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The valuation of deposit insurance allowing for the interest rate spread and early-bankruptcy risk
Journal Pre-proof The Valuation of Deposit Insurance Allowing for the Interest Rate Spread and Early-Bankruptcy Risk Shu Ling Chiang, Ming Shann Tsai
PII:
S1062-9769(19)30065-1
DOI:
https://doi.org/10.1016/j.qref.2019.09.008
Reference:
QUAECO 1301
To appear in:
Quarterly Review of Economics and Finance
Received Date:
18 February 2019
Revised Date:
7 July 2019
Accepted Date:
10 September 2019
Please cite this article as: Chiang SL, Tsai MS, The Valuation of Deposit Insurance Allowing for the Interest Rate Spread and Early-Bankruptcy Risk, Quarterly Review of Economics and Finance (2019), doi: https://doi.org/10.1016/j.qref.2019.09.008
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The Valuation of Deposit Insurance Allowing for the Interest Rate Spread and EarlyBankruptcy Risk
Shu Ling Chiang Professor Department of Business Management National Kaohsiung Normal University, Kaohsiung, Taiwan
Ming Shann Tsai Professor Department of Finance National University of Kaohsiung, Kaohsiung, Taiwan
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E-mail: [email protected].
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E-mail: [email protected].
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Highlights
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We derivea closed-form formula of deposit insurance premiums with the two important factors. We derive new formulas with the risk of early bankruptcy for estimating the necessary parameters. Our empirical results show the spread of the interest rate is negatively correlated with the premium. The traditional model underprices the premium by 20.81% if it ignores
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the risk of early bankruptcy.
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Abstract
Our proposed model allows for inclusion of the interest rate spread (defined as the difference between the lending rate and the borrowing rate) and the risk of a bank’s early bankruptcy in the derivation of a closed-form pricing formula for calculating deposit insurance premiums. This is important because, in practice, these two factors significantly influence the expected value of a bank’s assets. Also, we use a method that considers the
possibility of early bankruptcy to derive new formulas for calibrating the necessary parameters. Data from Taiwanese banks are used to illustrate the application of our model. Our empirical results show that the spread of the interest rate is negatively correlated with the premium. This result is consistent with our theoretical inferences. Moreover, premiums associated with a risk of early bankruptcy are always higher than premiums not associated with such a risk. Using the example of Taiwanese banks, we show that the traditional model
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underprices the premium by 20.81% if it does not consider the risk of early bankruptcy.
Keywords: Deposit Insurance, Premiums, Interest Rate Spread, Early Bankruptcy
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1. Introduction
A well-designed deposit insurance system strengthens the depositor’s confidence in
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financial institutions, and this confidence in turn stabilizes the financial condition of a country. Therefore, constructing a good deposit insurance system is crucial for a country’s
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economic development. A fair deposit insurance premium is important for constructing a good deposit insurance system. If the premium is lower than what a bank should pay, the
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bank is likely to shift its risk to the deposit insurer (Peltzman, 1970; Santomero and Vinso, 1977; Marcus and Shaked, 1984; Ronn and Verma, 1986; Duan, Moreau and Sealey, 1992;
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Shyu and Tsai, 1999a, b). In contrast, if the calculated premium is too high, the bank may be unwilling to participate in the deposit insurance program at all. These two situations
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could even result in the collapse of the entire system. Therefore, accurately and fairly valuating the deposit insurance premium is an important consideration in researching how banks manage risk. Merton (1977) was the first to derive a formula for pricing risk-based deposit insurance premiums. He argued that this premium is the same as the initial value of a European put option for which the strike price and underlying assets are respectively the 2
value of the deposits and the value of the bank’s total assets. Numerous researchers have investigated how Merton’s valuation framework can be used to more precisely determine deposit insurance premiums (Ronn and Verma, 1986; Allen and Saunders, 1993; Duan and Yu, 1994, Duan and Simonato, 2002; Lee, Lee and Yu, 2005; Chen, Ju, Mazumdar and Verma, 2006; Chuang, Lee, Lin and Yu, 2009). For example, in Merton’s model, the value of the bank’s assets and the standard deviation of its returns are two important but unobserved parameters. Some authors have used simultaneous equations of the bank’s
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equity and the standard deviation of its equity returns to determine these two parameters
(Ronn and Verma,1986; Giammarino, Schwarz and Zechner, 1989). Duan and Yu (1994) used Maximum Likelihood Estimation to estimate them. Recently, some scholars have
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derived a formula for calculating a deposit insurance premium when the stochastic process
of the interest rate is included in the model (Duan and Simonato, 2002; Chuang et al, 2009).
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Pennacchi (1987a) made the point that the magnitude of an insurance premium depends on whether the deposit insurer is offering a fixed-rate, unlimited-term contract or a variable-
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rate, limited-term contract. Furthermore, in several studies, estimating a deposit insurance premium took into account the default risk of the guaranty fund (Episcopos, 2004).
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An important factor in the determination and valuation of deposit insurance premiums is whether the effect of the difference between the lending rate and the borrowing rate (i.e.,
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the interest rate spread) is incorporated in the model. As we know, a commercial bank earns its profits by paying a lower borrowing rate and receiving a higher lending rate. Recently,
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because of the very competitive financial market, banks have generally either increased the borrowing rate to attract depositors or decreased the lending rate to attract lenders. Both of these actions decrease the interest rate spread. As shown in Figure 1, this spread in the Taiwanese financial market had great volatility from 1981 to 2010. The minimum value of the interest rate spread was 1.901% in 1998 and the maximum value was 5.4298% in 2002. This spread sharply decreased from 2002 to 2010 and in turn decreased the profits of 3
Taiwanese banks. On the other hand, the profits of commercial banks, and in turn the value of their assets, come from exploiting the interest rate spread. The size of this spread can affect the likelihood of bankruptcy. Merton (1978) was the first to incorporate an interest rate spread into a valuation model of deposit insurance. He modeled the spread as a lowerthan-competitive interest rate paid to depositors. Pennacchi (1987a, 1987b) allowed for a spread in the form of a lower-than-competitive deposit interest rate. We also consider in this paper the magnitude of this spread when determining the optimal deposit insurance
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premium.
In addition, valuation models such as Merton’s are feasible only when a bank goes
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into bankruptcy at the maturity date of the insurance contract (defined as the auditing date in Merton’s model). Generally, banks are involuntarily shut down by the decision of a
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government regulator or deposit insurer. However, from the financial viewpoint, a bank is in bankruptcy if its assets fall below its debt at any time point before the maturity date. In
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such a case, using Merton’s model could make calculation of the premium inaccurate, because it uses only one time point (i.e., the auditing time in Merton’s model) to determine
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the probability of bankruptcy and the expected loss. To accurately determine a fair deposit insurance premium, the model should also consider the risk of early bankruptcy at any time
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point before the maturity date stipulated in the insurance contract. We calculate the deposit insurance premium based on this consideration.
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When estimating the risk of early bankruptcy, one needs to consider the probability of
early bankruptcy and recovery given bankruptcy. The first-passage time model has been adopted in many traditional studies aimed at valuating a security when there is an early termination risk (Black and Cox, 1976; Finger, 2002). When using this model to investigate the risk of bankruptcy, one must keep in mind that the bankruptcy occurs as soon as the bank’s assets are less than its debt (i.e., the first absorption situation). As mentioned by 4
Black and Cox (1976), the first-passage time model cannot be applied directly when the termination payoff is found to endogenously contribute to an optimal-stopping problem. In the papers reporting this use of the first-passage time model, the focus has been exclusively on determining the termination probability (i.e., the probability of bankruptcy); moreover, these authors usually specified this termination payoff (i.e., the recovery given bankruptcy) to be uncorrelated with the bank’s underlying assets. In such cases, they must assume that the termination payoff is either an exogenous variable (see Finger, 2002) or an endogenous
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variable that is uncorrelated with the underlying assets (see Black and Cox, 1976).
However, when dealing with the risk of bankruptcy in valuating deposit insurance, the probability of early bankruptcy and the recovery given bankruptcy should be
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simultaneously determined by the same factor: the bank’s assets. Thus, treating these two factors separately may result in incorrect pricing of the deposit insurance premium. We
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support a new method for accurately pricing a deposit insurance premium, one that takes account of the possibility of the bank’s early bankruptcy. The main difference between our
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model and the first-passage time models is that we do not separate consideration of the probability of early bankruptcy from consideration of recovery given bankruptcy;
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specifically, we let them both be endogenous variables, simultaneously determined by the bank’s assets. In real applications, our specifications yield more accurate pricing than
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traditional specifications. To the best of our knowledge, ours is the first paper to report a closed-form deposit insurance pricing formula that lets the probability of early bankruptcy
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and the recovery given bankruptcy be simultaneously determined by the bank’s asset value when the model takes account of the bank’s early bankruptcy risk. Note that our closed-form pricing formula can also be applied to the valuation of an
American option, because the early exercise probability and the payoff of this option are both determined by the same underlying asset value, as is most likely the case when one uses the specifications of our model. Our formulas give values similar to those reported by 5
Kim (1990) for the valuation of an American option. However, the valuation problems of Kim’s (1990) model and our model are different. In Kim (1990), the early exercise premium corresponds to the compensation that an American-style option holder would require (in the stopping region) in order to postpone exercising the underlying asset until the maturity date of the option contract. For deposit insurance contracts, the valuation model in our study estimates an early bankruptcy premium. Recognizing that both the interest rate spread and the bank’s risk of early bankruptcy
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significantly influence the value of the deposit insurance, we provide in this paper a pricing framework that takes account of these two important factors in accurately valuating deposit insurance premiums. Pricing a deposit insurance premium also requires estimating the
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bank’s assets and the standard deviation of its returns. Many researchers have estimated these parameters from equity data (Marcus and Shaked, 1984; Ronn and Verma, 1986).
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However, if a bank goes bankrupt before the maturity date of the insurance contract, the estimates using the formulas in traditional studies (i.e., Ronn and Vermas, 1986) are
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inaccurate. To more accurately estimate the assets and the standard deviation of the returns, we need a method that considers the possibility of early bankruptcy. A bank’s stockholders
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often receive nothing when the bank goes bankrupt. The bank’s equity, from which the likelihood of early bankruptcy can be inferred, can be determined by employing a down-
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and-out barrier option with a zero rebate. We therefore use this option to derive new formulas for estimating the two necessary parameters, the values of which are derived from
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the first-passage time model.
For our empirical analysis we collected data from 32 banks listed on the Taiwanese
stock exchange. After applying the new formulas described above, we calculated a fair, risk-based deposit insurance premium, taking into account the interest rate spread and the risk of early bankruptcy. We discuss below the differences between calculations using our model and the traditional model. 6
The remainder of the paper is organized as follows. Section 2 presents our formula that incorporates the interest rate spread and the early-bankruptcy risk to obtain an approximate price for a deposit insurance premium. Section 3 shows how we utilize the down-and-out barrier option to develop empirical formulas for calibrating the bank’s assets and the standard deviation of its returns. In this section, we also report the results of our analysis of data from the 32 banks listed on the Taiwanese stock exchange, including a comparison between the results calculated using our model and those calculated using the
2. The valuation model for a deposit insurance premium
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traditional model. Our conclusions are summarized in Section 4.
In this section, we present the theoretical valuation model for a deposit insurance (hereafter
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denoted as DI) premium. In Subsection 2.1, we describe a model that takes account of the interest rate spread and only one auditing time based on the Merton framework. In
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Subsection 2.2, we extend the model in Subsection 2.1 to take account of the likelihood of a bank’s early bankruptcy.
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2.1 A model for valuating a DI premium considering the interest rate spread and only one auditing time (Merton framework)
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In valuation theory, the two most common methods for evaluating a financial contract with default events are the structural-form model and the reduced-form model. The idea of
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pricing defaultable securities began with a structural-form model (Merton 1974, Black and Cox 1976, Leland 1994). The structural-form model uses market data to estimate a bank’s
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asset value and its asset volatility, and then the bank’s probability of default is calculated. Merton (1977) was the first to derive a DI formula using a structural-form model. Recently, reduced-form model has been applied to price defaultable securities and to calculate the termination probabilities (Jarrow and Turnbull 1995, Jarrow 2001). The reduced-form model uses market information on the hazard rate to calculate the probability of default. Duffie, Jarrow and Purnanandam (2003) give a DI formula using a reduced-form model. 7
Thus, they obtained the termination probability from market data using the default hazard rate and then calculated the DI premiums. Our model, an extension of Merton’s model, is a structural-form model which obtains the DI formula with the early bankruptcy probability by estimating the bank’s asset value and its asset volatility. Let the underlying probability space be labeled (, F , Q) , where is the state space, F is the filtration generated by the information set F ( Ft ) 0t T , and Q is the risk-neutral probability. Let V be the unobserved value of the bank’s assets. In Merton
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(1977), the evolution of V is specified as following a geometric Brownian motion. Given the risk-neutral probability, V can be expressed as
dV (r f (t ) )dt V dZV (t ) , V
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(1)
is the dividend yield;
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r f (t ) is the risk-free interest rate;
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where
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V is the instantaneous standard deviation of bank’s asset value return; and ZV (t ) is a standard Brownian motion under the risk-neutral measure.
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In the above equation, the bank’s assets increase by the rate r f (t ) , which is decreased by the dividend yield . How much the assets increase depends on what the bank earns from
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its loans, which, in practice, is determined by the lending rate. Accordingly, Merton’s model specifies that the lending rate equals the risk-free interest rate under the risk-neutral measure.
According to Merton (1977), the bank’s assets are assumed to grow at the rate r f (t ) . Because its debt also grows at this rate, the effect of the interest rate disappears in Merton’s
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formula if it is a constant. However, this assumption is unrealistic. The bank obtains funds from its depositors and then loans these funds to the borrowers. The bank’s profits come mainly from managing the deposits and the lending. To earn a profit, the bank’s deposit rate must generally be less than its lending rate. The interest rate spread is defined in our model as the difference between these two rates. Because we assume that the average cost of the debt paid to the depositors is the risk-free interest rate, we set the lending rate as
r f (t ) (t ) , where (t ) is the interest rate spread under the assumption of risk neutrality.
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The value of V can be expressed as
dV (rf (t ) (t ) )dt V dZV (t ) . V
(2)
It is worth noting that if we let (t ) 0 , our model is identical to Merton’s pricing formula.
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In general, DI contracts provide depositors with minimal protection against the risk of
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the bank going bankrupt. We let B1 (t ) denote the protected deposit amount at time t , and
B2 (t ) the unprotected deposit amount at time t , the latter of which is equivalent to the
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value of all the debt liabilities other than the insured deposit. Thus, the total value of the deposit at time t can be expressed as B(t ) B1 (t ) B2 (t ) . The interval [t , T ] stands for the
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period of the DI contract, with T indicating the maturity date. In Merton’s model, the maturity date is specified as the auditing time at which one can judge whether or not the
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bank is bankrupt. At the maturity date, the total value of the bank’s assets is V (T ) and the total value of the deposits is B(T ) .
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In pricing theory, any asset process can be chosen as the numeraire asset (Davis,
1998). We chose a money market account to serve as the numeraire asset for our model. Let the money market account (defined as M (t , v) ) be expressed as follows: v
M (t , v) exp( rf (u )du) . t
(3)
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The process of the money market account is stochastic if we specify r f (t ) as following a stochastic process. Under the risk-neutral measure, the drift term of the return of money market account is rf (t )dt and the standard deviation of the return of the money market account is b(t , u ) , its form dependent on the specification of the risk-free interest rate process.1 For example, if the interest rate is specified as of the Vasciek (1977) type, the standard deviation of the return of the money market account can be obtained as follows
b(t , u )
r a
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(Heath, Jarrow and Morton, 1992):
(1 e a (ut ) ) ,
(4)
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where a is the speed of adjustment and r is the volatility of the risk-free interest rate,
have the following equation: T
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rf (u ) du B(t )M (t , T ) . B(T ) B(t )e t
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both being positive constants. Given this specification of the money market account, we
(5)
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We then used the previous specifications to price the DI premium. Several studies have considered the influences of delayed resolution or capital forbearance on the valuation of DI premiums (Duan and Yu, 1994, 1999; Nagarajan and Sealey, 1995; Lee, Lee and Yu,
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2005; Shie, 2012). In addition, depositor preference arrangements2 can increase the
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recovery rate for DI institutions because depositor claims get paid before those of general, 1
We assume that b(t , u) is a deterministic value.
2
In the U.S., the national depositor preference law of 1993 requires that all the domestic depositors, insured
and uninsured, be paid in full before the general, unsecured creditors. It gives a statutory priority to depositors over other unsecured creditors. In Taiwan, Article 64-1of the Banking Act of The Republic of China also stipulates that if a bank or a financial organization is run poorly and needs to cease its operations and liquidate its debts, deposit debts shall precede non-deposit debts. Such depositor preference laws have existed for decades in a number of countries (such as Australia, the USA, Russia, and Taiwan). 10
unsecured creditors. Some studies have considered the effects of the seniority of insured deposits in the DI valuation models (Davis, 2015; Chang, Chung, Ho and Hsiao, 2018). The main purpose of our manuscript is to provide a DI premium model allowing for the interest rate spread and early-bankruptcy risk. To avoid complicated specifications that would blur our point, our model assumes that there are no effects caused by the policies of the delayed resolution or capital forbearance for defaulted banks and the seniority of insured deposits. As noted by Merton (1977), if V (T ) B(T ) at the maturity date, the protected amount
protected amount of the deposit becomes V (T )
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of the deposit is B1 (T ) . In contrast, if V (T ) B(T ) , the bank goes into bankruptcy and the B1 (T ) 3 . The value of the DI contract at T is B(T )
then
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(6)
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B (T ) Max 0, B1 (T ) V (T ) 1 . B(T )
V (T ) ). B(T )
V (T ) ) is the loss rate given bankruptcy at the maturity date T . Thus, in Merton’s B(T )
model
V (T ) is the recovery rate given bankruptcy at the maturity date T . B(T )
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(1
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In view of that, we can obtain the loss amount given default to be B1 (T )(1
According to Equation (2), the change in the bank’s assets is a random variable that
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follows a lognormal distribution. Thus, the DI premium can be calculated based on the
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theory of that distribution. In light of the above equation, the contract value can be regarded as a classic European put option with strike price B1 (T ) and underlying asset value V (T )
B1 (T ) . The DI premium, denoted as IP(t , T ) , is specified under the assumption of B(T )
only one auditing time (at the maturity date) and a given interest rate spread. We can obtain the following result by extending the classical Black-Scholes-Merton model (Black and 3
For simplicity, we assume that B1 (T ) and B2 (T ) have the same seniority. 11
Scholes, 1973; Merton, 1973) (see Appendix A): IP(t , T ) B1 (t ) N ( y2 (t , T )) V (t )e( s )( T t )
B1 (t ) N ( y1 (t , T )) , B(t )
(7) where T
s (T t ) 1 (u )du , defined as the average interest rate spread; t
N () is the cumulative density of a standard random variable;
ln( y 2 (t , T )
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y1 (t , T )
B(t ) ) 1 (t , T ) V (t ) ; v(t , T ) B(t ) ) 2 (t , T ) V (t ) ; v(t , T ) 1 2
1 (t , T ) ( s )(T t ) v 2 (t ,T ) ; 1 2
T
v 2 (t , T ) V b(t , u ) du . 2
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t
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2 (t ,T ) ( s )(T t ) v 2 (t , T ) ; and
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ln(
If we assume correlation between the returns of the bank’s assets and the returns of
T
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the money market account, we can obtain the following equation:
v 2 (t , T ) ( V2 2 V b(t , u) b 2 (t , u))du . t
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(8)
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Equation (8) is a general formula for calculating the total variance of a discounted bank’s assets in the DI model. One can obtain a closed-form expression of v 2 (t , T ) if the standard deviation of the interest rate has been determined. For example, if the interest rate is specified is of the Vasciek (1977) type, according to Equation (4) we have
v (t , T ) (T t ) 2 V 2
2 V
r
1 e a (T t ) ((T t ) ) a a
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r2 a2
((T t ) 2
1 e a (T t ) 1 e 2 a (T t ) ). a 2a
(9) Thus, our pricing formula gives the same results as those reported in Rabinovitch (1989) and Chung (1999). The main difference between our formulas and theirs is that our formulas valuate a DI contract under the assumption that the strike price increases as the interest rate increases, whereas their pricing formulas are applied to vanilla options, which have a constant strike price.
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As mentioned previously, one can easily show that Equation (7) becomes the Merton pricing formula for DI if s 0 and r (t ) is a deterministic value. To prove this, we let
r 0 when assuming a deterministic interest rate, in which case b(t , u) 0 according to
T
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Equation (4). We then have T
v 2 (t , T ) V b(t , u) du V2 du V2 (T t ) . 2
t
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t
(10)
shown in Merton (1977).
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Thus, our Equation (7) given a deterministic interest rate is the same as the DI model
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According to Equation (7), the premium decreases as the interest rate spread increases. We infer from this conclusion that the probability of bankruptcy decreases
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because the increase in the interest rate spread leads to a rise in the bank’s profits. Thus, there is a negative relationship between the interest rate spread and the DI premium.
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The per-monetary-unit DI premium, IPP(t , T ) , can be obtained by dividing Equation
(7) by B1 (t ) as follows: IPP (t , T ) N y 2 (t , T ) e ( s )( T t )
V (t ) N y1 (t , T ) . B(t )
(11) In Merton’s model, there is only one time point (i.e., the auditing time in Merton’s model) for judging whether the bank is in bankruptcy. However, this specification is not realistic. 13
The bankruptcy of a bank can occur at any time before the maturity of the DI contract. To accurately price DI, the associated premium, which must reflect the early-bankruptcy risk, should be modeled precisely. 2.2 A model for valuating a DI premium taking account of the interest rate spread and the likelihood of a bank’s early bankruptcy We derive below closed-form formulas for DI considering the likelihood of a bank’s early bankruptcy using both discrete-time and continuous-time frameworks. We first construct
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the model using a discrete-time framework. Then we discuss our continuous-time model,
assuming that the time interval between the valuation points approaches zero. This discretetime approach is the method that is usually used in the valuation of time-dependent
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securities in a continuous-time framework (Kim, 1990). We limit the occurrence of a
possible bankruptcy to a finite number of time points during the maturity of the DI contract.
T t be the number of time points. For example, d 12 , t 0 , and T 1 -year t
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Let d
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mean that it is possible for the bank to enter bankruptcy at the end of each month. Letting t i represent the date of the i-th time point, we have t t 0 t1 t d T . Then V (ti ) and
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B(ti ) , with i 0,1,, d , denote respectively the value of the bank’s total assets and
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deposits at time t i . Therefore, at every possible bankruptcy time point, there is a corresponding value for the bank’s assets, that is, V (t 0 ) , V (t1 ) , ,V (t d ) . Note that the
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maturity date is always the expiration date of the DI contract. If the contract is perpetual, this date can be specified as an infinite variable. If we are to take the probability of early bankruptcy into account, calculation of the DI
premium requires consideration of the expected value of a non-zero random loss, that is, the payoff shown in Equation (6). We denote IP A (t , T ) as the DI premium when there is a risk
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of early bankruptcy. At the maturity date, the payoffs of IP A (t , T ) and IP(t , T ) are the same; therefore, we have IP A (t d 1 , T ) IP(t d 1 , T ) . Except for this situation, the payoff of the DI contract at time t i can be described as follows: The payoff of the DI contract at time t i
if V (ti ) B(ti ) The live value of the deposit insurance contract, . The termination value of deposit insurance contract, if V (ti ) B(ti )
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(12)
Therefore, at time t d 2 , the time until maturity is 2t . Thus, IP A (t d 2 , T ) can be expressed
IP A (t d 1 , T ) (1 I D (td 1 ) ) | Ftd 2 ] M (t d 2 , t d 1 )
V (t d 1 ) ) B(t d 1 ) I D (td 1 ) | Ftd 2 ] , M (t d 2 , t d 1 )
B1 (t d 1 )(1
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E[
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IP A (t d 2 , T ) E[
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as follows:
(13)
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where E[ | Ft ] is an expected operator, conditional on information set Ft with a risk-
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neutral probability, and I D (td 1 ) is an indicator function. This gives us I D (td 1 ) 1 if
D(td 1 ) {V (td 1 ) B(td 1 )} and I D (td 1 ) 0 otherwise. The first and second terms on the
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right hand side of Equation (13) are the expected survival and termination values respectively of the DI contract at time t d 2 . On the right hand side of this equation, the second expectation operator ( E[ | Ft ] ) represents the expected loss when bankruptcy occurs at time t d 2 . With this expectation, the value
V (td 1 ) denotes the recovery rate given B(td 1 )
bankruptcy at time t d 2 . From Equation (13), one can find the probability of early 15
bankruptcy ( E[ I D (td 1 ) | Ft ] ), where D(td 1 ) {V (td 1 ) B(td 1 )} and the recovery rate given bankruptcy (
V (t d 1 ) ) are simultaneously determined by the bank’s assets V (td 1 ) , an B(t d 1 )
endogenous and random variable. Based on the previous specifications, IP A (t d 2 , T ) can be described as follows (see Appendix B):
IP A (t d 2 , T ) IP(t d 2 , T ) E[
B1 (t d 1 ) t B(td 1 ) I D (td 1 ) | Ftd 2 ] . M (t d 2 , td 1 )
( s ) V (t d 1 )
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(14)
We define the second term on the right side of Equation (14) as the premium associated
with the risk of early bankruptcy (i.e., the early-bankruptcy premium). At each time point,
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IP A (, T ) is the sum of the corresponding IP(, T ) and the early-bankruptcy premium. As shown by Kim (1990), if we work backwards recursively, the value of the DI contract at
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time t , taking account of an early-bankruptcy risk, can be described as follows (see also
IP A (t , T ) IP(t , T )
d 1
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Appendix B):
(s )t e i 1
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(15)
( s )( ti t )
V (t )
B1 (t ) N ( y1 (t , ti )) . B(t )
The per-monetary-unit DI premium IPP A (t , T ) at time t can be represented as follows: d 1
(s )t e
( s )( ti t )
i 1
V (t ) N ( y1 (t , ti )) . B(t )
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(16)
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IPP A (t , T ) IPP(t , T )
We now move from the discrete-time framework to the continuous-time framework.
Thus, we let the number of possible bankruptcies be infinite; that is, d tends toward infinity because t tends toward zero. In view of that, we have: d 1
IP A (t , T ) IP(t , T ) lim ( s )t e( s )( ti t )V (t ) t 0
i 1
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B1 (t ) N ( y1 (t , ti )) B(t )
IP(t , T )
T
t
( s ) e( s )( u t )V (t )
B1 (t ) N ( y1 (t , u ))du . B(t )
(17) The second equality of Equation (17) is obtained by the Riemann integration, since all the stochastic processes almost certainly have continuous sample paths. In addition, we have
IPP A (t , T ) IPP(t , T )
T
t
( s ) e( s )( u t )
V (t ) N ( y1 (t , u ))du . B(t )
(18)
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It is important to note that although we model the risk to the bank’s assets to include only one Brownian motion, as shown in Equation (2), the loss, V (t ) B(t ) , incurred from going into bankruptcy is not always zero, even using the continuous-time framework.
Given the properties of Brownian motion, a large increase or a large decrease in a bank’s
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assets may occur at the next time point.4 Therefore, a sudden and large decrease in a bank’s
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assets may cause the assets to drop below the bank’s debt, V (t ) B(t ) , even if the valuation model is using the continuous-time framework.
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2.3 Discussions of the effect of the interest rate spread on the DI premium One of our contributions is to incorporate the effect of the interest rate spread into the
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valuation model for DI premiums. We then provide two discussions of the influences of the interest rate spread on DI premiums. The first discussion is focused on insured and
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uninsured deposits. Uninsured deposits in a bank with a default risk should be priced in a similar way to that used by the deposit insurance authority. The spreads for these two types
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of deposits should be different because they should be adjusted according to the auditing rule of the deposit insurance fund. Thus, the DI premium should consider this practice. In 4
Using the continuous-time framework, Brownian motion is usually used to describe the diffusion parts of an asset return. However, Brownian motion is a random variable and its values are normally distributed at the next time point. The domain of a normal distribution ranges from negative infinity to positive infinity. In this paper, therefore, the properties of Brownian motion refer to a value of Brownian motion that might be extreme (i.e., negative infinity or positive infinity) at the next time point. Thus, the bank’s assets could show a sudden and large decrease if the value of the Brownian motion is an extreme value on the left hand side of the normal distribution at the next time point.
17
the discussion, we note that the spreads for insured deposits and uninsured deposits are different. We then use a weighted interest rate spread to address the influence of the different spreads of these two types of deposits on DI premiums. The weighted interest rate spread can be expressed as follows:
(t ) w1 (t )1 (t ) w2 (t ) 2 (t ) ,
(19)
where
deposits, respectively;
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1 (t ) and 2 (t ) are the interest rate spreads for insured deposits and uninsured
w1 (t ) and w2 (t ) are the weights for insured deposits and uninsured deposits, respectively; we have w1 (t )
B1 (t ) B (t ) and w2 (t ) 2 . B(t ) B(t )
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Putting the weighted interest rate spread into Equation (2), we can still obtain the formulas for IPP(t , T ) and IPP A (t , T ) , shown as Equations (11) and (18), respectively.
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Thus, one can also address the influence of uninsured deposits on DI premiums using our
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formulas. For example, if one wants to analyze the influence of 2 (t ) and w2 (t ) on
IPP A (t , T ) , (s)he can use the following equation:
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IPP A (t , T ) IPP A (t , T ) s IPP A (t , T ) IPP A (t , T ) s , and . 2 (t ) s 2 (t ) w2 (t ) s w2 (t ) (20)
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Because the changes of the interest rate spread can be influenced by a variety of
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factors, our second discussion is to specify the interest rate spread as a multi-factor function, such as the exogenous variables (i.e., the bank’s monopoly power, the competitive factors, and so on) and the original influential variables, on a DI premium (i.e., the asset risk V , the bank’s leverage ratio
B (t ) , and the deposit insurance premium IPP A (t , T ) ). V (t )
We can express the function of the interest rate spreads as follows:
18
(t ) f (,
B(t ) , V , IPP A (t , T )) , V (t )
(21) where f () is an operator of the function; represents the exogenous variable, such as the bank’s monopoly power, the competitive factors, and so on. In such a specification, we let
s E[ (t )] . The simplest assumption for describing this function is a linear relationship. Under
(t ) a0 a1 a2
B(t ) a3 V a4 IPP A (t , T ) , V (t )
(22)
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this assumption, an affine function of the interest rate spreads can be described as follows:
where a0 is the drift term and ai is the relative magnitude of variable i ’s effect on the
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interest rate spread for i =1, 2, 3, and 4.
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The influence of each exogenous variable on the DI premium depends on its coefficients in this regression. For example, one can let represent the bank’s monopoly
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power. If the empirical result of Equation (22) shows a1 0 , we can infer that the bank’s monopoly power will increase the interest rate spread. Because an increase in the interest
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rate spread decreases the DI premium, one can infer that an increase in the bank’s monopoly power will decrease the DI premium. In contrast, if a1 0 , one can infer that the
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decrease in the bank’s monopoly power will increase the DI premium. To analyze the influences of
B (t ) and V on the premium, one must consider not V (t )
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only the effects of their coefficients on the interest rate spread, but also the original influences of these variables on the DI premium formula. According to our DI valuation formula, an increase in the bank’s leverage ratio
B (t ) will increase the DI premium but an V (t )
increase in the interest rate spread may decrease the DI premium. In general, if a bank intends to attract more and more depositors, it should increase the deposit rate. Then the
19
interest rate spread will decrease in this case. Thus, we expect to have a2 0 in Equation (22) since an increase in the leverage ratio
B (t ) will decrease the interest rate spread. V (t )
Accordingly, if we have a2 0 , one can infer that an increase in the bank’s leverage ratio will increase the DI premium. However, if the empirical result of Equation (22) shows that the bank’s leverage ratio will increase the interest rate spread (i.e., a2 0 ), there is a tradeoff between the effects of the bank’s leverage ratio and the interest rate spread on the DI premium. The finally effects of the bank’s leverage ratio on the DI premium will be
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ambiguous.
If the interest rate spread includes asset risk factor V , a risk-return trade-off may
also be found in the model. In general, the determination of the interest rate spread usually
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includes the compensation of risks (i.e., default risk). In other words, if the bank intends to have a higher interest spread rate, it may undertake more risky investments and this may
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increase its asset risk. In such a case, we expect a3 0 since there is a positive relationship
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between the interest rate spread and the asset risk. However, we know that increasing the spread rate will reduce the default risk, but increasing the asset risk will increase the default
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risk. The former will decrease the DI premium but the later will increase it. Thus, the finally effects of the asset risk on the DI premium are also ambiguous because of the riskreturn trade-off.
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Moreover, if the DI premiums ( IPP(t , T ) and IPP A (t , T ) ) are specified as a factor of
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the interest rate spread, the formulas for the DI premiums will be implicit functions. For example, IPP A (t , T ) g ( IPP A (t , T )) 0 , where g () is an operator of the function (i.e., Equation (18) with s E[ (t )] ). Thus, we cannot obtain the explicit closed-form formula for a DI premium. However, one can use a numerical method to solve for the DI premiums. If a4 0 , we may expect the DI premium to be greater than that calculated by our model, which assumes that the interest spread rate is a deterministic variable.
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Finally, Equation (16) shows that if a bank’s earnings from the interest rate spread are larger than its payout rate (i.e., the dividend rate) and the possibility of early bankruptcy is taken into account, the DI premium the bank should pay is larger if estimated from our model than from the traditional model. On the other hand, the DI premium is smaller from our model than from the traditional model if the interest rate spread is smaller than the dividend rate. These results are reasonable, because the DI premium is the insurer’s expected loss. If the drift of the asset return is positive (e.g., the earnings are greater than
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the payouts), the expected value of the bank’s assets increases during the contract period. In this situation, the likelihood of bankruptcy is determined only by the volatility of the assets. The expected loss calculated from our formula is larger than that calculated from the
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traditional formula, because our formula takes account of many possible losses for the
insurer before the maturity date of the contract. By contrast, if the bank’s earnings are less
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than its payouts, the likelihood of bankruptcy increases, not only because of the volatility of the bank’s assets, but also because of the decrease in their value. Because the drift of the
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asset return is a decreasing function of time, the expected value of the assets eventually falls below the value of the debt (i.e., bankruptcy) if there is a long time before the contract
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reaches maturity. The insurer’s loss calculated from a model that includes many possible bankruptcy time points (i.e., our model) is smaller than that calculated by a model that
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includes only one auditing time (i.e., the traditional model). This is because our model
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identifies the date of the bankruptcy as earlier than the traditional model can, and its application therefore can forestall an increase of the loss. Thus, IPP A (t , T ) IPP(t , T ) when s .
3. Calibration of the model’s parameters When valuating a DI premium, the bank’s total assets and the standard deviation of its returns must be calibrated. Some authors have used equity data to calibrate these two
21
parameters (Marcus and Shaked, 1984; Ronn and Verma, 1986). For simplicity, we let the interest rate spread be a constant or a deterministic variable. The following simultaneous equations reflect the method introduced by Ronn and Verma (1986):
E (t ) V (t )e ( s )(T t ) N ( x1 ) B(t ) N ( x2 ) , and (23)
E
V (t ) E (t ) V (t ) ( s )(T t ) V e N ( x1 ) V , E (t ) V (t ) E (t )
where
E (t ) is the value of a bank’s equity at the time t ;
E is the standard deviation of a bank’s equity return;
ln( x2
V (t ) ) 2 (t , T ) B(t ) . V2 (T t )
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x1
V (t ) ) 1 (t , T ) B(t ) ; and V2 (T t )
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ln(
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is the level of declaring bankruptcy;
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(24)
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Equation (23) shows that the value of the equity equals the present value of the surplus assets after the debt is paid off at maturity. The payoff of the equity is (V (T ) B(T )) at
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the maturity date.5 Equation (24) gives the standard deviation of the bank’s equity returns, derived using Ito’s Lemma.
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However, the calibrations of the bank’s assets and the standard deviation of its returns
5
In the DI valuation model (e.g. Merton, 1977), the DI premium is compensation to the deposit insurance company, which bears the possible loss. Thus, we must calculate the full possible amount for insured depositors if the value of the bank’s assets ( V (t ) ) is less than the total value of its deposits ( B(t ) ). The empirical implementation of the DI pricing model relies mostly on the Ronn and Verma (1986) estimation method. Since this estimation method considers the concept of tolerance for allowing the bank’s bankruptcy, the bank may go into default if V (t ) > B(t ) , where denotes the market tolerance for allowing the bank’s bankruptcy. These two thresholds for the theoretical model ( B(t ) ) and empirical method ( B(t ) ) in our study both refer to the traditional studies (Merton, 1977; Ronn and Verma, 1986).
22
calculated from the above equations may be inaccurate, because they do not take into account the probability of bankruptcy prior to maturity. We therefore revise the formulas to address this problem. As is well known, a bank’s stockholders often receive nothing when the bank goes bankrupt. Therefore, the bank’s equity can be regarded as the value of a down-and-out barrier option with zero rebate. Thus, we adopt the first-passage time model to derive the revised formulas. Based on this model, the closed-form formula for the equity
E (t ) V (t )e ( s )( T t ) [ N ( x1 ) ( B(t )[ N ( x2 ) (
B(t ) V (t )
B(t ) V (t )
) 2( s ) v
) 2( s ) v
2
2
( t ,T ) 1
( t ,T ) 1
N (h1 )]
N (h2 )] ,
(25) 1 (t , T ) V (t ) , and h2 h1 V2 (T t ) . V2 (T t )
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where h1
B(t )
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ln
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can be expressed as follows (Musiela and Rutkowski, 2002):
as follows: V (t ) (t ) V , E (t )
(26) where
E (t ) V (t )
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(t )
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E
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Moreover, using Ito’s Lemma, the standard deviation of the equity return is obtained
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e( s )(T t ) ( N ( x1 ) 2( s )( V2 (T t ))1 ( (2( s )( V2 (T t )) 1 1)(
B(t ) V (t )
B(t ) V (t )
)2( s )(V (T t )) 2
1
1
N (h1 ))
1
) 2( s )(V (T t )) N (h2 ) . 2
We then use these formulas to calibrate the bank’s assets and the standard deviation of its returns. One can obtain more reasonable DI premiums by including the parameters calibrated from Equations (25) and (26) in our pricing formula. 4. Empirical methods and results 23
We use data from banks listed on the Taiwanese stock exchange to illustrate the application of our model. At the end of 2010, 32 banks were publicly offered in Taiwan.6 The yearly data, including the banks’ equities, debts, and cash dividends, were taken from the Taiwan Economic Journal (TEJ) Databank.7 The sample period is 1980 to 2010. Some banks were integrated during the sample period, and their data are missing from the TEJ Databank. For estimating the insurance premiums of these integrated banks, we use only data that can be observed before the integrated date. In Table 1, the sample number is the number of years
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for which data were used in the empirical studies. For example, the sample number 27 for Bank 1 means that data for that bank were used for each of 27 years.
We use the daily stock return to calculate the yearly standard deviation of the equity
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returns for each bank. The daily stock return is calculated using the end-of-day stock price,
also obtained from the TEJ Databank. We use the daily stock returns for each year to obtain
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the variance of the daily stock returns in the corresponding year. Then we multiply the calculated value by the number of actual trading dates in the year to obtain the standard
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deviation of equity for the year.8 During the sample period, the Taiwanese government encouraged financial institutions to merge, creating a single financial holding company. For
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these banks, we use the holding company’s stock returns as the bank’s stock returns after the date of the merger to calculate the variance of the daily stock returns. The main reason
6
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for doing so is that only the holding company’s stock prices after that time are available.9
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Banks in Taiwan are small compared with those in the United States. At the end of 2015, there were 6,184 banks and 69 banks in the United States and Taiwan respectively. The average bank assets were 2.57 trillion USD and 0.2045 trillion USD in the United States and Taiwan respectively. These data were retrieved from the website of the Federal Deposit Insurance Corporation (FDIC, https://www5.fdic.gov/) and the Taiwanese Banking Bureau, Financial Supervisory Commission (FSC, https://www.banking.gov.tw/). The exchange rate on Dec. 31, 2015 was 33.066 NTD/USD, obtained from the Central Bank of Taiwan (http://www.cbc.gov.tw/). 7 TEJ Databank is a well-known databank containing financial information on Taiwan. Its website is http://www.tej.com.tw/. 8 In financial theory, the variance for day T ( T2 ) is usually calculated by the formula T2 T D2 , where D is the daily standard deviation. 9 For each merger case, there is a constant conversion ratio of the stock for the bank to that of the holding company at the merger time. Thus, for such a constant ratio, the bank’s returns are the same as the returns of the holding company.
24
Table 1 presents descriptive statistics for each bank’s yearly data, specifically, the mean and standard deviation of the equity, mean debt, mean cash dividends, and the sample number for the bank. Next, we compare results from the different models. The assets and the standard deviations of the asset returns are calibrated from Equations (25) and (26), hereafter referred to as the FPT formulas, and from Equations (23) and (24) from Ronn and Verma
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(1986), hereafter referred to as the RV formulas. We use the grid search method to calibrate the assets and the standard deviations of the asset returns. We begin by generating 1000
possible asset values within the range [ A0 (t ), A0 (t ) 2B(t )] , where A0 (t ) E (t ) B(t ) , as
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well as 200 corresponding standard deviations within the range [0.01 E , 2 E ] .10 The
squares error, which is defined as follows:
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(27)
Eˆ (t ) E (t ) 2 ˆ E2 (t ) E2 (t ) 2 ) ( ) , E (t ) E2 (t )
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sum of squares error (
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optimal estimated values of V (t ) and V are based on the minimum values of the sum of
where Eˆ (t ) and ˆ E2 (t ) are the respective mean and standard deviation of the equity,
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calculated from the FPT and RV formulas, given different values of V (t ) and V .
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We let s 0.02 , t 0 , T 1 year, and 0.97 , the latter based on Ronn and Verma (1986). Table 2 shows the estimated values of the bank’s assets and the standard deviations of its returns, based on the FPT and RV formulas. 10
In general, the volatility of assets is smaller than the volatility of equity in real cases. This small range is
used to reduce the time needed to calculate the estimates.
25
Table 2 shows that most of the estimates from the RV formulas are equal to or less than those from the FPT formulas. For example, the assets of Bank 1 are estimated as 891.250 billion from the FPT formulas and 888.840 billion from the RV formulas. Likewise, the estimated standard deviation of the returns is 4.418% from the FPT formulas and 4.330% from the RV formulas. However, in some cases (e.g., Banks 12, 13, 21, 22, 31, and 32) the two formulas give the same estimates. Therefore, we conclude that the model that takes account of a possible early bankruptcy gives equal or lower estimates for the
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assets and returns than the model that does not take this into account. In theory, we can infer that the lower the assets, the higher the DI premium; but the lower the standard
deviation of the returns, the lower the DI premium. Therefore, we cannot determine from
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the results based on the FPT formulas whether DI premiums increase or decrease in the tested circumstances.
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Table 3 shows these estimated DI premiums, based on the above parameters as estimated by the various formulas. The table shows premium estimates incorporating the
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interest rate spread alone, IPP from Equation (11), as well as estimates incorporating both the interest rate spread and the early-bankruptcy risk, IPP A from Equation (18). For
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example, the IPP values for Bank 1 (Chang Hwa Bank), calculated in accordance with the parameters estimated by the FPT and RV formulas, are 3.0400 103 and 3.6301 103
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respectively; the corresponding IPP A values are 3.9929 103 and 5.0657 103 . Table 3
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also shows that IPP A is always higher than the corresponding IPP , regardless of the parameters estimated by the FPT and RV formulas. The means of the IPP / IPP A ratios ( IPP divided by IPP A ) are nearly 74% for the FPT formulas and 71% for the RV formulas. The DI premiums in the second and fifth columns are calculated using both our model (i.e., the FPT formulas and our pricing formula) and the traditional model (i.e., the RV formulas and the European pricing formula). Using the values in these two columns, we find that the mean IPP / IPP A ratio is 79.19%. In other words, on average, DI premiums for Taiwanese 26
banks are underestimated by 20.81% (i.e., 1-79.19%) using the traditional pricing formula. This result suggests that insurers who use the traditional model are likely to conclude that they have insufficient funds to cover the bank’s default risk, because they ignore the possibility of early bankruptcy in calculating the DI premium. All banks in Taiwan are forced to participate in the DI system. The maximum insured DI is 3 million (NT dollars) for each depositor. That is, for each depositor, if the total
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amount of the deposit account (of any kind, such as demand deposit, fixed deposit, and so
on) is less than 3 million, the DI insurer will protect all the losses from the deposit account. Moreover, since 2013, all insured banks are subject to a risk-based premium system that has
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five premium rates: 5bp, 6bp, 8bp, 11bp, and 15bp. For a bank, the DI premium is
determined by its risk grading indicator, calculated by two methods: “The BIS Capital
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Adequacy Ratio” and “The Composite Score of the Risk-based Premium Rating System”.11 In this paper, the DI premiums range from 0.4bp to 178bp, as calculated by Merton’s
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model. Moreover, the calculated premiums are larger than 15bp for 15 of the 32 banks. The current DI premium in Taiwan seems to be underestimated compared to the theoretical DI
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premium from Merton’s model. The same conclusion can be reached by using our model, since the DI premiums estimated by our model are usually larger than those estimated by
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Merton’s model. This conclusion is not surprising, because many previous studies have found that the theoretical risk-based premium is usually higher than the actual DI premium
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(e.g., Ronn and Verma, 1986; Chuang, Lee, Lin and Yu, 2009). We used our model to analyze the changes in the DI premium due to specific events
in the 1997 Asia financial crisis. We divided the sample period into three sub-periods: 1996, 1997, and 1998. To avoid confusion on the effects of the IPO (Initial Public Offer), we
11
See the data in Central Deposit Insurance Corporation, Taiwan. Retrieve from: http://www.cdic.gov.tw/.
27
selected the 11 banks (see Table 1) which were traded in the stock market for more than 2 years as of 1996. We estimated the parameters (i.e., s ,
B ( 0) and V ) using the FPT V ( 0)
formula and calculated the DI premiums (i.e., IPP and IPP A ) for each bank in each subperiod. Table 4 shows the means of the estimated parameters and of the calculated DI premiums. As shown in Table 4, the mean V was 12.7769% in 1997. It was 9.3790% and
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9.6985% in 1996 and 1998, respectively. This result implies that the bank’s asset volatility increased due to the 1997 Asian financial crisis. The interest rate spreads were 0.0242,
0.0233, and 0.0190 in 1996, 1997, and 1998, respectively. The mean debt/asset ratios were
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0.7240, 0.7118, and 0.7865. Based on these parameters, the mean DI premiums ( IPP A )
were 0.0276bp, 0.6787bp, and 1.0117bp in 1996, 1997, and 1998, respectively. Compared
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with the results in 1996 and 1997, the DI premium increased during 1997 due to the
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increases in asset volatility. The DI premium for 1998 shows an increase after the financial crisis, even though the banks’ asset volatility decreased. We can infer that this could be
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caused by both the decreases in the interest rate spread and the increases in the debt/asset ratio.
Since 2001, the Taiwanese government has encouraged financial institutions to
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merge, creating a single financial holding company. As shown in Table 1, there are thirteen
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financial holding companies (Banks 2, 3, 4, 5, 11, 12, 13, 19, 20, 21, 22, 29 and 31) in our sample. Only Bank 31 merged in 2010. The others merged in 2001 or 2002. We therefore chose the banks which became financial holding companies in 2001 or 2002 to analyze the consolidation effects. In the sample, eight banks (3, 4, 11, 19, 20, 21, 22 and 29) became financial holding companies in 2001, and four banks (Banks 2, 5, 12 and 13) became financial holding companies in 2002. For purpose of the analyses, we defined three sub-
28
samples: Sub-sample 1 contains the bank’s data from the year before consolidation; Subsample 2 contains the bank’s data from the year of consolidation; and Sub-sample 3 contains the bank’s data from the year after consolidation. We assigned each bank’s data to the corresponding sub-sample based on the bank’s real consolidation year. For example, Bank 2 became a financial holding company in 2002. Thus, for Bank 2, Sub-sample 1 contains its data on 2001; Sub-sample 2 contains its data on 2002; and Sub-sample 3 contains its data on 2003. For Bank 3, its consolidation year is 2001. Thus, for Bank 3, Sub-
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sample 1, Sub-sample 2, and Sub-sample 3 contain data on years 2000, 2001, and 2002, respectively. For each bank in each sub-sample, we estimated its parameters (i.e., s ,
B ( 0) V ( 0)
and V ) by the FPT formula, and then we calculated its DI premiums (i.e., IPP and IPP A )
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using our model. The means of each parameter and DI premium for each sub-period are
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shown in Table 4 for the analyses of the consolidation effect.
As shown in Table 4, the mean interest rate spreads are 0.0356, 0.0456, and 0.0456 in
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Sub-sample 1, Sub-sample 2, and Sub-sample 3, respectively. Accordingly, the interest rate spreads for these banks increased due to the consolidation effect. Moreover, in Sub-sample
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1, the means of the debt/asset ratios and asset volatility are 0.9698 and 5.0483%, respectively. In Sub-sample 2, they are 0.9674 and 4.8653%. In Sub-sample 3, they are
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0.9610 and 4.5170%. These results show that the debt/asset ratios and asset volatility decreased due to the consolidation effect. Accordingly, the means of the DI premiums
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( IPP A ) are 50.7015bp, 31.8128bp, and 17.4516bp in Sub-sample 1, Sub-sample 2, and Sub-sample 3, respectively. Thus, we can infer that the increases of the interest rate spreads and the decreases of both the debt/asset ratios and asset volatility resulted from the consolidation effect, which induced a decrease in the DI premiums for the consolidated banks. As mentioned in a previous section, our model indicates that if a bank’s interest rate 29
spread is smaller than its dividend yield, the DI premium is higher using our model than using the traditional model. Figure 2 displays the empirical evidence for the sensitivity of DI premiums to changes in the interest rate spread. We take Bank 1 as our example. As shown in Table 2, its average estimated dividend yield, computed by the FPT formulas, is 0.16%.12 Therefore, we let the interest rate spread range from 0.1% to 1%. As shown in Figure 2, both IPP and IPP A are negatively correlated with the interest rate spread. In other words, if a bank’s earnings increase because of the interest rate spread, the DI
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premium decreases. We have also proven that if the interest rate spread is smaller (larger) than the dividend yield, IPP A is smaller (larger) than IPP . Figure 2 also shows that the
IPP A curve is flatter than the IPP curve. In other words, compared with a model that takes
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into account only the likelihood of bankruptcy at the maturity date, the sensitivity of the DI
of a possible early bankruptcy.
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5. Conclusion
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premium to the change in the interest rate spread is less using the model that takes account
The DI system plays an important role in a country’s financial market. A well-designed
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system can facilitate the country’s economic development because it can prevent an economic crisis from spreading if the banks go bankrupt. The maintenance of a good DI
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system depends heavily on the determination of a fair DI premium. Therefore, the question of how best to valuate a DI premium has received a great deal of attention, not only from
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DI companies but also from academic researchers. The main objective of the present study was to provide an accurate closed-form pricing formula and optimal empirical methods for determining fair DI premiums. In practice, the interest rate spread affects a bank’s earnings, and this will in turn 12
The estimated dividend yield equals the cash dividend divided by the estimated assets.
30
further influence the value of the bank’s assets and the likelihood of it going bankrupt. In addition, it must also be kept in mind that a bank can go bankrupt prior to the maturity date of the DI contract. These two factors should influence whether a DI premium is accurately valuated. Therefore, we incorporated these two factors into our model for deriving a pricing formula for a DI premium. Our consideration of the early bankruptcy risk also led us to introduce new empirical formulas derived from the first-passage time model to calibrate the necessary parameters, namely, the bank’s assets and the standard deviation of its returns.
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Therefore, our model should help DI institutions more precisely determine fair premiums.
We collected data from the 32 banks listed on the Taiwan stock exchange to illustrate the operation of our model. We have shown that, regardless of the theoretical inferences or
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the empirical evidence, DI premiums are negatively correlated with the interest rate spread. We also used numerical results to prove that the sensitivity of DI premiums to changes in
possibility of early bankruptcy.
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the interest rate spread is lessened if the premium is priced by a model that allows for the
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Our study also compared results that were calculated based on the parameters calibrated by our formulas (FPT formulas) and by Ronn and Verma’s (1986) formulas (RV
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formulas). Our results indicate that most of the values calibrated by the RV formulas are equal to or lower than those calibrated by the FPT formulas. In other words, the RV
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formulas can underestimate the bank’s assets as well as its risk. In addition, DI premiums based on the possibility of early bankruptcy ( IPP A ) are always higher than premiums not
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based on a consideration of this risk ( IPP ), regardless of whether the FPT or RV formulas are used to calibrate the parameters. Our results indicate that DI premiums calculated by the model that does not take account of the early bankruptcy risk are underpriced by 20.81%. If a DI company prices its premiums using the traditional model and ignores the risk of early bankruptcy, the premiums will be too low. Currently, we have available for use in our model only the DI values for each bank in 31
our sample. In future research, it may be interesting to examine how these values vary over time and whether their dynamics provide a good match to important news events. It may also be interesting to ask how likely it is that banks will transfer their risks to DI companies if the premium is calculated by our model. In addition, our model can be enriched and enhanced, and allowing for discrete jumps in assets would increase its practical relevance. Finally, there are some common phenomena that could influence the valuation of a DI premium. For example, if a bank fails or goes bankrupt, government regulators usually
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delay resolving the default, using the policies of delayed resolution and capital forbearance. Depositor preference arrangements can also influence the DI premium. If the DI valuation model considers the effects caused by the policies of delayed resolution or capital
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forbearance for defaulted banks and the seniority of insured deposits, the PD (probability of default) and LGD (loss given default) for banks may be altered. In future research, it may
their influence on DI premiums.
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Appendix A
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be interesting to incorporate these phenomena into our DI valuation model to understand
This appendix shows how to obtain Equation (7). In pricing theory, any asset process can be
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chosen as the numeraire asset (Davis, 1998). For this paper, we chose a money market account to serve as the numeraire asset. By doing it, the strike price becomes a constant in
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the valuation. The money market account is defined as follows: T
M (t ,T ) exp( r f (u )du ) and M (t , t ) 1.
(A1)
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t
Accordingly, the dynamic process of M (t , u) under the risk-neutral probability measure can be expressed as follows: dM (t , u ) r f (u )du b(t , u )dWr (u ) , M (t , u )
(A2)
where dWr (u) is a Brownian motion under the risk-neutral probability measure. When choosing a money market account M (t , u) to serve as the numeraire asset, 32
V (u ) , B1 (u ) , and B(u ) can be transferred as the new processes of FMV (t , u ) , FMB1 (t , u) , and FMB (t , u) . They can be represented as follows: FMV (t , u )
V (u ) ; M (t , u )
(A3)
FMB1 (t , u )
B1 (u ) ; and M (t , u )
(A4)
FMB (t , u )
B(u ) . M (t , u )
(A5)
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According to Equation (A3), using Taylor’s polynomials and Itoˆ ’s Lemma, we have:
dFMV (t , u ) 1 ( (u ) )du | V b(t , u ) |2 du | V b(t , u ) | dW F (u ) , V 2 FM (t , u )
(A6)
-p
where | V b(t , u) |2 V2 2V b(t , u) b2 (t , u) and dW F (u ) is a standard Brownian
FMV (t , T ) FMV (t , t ) exp(( s )(T t ))
T 1 T | V b(t , u ) |2 du | V b(t , u ) |dW F (u )) t 2 t
lP
exp(
re
motion under the risk-neutral probability measure. Thus, we have
(A7) where
T 1 T | V b(t , u ) |2 du | V b(t , u ) |dW F (u )) . t 2 t
ur
exp(
na
FMV (t , t ) exp(( s )(T t )) ,
Jo
We have E[ ] 1, where E[] is an expectation operator. Thus is a Radon-Nikodym derivative that can be used for changing the measure. Because we have
FMV (t , t ) FMV (t , t ) exp( (s )(t t )) and
E[exp(
T 1 T | V b(t , u ) |2 du | V b(t , u ) |dW F (u ))] 1 , t 2 t
we can obtain
33
E[ FMV (t , T ) exp( (s )(T t ))] FMV (t , t ) exp( (s )(t t )) . Accordingly, FMV (t , u) exp( (s )(u t )) is a martingale process under the risk-neutral measure. Moreover, according to Equation (A4), we have B1 (u ) B (t ) M (t , u ) 1 B1 (t ) , for t u T . M (t , u ) M (t , u )
FMB1 (t , u )
(A8)
By the same token, we have
FMB (t , u) B(t ) . Thus, FMB1 (t , T ) and FMB (t , T ) are deterministic values.
FMIP (t , t )
IP(t , T ) IP(T , T ) and FMIP (t , T ) , M (t , t ) M (t , T )
-p
We now define the following:
ro of
(A9)
(A10)
re
where FMIP (t , u) is a new processes transferred from IP(u, T ) by choosing money market
lP
account M (t , u) as the numeraire asset.
In pricing theory, the asset values discounted by money market account are
na
martingales (Duffie, 1996). We have:
FMIP (t , t ) E[ FMIP (t , T )] E[
B1 (T ) V (T )) B(T ) ] M (t , T )
( B1 (T )
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ur
B1 (T ) V (T ) B1 (T ) B(T ) E[( ) ] M (t , T ) M (t , T )
E[( FMB1 (t , T )
FMB1 (t , T ) V FM (t , T )) ] B FM (t , T )
FMB1 (t , T ) V E[ F (t , T ) I D B FM (t , T ) I D ] FM (t , T ) B1 M
34
FMB1 (t , t ) E[ I D ]
FMB1 (t , t ) E[ FMV (t , T ) I D ] , FMB (t , t )
(A11)
where I {} is an indicator function and D {FMV (t , T ) FMB (t , T )} . Because E[ I D ] is the default probability, we have:
E[ I D ] P( FMV (t ,T ) FMB (t ,T )) , where P() is the probability operator. (A12)
E[ I D ] P( FMV (t , t ) exp(( s )(T t )) exp(
ln( P(
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FMB (t , t ) 1 ) ( s )(T t ) v 2 (t , T ) V FM (t , t ) 2 ) v(t , T )
B(t ) 1 ) ( s )(T t ) v 2 (t , T ) V (t ) 2 ) v(t , T )
N ( y2 (t , T )) ,
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P(
T 1 T 2 | b ( t , u ) | du | V b(t , u ) |dW F (u )) FMB (t , T )) V t t 2
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ln(
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By substituting Equation (A7) into Equation (A12), we get the following:
(A13)
T
where ~ N (0,1) and v 2 (t , u ) | V b(t , u ) |2 du . Moreover, we have
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t
E[ FMV (t , T ) I D ] FMV (t , t ) exp(( s )(T t )) E[I D ] .
(A14)
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As shown in Equation (A7), is a Radon-Nikodym derivative used for changing the measure. Thus, we have
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E[I D ] N ( y1 (t , T )) , and y1 (t , T ) y2 (t , T ) v(t , T ) .
(A15)
According to Equations (A11)-(A15), FMIP (t , t ) can be rewritten as follows:
FMB1 (t , t ) V FM (t , t )e( s )( T t ) N ( y1 (t , T )) . F (t , t ) F (t , t ) N ( y2 (t , T )) B FM (t , t ) IP M
B1 M
Using Equations (A1), (A3)-(A5), we can obtain Equation (7).
35
(A16)
Appendix B This appendix gives the derivation of Equations (14) and (15). According to Equation (13), we have the following:
IP A (t d 2 , T ) E[
IP A (t d 1 , T ) (1 I D (td 1 ) ) | Ftd 2 ] M (t d 2 , t d 1 ) V (t d 1 ) ) B(t d 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , t d 1 )
E[ E[
IP(t d 1 , T ) (1 I D (td 1 ) ) | Ftd 2 ] M (t d 2 , t d 1 )
V (t d 1 ) ) B(t d 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , t d 1 )
IP(td 2 , T ) E[
IP(td 1, T ) I D (td 1 ) | Ftd 2 ] M (td 2 , td 1 )
V (td 1 ) ) B(td 1 ) I D (td 1 ) | Ftd 2 ] . M (t d 2 , t d 1 )
B1 (t d 1 )(1
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E[
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E[
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B1 (t d 1 )(1
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B1 (t d 1 )(1
(B1)
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For a traditional DI pricing formula, we have
IP(td 1 , T ) B1 (td 1 ) E[ I D(td ) | Ftd 1 ] B1 (td 1 )
V (td 1 ) ( s ) t * e E [ I D (td ) | Ftd 1 ] B(td 1 )
ur
By substituting IP(td 1 , T ) into Equation (B1), we get the following:
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IP A (t d 2 , T ) IP(td 2 , T ) E[
E[
B1 (td 1 )
B1 (td 1 )(1 E[ I D (td ) | Ftd 1 ]) M (td 2 , td 1 )
I D (td 1 ) | Ftd 2 ]
V (td 1 ) (1 e( s ) t E *[ I D (td ) | Ftd 1 ]) B(td 1 ) I D (td 1 ) | Ftd 2 ] , M (td 2 , td 1 )
where E *[ I D (td ) | Ftd 1 ] E[ I D(td ) ] , and is a Radon-Nikodym derivative used for changing the measure. 36
(B2)
If we let e ( s ) t 1 ( s)t (( s)t ) 2 , we have:
1 e( s ) t E *[ I D(td ) | Ftd 1 ] 1 (1 ( s)t ) E *[ I D(td ) | Ftd 1 ] 1 E *[ I D(td ) | Ftd 1 ] (( s)t ) E *[ I D(td ) | Ftd 1 ] O(t ) , (B3) where O(t ) is a function of the higher order of t . If we let t tend toward zero, we
IP A (t d 2 , T ) IP(td 2 , T ) E[
E[ B1 (td 1 )
I D (td 1 ) | Ftd 2 ]
V (td 1 ) * E [1 I D (td ) | Ftd 1 ] B(td 1 ) I D (td 1 ) | Ftd 2 ] M (td 2 , td 1 )
V (td 1 ) ( s)tE *[ I D (td ) | Ftd 1 ] B(td 1 ) I D (td 1 ) | Ftd 2 ] . M (td 2 , t d 1 )
re
E[
M (td 2 , td 1 )
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B1 (t d 1 )
B1 (td 1 ) E[1 I D (td ) | Ftd 1 ]
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obtain O(t ) 0 . Thus, Equation (B1) can be rewritten as follows:
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(B4)
According to our specifications, I D (td 1 ) means that the bank defaulted at time t d 1 .
na
Because td 2 td 1 td , if the bank defaulted at time t d 1 , it will still be in default at time
t d . That is, I D (td ) 1 when I D (td 1 ) 1 . Therefore, we have E[(1 I D(td ) )]I D(td 1 ) 0 , no
ur
matter whether I D (td 1 ) 1 or 0. By the same token, we have E *[(1 I D(td ) )]I D (td 1 ) 0 . We
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therefore have
E[
B1 (td 1 ) E[1 I D (td ) | Ftd 1 ] M (td 2 , td 1 )
I D (td 1 ) | Ftd 2 ] 0 ; and
(B5) B1 (td 1 ) E[
V (t d 1 ) * E [1 I D (td ) | Ftd 1 ] B(t d 1 ) I D (td 1 ) | Ftd 2 ] 0 . M (td 2 , td 1 ) 37
(B6) In addition, we have E *[ I D (td ) ] 1 , if I D (td 1 ) equals 1. We conclude that E *[ I D (td ) ]I D (td 1 )
I D (td 1 ) . We therefore have B1 (t d 1 ) E[
V (td 1 ) ( s)tE *[ I D (td ) | Ftd 1 ] B(td 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , td 1 ) V (td 1 ) ( s)t B(td 1 ) I D (td 1 ) | Ftd 2 ]) . M (t d 2 , td 1 )
B1 (t d 1 )
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E[
(B7)
Using the results of Equations (B4)-(B7), we find that the sum of the second and the third terms on the right hand side of Equation (B1) can be expressed as follows:
V (td 1 ) ( s)t B(td 1 ) I D (td 1 ) | Ftd 2 ] . M (td 2 , td 1 )
B1 (td 1 )
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E[
-p
IP(td 1, T ) I D (td 1 ) | Ftd 2 ] E[ M (td 2 , td 1 )
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E[
V (td 1 ) ) B(td 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , t d 1 )
B1 (t d 1 )(1
(B8)
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Using the previous results, we can rewrite Equation (B4) as follows:
(B9)
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IP A (t d 2 , T ) IP(td 2 , T ) E[
B1 (td 1 ) t B(t d 1 ) I D (td 1 ) | Ftd 2 ] . M (t d 2 , td 1 )
( s )V (t d 1 )
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This is the same as Equation (14). According to Equation (13), the following result can be obtained for the time t d 3 :
IP A (td 2 , T ) IP (td 3 , T ) E[ (1 I D (td 2 ) ) | Ftd 3 ] M (td 3 , td 2 ) A
38
V (t d 2 ) ) B(t d 2 ) I D (td 2 ) | Ftd 3 ] . M (t d 3 , t d 2 )
B1 (td 2 )(1 E[
(B10) According to Equation (B9), the first term on the right hand side of Equation (B10) can be expressed as follows:
E[
IP A (td 2 , T ) (1 I D (td 2 ) ) | Ftd 3 ] M (td 3 , td 2 ) B1 (t d 1 ) t B(t d 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , t d 1 ) (1 I D (td 2 ) ) | Ftd 3 ] M (t d 3 , t d 2 )
IP(t d 2 , T ) E[ E[
IP(td 2 , T ) IP(td 2 , T ) | Ftd 3 ] E[ I D (td 2 ) | Ftd 3 ] M (t d 3 , td 2 ) M (td 3 , td 2 )
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E[
ro of
( s )V (td 1 )
B1 (td 1 ) t B(td 1 ) I D (td 1 ) (1 I D (td 2 ) ) | Ftd 3 ] M (td 3 , t d 1 )
E[
IP(td 2 , T ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
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IP(td 3 , T ) E[
re
( s )V (td 1 )
B1 (td 1 ) t B(td 1 ) I D (td 1 ) (1 I D (td 2 ) ) | Ftd 3 ] . M (td 3 , t d 1 )
( s )V (td 1 )
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E[
(B11)
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Again, we have I D (td 1 ) I D (td 2 ) I D (td 2 ) . The following result can be obtained: B1 (t d 1 ) t B(t d 1 ) I D (td 1 ) (1 I D (td 2 ) ) | Ftd 3 ] M (t d 3 , t d 1 )
( s )V (t d 1 )
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E[
B1 (t d 1 ) t B(t d 1 ) I D (td 1 ) | Ftd 3 ] M (td 3 , td 1 )
( s )V (td 1 )
E[
E[
B1 (td 2 ) t B(td 2 ) I D (td 2 ) | Ftd 3 ] . M (td 3 , td 2 )
( s )V (td 2 )
(B12) 39
In Equation (12), we let E[
V (td 1 ) V (td 2 ) I D (td 2 ) | Ftd 3 ] E[ I D (td 2 ) | Ftd 3 ] . In the M (td 3 , td 1 ) M (td 3 , td 2 )
specification of the model, td 3 td 2 td 1 . If bank defaulted at time t d 2 (i.e., I D (td 2 ) 1 ), there is no bank value (i.e., V (td 1 ) ) at time td 1 . Thus, we only need to calculate the expected bank value at defaulted time t d 2 . The pricing formula for time t d 3 can be expressed as follows:
E[
IP(td 2 , T ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
-p
E[
B1 (td 2 ) t B(td 2 ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
( s )V (td 2 )
ro of
IP A (td 3 , T ) IP(td 3 , T ) E[
B1 (t d 1 ) t B(td 1 ) I D (td 1 ) | Ftd 3 ] M (t d 3 , t d 1 )
( s )V (t d 1 )
V (t d 2 ) ) B(t d 2 ) I D (td 2 ) | Ftd 3 ] . M (t d 3 , t d 2 )
B1 (td 2 )(1
re
E[
lP
(B13) Because we have:
na
IP(td 2 , T ) B1 (td 2 ) E[ I D(td ) | Ftd 2 ] B1 (td 2 )
V (td 2 ) ( s ) 2 t * e E [ I D (td ) | Ftd 2 ] . B(td 2 )
Thus, the last two terms on the right hand side of Equation (B13) can be expressed as
ur
follows:
IP(td 2 , T ) I D (td 2 ) | Ftd 3 ] E[ M (td 3 , td 2 )
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E[
E[
E[
B1 (td 2 )(1 E[ I D (td ) | Ftd 2 ]) M (td 3 , td 2 )
B1 (td 2 )
V (t d 2 ) ) B(t d 2 ) I D (td 2 ) | Ftd 3 ] M (t d 3 , td 2 )
B1 (t d 2 )(1
I D (td 2 ) | Ftd 3 ]
V (td 2 ) (1 e( s ) 2 t E *[ I D (td ) | Ftd 2 ]) B(td 2 ) I D (td 2 ) | Ftd 3 ] . M (td 3 , td 2 )
Again, we have: 40
(B14)
1 e( s ) 2t E *[ I D (td ) | Ftd 1 ] 1 E *[ I D(td ) | Ftd 1 ] (2( s)t ) E *[ I D(td ) | Ftd 1 ] O(t ) . If the bank defaulted at time t d 1 or t d 2 , it will still be in default at time t d . Thus, we also have E[(1 I D(td ) )]I D(td 2 ) 0 and E *[ I D (td ) ]I D (td 1 ) I D (td 1 ) . Using the results shown in Equations (B4)-(B7), we have: IP(td 2 , T ) I D (td 2 ) | Ftd 3 ] E[ M (td 3 , td 2 )
V (t d 2 ) t B(t d 2 ) I D (t d 2 ) | Ft d 3 ] . M (t d 3 , t d 2 )
( s) B1 (t d 2 ) 2 E[
(B15)
re
E[
B1 (t d 1 ) t B(td 1 ) I D (td 1 ) | Ftd 3 ] M (t d 3 , t d 1 )
( s )V (t d 1 )
B1 (td 2 ) t B(td 2 ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
( s )V (td 2 )
lP
IP A (td 3 , T ) IP(td 3 , T ) E[
B1 (td 2 ) t B(td 2 ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
( s )V (td 2 )
na
2 E[
(B16)
ur
IP(td 3 , T )
-p
Therefore, we have
ro of
E[
V (t d 2 ) ) B(t d 2 ) I D (td 2 ) | Ftd 3 ] M (t d 3 , td 2 )
B1 (t d 2 )(1
d 1
E[
i d 2
B1 (ti ) B(ti ) I D (ti ) | Ftd 3 ] . M (t d 3 , ti )
( s )V (ti )
Jo
Going backwards recursively from Equation (B16) to obtain Equation (15), we get d 1
IP A (t , T ) IP(t , T ) E[ i 1
B1 (ti ) B(ti ) I D (ti ) | Ft ] . M (t , ti )
( s )V (ti )
(B17) Then we obtain
41
B1 (u ) B (t ) V (u ) B(u ) I D (u ) | Ft ] ( s ) 1 E[ I D (u ) | Ft ] M (t , u ) B(t ) M (t , u )
( s )V (u ) E[
( s ) e( s )( u t )V (t )
B1 (t ) E[ I D* (u ) | Ft ] B(t )
( s ) e( s )( ut )V (t )
B1 (t ) N ( y1 (t , u )) . B(t )
(B18)
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By substituting this result into Equation (B17), we get Equation (15).
42
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Chang, C. and S. Chung, R.-J. Ho, and Y.-J. Hsiao, 2018, Revisiting the Valuation of
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Chen, A. H., N. Ju, S. C. Mazumdar and A. Verma, 2006, Correlated Default Risks and Bank Regulations, Journal of Money, Credit, and Banking 38(2), 375-398.
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Chuang, H. L., S. C. Lee, Y. C. Lin and M. T. Yu, 2009, Estimating the Cost of Deposit Insurance with Stochastic Interest rates: the Case of Taiwan, Quantitative Finance 9(1),
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Chung, S., 1999, American Option Valuation under Stochastic Interest Rates, Review of
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Davis, K., 2015, Depositor preference, bail-in, and deposit insurance pricing and design,
ur
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Davis, M. H. A., 1998, A Note on the Forward Measure, Finance Stochastic 2(1), 19-28.
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Duan, J. C., A. F. Moreau and C.W. Sealey, 1992, Fixed Rate Deposit Insurance and RiskShifting Behavior at Commercial Banks, Journal of Banking and Finance 16(4), 715742.
Duan, J. C. and J. G. Simonato, 2002, Maximum Likelihood Estimation of Deposit Insurance Value with Interest Rate Risk, Journal of Empirical Finance 9(1), 109-132. Duan, J. C. and M. T. Yu, 1994, Assessing the Cost of Taiwan’s Deposit Insurance, Pacific43
Basin Finance Journal 2(1), 73-90. Duan, J. and M. Yu, 1999, Capital standard, forbearance and deposit insurance pricing under GARCH, Journal of Banking and Finance, 23, 1691- 1706. Duffie, D., 1996, Dynamic Asset Pricing Theory, published by Princeton University. Episcopos, A., 2004, The Implied Reserve of the Bank Insurance Funds, Journal of Banking and Finance 28, 1617-1635. Finger, C. C., 2002, CreditGrades: Technical Document, Risk Metrics Group, Inc. pressed.
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http://www.creditrisk.ru/publications/files_attached/cgtechdoc.pdf.
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Heath, D., R. Jarrow and A. Morton, 1992, Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation, Econometrica 60(1), 77-
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Hwang, D. Y., F. S. Shie, K. Wang and J. C. Lin, 2009, The Pricing of Deposit Insurance
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Considering Bankruptcy Costs and Closure Policies, Insurance: Mathematics and Economics 33(10), 1909-1919.
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Kim, I. J., 1990, The Analytic Valuation of American Options, The Review of Financial Studies 3(4), 547-572.
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Leland, H. E., 1994, Corporate Debt Value, Bond Covenants, and Optimal Capital Structure, Journal of Finance, 49, 1213-1252.
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Lee, S. C, J. P. Lee and M. T. Yu, 2005, Bank Capital Forebearance and Valuation of Deposit Insurance, Canadian Journal of Administrative Sciences 22(3), 220-229.
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________, 1977, An Analytic Derivation of the Cost of Deposit Insurance and Loan Guarantee, Journal of Banking and Finance 1(1), 3-11. ________, 1978, On the Cost of Deposit Insurance When There are Surveillance Costs, Journal of Business 51, 439-452. Musiela, M. and M. Rutkowski, 2002, Martingale Methods in Financial Modelling, Springer. Nagarajan, S. and C.W. Sealey, 1995, Forbearance, deposit insurance pricing, and incentive
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Pennacchi, G. G., 1987a, Alternative Forms of Deposit Insurance: Pricing and Bank Incentive Issues, Journal of Banking and Finance 11(2), 291-312.
re
________, 1987b, A Reexamination of the Over (or Under) Pricing of Deposit Insurance, Journal of Money, Credit and Banking 19, 340-360.
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Rabinovitch, R., 1989, Pricing Stock and Bond Options When the Default Free Rate is Stochastic, Journal of Financial and Quantitative Analysis 24, 447-457.
na
Ronn, E. I. and A. K. Verma, 1986, Pricing Risk-adjusted Deposit Insurance: An Optionbased Model, Journal of Finance 41(4), 871-895.
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Santomero, A. M. and J. D. Vinso, 1977, Estimating the Probability of Failure for Commercial Banks and the Banking System, Journal of Banking and Finance 1(2),
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Shie, F. S., 2012, Pricing of Deposit Insurance Considering Capital Forbearance and Jump Risk, International Journal of Information and Management Sciences 23 (1), 1-18.
Shyu, D. and M. S. Tsai, 1999a, Deposit Insurance Rate and Risk-Shifting Behavior of Taiwan’s Commercial Banks, Review of Securities and Futures Markets 11, 1-28. _______, 1999b, Deposit Insurance Rate and Risk-Shifting Behavior of Taiwan’s New 45
Commercial Banks, Sun Yat-sen Management Review 8, 691-708. Vasicek, O.. 1977. An Equilibrium Characterization of the Term Structure, Journal of
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Financial Economics 5(2), 177-188.
46
Figure 1: Trend of the Interest Rate Spread in Taiwan Financial Markets 5.5 5
4 3.5 3
ro of
Interest Rate Spread (%)
4.5
2.5
1985
1990
1995 years
2000
re
1.5 1980
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2
2005
2010
Note: This figure shows the trend of the average interest rate spread for the top five Taiwanese banks. The data were obtained from the Central Bank of Taiwan. The website for the data is expressed in the following:
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https://www.cbc.gov.tw/ct.asp?xItem=995&ctNode=523&mp=1 .
47
Figure 2: Relationship Between Deposit Insurance Premiums and Interest Rate Spreads 66 64
60 58 56
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Deposit Insurance Premium (bp)
62
54 52 50
0.2
0.3
0.4 0.5 0.6 0.7 Interest Rate Spread (%)
0.8
0.9
1
re
46 0.1
-p
48
Note: The vertical axis denotes the deposit insurance premiums and the horizontal axis denotes the interest
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rate spreads for Bank 1. The dividend yield for the bank was 0.16% and the interest rate spreads ranged from 0.1% to 1%. The broken line represents premiums calculated using the traditional model. The asterisk line
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represents premiums calculated using our model.
48
Table 1: Basic information about the bank sample Symbol
Description
Equity Equity Std (billion) (%)
Bank 1
Chang Hwa Bank
48.574
43.299
839.430
1.448
27
Bank 2**
First Bank
64.967
41.992
968.030
2.098
27
Bank 3**
Hua Nan Bank
55.745
44.227
905.450
1.861
29
Bank 4**
China Development Industrial Bank
48.398
44.696
43.847
1.378
30
40.818
40.217
362.500
1.128
22
Mega International Commercial Bank
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Bank 5**
Debt Dividend Sample (billion) (billion) Number
Standard Chartered Bank (Taiwan)
10.709
42.127
179.260
0.101
24
Bank 7
King's Town Bank
7.544
42.569
93.114
0.011
28
Bank 8*
Kaohsiung Business Bank
108.530
52.307
55.669
0.009
18
Bank 9*
Taitung Business Bank
7.733
54.287
0.001
23
Bank 10
Taichung Commercial Bank
16.155
167.400
0.051
27
Bank 11**
Chinatrust bank
Bank 12**
Cathay United Bank
Bank 13**
Taipei Fubon Bank
Bank 14*
The Chinese Bank
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Bank 6
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43.592
69.951
39.823
825.480
2.510
20
165.120
37.726
908.730
3.287
13
77.592
34.611
860.080
2.292
12
12.378
29.813
170.020
0.026
12
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ur
25.740
Taiwan Business Bank
38.178
40.665
968.600
0.214
13
Bank 16
Bank of Kaohsiung
5.522
37.561
161.610
0.216
13
Bank 17
Cosmos Bank
131.160
42.851
169.460
0.020
16
Bank 18
Union Bank of Taiwan
18.130
33.742
224.460
0.045
16
Bank 19**
Bank of SinoPac
34.120
36.040
496.500
0.800
15
Bank 20**
E.SUN Commercial Bank
28.118
31.544
419.280
0.540
16
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Bank 15
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Note: This table gives data from the 32 banks listed on the Taiwanese stock exchange. The sample period is 1980 to 2010. The first column displays the bank symbols and the second column gives their names. The third and fifth columns give the means of the bank’s equity and debt during the sample period. The unit of currency is NTD (New Taiwanese Dollar). The fourth column gives the standard deviations of the bank’s equity returns during the sample period, calculated from the daily equity values from 1980 to 2010. The sixth column gives the means of the bank’s cash dividends. The final column gives the annual data for the bank. “*” means that the bank became integrated during the sample period. “**” means that the bank became a financial holding company.
Table 1: Basic information about the bank sample (continuously) Symbol
Description
Equity
Equity
Debt
Sample
(billion)
Number
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(billion) Std (%) (billion)
Dividend
Yuanta Bank
20.609
37.587
224.130
0.063
15
Bank 22**
TaiShin Bank
18.507
40.791
190.560
0.111
7
Bank 23
Far Eastern International Bank
17.819
39.171
223.580
0.185
16
Bank 24*
Chung Shing Bank 193.420
80.856
148.130
0.058
7
Bank 25
Public Bank
17.730
39.133
240.090
0.037
15
Bank 26
Entie Commercial Bank
30.034
39.338
205.790
0.094
16
50.428
85.775
0.693
7
Taiwan Bowa Bank
18.504
44.090
147.620
0.016
12
Bank 29**
JihSun Bank
20.973
40.969
189.340
0.000
15
Bank 30*
Bank of Overseas Chinese citi bank
18.488
49.510
258.110
0.000
9
Bank 31**
Taiwan Cooperative 72.674 Bank
34.270
2263.000
2.164
7
64.243
38.254
0.074
4
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Bank 28*
Bank 32
15.009
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Industrial bank of
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Bank 24
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Bank 21**
Bank of Taipei
1.834
Note: This table gives data from the 32 banks listed on the Taiwanese stock exchange. The sample period is
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1980 to 2010. The first column displays the bank symbols and the second column gives their names. The third and fifth columns give the means of the bank’s equity and debt during the sample period. The unit of currency is NTD (New Taiwanese Dollar). The fourth column gives the standard deviations of the bank’s equity returns during the sample period, calculated from the daily equity values from 1980 to 2010. The sixth column gives the means of the bank’s cash dividends. The final column gives the annual data for the bank. “*” means that the bank became integrated during the sample period. “**” means that the bank became a financial holding company.
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Table 2: Estimates of each bank’s assets and the standard deviation of its returns FPT formulas RV formulas Symbol Bank value Value Std Bank value Value Std (billion) (%) (billion) (%) 891.250 4.418 888.840 4.330 Bank 1 1035.400 4.415 1034.000 4.199 Bank 2 964.940 4.768 962.100 4.423 Bank 3 98.869 15.232 96.492 14.237 Bank 4 404.070 4.661 403.680 4.315 Bank 5 190.990 4.213 190.150 4.213 Bank 6 101.000 4.630 100.750 4.257 Bank 7 164.250 18.468 164.250 18.538 Bank 8 33.534 11.812 33.498 10.050 Bank 9 184.030 4.880 183.720 4.419 Bank 10 896.250 4.238 896.250 3.982 Bank 11 1074.800 4.447 1074.800 4.447 Bank 12 938.530 3.461 938.530 3.461 Bank 13 182.810 3.104 182.570 3.104 Bank 14 1011.300 4.067 1007.700 4.067 Bank 15 168.380 3.756 167.300 3.756 Bank 16 300.790 10.555 300.790 10.772 Bank 17 242.880 3.594 242.820 3.594 Bank 18 531.120 3.704 531.120 3.604 Bank 19 448.400 3.154 447.820 3.154 Bank 20 244.960 3.862 244.960 3.862 Bank 21 209.260 4.317 209.260 4.317 Bank 22 241.620 4.020 241.620 3.917 Bank 23 341.700 19.077 341.700 19.322 Bank 24 258.090 4.017 258.060 4.017 Bank 25 236.030 4.778 236.030 4.824 Bank 26 100.870 7.129 100.870 7.356 Bank 24 166.280 5.635 166.280 5.697 Bank 28 210.760 4.350 210.500 4.237 Bank 29 276.850 5.149 276.850 5.149 Bank 30 2339.300 3.427 2337.900 3.427 Bank 31 40.127 6.424 40.127 6.424 Bank 32
Note: The first column displays the bank symbols. The second and third columns give the estimates of the
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bank’s assets and the standard deviation of its returns (denoted as Value Std), calculated using the FPT formulas expressed in Equations (25) and (26). The fourth and fifth columns give the corresponding estimates using the RV formulas expressed in Equations (23) and (24). The interest rate spread is set at 0.02.
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Table 3: Estimates of the deposit insurance premiums on per deposit amount FPT formulas RV formulas A Symbol IPP (bp) IPP (bp) Ratios (%) IPP (bp) IPP A (bp) Ratios (%) 30.4001 39.9292 0.7613 36.3014 50.6575 0.7166 Bank 1 29.4050 37.1271 0.7920 29.9086 41.8513 0.7146 Bank 2 32.6478 41.5981 0.7848 36.0709 51.4171 0.7015 Bank 3 27.0516 28.5727 0.9468 17.7397 22.4546 0.7900 Bank 4 2.6696 3.8861 0.6870 3.6733 6.8676 0.5349 Bank 5 21.2569 30.0496 0.7074 29.7321 45.7347 0.6501 Bank 6 14.3088 19.2481 0.7434 14.0824 21.2620 0.6623 Bank 7 21.2507 26.2612 0.8092 21.1049 26.1115 0.8083 Bank 8 19.5293 26.2752 0.7433 20.5436 28.5467 0.7196 Bank 9 17.8828 0.7190 16.3412 25.4880 0.6411 Bank 10 12.8572 5.3693 0.8017 2.4195 3.2346 0.7480 Bank 11 4.3048 0.4871 0.8259 0.4023 0.4871 0.8259 Bank 12 0.4023 4.7749 0.7395 3.5311 4.7749 0.7395 Bank 13 3.5311 3.7177 0.6195 3.3884 5.6246 0.6024 Bank 14 2.3033 14.8959 22.4010 0.6650 18.8812 29.1027 0.6488 Bank 15 23.3836 0.6709 23.7056 37.0693 0.6395 Bank 16 15.6872 7.1569 0.8050 5.6557 7.0465 0.8026 Bank 17 5.7615 3.7647 0.6990 2.7285 3.9336 0.6936 Bank 18 2.6316 6.1122 0.6890 4.2108 6.1117 0.6890 Bank 19 4.2112 3.5587 0.6645 3.1912 4.9297 0.6473 Bank 20 2.3646 4.2217 0.6668 2.8148 4.2217 0.6668 Bank 21 2.8148 1.9772 0.7435 1.4701 1.9772 0.7435 Bank 22 1.4701 10.0246 0.7045 7.0591 10.0206 0.7045 Bank 23 7.0623 177.5118 190.3193 0.9327 Bank 24 177.5058 190.3130 0.9327 11.4376 0.6792 7.8792 11.6191 0.6781 Bank 25 7.7684 6.2479 0.7523 4.7006 6.2480 0.7523 Bank 26 4.7006 4.9198 0.8336 4.3010 5.1242 0.8393 Bank 24 4.1013 33.5770 0.6989 23.4690 33.5783 0.6989 Bank 28 23.4677 7.8569 0.6718 6.9169 10.6584 0.6490 Bank 29 5.2783 30.8275 0.7068 21.7884 30.8275 0.7068 Bank 30 21.7884 30.2603 0.6413 19.7935 31.2952 0.6325 Bank 31 19.4057 80.1044 0.7862 62.9782 80.1044 0.7862 Bank 32 62.9782 18.9316 23.8539 0.7404 19.8217 26.2094 0.7115 mean
Note: This table gives the estimated deposit insurance premiums based on the parameter values displayed in
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Table 2. The second and third columns give the means of the estimated premiums using parameter values calculated from the FPT formulas. The symbols IPP and IPP A are the per-monetary-unit deposit insurance premium assuming no early bankruptcy and early bankruptcy, respectively. The fourth column gives the means of the IPP / IPP A ratios. The symbol “bp” is 0.01%. The fifth to seventh columns give the corresponding estimates based on the parameter values obtained from the RV formulas.
52
Table 4: Analyses of the DI premium for the Asian financial crisis effects and consolidation effects Mean Values
Analyses for the consolidation effect
1996 1997 1998 Before consolidation During consolidation After consolidation
0.0242 0.0233 0.0190 0.0356
0.9698 5.0483
24.7390 50.7015
0.0456
0.9674 4.8653
11.0878 31.8128
0.0456
0.9610 4.5170
5.6422
(%) 9.3790 12.7769 9.6985
IPP A (bp) 0.0219 0.0276 0.5247 0.6787 0.7546 1.0117
IPP (bp)
17.4516
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Analyses for the Asian financial crisis
V
B ( 0) V (0) 0.7240 0.7118 0.7865
s
Note: This table shows the analyses for the DI premiums influenced by the 1997 Asian financial crisis and the bank’s consolidations in 2001 to 2002. For the analyses of the Asian financial crisis, we chose three sub-
periods: 1996, 1997, and 1998. For the analyses of the consolidation effects, we defined three sub-samples: Sub-sample 1 contains the bank’s data on the year before consolidation; Sub-sample 2 contains the bank’s
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data on the year of consolidation; and Sub-sample 3 contains the bank’s data on the year after consolidation. We assigned the sub-samples for each bank based on the bank’s consolidation year. For each bank in each
s,
B (0) and ) using the FPT formula, and then we V V ( 0)
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sub-period, we also estimated its parameters (i.e.,
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calculated its DI premiums (i.e., IPP and IPP A ) using our model. For other definitions, see Table 3.
53
PII:
S1062-9769(19)30065-1
DOI:
https://doi.org/10.1016/j.qref.2019.09.008
Reference:
QUAECO 1301
To appear in:
Quarterly Review of Economics and Finance
Received Date:
18 February 2019
Revised Date:
7 July 2019
Accepted Date:
10 September 2019
Please cite this article as: Chiang SL, Tsai MS, The Valuation of Deposit Insurance Allowing for the Interest Rate Spread and Early-Bankruptcy Risk, Quarterly Review of Economics and Finance (2019), doi: https://doi.org/10.1016/j.qref.2019.09.008
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
The Valuation of Deposit Insurance Allowing for the Interest Rate Spread and EarlyBankruptcy Risk
Shu Ling Chiang Professor Department of Business Management National Kaohsiung Normal University, Kaohsiung, Taiwan
Ming Shann Tsai Professor Department of Finance National University of Kaohsiung, Kaohsiung, Taiwan
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E-mail: [email protected].
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E-mail: [email protected].
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Highlights
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We derivea closed-form formula of deposit insurance premiums with the two important factors. We derive new formulas with the risk of early bankruptcy for estimating the necessary parameters. Our empirical results show the spread of the interest rate is negatively correlated with the premium. The traditional model underprices the premium by 20.81% if it ignores
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the risk of early bankruptcy.
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Abstract
Our proposed model allows for inclusion of the interest rate spread (defined as the difference between the lending rate and the borrowing rate) and the risk of a bank’s early bankruptcy in the derivation of a closed-form pricing formula for calculating deposit insurance premiums. This is important because, in practice, these two factors significantly influence the expected value of a bank’s assets. Also, we use a method that considers the
possibility of early bankruptcy to derive new formulas for calibrating the necessary parameters. Data from Taiwanese banks are used to illustrate the application of our model. Our empirical results show that the spread of the interest rate is negatively correlated with the premium. This result is consistent with our theoretical inferences. Moreover, premiums associated with a risk of early bankruptcy are always higher than premiums not associated with such a risk. Using the example of Taiwanese banks, we show that the traditional model
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underprices the premium by 20.81% if it does not consider the risk of early bankruptcy.
Keywords: Deposit Insurance, Premiums, Interest Rate Spread, Early Bankruptcy
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1. Introduction
A well-designed deposit insurance system strengthens the depositor’s confidence in
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financial institutions, and this confidence in turn stabilizes the financial condition of a country. Therefore, constructing a good deposit insurance system is crucial for a country’s
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economic development. A fair deposit insurance premium is important for constructing a good deposit insurance system. If the premium is lower than what a bank should pay, the
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bank is likely to shift its risk to the deposit insurer (Peltzman, 1970; Santomero and Vinso, 1977; Marcus and Shaked, 1984; Ronn and Verma, 1986; Duan, Moreau and Sealey, 1992;
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Shyu and Tsai, 1999a, b). In contrast, if the calculated premium is too high, the bank may be unwilling to participate in the deposit insurance program at all. These two situations
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could even result in the collapse of the entire system. Therefore, accurately and fairly valuating the deposit insurance premium is an important consideration in researching how banks manage risk. Merton (1977) was the first to derive a formula for pricing risk-based deposit insurance premiums. He argued that this premium is the same as the initial value of a European put option for which the strike price and underlying assets are respectively the 2
value of the deposits and the value of the bank’s total assets. Numerous researchers have investigated how Merton’s valuation framework can be used to more precisely determine deposit insurance premiums (Ronn and Verma, 1986; Allen and Saunders, 1993; Duan and Yu, 1994, Duan and Simonato, 2002; Lee, Lee and Yu, 2005; Chen, Ju, Mazumdar and Verma, 2006; Chuang, Lee, Lin and Yu, 2009). For example, in Merton’s model, the value of the bank’s assets and the standard deviation of its returns are two important but unobserved parameters. Some authors have used simultaneous equations of the bank’s
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equity and the standard deviation of its equity returns to determine these two parameters
(Ronn and Verma,1986; Giammarino, Schwarz and Zechner, 1989). Duan and Yu (1994) used Maximum Likelihood Estimation to estimate them. Recently, some scholars have
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derived a formula for calculating a deposit insurance premium when the stochastic process
of the interest rate is included in the model (Duan and Simonato, 2002; Chuang et al, 2009).
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Pennacchi (1987a) made the point that the magnitude of an insurance premium depends on whether the deposit insurer is offering a fixed-rate, unlimited-term contract or a variable-
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rate, limited-term contract. Furthermore, in several studies, estimating a deposit insurance premium took into account the default risk of the guaranty fund (Episcopos, 2004).
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An important factor in the determination and valuation of deposit insurance premiums is whether the effect of the difference between the lending rate and the borrowing rate (i.e.,
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the interest rate spread) is incorporated in the model. As we know, a commercial bank earns its profits by paying a lower borrowing rate and receiving a higher lending rate. Recently,
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because of the very competitive financial market, banks have generally either increased the borrowing rate to attract depositors or decreased the lending rate to attract lenders. Both of these actions decrease the interest rate spread. As shown in Figure 1, this spread in the Taiwanese financial market had great volatility from 1981 to 2010. The minimum value of the interest rate spread was 1.901% in 1998 and the maximum value was 5.4298% in 2002. This spread sharply decreased from 2002 to 2010 and in turn decreased the profits of 3
Taiwanese banks. On the other hand, the profits of commercial banks, and in turn the value of their assets, come from exploiting the interest rate spread. The size of this spread can affect the likelihood of bankruptcy. Merton (1978) was the first to incorporate an interest rate spread into a valuation model of deposit insurance. He modeled the spread as a lowerthan-competitive interest rate paid to depositors. Pennacchi (1987a, 1987b) allowed for a spread in the form of a lower-than-competitive deposit interest rate. We also consider in this paper the magnitude of this spread when determining the optimal deposit insurance
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premium.
In addition, valuation models such as Merton’s are feasible only when a bank goes
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into bankruptcy at the maturity date of the insurance contract (defined as the auditing date in Merton’s model). Generally, banks are involuntarily shut down by the decision of a
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government regulator or deposit insurer. However, from the financial viewpoint, a bank is in bankruptcy if its assets fall below its debt at any time point before the maturity date. In
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such a case, using Merton’s model could make calculation of the premium inaccurate, because it uses only one time point (i.e., the auditing time in Merton’s model) to determine
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the probability of bankruptcy and the expected loss. To accurately determine a fair deposit insurance premium, the model should also consider the risk of early bankruptcy at any time
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point before the maturity date stipulated in the insurance contract. We calculate the deposit insurance premium based on this consideration.
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When estimating the risk of early bankruptcy, one needs to consider the probability of
early bankruptcy and recovery given bankruptcy. The first-passage time model has been adopted in many traditional studies aimed at valuating a security when there is an early termination risk (Black and Cox, 1976; Finger, 2002). When using this model to investigate the risk of bankruptcy, one must keep in mind that the bankruptcy occurs as soon as the bank’s assets are less than its debt (i.e., the first absorption situation). As mentioned by 4
Black and Cox (1976), the first-passage time model cannot be applied directly when the termination payoff is found to endogenously contribute to an optimal-stopping problem. In the papers reporting this use of the first-passage time model, the focus has been exclusively on determining the termination probability (i.e., the probability of bankruptcy); moreover, these authors usually specified this termination payoff (i.e., the recovery given bankruptcy) to be uncorrelated with the bank’s underlying assets. In such cases, they must assume that the termination payoff is either an exogenous variable (see Finger, 2002) or an endogenous
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variable that is uncorrelated with the underlying assets (see Black and Cox, 1976).
However, when dealing with the risk of bankruptcy in valuating deposit insurance, the probability of early bankruptcy and the recovery given bankruptcy should be
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simultaneously determined by the same factor: the bank’s assets. Thus, treating these two factors separately may result in incorrect pricing of the deposit insurance premium. We
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support a new method for accurately pricing a deposit insurance premium, one that takes account of the possibility of the bank’s early bankruptcy. The main difference between our
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model and the first-passage time models is that we do not separate consideration of the probability of early bankruptcy from consideration of recovery given bankruptcy;
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specifically, we let them both be endogenous variables, simultaneously determined by the bank’s assets. In real applications, our specifications yield more accurate pricing than
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traditional specifications. To the best of our knowledge, ours is the first paper to report a closed-form deposit insurance pricing formula that lets the probability of early bankruptcy
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and the recovery given bankruptcy be simultaneously determined by the bank’s asset value when the model takes account of the bank’s early bankruptcy risk. Note that our closed-form pricing formula can also be applied to the valuation of an
American option, because the early exercise probability and the payoff of this option are both determined by the same underlying asset value, as is most likely the case when one uses the specifications of our model. Our formulas give values similar to those reported by 5
Kim (1990) for the valuation of an American option. However, the valuation problems of Kim’s (1990) model and our model are different. In Kim (1990), the early exercise premium corresponds to the compensation that an American-style option holder would require (in the stopping region) in order to postpone exercising the underlying asset until the maturity date of the option contract. For deposit insurance contracts, the valuation model in our study estimates an early bankruptcy premium. Recognizing that both the interest rate spread and the bank’s risk of early bankruptcy
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significantly influence the value of the deposit insurance, we provide in this paper a pricing framework that takes account of these two important factors in accurately valuating deposit insurance premiums. Pricing a deposit insurance premium also requires estimating the
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bank’s assets and the standard deviation of its returns. Many researchers have estimated these parameters from equity data (Marcus and Shaked, 1984; Ronn and Verma, 1986).
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However, if a bank goes bankrupt before the maturity date of the insurance contract, the estimates using the formulas in traditional studies (i.e., Ronn and Vermas, 1986) are
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inaccurate. To more accurately estimate the assets and the standard deviation of the returns, we need a method that considers the possibility of early bankruptcy. A bank’s stockholders
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often receive nothing when the bank goes bankrupt. The bank’s equity, from which the likelihood of early bankruptcy can be inferred, can be determined by employing a down-
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and-out barrier option with a zero rebate. We therefore use this option to derive new formulas for estimating the two necessary parameters, the values of which are derived from
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the first-passage time model.
For our empirical analysis we collected data from 32 banks listed on the Taiwanese
stock exchange. After applying the new formulas described above, we calculated a fair, risk-based deposit insurance premium, taking into account the interest rate spread and the risk of early bankruptcy. We discuss below the differences between calculations using our model and the traditional model. 6
The remainder of the paper is organized as follows. Section 2 presents our formula that incorporates the interest rate spread and the early-bankruptcy risk to obtain an approximate price for a deposit insurance premium. Section 3 shows how we utilize the down-and-out barrier option to develop empirical formulas for calibrating the bank’s assets and the standard deviation of its returns. In this section, we also report the results of our analysis of data from the 32 banks listed on the Taiwanese stock exchange, including a comparison between the results calculated using our model and those calculated using the
2. The valuation model for a deposit insurance premium
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traditional model. Our conclusions are summarized in Section 4.
In this section, we present the theoretical valuation model for a deposit insurance (hereafter
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denoted as DI) premium. In Subsection 2.1, we describe a model that takes account of the interest rate spread and only one auditing time based on the Merton framework. In
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Subsection 2.2, we extend the model in Subsection 2.1 to take account of the likelihood of a bank’s early bankruptcy.
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2.1 A model for valuating a DI premium considering the interest rate spread and only one auditing time (Merton framework)
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In valuation theory, the two most common methods for evaluating a financial contract with default events are the structural-form model and the reduced-form model. The idea of
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pricing defaultable securities began with a structural-form model (Merton 1974, Black and Cox 1976, Leland 1994). The structural-form model uses market data to estimate a bank’s
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asset value and its asset volatility, and then the bank’s probability of default is calculated. Merton (1977) was the first to derive a DI formula using a structural-form model. Recently, reduced-form model has been applied to price defaultable securities and to calculate the termination probabilities (Jarrow and Turnbull 1995, Jarrow 2001). The reduced-form model uses market information on the hazard rate to calculate the probability of default. Duffie, Jarrow and Purnanandam (2003) give a DI formula using a reduced-form model. 7
Thus, they obtained the termination probability from market data using the default hazard rate and then calculated the DI premiums. Our model, an extension of Merton’s model, is a structural-form model which obtains the DI formula with the early bankruptcy probability by estimating the bank’s asset value and its asset volatility. Let the underlying probability space be labeled (, F , Q) , where is the state space, F is the filtration generated by the information set F ( Ft ) 0t T , and Q is the risk-neutral probability. Let V be the unobserved value of the bank’s assets. In Merton
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(1977), the evolution of V is specified as following a geometric Brownian motion. Given the risk-neutral probability, V can be expressed as
dV (r f (t ) )dt V dZV (t ) , V
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(1)
is the dividend yield;
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r f (t ) is the risk-free interest rate;
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where
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V is the instantaneous standard deviation of bank’s asset value return; and ZV (t ) is a standard Brownian motion under the risk-neutral measure.
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In the above equation, the bank’s assets increase by the rate r f (t ) , which is decreased by the dividend yield . How much the assets increase depends on what the bank earns from
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its loans, which, in practice, is determined by the lending rate. Accordingly, Merton’s model specifies that the lending rate equals the risk-free interest rate under the risk-neutral measure.
According to Merton (1977), the bank’s assets are assumed to grow at the rate r f (t ) . Because its debt also grows at this rate, the effect of the interest rate disappears in Merton’s
8
formula if it is a constant. However, this assumption is unrealistic. The bank obtains funds from its depositors and then loans these funds to the borrowers. The bank’s profits come mainly from managing the deposits and the lending. To earn a profit, the bank’s deposit rate must generally be less than its lending rate. The interest rate spread is defined in our model as the difference between these two rates. Because we assume that the average cost of the debt paid to the depositors is the risk-free interest rate, we set the lending rate as
r f (t ) (t ) , where (t ) is the interest rate spread under the assumption of risk neutrality.
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The value of V can be expressed as
dV (rf (t ) (t ) )dt V dZV (t ) . V
(2)
It is worth noting that if we let (t ) 0 , our model is identical to Merton’s pricing formula.
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In general, DI contracts provide depositors with minimal protection against the risk of
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the bank going bankrupt. We let B1 (t ) denote the protected deposit amount at time t , and
B2 (t ) the unprotected deposit amount at time t , the latter of which is equivalent to the
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value of all the debt liabilities other than the insured deposit. Thus, the total value of the deposit at time t can be expressed as B(t ) B1 (t ) B2 (t ) . The interval [t , T ] stands for the
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period of the DI contract, with T indicating the maturity date. In Merton’s model, the maturity date is specified as the auditing time at which one can judge whether or not the
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bank is bankrupt. At the maturity date, the total value of the bank’s assets is V (T ) and the total value of the deposits is B(T ) .
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In pricing theory, any asset process can be chosen as the numeraire asset (Davis,
1998). We chose a money market account to serve as the numeraire asset for our model. Let the money market account (defined as M (t , v) ) be expressed as follows: v
M (t , v) exp( rf (u )du) . t
(3)
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The process of the money market account is stochastic if we specify r f (t ) as following a stochastic process. Under the risk-neutral measure, the drift term of the return of money market account is rf (t )dt and the standard deviation of the return of the money market account is b(t , u ) , its form dependent on the specification of the risk-free interest rate process.1 For example, if the interest rate is specified as of the Vasciek (1977) type, the standard deviation of the return of the money market account can be obtained as follows
b(t , u )
r a
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(Heath, Jarrow and Morton, 1992):
(1 e a (ut ) ) ,
(4)
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where a is the speed of adjustment and r is the volatility of the risk-free interest rate,
have the following equation: T
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rf (u ) du B(t )M (t , T ) . B(T ) B(t )e t
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both being positive constants. Given this specification of the money market account, we
(5)
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We then used the previous specifications to price the DI premium. Several studies have considered the influences of delayed resolution or capital forbearance on the valuation of DI premiums (Duan and Yu, 1994, 1999; Nagarajan and Sealey, 1995; Lee, Lee and Yu,
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2005; Shie, 2012). In addition, depositor preference arrangements2 can increase the
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recovery rate for DI institutions because depositor claims get paid before those of general, 1
We assume that b(t , u) is a deterministic value.
2
In the U.S., the national depositor preference law of 1993 requires that all the domestic depositors, insured
and uninsured, be paid in full before the general, unsecured creditors. It gives a statutory priority to depositors over other unsecured creditors. In Taiwan, Article 64-1of the Banking Act of The Republic of China also stipulates that if a bank or a financial organization is run poorly and needs to cease its operations and liquidate its debts, deposit debts shall precede non-deposit debts. Such depositor preference laws have existed for decades in a number of countries (such as Australia, the USA, Russia, and Taiwan). 10
unsecured creditors. Some studies have considered the effects of the seniority of insured deposits in the DI valuation models (Davis, 2015; Chang, Chung, Ho and Hsiao, 2018). The main purpose of our manuscript is to provide a DI premium model allowing for the interest rate spread and early-bankruptcy risk. To avoid complicated specifications that would blur our point, our model assumes that there are no effects caused by the policies of the delayed resolution or capital forbearance for defaulted banks and the seniority of insured deposits. As noted by Merton (1977), if V (T ) B(T ) at the maturity date, the protected amount
protected amount of the deposit becomes V (T )
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of the deposit is B1 (T ) . In contrast, if V (T ) B(T ) , the bank goes into bankruptcy and the B1 (T ) 3 . The value of the DI contract at T is B(T )
then
re
(6)
-p
B (T ) Max 0, B1 (T ) V (T ) 1 . B(T )
V (T ) ). B(T )
V (T ) ) is the loss rate given bankruptcy at the maturity date T . Thus, in Merton’s B(T )
model
V (T ) is the recovery rate given bankruptcy at the maturity date T . B(T )
na
(1
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In view of that, we can obtain the loss amount given default to be B1 (T )(1
According to Equation (2), the change in the bank’s assets is a random variable that
ur
follows a lognormal distribution. Thus, the DI premium can be calculated based on the
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theory of that distribution. In light of the above equation, the contract value can be regarded as a classic European put option with strike price B1 (T ) and underlying asset value V (T )
B1 (T ) . The DI premium, denoted as IP(t , T ) , is specified under the assumption of B(T )
only one auditing time (at the maturity date) and a given interest rate spread. We can obtain the following result by extending the classical Black-Scholes-Merton model (Black and 3
For simplicity, we assume that B1 (T ) and B2 (T ) have the same seniority. 11
Scholes, 1973; Merton, 1973) (see Appendix A): IP(t , T ) B1 (t ) N ( y2 (t , T )) V (t )e( s )( T t )
B1 (t ) N ( y1 (t , T )) , B(t )
(7) where T
s (T t ) 1 (u )du , defined as the average interest rate spread; t
N () is the cumulative density of a standard random variable;
ln( y 2 (t , T )
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y1 (t , T )
B(t ) ) 1 (t , T ) V (t ) ; v(t , T ) B(t ) ) 2 (t , T ) V (t ) ; v(t , T ) 1 2
1 (t , T ) ( s )(T t ) v 2 (t ,T ) ; 1 2
T
v 2 (t , T ) V b(t , u ) du . 2
lP
t
re
2 (t ,T ) ( s )(T t ) v 2 (t , T ) ; and
-p
ln(
If we assume correlation between the returns of the bank’s assets and the returns of
T
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the money market account, we can obtain the following equation:
v 2 (t , T ) ( V2 2 V b(t , u) b 2 (t , u))du . t
ur
(8)
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Equation (8) is a general formula for calculating the total variance of a discounted bank’s assets in the DI model. One can obtain a closed-form expression of v 2 (t , T ) if the standard deviation of the interest rate has been determined. For example, if the interest rate is specified is of the Vasciek (1977) type, according to Equation (4) we have
v (t , T ) (T t ) 2 V 2
2 V
r
1 e a (T t ) ((T t ) ) a a
12
r2 a2
((T t ) 2
1 e a (T t ) 1 e 2 a (T t ) ). a 2a
(9) Thus, our pricing formula gives the same results as those reported in Rabinovitch (1989) and Chung (1999). The main difference between our formulas and theirs is that our formulas valuate a DI contract under the assumption that the strike price increases as the interest rate increases, whereas their pricing formulas are applied to vanilla options, which have a constant strike price.
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As mentioned previously, one can easily show that Equation (7) becomes the Merton pricing formula for DI if s 0 and r (t ) is a deterministic value. To prove this, we let
r 0 when assuming a deterministic interest rate, in which case b(t , u) 0 according to
T
-p
Equation (4). We then have T
v 2 (t , T ) V b(t , u) du V2 du V2 (T t ) . 2
t
re
t
(10)
shown in Merton (1977).
lP
Thus, our Equation (7) given a deterministic interest rate is the same as the DI model
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According to Equation (7), the premium decreases as the interest rate spread increases. We infer from this conclusion that the probability of bankruptcy decreases
ur
because the increase in the interest rate spread leads to a rise in the bank’s profits. Thus, there is a negative relationship between the interest rate spread and the DI premium.
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The per-monetary-unit DI premium, IPP(t , T ) , can be obtained by dividing Equation
(7) by B1 (t ) as follows: IPP (t , T ) N y 2 (t , T ) e ( s )( T t )
V (t ) N y1 (t , T ) . B(t )
(11) In Merton’s model, there is only one time point (i.e., the auditing time in Merton’s model) for judging whether the bank is in bankruptcy. However, this specification is not realistic. 13
The bankruptcy of a bank can occur at any time before the maturity of the DI contract. To accurately price DI, the associated premium, which must reflect the early-bankruptcy risk, should be modeled precisely. 2.2 A model for valuating a DI premium taking account of the interest rate spread and the likelihood of a bank’s early bankruptcy We derive below closed-form formulas for DI considering the likelihood of a bank’s early bankruptcy using both discrete-time and continuous-time frameworks. We first construct
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the model using a discrete-time framework. Then we discuss our continuous-time model,
assuming that the time interval between the valuation points approaches zero. This discretetime approach is the method that is usually used in the valuation of time-dependent
-p
securities in a continuous-time framework (Kim, 1990). We limit the occurrence of a
possible bankruptcy to a finite number of time points during the maturity of the DI contract.
T t be the number of time points. For example, d 12 , t 0 , and T 1 -year t
re
Let d
lP
mean that it is possible for the bank to enter bankruptcy at the end of each month. Letting t i represent the date of the i-th time point, we have t t 0 t1 t d T . Then V (ti ) and
na
B(ti ) , with i 0,1,, d , denote respectively the value of the bank’s total assets and
ur
deposits at time t i . Therefore, at every possible bankruptcy time point, there is a corresponding value for the bank’s assets, that is, V (t 0 ) , V (t1 ) , ,V (t d ) . Note that the
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maturity date is always the expiration date of the DI contract. If the contract is perpetual, this date can be specified as an infinite variable. If we are to take the probability of early bankruptcy into account, calculation of the DI
premium requires consideration of the expected value of a non-zero random loss, that is, the payoff shown in Equation (6). We denote IP A (t , T ) as the DI premium when there is a risk
14
of early bankruptcy. At the maturity date, the payoffs of IP A (t , T ) and IP(t , T ) are the same; therefore, we have IP A (t d 1 , T ) IP(t d 1 , T ) . Except for this situation, the payoff of the DI contract at time t i can be described as follows: The payoff of the DI contract at time t i
if V (ti ) B(ti ) The live value of the deposit insurance contract, . The termination value of deposit insurance contract, if V (ti ) B(ti )
ro of
(12)
Therefore, at time t d 2 , the time until maturity is 2t . Thus, IP A (t d 2 , T ) can be expressed
IP A (t d 1 , T ) (1 I D (td 1 ) ) | Ftd 2 ] M (t d 2 , t d 1 )
V (t d 1 ) ) B(t d 1 ) I D (td 1 ) | Ftd 2 ] , M (t d 2 , t d 1 )
B1 (t d 1 )(1
lP
E[
re
IP A (t d 2 , T ) E[
-p
as follows:
(13)
na
where E[ | Ft ] is an expected operator, conditional on information set Ft with a risk-
ur
neutral probability, and I D (td 1 ) is an indicator function. This gives us I D (td 1 ) 1 if
D(td 1 ) {V (td 1 ) B(td 1 )} and I D (td 1 ) 0 otherwise. The first and second terms on the
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right hand side of Equation (13) are the expected survival and termination values respectively of the DI contract at time t d 2 . On the right hand side of this equation, the second expectation operator ( E[ | Ft ] ) represents the expected loss when bankruptcy occurs at time t d 2 . With this expectation, the value
V (td 1 ) denotes the recovery rate given B(td 1 )
bankruptcy at time t d 2 . From Equation (13), one can find the probability of early 15
bankruptcy ( E[ I D (td 1 ) | Ft ] ), where D(td 1 ) {V (td 1 ) B(td 1 )} and the recovery rate given bankruptcy (
V (t d 1 ) ) are simultaneously determined by the bank’s assets V (td 1 ) , an B(t d 1 )
endogenous and random variable. Based on the previous specifications, IP A (t d 2 , T ) can be described as follows (see Appendix B):
IP A (t d 2 , T ) IP(t d 2 , T ) E[
B1 (t d 1 ) t B(td 1 ) I D (td 1 ) | Ftd 2 ] . M (t d 2 , td 1 )
( s ) V (t d 1 )
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(14)
We define the second term on the right side of Equation (14) as the premium associated
with the risk of early bankruptcy (i.e., the early-bankruptcy premium). At each time point,
-p
IP A (, T ) is the sum of the corresponding IP(, T ) and the early-bankruptcy premium. As shown by Kim (1990), if we work backwards recursively, the value of the DI contract at
re
time t , taking account of an early-bankruptcy risk, can be described as follows (see also
IP A (t , T ) IP(t , T )
d 1
lP
Appendix B):
(s )t e i 1
na
(15)
( s )( ti t )
V (t )
B1 (t ) N ( y1 (t , ti )) . B(t )
The per-monetary-unit DI premium IPP A (t , T ) at time t can be represented as follows: d 1
(s )t e
( s )( ti t )
i 1
V (t ) N ( y1 (t , ti )) . B(t )
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(16)
ur
IPP A (t , T ) IPP(t , T )
We now move from the discrete-time framework to the continuous-time framework.
Thus, we let the number of possible bankruptcies be infinite; that is, d tends toward infinity because t tends toward zero. In view of that, we have: d 1
IP A (t , T ) IP(t , T ) lim ( s )t e( s )( ti t )V (t ) t 0
i 1
16
B1 (t ) N ( y1 (t , ti )) B(t )
IP(t , T )
T
t
( s ) e( s )( u t )V (t )
B1 (t ) N ( y1 (t , u ))du . B(t )
(17) The second equality of Equation (17) is obtained by the Riemann integration, since all the stochastic processes almost certainly have continuous sample paths. In addition, we have
IPP A (t , T ) IPP(t , T )
T
t
( s ) e( s )( u t )
V (t ) N ( y1 (t , u ))du . B(t )
(18)
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It is important to note that although we model the risk to the bank’s assets to include only one Brownian motion, as shown in Equation (2), the loss, V (t ) B(t ) , incurred from going into bankruptcy is not always zero, even using the continuous-time framework.
Given the properties of Brownian motion, a large increase or a large decrease in a bank’s
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assets may occur at the next time point.4 Therefore, a sudden and large decrease in a bank’s
re
assets may cause the assets to drop below the bank’s debt, V (t ) B(t ) , even if the valuation model is using the continuous-time framework.
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2.3 Discussions of the effect of the interest rate spread on the DI premium One of our contributions is to incorporate the effect of the interest rate spread into the
na
valuation model for DI premiums. We then provide two discussions of the influences of the interest rate spread on DI premiums. The first discussion is focused on insured and
ur
uninsured deposits. Uninsured deposits in a bank with a default risk should be priced in a similar way to that used by the deposit insurance authority. The spreads for these two types
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of deposits should be different because they should be adjusted according to the auditing rule of the deposit insurance fund. Thus, the DI premium should consider this practice. In 4
Using the continuous-time framework, Brownian motion is usually used to describe the diffusion parts of an asset return. However, Brownian motion is a random variable and its values are normally distributed at the next time point. The domain of a normal distribution ranges from negative infinity to positive infinity. In this paper, therefore, the properties of Brownian motion refer to a value of Brownian motion that might be extreme (i.e., negative infinity or positive infinity) at the next time point. Thus, the bank’s assets could show a sudden and large decrease if the value of the Brownian motion is an extreme value on the left hand side of the normal distribution at the next time point.
17
the discussion, we note that the spreads for insured deposits and uninsured deposits are different. We then use a weighted interest rate spread to address the influence of the different spreads of these two types of deposits on DI premiums. The weighted interest rate spread can be expressed as follows:
(t ) w1 (t )1 (t ) w2 (t ) 2 (t ) ,
(19)
where
deposits, respectively;
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1 (t ) and 2 (t ) are the interest rate spreads for insured deposits and uninsured
w1 (t ) and w2 (t ) are the weights for insured deposits and uninsured deposits, respectively; we have w1 (t )
B1 (t ) B (t ) and w2 (t ) 2 . B(t ) B(t )
-p
Putting the weighted interest rate spread into Equation (2), we can still obtain the formulas for IPP(t , T ) and IPP A (t , T ) , shown as Equations (11) and (18), respectively.
re
Thus, one can also address the influence of uninsured deposits on DI premiums using our
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formulas. For example, if one wants to analyze the influence of 2 (t ) and w2 (t ) on
IPP A (t , T ) , (s)he can use the following equation:
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IPP A (t , T ) IPP A (t , T ) s IPP A (t , T ) IPP A (t , T ) s , and . 2 (t ) s 2 (t ) w2 (t ) s w2 (t ) (20)
ur
Because the changes of the interest rate spread can be influenced by a variety of
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factors, our second discussion is to specify the interest rate spread as a multi-factor function, such as the exogenous variables (i.e., the bank’s monopoly power, the competitive factors, and so on) and the original influential variables, on a DI premium (i.e., the asset risk V , the bank’s leverage ratio
B (t ) , and the deposit insurance premium IPP A (t , T ) ). V (t )
We can express the function of the interest rate spreads as follows:
18
(t ) f (,
B(t ) , V , IPP A (t , T )) , V (t )
(21) where f () is an operator of the function; represents the exogenous variable, such as the bank’s monopoly power, the competitive factors, and so on. In such a specification, we let
s E[ (t )] . The simplest assumption for describing this function is a linear relationship. Under
(t ) a0 a1 a2
B(t ) a3 V a4 IPP A (t , T ) , V (t )
(22)
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this assumption, an affine function of the interest rate spreads can be described as follows:
where a0 is the drift term and ai is the relative magnitude of variable i ’s effect on the
-p
interest rate spread for i =1, 2, 3, and 4.
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The influence of each exogenous variable on the DI premium depends on its coefficients in this regression. For example, one can let represent the bank’s monopoly
lP
power. If the empirical result of Equation (22) shows a1 0 , we can infer that the bank’s monopoly power will increase the interest rate spread. Because an increase in the interest
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rate spread decreases the DI premium, one can infer that an increase in the bank’s monopoly power will decrease the DI premium. In contrast, if a1 0 , one can infer that the
ur
decrease in the bank’s monopoly power will increase the DI premium. To analyze the influences of
B (t ) and V on the premium, one must consider not V (t )
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only the effects of their coefficients on the interest rate spread, but also the original influences of these variables on the DI premium formula. According to our DI valuation formula, an increase in the bank’s leverage ratio
B (t ) will increase the DI premium but an V (t )
increase in the interest rate spread may decrease the DI premium. In general, if a bank intends to attract more and more depositors, it should increase the deposit rate. Then the
19
interest rate spread will decrease in this case. Thus, we expect to have a2 0 in Equation (22) since an increase in the leverage ratio
B (t ) will decrease the interest rate spread. V (t )
Accordingly, if we have a2 0 , one can infer that an increase in the bank’s leverage ratio will increase the DI premium. However, if the empirical result of Equation (22) shows that the bank’s leverage ratio will increase the interest rate spread (i.e., a2 0 ), there is a tradeoff between the effects of the bank’s leverage ratio and the interest rate spread on the DI premium. The finally effects of the bank’s leverage ratio on the DI premium will be
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ambiguous.
If the interest rate spread includes asset risk factor V , a risk-return trade-off may
also be found in the model. In general, the determination of the interest rate spread usually
-p
includes the compensation of risks (i.e., default risk). In other words, if the bank intends to have a higher interest spread rate, it may undertake more risky investments and this may
re
increase its asset risk. In such a case, we expect a3 0 since there is a positive relationship
lP
between the interest rate spread and the asset risk. However, we know that increasing the spread rate will reduce the default risk, but increasing the asset risk will increase the default
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risk. The former will decrease the DI premium but the later will increase it. Thus, the finally effects of the asset risk on the DI premium are also ambiguous because of the riskreturn trade-off.
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Moreover, if the DI premiums ( IPP(t , T ) and IPP A (t , T ) ) are specified as a factor of
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the interest rate spread, the formulas for the DI premiums will be implicit functions. For example, IPP A (t , T ) g ( IPP A (t , T )) 0 , where g () is an operator of the function (i.e., Equation (18) with s E[ (t )] ). Thus, we cannot obtain the explicit closed-form formula for a DI premium. However, one can use a numerical method to solve for the DI premiums. If a4 0 , we may expect the DI premium to be greater than that calculated by our model, which assumes that the interest spread rate is a deterministic variable.
20
Finally, Equation (16) shows that if a bank’s earnings from the interest rate spread are larger than its payout rate (i.e., the dividend rate) and the possibility of early bankruptcy is taken into account, the DI premium the bank should pay is larger if estimated from our model than from the traditional model. On the other hand, the DI premium is smaller from our model than from the traditional model if the interest rate spread is smaller than the dividend rate. These results are reasonable, because the DI premium is the insurer’s expected loss. If the drift of the asset return is positive (e.g., the earnings are greater than
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the payouts), the expected value of the bank’s assets increases during the contract period. In this situation, the likelihood of bankruptcy is determined only by the volatility of the assets. The expected loss calculated from our formula is larger than that calculated from the
-p
traditional formula, because our formula takes account of many possible losses for the
insurer before the maturity date of the contract. By contrast, if the bank’s earnings are less
re
than its payouts, the likelihood of bankruptcy increases, not only because of the volatility of the bank’s assets, but also because of the decrease in their value. Because the drift of the
lP
asset return is a decreasing function of time, the expected value of the assets eventually falls below the value of the debt (i.e., bankruptcy) if there is a long time before the contract
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reaches maturity. The insurer’s loss calculated from a model that includes many possible bankruptcy time points (i.e., our model) is smaller than that calculated by a model that
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includes only one auditing time (i.e., the traditional model). This is because our model
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identifies the date of the bankruptcy as earlier than the traditional model can, and its application therefore can forestall an increase of the loss. Thus, IPP A (t , T ) IPP(t , T ) when s .
3. Calibration of the model’s parameters When valuating a DI premium, the bank’s total assets and the standard deviation of its returns must be calibrated. Some authors have used equity data to calibrate these two
21
parameters (Marcus and Shaked, 1984; Ronn and Verma, 1986). For simplicity, we let the interest rate spread be a constant or a deterministic variable. The following simultaneous equations reflect the method introduced by Ronn and Verma (1986):
E (t ) V (t )e ( s )(T t ) N ( x1 ) B(t ) N ( x2 ) , and (23)
E
V (t ) E (t ) V (t ) ( s )(T t ) V e N ( x1 ) V , E (t ) V (t ) E (t )
where
E (t ) is the value of a bank’s equity at the time t ;
E is the standard deviation of a bank’s equity return;
ln( x2
V (t ) ) 2 (t , T ) B(t ) . V2 (T t )
re
x1
V (t ) ) 1 (t , T ) B(t ) ; and V2 (T t )
lP
ln(
-p
is the level of declaring bankruptcy;
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(24)
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Equation (23) shows that the value of the equity equals the present value of the surplus assets after the debt is paid off at maturity. The payoff of the equity is (V (T ) B(T )) at
ur
the maturity date.5 Equation (24) gives the standard deviation of the bank’s equity returns, derived using Ito’s Lemma.
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However, the calibrations of the bank’s assets and the standard deviation of its returns
5
In the DI valuation model (e.g. Merton, 1977), the DI premium is compensation to the deposit insurance company, which bears the possible loss. Thus, we must calculate the full possible amount for insured depositors if the value of the bank’s assets ( V (t ) ) is less than the total value of its deposits ( B(t ) ). The empirical implementation of the DI pricing model relies mostly on the Ronn and Verma (1986) estimation method. Since this estimation method considers the concept of tolerance for allowing the bank’s bankruptcy, the bank may go into default if V (t ) > B(t ) , where denotes the market tolerance for allowing the bank’s bankruptcy. These two thresholds for the theoretical model ( B(t ) ) and empirical method ( B(t ) ) in our study both refer to the traditional studies (Merton, 1977; Ronn and Verma, 1986).
22
calculated from the above equations may be inaccurate, because they do not take into account the probability of bankruptcy prior to maturity. We therefore revise the formulas to address this problem. As is well known, a bank’s stockholders often receive nothing when the bank goes bankrupt. Therefore, the bank’s equity can be regarded as the value of a down-and-out barrier option with zero rebate. Thus, we adopt the first-passage time model to derive the revised formulas. Based on this model, the closed-form formula for the equity
E (t ) V (t )e ( s )( T t ) [ N ( x1 ) ( B(t )[ N ( x2 ) (
B(t ) V (t )
B(t ) V (t )
) 2( s ) v
) 2( s ) v
2
2
( t ,T ) 1
( t ,T ) 1
N (h1 )]
N (h2 )] ,
(25) 1 (t , T ) V (t ) , and h2 h1 V2 (T t ) . V2 (T t )
re
where h1
B(t )
-p
ln
ro of
can be expressed as follows (Musiela and Rutkowski, 2002):
as follows: V (t ) (t ) V , E (t )
(26) where
E (t ) V (t )
ur
(t )
na
E
lP
Moreover, using Ito’s Lemma, the standard deviation of the equity return is obtained
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e( s )(T t ) ( N ( x1 ) 2( s )( V2 (T t ))1 ( (2( s )( V2 (T t )) 1 1)(
B(t ) V (t )
B(t ) V (t )
)2( s )(V (T t )) 2
1
1
N (h1 ))
1
) 2( s )(V (T t )) N (h2 ) . 2
We then use these formulas to calibrate the bank’s assets and the standard deviation of its returns. One can obtain more reasonable DI premiums by including the parameters calibrated from Equations (25) and (26) in our pricing formula. 4. Empirical methods and results 23
We use data from banks listed on the Taiwanese stock exchange to illustrate the application of our model. At the end of 2010, 32 banks were publicly offered in Taiwan.6 The yearly data, including the banks’ equities, debts, and cash dividends, were taken from the Taiwan Economic Journal (TEJ) Databank.7 The sample period is 1980 to 2010. Some banks were integrated during the sample period, and their data are missing from the TEJ Databank. For estimating the insurance premiums of these integrated banks, we use only data that can be observed before the integrated date. In Table 1, the sample number is the number of years
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for which data were used in the empirical studies. For example, the sample number 27 for Bank 1 means that data for that bank were used for each of 27 years.
We use the daily stock return to calculate the yearly standard deviation of the equity
-p
returns for each bank. The daily stock return is calculated using the end-of-day stock price,
also obtained from the TEJ Databank. We use the daily stock returns for each year to obtain
re
the variance of the daily stock returns in the corresponding year. Then we multiply the calculated value by the number of actual trading dates in the year to obtain the standard
lP
deviation of equity for the year.8 During the sample period, the Taiwanese government encouraged financial institutions to merge, creating a single financial holding company. For
na
these banks, we use the holding company’s stock returns as the bank’s stock returns after the date of the merger to calculate the variance of the daily stock returns. The main reason
6
ur
for doing so is that only the holding company’s stock prices after that time are available.9
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Banks in Taiwan are small compared with those in the United States. At the end of 2015, there were 6,184 banks and 69 banks in the United States and Taiwan respectively. The average bank assets were 2.57 trillion USD and 0.2045 trillion USD in the United States and Taiwan respectively. These data were retrieved from the website of the Federal Deposit Insurance Corporation (FDIC, https://www5.fdic.gov/) and the Taiwanese Banking Bureau, Financial Supervisory Commission (FSC, https://www.banking.gov.tw/). The exchange rate on Dec. 31, 2015 was 33.066 NTD/USD, obtained from the Central Bank of Taiwan (http://www.cbc.gov.tw/). 7 TEJ Databank is a well-known databank containing financial information on Taiwan. Its website is http://www.tej.com.tw/. 8 In financial theory, the variance for day T ( T2 ) is usually calculated by the formula T2 T D2 , where D is the daily standard deviation. 9 For each merger case, there is a constant conversion ratio of the stock for the bank to that of the holding company at the merger time. Thus, for such a constant ratio, the bank’s returns are the same as the returns of the holding company.
24
Table 1 presents descriptive statistics for each bank’s yearly data, specifically, the mean and standard deviation of the equity, mean debt, mean cash dividends, and the sample number for the bank.
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(1986), hereafter referred to as the RV formulas. We use the grid search method to calibrate the assets and the standard deviations of the asset returns. We begin by generating 1000
possible asset values within the range [ A0 (t ), A0 (t ) 2B(t )] , where A0 (t ) E (t ) B(t ) , as
-p
well as 200 corresponding standard deviations within the range [0.01 E , 2 E ] .10 The
squares error, which is defined as follows:
na
(27)
Eˆ (t ) E (t ) 2 ˆ E2 (t ) E2 (t ) 2 ) ( ) , E (t ) E2 (t )
lP
sum of squares error (
re
optimal estimated values of V (t ) and V are based on the minimum values of the sum of
where Eˆ (t ) and ˆ E2 (t ) are the respective mean and standard deviation of the equity,
ur
calculated from the FPT and RV formulas, given different values of V (t ) and V .
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We let s 0.02 , t 0 , T 1 year, and 0.97 , the latter based on Ronn and Verma (1986). Table 2 shows the estimated values of the bank’s assets and the standard deviations of its returns, based on the FPT and RV formulas.
In general, the volatility of assets is smaller than the volatility of equity in real cases. This small range is
used to reduce the time needed to calculate the estimates.
25
Table 2 shows that most of the estimates from the RV formulas are equal to or less than those from the FPT formulas. For example, the assets of Bank 1 are estimated as 891.250 billion from the FPT formulas and 888.840 billion from the RV formulas. Likewise, the estimated standard deviation of the returns is 4.418% from the FPT formulas and 4.330% from the RV formulas. However, in some cases (e.g., Banks 12, 13, 21, 22, 31, and 32) the two formulas give the same estimates. Therefore, we conclude that the model that takes account of a possible early bankruptcy gives equal or lower estimates for the
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assets and returns than the model that does not take this into account. In theory, we can infer that the lower the assets, the higher the DI premium; but the lower the standard
deviation of the returns, the lower the DI premium. Therefore, we cannot determine from
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the results based on the FPT formulas whether DI premiums increase or decrease in the tested circumstances.
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Table 3 shows these estimated DI premiums, based on the above parameters as estimated by the various formulas. The table shows premium estimates incorporating the
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interest rate spread alone, IPP from Equation (11), as well as estimates incorporating both the interest rate spread and the early-bankruptcy risk, IPP A from Equation (18). For
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example, the IPP values for Bank 1 (Chang Hwa Bank), calculated in accordance with the parameters estimated by the FPT and RV formulas, are 3.0400 103 and 3.6301 103
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respectively; the corresponding IPP A values are 3.9929 103 and 5.0657 103 . Table 3
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also shows that IPP A is always higher than the corresponding IPP , regardless of the parameters estimated by the FPT and RV formulas. The means of the IPP / IPP A ratios ( IPP divided by IPP A ) are nearly 74% for the FPT formulas and 71% for the RV formulas. The DI premiums in the second and fifth columns are calculated using both our model (i.e., the FPT formulas and our pricing formula) and the traditional model (i.e., the RV formulas and the European pricing formula). Using the values in these two columns, we find that the mean IPP / IPP A ratio is 79.19%. In other words, on average, DI premiums for Taiwanese 26
banks are underestimated by 20.81% (i.e., 1-79.19%) using the traditional pricing formula. This result suggests that insurers who use the traditional model are likely to conclude that they have insufficient funds to cover the bank’s default risk, because they ignore the possibility of early bankruptcy in calculating the DI premium.
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amount of the deposit account (of any kind, such as demand deposit, fixed deposit, and so
on) is less than 3 million, the DI insurer will protect all the losses from the deposit account. Moreover, since 2013, all insured banks are subject to a risk-based premium system that has
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five premium rates: 5bp, 6bp, 8bp, 11bp, and 15bp. For a bank, the DI premium is
determined by its risk grading indicator, calculated by two methods: “The BIS Capital
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Adequacy Ratio” and “The Composite Score of the Risk-based Premium Rating System”.11 In this paper, the DI premiums range from 0.4bp to 178bp, as calculated by Merton’s
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model. Moreover, the calculated premiums are larger than 15bp for 15 of the 32 banks. The current DI premium in Taiwan seems to be underestimated compared to the theoretical DI
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premium from Merton’s model. The same conclusion can be reached by using our model, since the DI premiums estimated by our model are usually larger than those estimated by
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Merton’s model. This conclusion is not surprising, because many previous studies have found that the theoretical risk-based premium is usually higher than the actual DI premium
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(e.g., Ronn and Verma, 1986; Chuang, Lee, Lin and Yu, 2009). We used our model to analyze the changes in the DI premium due to specific events
in the 1997 Asia financial crisis. We divided the sample period into three sub-periods: 1996, 1997, and 1998. To avoid confusion on the effects of the IPO (Initial Public Offer), we
11
See the data in Central Deposit Insurance Corporation, Taiwan. Retrieve from: http://www.cdic.gov.tw/.
27
selected the 11 banks (see Table 1) which were traded in the stock market for more than 2 years as of 1996. We estimated the parameters (i.e., s ,
B ( 0) and V ) using the FPT V ( 0)
formula and calculated the DI premiums (i.e., IPP and IPP A ) for each bank in each subperiod. Table 4 shows the means of the estimated parameters and of the calculated DI premiums.
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9.6985% in 1996 and 1998, respectively. This result implies that the bank’s asset volatility increased due to the 1997 Asian financial crisis. The interest rate spreads were 0.0242,
0.0233, and 0.0190 in 1996, 1997, and 1998, respectively. The mean debt/asset ratios were
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0.7240, 0.7118, and 0.7865. Based on these parameters, the mean DI premiums ( IPP A )
were 0.0276bp, 0.6787bp, and 1.0117bp in 1996, 1997, and 1998, respectively. Compared
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with the results in 1996 and 1997, the DI premium increased during 1997 due to the
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increases in asset volatility. The DI premium for 1998 shows an increase after the financial crisis, even though the banks’ asset volatility decreased. We can infer that this could be
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caused by both the decreases in the interest rate spread and the increases in the debt/asset ratio.
Since 2001, the Taiwanese government has encouraged financial institutions to
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merge, creating a single financial holding company. As shown in Table 1, there are thirteen
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financial holding companies (Banks 2, 3, 4, 5, 11, 12, 13, 19, 20, 21, 22, 29 and 31) in our sample. Only Bank 31 merged in 2010. The others merged in 2001 or 2002. We therefore chose the banks which became financial holding companies in 2001 or 2002 to analyze the consolidation effects. In the sample, eight banks (3, 4, 11, 19, 20, 21, 22 and 29) became financial holding companies in 2001, and four banks (Banks 2, 5, 12 and 13) became financial holding companies in 2002. For purpose of the analyses, we defined three sub-
28
samples: Sub-sample 1 contains the bank’s data from the year before consolidation; Subsample 2 contains the bank’s data from the year of consolidation; and Sub-sample 3 contains the bank’s data from the year after consolidation. We assigned each bank’s data to the corresponding sub-sample based on the bank’s real consolidation year. For example, Bank 2 became a financial holding company in 2002. Thus, for Bank 2, Sub-sample 1 contains its data on 2001; Sub-sample 2 contains its data on 2002; and Sub-sample 3 contains its data on 2003. For Bank 3, its consolidation year is 2001. Thus, for Bank 3, Sub-
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sample 1, Sub-sample 2, and Sub-sample 3 contain data on years 2000, 2001, and 2002, respectively. For each bank in each sub-sample, we estimated its parameters (i.e., s ,
B ( 0) V ( 0)
and V ) by the FPT formula, and then we calculated its DI premiums (i.e., IPP and IPP A )
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using our model. The means of each parameter and DI premium for each sub-period are
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shown in Table 4 for the analyses of the consolidation effect.
As shown in Table 4, the mean interest rate spreads are 0.0356, 0.0456, and 0.0456 in
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Sub-sample 1, Sub-sample 2, and Sub-sample 3, respectively. Accordingly, the interest rate spreads for these banks increased due to the consolidation effect. Moreover, in Sub-sample
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1, the means of the debt/asset ratios and asset volatility are 0.9698 and 5.0483%, respectively. In Sub-sample 2, they are 0.9674 and 4.8653%. In Sub-sample 3, they are
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0.9610 and 4.5170%. These results show that the debt/asset ratios and asset volatility decreased due to the consolidation effect. Accordingly, the means of the DI premiums
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( IPP A ) are 50.7015bp, 31.8128bp, and 17.4516bp in Sub-sample 1, Sub-sample 2, and Sub-sample 3, respectively. Thus, we can infer that the increases of the interest rate spreads and the decreases of both the debt/asset ratios and asset volatility resulted from the consolidation effect, which induced a decrease in the DI premiums for the consolidated banks. As mentioned in a previous section, our model indicates that if a bank’s interest rate 29
spread is smaller than its dividend yield, the DI premium is higher using our model than using the traditional model. Figure 2 displays the empirical evidence for the sensitivity of DI premiums to changes in the interest rate spread. We take Bank 1 as our example. As shown in Table 2, its average estimated dividend yield, computed by the FPT formulas, is 0.16%.12 Therefore, we let the interest rate spread range from 0.1% to 1%. As shown in Figure 2, both IPP and IPP A are negatively correlated with the interest rate spread. In other words, if a bank’s earnings increase because of the interest rate spread, the DI
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premium decreases. We have also proven that if the interest rate spread is smaller (larger) than the dividend yield, IPP A is smaller (larger) than IPP . Figure 2 also shows that the
IPP A curve is flatter than the IPP curve. In other words, compared with a model that takes
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into account only the likelihood of bankruptcy at the maturity date, the sensitivity of the DI
of a possible early bankruptcy.
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5. Conclusion
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premium to the change in the interest rate spread is less using the model that takes account
The DI system plays an important role in a country’s financial market. A well-designed
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system can facilitate the country’s economic development because it can prevent an economic crisis from spreading if the banks go bankrupt. The maintenance of a good DI
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system depends heavily on the determination of a fair DI premium. Therefore, the question of how best to valuate a DI premium has received a great deal of attention, not only from
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DI companies but also from academic researchers. The main objective of the present study was to provide an accurate closed-form pricing formula and optimal empirical methods for determining fair DI premiums. In practice, the interest rate spread affects a bank’s earnings, and this will in turn 12
The estimated dividend yield equals the cash dividend divided by the estimated assets.
30
further influence the value of the bank’s assets and the likelihood of it going bankrupt. In addition, it must also be kept in mind that a bank can go bankrupt prior to the maturity date of the DI contract. These two factors should influence whether a DI premium is accurately valuated. Therefore, we incorporated these two factors into our model for deriving a pricing formula for a DI premium. Our consideration of the early bankruptcy risk also led us to introduce new empirical formulas derived from the first-passage time model to calibrate the necessary parameters, namely, the bank’s assets and the standard deviation of its returns.
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Therefore, our model should help DI institutions more precisely determine fair premiums.
We collected data from the 32 banks listed on the Taiwan stock exchange to illustrate the operation of our model. We have shown that, regardless of the theoretical inferences or
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the empirical evidence, DI premiums are negatively correlated with the interest rate spread. We also used numerical results to prove that the sensitivity of DI premiums to changes in
possibility of early bankruptcy.
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the interest rate spread is lessened if the premium is priced by a model that allows for the
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Our study also compared results that were calculated based on the parameters calibrated by our formulas (FPT formulas) and by Ronn and Verma’s (1986) formulas (RV
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formulas). Our results indicate that most of the values calibrated by the RV formulas are equal to or lower than those calibrated by the FPT formulas. In other words, the RV
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formulas can underestimate the bank’s assets as well as its risk. In addition, DI premiums based on the possibility of early bankruptcy ( IPP A ) are always higher than premiums not
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based on a consideration of this risk ( IPP ), regardless of whether the FPT or RV formulas are used to calibrate the parameters. Our results indicate that DI premiums calculated by the model that does not take account of the early bankruptcy risk are underpriced by 20.81%. If a DI company prices its premiums using the traditional model and ignores the risk of early bankruptcy, the premiums will be too low. Currently, we have available for use in our model only the DI values for each bank in 31
our sample. In future research, it may be interesting to examine how these values vary over time and whether their dynamics provide a good match to important news events. It may also be interesting to ask how likely it is that banks will transfer their risks to DI companies if the premium is calculated by our model. In addition, our model can be enriched and enhanced, and allowing for discrete jumps in assets would increase its practical relevance. Finally, there are some common phenomena that could influence the valuation of a DI premium. For example, if a bank fails or goes bankrupt, government regulators usually
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delay resolving the default, using the policies of delayed resolution and capital forbearance. Depositor preference arrangements can also influence the DI premium. If the DI valuation model considers the effects caused by the policies of delayed resolution or capital
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forbearance for defaulted banks and the seniority of insured deposits, the PD (probability of default) and LGD (loss given default) for banks may be altered. In future research, it may
their influence on DI premiums.
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Appendix A
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be interesting to incorporate these phenomena into our DI valuation model to understand
This appendix shows how to obtain Equation (7). In pricing theory, any asset process can be
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chosen as the numeraire asset (Davis, 1998). For this paper, we chose a money market account to serve as the numeraire asset. By doing it, the strike price becomes a constant in
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the valuation. The money market account is defined as follows: T
M (t ,T ) exp( r f (u )du ) and M (t , t ) 1.
(A1)
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t
Accordingly, the dynamic process of M (t , u) under the risk-neutral probability measure can be expressed as follows: dM (t , u ) r f (u )du b(t , u )dWr (u ) , M (t , u )
(A2)
where dWr (u) is a Brownian motion under the risk-neutral probability measure. When choosing a money market account M (t , u) to serve as the numeraire asset, 32
V (u ) , B1 (u ) , and B(u ) can be transferred as the new processes of FMV (t , u ) , FMB1 (t , u) , and FMB (t , u) . They can be represented as follows: FMV (t , u )
V (u ) ; M (t , u )
(A3)
FMB1 (t , u )
B1 (u ) ; and M (t , u )
(A4)
FMB (t , u )
B(u ) . M (t , u )
(A5)
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According to Equation (A3), using Taylor’s polynomials and Itoˆ ’s Lemma, we have:
dFMV (t , u ) 1 ( (u ) )du | V b(t , u ) |2 du | V b(t , u ) | dW F (u ) , V 2 FM (t , u )
(A6)
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where | V b(t , u) |2 V2 2V b(t , u) b2 (t , u) and dW F (u ) is a standard Brownian
FMV (t , T ) FMV (t , t ) exp(( s )(T t ))
T 1 T | V b(t , u ) |2 du | V b(t , u ) |dW F (u )) t 2 t
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exp(
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motion under the risk-neutral probability measure. Thus, we have
(A7) where
T 1 T | V b(t , u ) |2 du | V b(t , u ) |dW F (u )) . t 2 t
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exp(
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FMV (t , t ) exp(( s )(T t )) ,
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We have E[ ] 1, where E[] is an expectation operator. Thus is a Radon-Nikodym derivative that can be used for changing the measure. Because we have
FMV (t , t ) FMV (t , t ) exp( (s )(t t )) and
E[exp(
T 1 T | V b(t , u ) |2 du | V b(t , u ) |dW F (u ))] 1 , t 2 t
we can obtain
33
E[ FMV (t , T ) exp( (s )(T t ))] FMV (t , t ) exp( (s )(t t )) . Accordingly, FMV (t , u) exp( (s )(u t )) is a martingale process under the risk-neutral measure. Moreover, according to Equation (A4), we have B1 (u ) B (t ) M (t , u ) 1 B1 (t ) , for t u T . M (t , u ) M (t , u )
FMB1 (t , u )
(A8)
By the same token, we have
FMB (t , u) B(t ) . Thus, FMB1 (t , T ) and FMB (t , T ) are deterministic values.
FMIP (t , t )
IP(t , T ) IP(T , T ) and FMIP (t , T ) , M (t , t ) M (t , T )
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We now define the following:
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(A9)
(A10)
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where FMIP (t , u) is a new processes transferred from IP(u, T ) by choosing money market
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account M (t , u) as the numeraire asset.
In pricing theory, the asset values discounted by money market account are
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martingales (Duffie, 1996). We have:
FMIP (t , t ) E[ FMIP (t , T )] E[
B1 (T ) V (T )) B(T ) ] M (t , T )
( B1 (T )
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B1 (T ) V (T ) B1 (T ) B(T ) E[( ) ] M (t , T ) M (t , T )
E[( FMB1 (t , T )
FMB1 (t , T ) V FM (t , T )) ] B FM (t , T )
FMB1 (t , T ) V E[ F (t , T ) I D B FM (t , T ) I D ] FM (t , T ) B1 M
34
FMB1 (t , t ) E[ I D ]
FMB1 (t , t ) E[ FMV (t , T ) I D ] , FMB (t , t )
(A11)
where I {} is an indicator function and D {FMV (t , T ) FMB (t , T )} . Because E[ I D ] is the default probability, we have:
E[ I D ] P( FMV (t ,T ) FMB (t ,T )) , where P() is the probability operator. (A12)
E[ I D ] P( FMV (t , t ) exp(( s )(T t )) exp(
ln( P(
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FMB (t , t ) 1 ) ( s )(T t ) v 2 (t , T ) V FM (t , t ) 2 ) v(t , T )
B(t ) 1 ) ( s )(T t ) v 2 (t , T ) V (t ) 2 ) v(t , T )
N ( y2 (t , T )) ,
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P(
T 1 T 2 | b ( t , u ) | du | V b(t , u ) |dW F (u )) FMB (t , T )) V t t 2
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ln(
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By substituting Equation (A7) into Equation (A12), we get the following:
(A13)
T
where ~ N (0,1) and v 2 (t , u ) | V b(t , u ) |2 du . Moreover, we have
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t
E[ FMV (t , T ) I D ] FMV (t , t ) exp(( s )(T t )) E[I D ] .
(A14)
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As shown in Equation (A7), is a Radon-Nikodym derivative used for changing the measure. Thus, we have
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E[I D ] N ( y1 (t , T )) , and y1 (t , T ) y2 (t , T ) v(t , T ) .
(A15)
According to Equations (A11)-(A15), FMIP (t , t ) can be rewritten as follows:
FMB1 (t , t ) V FM (t , t )e( s )( T t ) N ( y1 (t , T )) . F (t , t ) F (t , t ) N ( y2 (t , T )) B FM (t , t ) IP M
B1 M
Using Equations (A1), (A3)-(A5), we can obtain Equation (7).
35
(A16)
Appendix B This appendix gives the derivation of Equations (14) and (15). According to Equation (13), we have the following:
IP A (t d 2 , T ) E[
IP A (t d 1 , T ) (1 I D (td 1 ) ) | Ftd 2 ] M (t d 2 , t d 1 ) V (t d 1 ) ) B(t d 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , t d 1 )
E[ E[
IP(t d 1 , T ) (1 I D (td 1 ) ) | Ftd 2 ] M (t d 2 , t d 1 )
V (t d 1 ) ) B(t d 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , t d 1 )
IP(td 2 , T ) E[
IP(td 1, T ) I D (td 1 ) | Ftd 2 ] M (td 2 , td 1 )
V (td 1 ) ) B(td 1 ) I D (td 1 ) | Ftd 2 ] . M (t d 2 , t d 1 )
B1 (t d 1 )(1
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E[
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E[
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B1 (t d 1 )(1
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B1 (t d 1 )(1
(B1)
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For a traditional DI pricing formula, we have
IP(td 1 , T ) B1 (td 1 ) E[ I D(td ) | Ftd 1 ] B1 (td 1 )
V (td 1 ) ( s ) t * e E [ I D (td ) | Ftd 1 ] B(td 1 )
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By substituting IP(td 1 , T ) into Equation (B1), we get the following:
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IP A (t d 2 , T ) IP(td 2 , T ) E[
E[
B1 (td 1 )
B1 (td 1 )(1 E[ I D (td ) | Ftd 1 ]) M (td 2 , td 1 )
I D (td 1 ) | Ftd 2 ]
V (td 1 ) (1 e( s ) t E *[ I D (td ) | Ftd 1 ]) B(td 1 ) I D (td 1 ) | Ftd 2 ] , M (td 2 , td 1 )
where E *[ I D (td ) | Ftd 1 ] E[ I D(td ) ] , and is a Radon-Nikodym derivative used for changing the measure. 36
(B2)
If we let e ( s ) t 1 ( s)t (( s)t ) 2 , we have:
1 e( s ) t E *[ I D(td ) | Ftd 1 ] 1 (1 ( s)t ) E *[ I D(td ) | Ftd 1 ] 1 E *[ I D(td ) | Ftd 1 ] (( s)t ) E *[ I D(td ) | Ftd 1 ] O(t ) , (B3) where O(t ) is a function of the higher order of t . If we let t tend toward zero, we
IP A (t d 2 , T ) IP(td 2 , T ) E[
E[ B1 (td 1 )
I D (td 1 ) | Ftd 2 ]
V (td 1 ) * E [1 I D (td ) | Ftd 1 ] B(td 1 ) I D (td 1 ) | Ftd 2 ] M (td 2 , td 1 )
V (td 1 ) ( s)tE *[ I D (td ) | Ftd 1 ] B(td 1 ) I D (td 1 ) | Ftd 2 ] . M (td 2 , t d 1 )
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E[
M (td 2 , td 1 )
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B1 (t d 1 )
B1 (td 1 ) E[1 I D (td ) | Ftd 1 ]
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obtain O(t ) 0 . Thus, Equation (B1) can be rewritten as follows:
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(B4)
According to our specifications, I D (td 1 ) means that the bank defaulted at time t d 1 .
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Because td 2 td 1 td , if the bank defaulted at time t d 1 , it will still be in default at time
t d . That is, I D (td ) 1 when I D (td 1 ) 1 . Therefore, we have E[(1 I D(td ) )]I D(td 1 ) 0 , no
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matter whether I D (td 1 ) 1 or 0. By the same token, we have E *[(1 I D(td ) )]I D (td 1 ) 0 . We
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therefore have
E[
B1 (td 1 ) E[1 I D (td ) | Ftd 1 ] M (td 2 , td 1 )
I D (td 1 ) | Ftd 2 ] 0 ; and
(B5) B1 (td 1 ) E[
V (t d 1 ) * E [1 I D (td ) | Ftd 1 ] B(t d 1 ) I D (td 1 ) | Ftd 2 ] 0 . M (td 2 , td 1 ) 37
(B6) In addition, we have E *[ I D (td ) ] 1 , if I D (td 1 ) equals 1. We conclude that E *[ I D (td ) ]I D (td 1 )
I D (td 1 ) . We therefore have B1 (t d 1 ) E[
V (td 1 ) ( s)tE *[ I D (td ) | Ftd 1 ] B(td 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , td 1 ) V (td 1 ) ( s)t B(td 1 ) I D (td 1 ) | Ftd 2 ]) . M (t d 2 , td 1 )
B1 (t d 1 )
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E[
(B7)
Using the results of Equations (B4)-(B7), we find that the sum of the second and the third terms on the right hand side of Equation (B1) can be expressed as follows:
V (td 1 ) ( s)t B(td 1 ) I D (td 1 ) | Ftd 2 ] . M (td 2 , td 1 )
B1 (td 1 )
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E[
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IP(td 1, T ) I D (td 1 ) | Ftd 2 ] E[ M (td 2 , td 1 )
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E[
V (td 1 ) ) B(td 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , t d 1 )
B1 (t d 1 )(1
(B8)
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Using the previous results, we can rewrite Equation (B4) as follows:
(B9)
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IP A (t d 2 , T ) IP(td 2 , T ) E[
B1 (td 1 ) t B(t d 1 ) I D (td 1 ) | Ftd 2 ] . M (t d 2 , td 1 )
( s )V (t d 1 )
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This is the same as Equation (14). According to Equation (13), the following result can be obtained for the time t d 3 :
IP A (td 2 , T ) IP (td 3 , T ) E[ (1 I D (td 2 ) ) | Ftd 3 ] M (td 3 , td 2 ) A
38
V (t d 2 ) ) B(t d 2 ) I D (td 2 ) | Ftd 3 ] . M (t d 3 , t d 2 )
B1 (td 2 )(1 E[
(B10) According to Equation (B9), the first term on the right hand side of Equation (B10) can be expressed as follows:
E[
IP A (td 2 , T ) (1 I D (td 2 ) ) | Ftd 3 ] M (td 3 , td 2 ) B1 (t d 1 ) t B(t d 1 ) I D (td 1 ) | Ftd 2 ] M (t d 2 , t d 1 ) (1 I D (td 2 ) ) | Ftd 3 ] M (t d 3 , t d 2 )
IP(t d 2 , T ) E[ E[
IP(td 2 , T ) IP(td 2 , T ) | Ftd 3 ] E[ I D (td 2 ) | Ftd 3 ] M (t d 3 , td 2 ) M (td 3 , td 2 )
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E[
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( s )V (td 1 )
B1 (td 1 ) t B(td 1 ) I D (td 1 ) (1 I D (td 2 ) ) | Ftd 3 ] M (td 3 , t d 1 )
E[
IP(td 2 , T ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
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IP(td 3 , T ) E[
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( s )V (td 1 )
B1 (td 1 ) t B(td 1 ) I D (td 1 ) (1 I D (td 2 ) ) | Ftd 3 ] . M (td 3 , t d 1 )
( s )V (td 1 )
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E[
(B11)
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Again, we have I D (td 1 ) I D (td 2 ) I D (td 2 ) . The following result can be obtained: B1 (t d 1 ) t B(t d 1 ) I D (td 1 ) (1 I D (td 2 ) ) | Ftd 3 ] M (t d 3 , t d 1 )
( s )V (t d 1 )
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E[
B1 (t d 1 ) t B(t d 1 ) I D (td 1 ) | Ftd 3 ] M (td 3 , td 1 )
( s )V (td 1 )
E[
E[
B1 (td 2 ) t B(td 2 ) I D (td 2 ) | Ftd 3 ] . M (td 3 , td 2 )
( s )V (td 2 )
(B12) 39
In Equation (12), we let E[
V (td 1 ) V (td 2 ) I D (td 2 ) | Ftd 3 ] E[ I D (td 2 ) | Ftd 3 ] . In the M (td 3 , td 1 ) M (td 3 , td 2 )
specification of the model, td 3 td 2 td 1 . If bank defaulted at time t d 2 (i.e., I D (td 2 ) 1 ), there is no bank value (i.e., V (td 1 ) ) at time td 1 . Thus, we only need to calculate the expected bank value at defaulted time t d 2 . The pricing formula for time t d 3 can be expressed as follows:
E[
IP(td 2 , T ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
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E[
B1 (td 2 ) t B(td 2 ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
( s )V (td 2 )
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IP A (td 3 , T ) IP(td 3 , T ) E[
B1 (t d 1 ) t B(td 1 ) I D (td 1 ) | Ftd 3 ] M (t d 3 , t d 1 )
( s )V (t d 1 )
V (t d 2 ) ) B(t d 2 ) I D (td 2 ) | Ftd 3 ] . M (t d 3 , t d 2 )
B1 (td 2 )(1
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E[
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(B13) Because we have:
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IP(td 2 , T ) B1 (td 2 ) E[ I D(td ) | Ftd 2 ] B1 (td 2 )
V (td 2 ) ( s ) 2 t * e E [ I D (td ) | Ftd 2 ] . B(td 2 )
Thus, the last two terms on the right hand side of Equation (B13) can be expressed as
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follows:
IP(td 2 , T ) I D (td 2 ) | Ftd 3 ] E[ M (td 3 , td 2 )
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E[
E[
E[
B1 (td 2 )(1 E[ I D (td ) | Ftd 2 ]) M (td 3 , td 2 )
B1 (td 2 )
V (t d 2 ) ) B(t d 2 ) I D (td 2 ) | Ftd 3 ] M (t d 3 , td 2 )
B1 (t d 2 )(1
I D (td 2 ) | Ftd 3 ]
V (td 2 ) (1 e( s ) 2 t E *[ I D (td ) | Ftd 2 ]) B(td 2 ) I D (td 2 ) | Ftd 3 ] . M (td 3 , td 2 )
Again, we have: 40
(B14)
1 e( s ) 2t E *[ I D (td ) | Ftd 1 ] 1 E *[ I D(td ) | Ftd 1 ] (2( s)t ) E *[ I D(td ) | Ftd 1 ] O(t ) . If the bank defaulted at time t d 1 or t d 2 , it will still be in default at time t d . Thus, we also have E[(1 I D(td ) )]I D(td 2 ) 0 and E *[ I D (td ) ]I D (td 1 ) I D (td 1 ) . Using the results shown in Equations (B4)-(B7), we have: IP(td 2 , T ) I D (td 2 ) | Ftd 3 ] E[ M (td 3 , td 2 )
V (t d 2 ) t B(t d 2 ) I D (t d 2 ) | Ft d 3 ] . M (t d 3 , t d 2 )
( s) B1 (t d 2 ) 2 E[
(B15)
re
E[
B1 (t d 1 ) t B(td 1 ) I D (td 1 ) | Ftd 3 ] M (t d 3 , t d 1 )
( s )V (t d 1 )
B1 (td 2 ) t B(td 2 ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
( s )V (td 2 )
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IP A (td 3 , T ) IP(td 3 , T ) E[
B1 (td 2 ) t B(td 2 ) I D (td 2 ) | Ftd 3 ] M (td 3 , td 2 )
( s )V (td 2 )
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2 E[
(B16)
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IP(td 3 , T )
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Therefore, we have
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E[
V (t d 2 ) ) B(t d 2 ) I D (td 2 ) | Ftd 3 ] M (t d 3 , td 2 )
B1 (t d 2 )(1
d 1
E[
i d 2
B1 (ti ) B(ti ) I D (ti ) | Ftd 3 ] . M (t d 3 , ti )
( s )V (ti )
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Going backwards recursively from Equation (B16) to obtain Equation (15), we get d 1
IP A (t , T ) IP(t , T ) E[ i 1
B1 (ti ) B(ti ) I D (ti ) | Ft ] . M (t , ti )
( s )V (ti )
(B17) Then we obtain
41
B1 (u ) B (t ) V (u ) B(u ) I D (u ) | Ft ] ( s ) 1 E[ I D (u ) | Ft ] M (t , u ) B(t ) M (t , u )
( s )V (u ) E[
( s ) e( s )( u t )V (t )
B1 (t ) E[ I D* (u ) | Ft ] B(t )
( s ) e( s )( ut )V (t )
B1 (t ) N ( y1 (t , u )) . B(t )
(B18)
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By substituting this result into Equation (B17), we get Equation (15).
42
References Allen, L. and A. Saunders, 1993, Forbearance and Valuation of Deposit Insurance as a Callable Put, Journal of Banking and Finance 17(4), 629-643. Black, F. and J. C. Cox, 1976, Valuing Corporate Securities: Some Effects of Bond Indenture Provisions, Journal of Finance 31(2), 351-367. Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81(3), 637-654.
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Chang, C. and S. Chung, R.-J. Ho, and Y.-J. Hsiao, 2018, Revisiting the Valuation of
Deposit Insurance. 27th Annual Meeting of the European Financial Management Association. Working paper.
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Chen, A. H., N. Ju, S. C. Mazumdar and A. Verma, 2006, Correlated Default Risks and Bank Regulations, Journal of Money, Credit, and Banking 38(2), 375-398.
re
Chuang, H. L., S. C. Lee, Y. C. Lin and M. T. Yu, 2009, Estimating the Cost of Deposit Insurance with Stochastic Interest rates: the Case of Taiwan, Quantitative Finance 9(1),
lP
1-8.
Chung, S., 1999, American Option Valuation under Stochastic Interest Rates, Review of
na
Derivatives Research 3, 283-307.
Davis, K., 2015, Depositor preference, bail-in, and deposit insurance pricing and design,
ur
Working Paper.
Davis, M. H. A., 1998, A Note on the Forward Measure, Finance Stochastic 2(1), 19-28.
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Duan, J. C., A. F. Moreau and C.W. Sealey, 1992, Fixed Rate Deposit Insurance and RiskShifting Behavior at Commercial Banks, Journal of Banking and Finance 16(4), 715742.
Duan, J. C. and J. G. Simonato, 2002, Maximum Likelihood Estimation of Deposit Insurance Value with Interest Rate Risk, Journal of Empirical Finance 9(1), 109-132. Duan, J. C. and M. T. Yu, 1994, Assessing the Cost of Taiwan’s Deposit Insurance, Pacific43
Basin Finance Journal 2(1), 73-90. Duan, J. and M. Yu, 1999, Capital standard, forbearance and deposit insurance pricing under GARCH, Journal of Banking and Finance, 23, 1691- 1706. Duffie, D., 1996, Dynamic Asset Pricing Theory, published by Princeton University. Episcopos, A., 2004, The Implied Reserve of the Bank Insurance Funds, Journal of Banking and Finance 28, 1617-1635. Finger, C. C., 2002, CreditGrades: Technical Document, Risk Metrics Group, Inc. pressed.
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http://www.creditrisk.ru/publications/files_attached/cgtechdoc.pdf.
Giammarino, R., E. Schwartz and J. Zechner, 1989, Market Valuation of Bank Assets and Deposit Insurance in Canada, Canadian Journal of Economics 22(1), 109-127.
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Heath, D., R. Jarrow and A. Morton, 1992, Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation, Econometrica 60(1), 77-
re
105.
Hwang, D. Y., F. S. Shie, K. Wang and J. C. Lin, 2009, The Pricing of Deposit Insurance
lP
Considering Bankruptcy Costs and Closure Policies, Insurance: Mathematics and Economics 33(10), 1909-1919.
na
Kim, I. J., 1990, The Analytic Valuation of American Options, The Review of Financial Studies 3(4), 547-572.
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Leland, H. E., 1994, Corporate Debt Value, Bond Covenants, and Optimal Capital Structure, Journal of Finance, 49, 1213-1252.
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Lee, S. C, J. P. Lee and M. T. Yu, 2005, Bank Capital Forebearance and Valuation of Deposit Insurance, Canadian Journal of Administrative Sciences 22(3), 220-229.
Marcus, A. J. and I. Shaked, 1984, The Valuation of FDIC Deposit Insurance Using Optionpricing Estimates, Journal of Money, Credit and Banking 16(4), 446-460. Merton, R., 1974, On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance, 29, 449-470. 44
________, 1977, An Analytic Derivation of the Cost of Deposit Insurance and Loan Guarantee, Journal of Banking and Finance 1(1), 3-11. ________, 1978, On the Cost of Deposit Insurance When There are Surveillance Costs, Journal of Business 51, 439-452. Musiela, M. and M. Rutkowski, 2002, Martingale Methods in Financial Modelling, Springer. Nagarajan, S. and C.W. Sealey, 1995, Forbearance, deposit insurance pricing, and incentive
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compatible bank regulation, Journal of Banking and Finance 19, 1109-1130.
Peltzman, S., 1970, Capital Investment in Commercial Banking and Its Relationship to Portfolio Regulation, Journal of Political Economy 78(1), 1-26.
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Pennacchi, G. G., 1987a, Alternative Forms of Deposit Insurance: Pricing and Bank Incentive Issues, Journal of Banking and Finance 11(2), 291-312.
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________, 1987b, A Reexamination of the Over (or Under) Pricing of Deposit Insurance, Journal of Money, Credit and Banking 19, 340-360.
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Rabinovitch, R., 1989, Pricing Stock and Bond Options When the Default Free Rate is Stochastic, Journal of Financial and Quantitative Analysis 24, 447-457.
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Ronn, E. I. and A. K. Verma, 1986, Pricing Risk-adjusted Deposit Insurance: An Optionbased Model, Journal of Finance 41(4), 871-895.
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Santomero, A. M. and J. D. Vinso, 1977, Estimating the Probability of Failure for Commercial Banks and the Banking System, Journal of Banking and Finance 1(2),
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185-205.
Shie, F. S., 2012, Pricing of Deposit Insurance Considering Capital Forbearance and Jump Risk, International Journal of Information and Management Sciences 23 (1), 1-18.
Shyu, D. and M. S. Tsai, 1999a, Deposit Insurance Rate and Risk-Shifting Behavior of Taiwan’s Commercial Banks, Review of Securities and Futures Markets 11, 1-28. _______, 1999b, Deposit Insurance Rate and Risk-Shifting Behavior of Taiwan’s New 45
Commercial Banks, Sun Yat-sen Management Review 8, 691-708. Vasicek, O.. 1977. An Equilibrium Characterization of the Term Structure, Journal of
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Financial Economics 5(2), 177-188.
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Figure 1: Trend of the Interest Rate Spread in Taiwan Financial Markets 5.5 5
4 3.5 3
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Interest Rate Spread (%)
4.5
2.5
1985
1990
1995 years
2000
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1.5 1980
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2
2005
2010
Note: This figure shows the trend of the average interest rate spread for the top five Taiwanese banks. The data were obtained from the Central Bank of Taiwan. The website for the data is expressed in the following:
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https://www.cbc.gov.tw/ct.asp?xItem=995&ctNode=523&mp=1 .
47
Figure 2: Relationship Between Deposit Insurance Premiums and Interest Rate Spreads 66 64
60 58 56
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Deposit Insurance Premium (bp)
62
54 52 50
0.2
0.3
0.4 0.5 0.6 0.7 Interest Rate Spread (%)
0.8
0.9
1
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46 0.1
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48
Note: The vertical axis denotes the deposit insurance premiums and the horizontal axis denotes the interest
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rate spreads for Bank 1. The dividend yield for the bank was 0.16% and the interest rate spreads ranged from 0.1% to 1%. The broken line represents premiums calculated using the traditional model. The asterisk line
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represents premiums calculated using our model.
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Table 1: Basic information about the bank sample Symbol
Description
Equity Equity Std (billion) (%)
Bank 1
Chang Hwa Bank
48.574
43.299
839.430
1.448
27
Bank 2**
First Bank
64.967
41.992
968.030
2.098
27
Bank 3**
Hua Nan Bank
55.745
44.227
905.450
1.861
29
Bank 4**
China Development Industrial Bank
48.398
44.696
43.847
1.378
30
40.818
40.217
362.500
1.128
22
Mega International Commercial Bank
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Bank 5**
Debt Dividend Sample (billion) (billion) Number
Standard Chartered Bank (Taiwan)
10.709
42.127
179.260
0.101
24
Bank 7
King's Town Bank
7.544
42.569
93.114
0.011
28
Bank 8*
Kaohsiung Business Bank
108.530
52.307
55.669
0.009
18
Bank 9*
Taitung Business Bank
7.733
54.287
0.001
23
Bank 10
Taichung Commercial Bank
16.155
167.400
0.051
27
Bank 11**
Chinatrust bank
Bank 12**
Cathay United Bank
Bank 13**
Taipei Fubon Bank
Bank 14*
The Chinese Bank
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Bank 6
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43.592
69.951
39.823
825.480
2.510
20
165.120
37.726
908.730
3.287
13
77.592
34.611
860.080
2.292
12
12.378
29.813
170.020
0.026
12
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25.740
Taiwan Business Bank
38.178
40.665
968.600
0.214
13
Bank 16
Bank of Kaohsiung
5.522
37.561
161.610
0.216
13
Bank 17
Cosmos Bank
131.160
42.851
169.460
0.020
16
Bank 18
Union Bank of Taiwan
18.130
33.742
224.460
0.045
16
Bank 19**
Bank of SinoPac
34.120
36.040
496.500
0.800
15
Bank 20**
E.SUN Commercial Bank
28.118
31.544
419.280
0.540
16
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Bank 15
49
Note: This table gives data from the 32 banks listed on the Taiwanese stock exchange. The sample period is 1980 to 2010. The first column displays the bank symbols and the second column gives their names. The third and fifth columns give the means of the bank’s equity and debt during the sample period. The unit of currency is NTD (New Taiwanese Dollar). The fourth column gives the standard deviations of the bank’s equity returns during the sample period, calculated from the daily equity values from 1980 to 2010. The sixth column gives the means of the bank’s cash dividends. The final column gives the annual data for the bank. “*” means that the bank became integrated during the sample period. “**” means that the bank became a financial holding company.
Table 1: Basic information about the bank sample (continuously) Symbol
Description
Equity
Equity
Debt
Sample
(billion)
Number
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(billion) Std (%) (billion)
Dividend
Yuanta Bank
20.609
37.587
224.130
0.063
15
Bank 22**
TaiShin Bank
18.507
40.791
190.560
0.111
7
Bank 23
Far Eastern International Bank
17.819
39.171
223.580
0.185
16
Bank 24*
Chung Shing Bank 193.420
80.856
148.130
0.058
7
Bank 25
Public Bank
17.730
39.133
240.090
0.037
15
Bank 26
Entie Commercial Bank
30.034
39.338
205.790
0.094
16
50.428
85.775
0.693
7
Taiwan Bowa Bank
18.504
44.090
147.620
0.016
12
Bank 29**
JihSun Bank
20.973
40.969
189.340
0.000
15
Bank 30*
Bank of Overseas Chinese citi bank
18.488
49.510
258.110
0.000
9
Bank 31**
Taiwan Cooperative 72.674 Bank
34.270
2263.000
2.164
7
64.243
38.254
0.074
4
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Bank 28*
Bank 32
15.009
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Industrial bank of
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Bank 24
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Bank 21**
Bank of Taipei
1.834
Note: This table gives data from the 32 banks listed on the Taiwanese stock exchange. The sample period is
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1980 to 2010. The first column displays the bank symbols and the second column gives their names. The third and fifth columns give the means of the bank’s equity and debt during the sample period. The unit of currency is NTD (New Taiwanese Dollar). The fourth column gives the standard deviations of the bank’s equity returns during the sample period, calculated from the daily equity values from 1980 to 2010. The sixth column gives the means of the bank’s cash dividends. The final column gives the annual data for the bank. “*” means that the bank became integrated during the sample period. “**” means that the bank became a financial holding company.
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Table 2: Estimates of each bank’s assets and the standard deviation of its returns FPT formulas RV formulas Symbol Bank value Value Std Bank value Value Std (billion) (%) (billion) (%) 891.250 4.418 888.840 4.330 Bank 1 1035.400 4.415 1034.000 4.199 Bank 2 964.940 4.768 962.100 4.423 Bank 3 98.869 15.232 96.492 14.237 Bank 4 404.070 4.661 403.680 4.315 Bank 5 190.990 4.213 190.150 4.213 Bank 6 101.000 4.630 100.750 4.257 Bank 7 164.250 18.468 164.250 18.538 Bank 8 33.534 11.812 33.498 10.050 Bank 9 184.030 4.880 183.720 4.419 Bank 10 896.250 4.238 896.250 3.982 Bank 11 1074.800 4.447 1074.800 4.447 Bank 12 938.530 3.461 938.530 3.461 Bank 13 182.810 3.104 182.570 3.104 Bank 14 1011.300 4.067 1007.700 4.067 Bank 15 168.380 3.756 167.300 3.756 Bank 16 300.790 10.555 300.790 10.772 Bank 17 242.880 3.594 242.820 3.594 Bank 18 531.120 3.704 531.120 3.604 Bank 19 448.400 3.154 447.820 3.154 Bank 20 244.960 3.862 244.960 3.862 Bank 21 209.260 4.317 209.260 4.317 Bank 22 241.620 4.020 241.620 3.917 Bank 23 341.700 19.077 341.700 19.322 Bank 24 258.090 4.017 258.060 4.017 Bank 25 236.030 4.778 236.030 4.824 Bank 26 100.870 7.129 100.870 7.356 Bank 24 166.280 5.635 166.280 5.697 Bank 28 210.760 4.350 210.500 4.237 Bank 29 276.850 5.149 276.850 5.149 Bank 30 2339.300 3.427 2337.900 3.427 Bank 31 40.127 6.424 40.127 6.424 Bank 32
Note: The first column displays the bank symbols. The second and third columns give the estimates of the
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bank’s assets and the standard deviation of its returns (denoted as Value Std), calculated using the FPT formulas expressed in Equations (25) and (26). The fourth and fifth columns give the corresponding estimates using the RV formulas expressed in Equations (23) and (24). The interest rate spread is set at 0.02.
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Table 3: Estimates of the deposit insurance premiums on per deposit amount FPT formulas RV formulas A Symbol IPP (bp) IPP (bp) Ratios (%) IPP (bp) IPP A (bp) Ratios (%) 30.4001 39.9292 0.7613 36.3014 50.6575 0.7166 Bank 1 29.4050 37.1271 0.7920 29.9086 41.8513 0.7146 Bank 2 32.6478 41.5981 0.7848 36.0709 51.4171 0.7015 Bank 3 27.0516 28.5727 0.9468 17.7397 22.4546 0.7900 Bank 4 2.6696 3.8861 0.6870 3.6733 6.8676 0.5349 Bank 5 21.2569 30.0496 0.7074 29.7321 45.7347 0.6501 Bank 6 14.3088 19.2481 0.7434 14.0824 21.2620 0.6623 Bank 7 21.2507 26.2612 0.8092 21.1049 26.1115 0.8083 Bank 8 19.5293 26.2752 0.7433 20.5436 28.5467 0.7196 Bank 9 17.8828 0.7190 16.3412 25.4880 0.6411 Bank 10 12.8572 5.3693 0.8017 2.4195 3.2346 0.7480 Bank 11 4.3048 0.4871 0.8259 0.4023 0.4871 0.8259 Bank 12 0.4023 4.7749 0.7395 3.5311 4.7749 0.7395 Bank 13 3.5311 3.7177 0.6195 3.3884 5.6246 0.6024 Bank 14 2.3033 14.8959 22.4010 0.6650 18.8812 29.1027 0.6488 Bank 15 23.3836 0.6709 23.7056 37.0693 0.6395 Bank 16 15.6872 7.1569 0.8050 5.6557 7.0465 0.8026 Bank 17 5.7615 3.7647 0.6990 2.7285 3.9336 0.6936 Bank 18 2.6316 6.1122 0.6890 4.2108 6.1117 0.6890 Bank 19 4.2112 3.5587 0.6645 3.1912 4.9297 0.6473 Bank 20 2.3646 4.2217 0.6668 2.8148 4.2217 0.6668 Bank 21 2.8148 1.9772 0.7435 1.4701 1.9772 0.7435 Bank 22 1.4701 10.0246 0.7045 7.0591 10.0206 0.7045 Bank 23 7.0623 177.5118 190.3193 0.9327 Bank 24 177.5058 190.3130 0.9327 11.4376 0.6792 7.8792 11.6191 0.6781 Bank 25 7.7684 6.2479 0.7523 4.7006 6.2480 0.7523 Bank 26 4.7006 4.9198 0.8336 4.3010 5.1242 0.8393 Bank 24 4.1013 33.5770 0.6989 23.4690 33.5783 0.6989 Bank 28 23.4677 7.8569 0.6718 6.9169 10.6584 0.6490 Bank 29 5.2783 30.8275 0.7068 21.7884 30.8275 0.7068 Bank 30 21.7884 30.2603 0.6413 19.7935 31.2952 0.6325 Bank 31 19.4057 80.1044 0.7862 62.9782 80.1044 0.7862 Bank 32 62.9782 18.9316 23.8539 0.7404 19.8217 26.2094 0.7115 mean
Note: This table gives the estimated deposit insurance premiums based on the parameter values displayed in
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Table 2. The second and third columns give the means of the estimated premiums using parameter values calculated from the FPT formulas. The symbols IPP and IPP A are the per-monetary-unit deposit insurance premium assuming no early bankruptcy and early bankruptcy, respectively. The fourth column gives the means of the IPP / IPP A ratios. The symbol “bp” is 0.01%. The fifth to seventh columns give the corresponding estimates based on the parameter values obtained from the RV formulas.
52
Table 4: Analyses of the DI premium for the Asian financial crisis effects and consolidation effects Mean Values
Analyses for the consolidation effect
1996 1997 1998 Before consolidation During consolidation After consolidation
0.0242 0.0233 0.0190 0.0356
0.9698 5.0483
24.7390 50.7015
0.0456
0.9674 4.8653
11.0878 31.8128
0.0456
0.9610 4.5170
5.6422
(%) 9.3790 12.7769 9.6985
IPP A (bp) 0.0219 0.0276 0.5247 0.6787 0.7546 1.0117
IPP (bp)
17.4516
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Analyses for the Asian financial crisis
V
B ( 0) V (0) 0.7240 0.7118 0.7865
s
Note: This table shows the analyses for the DI premiums influenced by the 1997 Asian financial crisis and the bank’s consolidations in 2001 to 2002. For the analyses of the Asian financial crisis, we chose three sub-
periods: 1996, 1997, and 1998. For the analyses of the consolidation effects, we defined three sub-samples: Sub-sample 1 contains the bank’s data on the year before consolidation; Sub-sample 2 contains the bank’s
-p
data on the year of consolidation; and Sub-sample 3 contains the bank’s data on the year after consolidation. We assigned the sub-samples for each bank based on the bank’s consolidation year. For each bank in each
s,
B (0) and ) using the FPT formula, and then we V V ( 0)
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sub-period, we also estimated its parameters (i.e.,
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calculated its DI premiums (i.e., IPP and IPP A ) using our model. For other definitions, see Table 3.
53