2)
where
ln( D / V ) + T σ 2 / 2
x1 =
,
σ T
x2 = x1 − σ T , and
σ = σ V sin ceσ D = 02.
Let d be the deposit-to-asset ratio at the present time and g(d, σ , T) be the
insurance cost per dollar of time deposit. That is, d = D/V and g(d, σ , T) =
G(T)/D. From Eq. (6), the formula for g can be written as:
(7)
g(d, σ , T)= N(x1)- d-1 N(x2),
where
x1 =
ln(d ) + T σ 2 / 2
, and
σ T
x2 = x1 − σ T
Note that σ = σ v since σ D= 0. Equation (7) is the insurance pricing scheme per
dollar of time deposit.
It can be shown that ∂ g/ ∂ d > 0. The proof is given in Appendix A. This
suggests that the insured bank with a higher deposit-to-asset ratio should pay a
higher deposit insurance premium. The underlying reason for this is that the bank
with a higher d has higher leverage and thus a higher default risk. In addition, it
can be shown that ∂ g/ ∂ v > 0. The proof is provided in Appendix A. This relation
implies that the higher the volatility of the value of the bank's asset, the higher is
the risk imposed on the insurance agency and thus the higher the insurance
premium.
The Insurance Cost for Demand Deposits
Unlike time deposits, demand deposits can be withdrawn at any point of time
within the period T. In the interest of tractability, it is assumed that once the funds
are needed, the deposits will be withdrawn entirely. Furthermore, it is assumed
that there are m discrete withdrawal points, and the interval between any two
points is the same and equal to T/m. Let be the probability that the withdrawal
occurs at point i, i = 1,...,m.
With these simplifying conditions, the insurance cost of demand deposits can be
determined as the sum of present values discounted from expected cash flows at
all possible withdrawal points. In the case of demand deposits, the expected cash
flows arise due to the probability of withdrawal of the demand deposits at each
point before the maturity date. At any withdrawal point, the insurer may suffer a
loss that is equivalent to the value of an option to exchange the bank's assets for
the demand deposits. A similar argument is used in Geske and Johnson (1984) to
value an American put option as a compound option.3 It should be added that the
insurance cost of demand deposits is not the same as the value of an American put
option although one is tempted to make such an analogy. For instance, the
premature exercise of an American put option is triggered by a sufficiently low
stock price, while a withdrawal of demand deposits is usually independent of the
value of bank's assets except when there is a run on the bank.
Let ti denote the interval from the present to time point i. Then ti = iT/m. Let
H(T,m) denote the expected cost of demand deposit insurance and h(T, σ , d, m)
denote the expected per dollar insurance cost for demand deposits, where T, σ , d,
and m are defined as before.4 We then have
m
^
H (T , m) = ∑ {e − rti E[π i max( Di − V , 0i )]}
i =1
m
^
= ∑ {π i e − rti E[max( Di − V , 0i )]}
i =1
m
= ∑ [π i G (ti )]
(8)
i =1
~
= E[G (t )],
^
where E is a risk neutral valuator, π i is the probability that the withdrawal
occurs at the time point i, and r is the risk-free rate of return. Thus,
h(T , σ , d , m) = H (T , m) / D
~
= E[G (t ) / D]
~
= E[ g (t )]
m
(9)
~
= ∑ [π i g (t )].
i =1
Substituting Eq.(7)into Eq.(9)gives
m
h(T , σ , d , m) = ∑ π i [ N ( x1i ) − d −1 N ( x2i )]
(10)
i =1
Again, for simplicity we assume that withdrawals follow a uniform discrete
distribution. This results in
π i = π 2 = ... = π m = 1/ m
Then, Eq. (10) becomes
h(T , σ , d , m) =
1 m
[ N ( x1i ) − d −1 N ( x2i )],
∑
m i =1
(11)
where
ln(d ) + tiσ 2 / 2
x1i =
,
σ ti
x2i = x1i − σ ti , and
ti = iT / m
Note that this model can be applied to both time deposits and demand deposits.
When m = 1, Eq. (11) reduces to Eq. (7). In other words, the pricing model (7) for
time deposits is a special case of the general pricing model (11).
A Numerical Illustration
From Eq.(11),deposit insurance cost is shown to be a function of T, σ ,d, and m. T
is a fixed length of time, and the bank is assumed to be dissolved at the end of T;
σ is a constant volatility per unit of time of the rate of return on the bank's assets;
d is the ratio of the deposits to the bank's assets; and m is the number of
withdrawal points. When m = 1, the deposit is labeled as 'time' since the possible
withdrawal of the deposits occurs only at the end of T. When m > 2, the deposit is
labeled as 'demand'. In reality, m is indefinite since the depositor could withdraw
the money at any point in time. The higher the value of m, the more closely the
simulated result approximates the actual insurance cost for demand deposits.
Table 1 shows the simulated results of the deposit insurance cost for a one-year
horizon (i.e., T = 1 year). Consider panel a of the table, where the volatility σ is
set equal to 0.10. The first column is the number of possible withdrawal points, m.
The second column records the deposit insurance cost when the bank's
deposit-to-asset ratio, d, is equal to 0.85. From the table, it is obvious that m is
negatively related to the insurance cost. The higher the value of m, the lower is the
insurance cost. For example, the annual insurance cost is 0.24 cent for one dollar
of time deposit when m = 1, while it is only 0.06 cent for one dollar of demand
deposit when m = 100.6
When the deposit-to-asset ratio, d, increases, the insurance cost rises
correspondingly. For instance, it is shown in panel a that the time deposit
insurance cost increases from 0.24 cent when d = 0.85 (column 2) to 0.79 cent
when d = 0.90 (column 3). Similarly, for demand deposits (m = 100)the insurance
cost increases from 0.06 cent to 0.28 cent. The increase in the insurance cost
reflects the increased default risk of the bank when the bank has larger liabilities.
The fourth and fifth columns are to be interpreted similarly but for more highly
leveraged banks.
Likewise, panels b and c of Table 1 report comparable results when the
volatility value is higher. It is evident from the three panels that the volatility of
the bank's assets is positively correlated with insurance cost. For example, at d =
0.90, the annual insurance cost for time deposits (m = 1) is 0.79 cent per dollar
when = 0.10, but rises to 2.25 cents per dollar when σ = 0.15, and 3.99 cents per
dollar when σ = 0.20.
Table 1:Insurance Cost for Demand Deposits
a) The insurance cost given T=1 year and volatility=0.01.d is the deposit-to-asset ratio.
No.of withdrawal points(m)
1
2
3
5
10
20
30
50
100
d=0.85
0.2224
0.0013
0.0011
0.0009
0.0007
0.0007
0.0007
0.0006
0.0006
d=0.90
d=0.95
d=1.00
0.0079
0.0051
0.0042
0.0036
0.0032
0.0030
0.0029
0.0029
0.0028
0.0199
0.0149
0.0132
0.0117
0.0107
0.0102
0.0100
0.0097
0.0098
0.0399
0.0340
0.0318
0.0299
0.0283
0.0275
0.0272
0.0270
0.0268
b)The insurance cost given T=1 year and volatility=0.15. d is the deposit-to-asset ratio.
No.of withdrawal points(m)
1
2
3
5
10
d=0.85
0.0115
0.0073
0.0061
0.0052
0.0046
d=0.90
d=0.95
d=1.00
0.0225
0.0159
0.0138
0.0121
0.0109
0.0386
0.0303
0.0273
0.0248
0.0228
0.0598
0.0510
0.0477
0.0448
0.0425
20
30
50
100
0.0043
0.0043
0.0041
0.0041
0.0103
0.0101
0.0100
0.0099
0.0218
0.0215
0.0212
0.0210
0.0412
0.0408
0.0404
0.0402
c) The insurance cost given T=1 year and volatility=0.20. d is the deposit-to-asset ratio.
No.of withdrawal points(m)
1
2
3
5
10
20
30
50
100
d=0.85
0.0254
0.0175
0.0150
0.0130
0.0117
0.0110
0.0108
0.0106
0.0105
d=0.90
d=0.95
d=1.00
0.0399
0.0298
0.0262
0.0234
0.0213
0.0203
0.0199
0.0196
0.0194
0.0581
0.0467
0.0425
0.0388
0.0360
0.0345
0.0340
0.0336
0.0333
0.0797
0.0680
0.0636
0.0598
0.0566
0.0550
0.0544
0.0539
0.0535
Another point illustrated in Table 1 is that with increasing deposit-to-asset
ratio, d, and increasing volatility, σ , the difference between the insurance costs
for time and demand deposits narrows. For example, when volatility is 0.10 as in
panel a, the insurance cost for time deposits (m = 1) is four times that for demand
deposits (m = 100) for d = 0.85 while the former is around twice as large as the
latter for d = 0.90. On the other hand, given d = 0.90, the insurance cost for time
deposits (m = 1) is 2.06 times that for demand deposits (m = 100) when σ = 0.20
while the former is 2.82 times the latter when σ = 0.10. These findings are
consistent with the comparative analysis in Section two that the cost of deposit
insurance is an increasing function of the volatility of the bank's assets and the
deposit-to-asset ratio.
Conclusion
This research has derived an insurance-pricing model which can be applied to both
time and demand deposits[ When the number of withdrawal points is taken as one,
the model gives the insurance cost for time deposits. When the number of
withdrawal points is taken as sufficiently large, such as 100, the model
approximates the insurance cost of demand deposits. The simulation results show
that the insurance cost of time deposits is much larger than that of demand deposit
insurance. However, such discrepancy narrows when the deposit-to-assets ratio
and/or volatility rise. Overall, this study shows that demand deposits impose a
lower level of risk than time deposits on the insurer and consequently justify a
lower insurance premium than do time deposits. This finding is consistent with the
fact that banks are more conservative in lending the proceeds of demand deposits
since they have to prepare for the withdrawal of demand deposits at any time.
A study like this would not be complete without discussing its limitations.
Two are noted here. First, the analysis assumes that the bank's liability consists
either entirely of time deposit or demand deposit. This, of course, is highly
stylized in order to highlight the issue at hand. In reality, all banks' liabilities
consist of a mixture of both deposits. Thus, the model is not readily applicable to
any bank or any particular type of deposit-taking institutions. Second, the analysis
assumes that the randomness related to demand deposit is only in terms of the time
of withdrawal. In this context, it assumes that if there is a withdrawal from a
demand deposit, the entire deposit is withdrawn. Clearly, this is an extreme
assumption and will make demand deposits appear less costly to insure than they
are. Consequently, the results are somewhat exaggerated regarding the difference
in fair insurance premium between the two types of deposit. Despite these
limitations, however, this study serves its primary aim of demonstrating the
differential risk imposed on the insurance agency by the two types of deposits and
makes a potential contribution toward more accurate pricing of deposit insurance
in accordance with risk.
Acknowledgments
The authors wish to thank Larry Merville and the two referees for their helpful
Comments. All errors remain our responsibility.
Notes
1. n some countries, time deposit is also called term deposit or fixed deposit.
2. The value of time deposits remains unchanged under an insured scheme.
Therefore, the volatility for the value of time deposits is zero.
3. An American option is an contract which gives the holder a right to buy or sell
an underlying asset at any time up until the maturity date for the prespecified
exercise price. In contrast, a European option allows the holder to exercise only at
the maturity date.
4. When T is the number of years, the insurance cost will be priced at dollar per
year.
5. This means that the bank's capital consists of eighty-five percent of deposits and
fifteen percent of equity.
6. It should be noted that the insurance cost changes very trivially when m > 100.
Thus, the simulated results for m = 100 can proxy for the insurance cost for
demand deposits.
References
Allen, L. and Sanders, A. (1993), ‘Forbearance and Valuation of Deposit Insurance as a
Callable Put', Journal of Banking and Finance, Vol. 17, No. 4, pp. 629-643.
Black, F. and Scholes, M. (1973), 'The Pricing of Options and Corporate Liabilities', Journal
of Political Economy, Vol. 81, No. 3, May-June, pp. 637-659.
Bruynell, A. and Miller, A. (1995), 'Belgium Implements Deposit-Guarantee Scheme', International Financial Law Review, Vol. 14, No. 6, June, pp. 18-20.
Buser, S., Chen, A. and Kane, E. (1981), 'Federal Deposit Insurance, Regulatory Policy, and
Optimal Bank Capital', Journal of Finance, Vol. 36, No. 1, March, pp. 51-60.
Carr, J., Mathewson, F. and Quigley N. (1995), 'Stability in the Absence of Deposit Insurance:
The Canadian Banking System, 1890-1966', Journal of Money, Credit and Banking, Vol. 27,
No. 4 (Part I), November, pp. 1137-1158.
Crane, R. (1995), 'Fairly Priced Deposit Insurance and Bank Charter Policy', Journal of
Finance, Vol. 50, No. 5, December, pp. 1735-1746.
Crouchy, M. and D. Galai (1991) 'A Contingent Claim Analysis of a Regulated Depository
Institution', Journal of Banking and Finance, Vol. 15, No. 1, March, pp. 73-90.
Daimon, S. (1995), 'New Wave of Financial Firm Bankruptcies Feared', Japan Times Weekly
International Edition, Vol. 35, No. 2, January 16-22, pp. 14.
Duan, J., Moreau, A. and Sealey, C. (1995), 'Deposit Insurance and Bank Interest Rate
Risk: Pricing and Regulatory Implications', Journal of Banking and Finance, Vol. 19, No. 6,
pp. 1091-1108.
Duan, J. and Yu, M. (1994) 'Forbearance and Pricing Deposit Insurance in a Multiperiod
Framework', The Journal of Risk and Insurance, Vol. 61, No. 4, pp. 575-591.
EI-Nafie, A. (1995), 'Oman-Banking Deposit Insurance: New Law Effective April 1 ', Middle
East Executive Reports, Vol. 18, No. 4, April, pp. 21-22.
Engbarth, D. (1995), 'Taipei's Missile Test', Banker, Vol. 145, No. 836, October, pp. 79-84.
Geske, R. (1979), 'The Valuation of Compound Options', Journal of Financial Economics,
Vol. 7, No. 1, March, pp. 63-81.
Geske, R. and H. Johnson (1984), 'The American Put Option Valued Analytically', Journal of
Finance, Vol. 39, No. 5, December, pp. 1511-24.
Ho, L.S. (1991)'A Proposal for Deposit Insurance in Hong Kong', Occasional Paper, Nol 9,
Hong Kong Institute of Asia-Pacific Studies, The Chinese University of Hong Kong.
Keeton, W. (1990), 'Small and Large Bank Views of Deposit Insurance: Today vs. the 1930's',
Economic Review, Federal Reserve Bank of Kansas City, Vol. 75, No. 5, September-October,
pp. 23-25.
Margrabe, W. (1978), 'The Valuation of an Option to Exchange One Asset for Another',
Journal of Finance, Vol. 33, No. 1, March, pp. 177-86.
Nagarajan, S. and sealey, C. (1995), 'Forbearance, Deposit Insurance Pricing, and Incentive
Compatible Bank Regulation', Journal of Banking and Finance, Vol. 19, No. 6, pp. 11091130.
Marcus, A. and I. Shaked (1984), 'The Valuation of FDIC Deposit Insurance Using Optionpricing Estimates', Journal of Money, Credit, and Banking, Vol. 14, No. 4, Part 1, pp. 446460.
Merton, R. (1977), 'An Analytic Derivation of the Cost of Deposit Insurance and Loan
Guarantee', Journal of Banking and Finance, Vol. 1, No. 1, June, pp. 3-11.
Pennacchi,G..(1987), 'A Reexamination of the Over-(or Under-) Pricing of Deposit Insurance',
Journal of Money, Credit, and Banking, Vol. 19, No. 3, August, pp. 340-360.
Ronn, E. and A. Verma (1986), 'Pricing Risk-Adjusted Deposit Insurance: An Option-based
Model', Journal of Finance, Vol. 41, No. 4, December, pp. 871-95.
Roth, M. (1994), 'Too-Big-to-Fail and the Stability of the Banking System: Some Insights
from Foreign Countries', Business Economics, Vol. 29, No. 4, October, pp. 43-49.
Rubinstein, M. (1983), 'Displaced Diffusion Option Pricing', Journal of Finance, Vol. 38, No.
1, March, pp. 213-217.
Shiers, A. (1994), 'Deposit Insurance and Banking System Risk: Some Empirical Evidence',
Quarterly Review of Economics and Finance, Vol. 34, No. 4, Winter, pp. 347-361.
Turner, G. (1996), 'An Idea Whose Time Has Gone', Canadian Business, Vol. 69, No. 2,
February, pp. 15.
Appendix A: Derivation of Two Comparative Statics
Proof for
∂g
> 0:
∂d
From Eq.(7),
g ( d , σ , T ) = N ( x1 ) − d −1 N ( x2 ) ,
where
Tσ 2
x1 = [ln d +
] / σ T , and
2
x2 = x1 − σ T .
Thus,
1
1
∂g
−2
−1
d
= n( x1 )
− [−d N ( x2 ) + d n( x2 ) d ]
∂d
σT
σT
= d −2 N ( x2 ) +
n( x 2 )
n( x1 )
[d −
]
2
n( x1 )
σd T
n( x1 )
1
= d N ( x2 ) +
[d −
2
σd T
1
−2
2π e − x2 / 2
2
2π e − x1 / 2
2
]
2
2
n( x1 )
[d − e1/ 2( x1 − x2 ) ]
2
σd T
n( x1 )
= d −2 N ( x2 ) +
[d − e1/ 2 (2ln d ) ]
2
σd T
−2
= d N ( x2 ) > 0.
= d −2 N ( x2 ) +
Note:n( )is the density function of the normal distribution.
Proof for
∂g
>0:
∂σ
∂g
σ 2T 2 − ln d
σ 2T 2 − ln d
−1
= n( x1 )
−
d
n
(
x
)
⋅
2
∂σ
σ2 T
σ2 T
= n( x1 ) ⋅
1
σ
2
T
[(σ 2T 2 − ln d ) + d −1
n( x2 ) 2
(σ T 2 − ln d )]
n( x1 )
n( x1 )
[(σ 2T 2 − ln d ) + d −1 ⋅ d ⋅ (σ 2T 2 − ln d )]
2
σ T
n( x )
= 2 1 ⋅ σ 2T
σ T
=
= n( x1 ) T > 0
Note:
n( x2 ) 1
=
n( x1 ) 1
2π e − X 2 / 2
2
2π e − X1 / 2
2
=d