Troubled Asset Relief Program, Bank Interest Margin and. Default Risk in Equity Return: An Option-Pricing Model

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1 Trouble Asset elief Program Bank Interest argin an Default isk in Equity eturn: An Option-Pricing oel JYH-JIUA I * CHIG-HUI CHAG 3 AD JYH-HOG I Department of tatistics Tamkang University 5 Ying-Chuan oa Tamsui Taipei County 5 TAIWA OC 764@mail.tku.eu.tw Department of Applie tatistics an Information cience ing Chuan University 5 De-ing oa Gui-han District Taoyuan 333 TAIWA OC chchang@mcu.eu.tw 3 Grauate Institute of International Business Tamkang University 5 Ying-Chuan oa Tamsui Taipei County 5 TAIWA OC lin95@mail.tku.eu.tw Abstract: Will banks be willing to sell their toxic loans with the help of the Trouble Asset elief Program TAP? The answer is yes as long as bis are high enough to tempt banks to eal. With the TAP s help an increase in the toxic loans sol to the government increases the bank s margin an ecreases the bank s efault probability in equity return when the bank encounters greater risk. This paper conclues that setting up the TAP for the ba bank solution may be a goo move for retail banking resulting in high margin an low efault risk when its target banks are willing sellers. Key-Wors: Toxic oans Interest argin Default isk Trouble Asset elief Program I: Issue 3 olume 8 arch 9

2 . Introuction * The U.. financial authority create the Trouble Asset elief Program TAP a $7 billion rescue fun in October 8 Economist 9a. The authority is a willing buyer of toxic loans but are banks willing sellers Economist 9b? Will setting up the TAP be a goo move for toxic loans? One solution is for the authority to pay more for the target toxic loans than their current market price assuming that values will eventually reboun. That is a subsiy for bank shareholers an ebtholers an may be one important reason why bank stocks went up % on January 8 9 Economist 9c. This paper evelops an option-pricing moel of bank behavior that is use to stuy bank interest margin with government help. Our results are largely supporte the above solution. The U.. banking inustry is experiencing a renewe focus on retail banking a tren often attribute to the stability an profitability of retail activities Hirtle an tiroh 7. Banking * Corresponing author. firms in retail banking are institutions that engage in two istinct types of activities eposit-taking an lening. Our primary emphasis on retail banking is the selection of the bank s optimal interest margin which is the ifference between the rate of interest the bank charges borrowers an the rate the bank pays to epositors. cshane an harpe 985 an Allen 988 have evelope moels of bank interest margins base on the bi-ask sprea moel of toll 978. Zarruk an aura 99 an Wong 997 also provie firm-theoretic moels to explain bank margin behavior. Unlike previous formulations the moel evelope here assumes an option-base setting in which the target bank is helpe by the TAP. The authority buys toxic loans hel by the bank when bis are high sufficit to tempt the bank to eal. Comparative static results show that the bank s optimal interest margin is an increasing function of the amount of toxic loans sol when the bank encounters greater risk. In aition the efault risk in the bank s equity return is a ecreasing function of the amount of toxic loans esults to be erive from our moel o not exten to the case the government s subsiy is the swappe high-yiel bon see ee an Cheng 8 The customer acceptance in the retail banking oes not consiere in our paper. Our results o not exten to this particular case Asosheha Bagherpour an Yahyapour 8. I: Issue 3 olume 8 arch 9

3 sol. The paper is organize as follows. ection evelops the basic structure of the moel. ection 3 erives the solution of the moel an the comparative static analysis. The final section conclues.. The oel We consier a single-perio moel of a banking firm. At the start of the perio the bank accepts D ollars of eposits. The bank provies epositors with a rate of return equal to the risk-free market rate D. Equity capital K hel by the bank is tie by regulation to be a fixe proportion q of the bank s eposits K qd. The require capital-to-eposits ratio q is assume to be an increasing function of the amount of the two types of loans for example iniviual loans an mortgage loans hel by the bank at the beginning of the perio q = q = q >. This assumption implies that the system of capital stanars is esigne to force banks capital positions to reflect their asset portfolio risks see Zarruk an aura 99. The bank makes the two loans which mature an are pai off at the en of the perio. Both the loan emans face by the bank are specifie as an an are iniviual loan interest rate an mortgage loan interest rate respectively. We assume that the bank has some market power in lening see Wong 997 which implies that < an <. Empirical evience by lovin an ushka 984 supports the presence of rate-setting behavior in loan market. In aition to term loans the bank can also hol an amount of B liqui assets for example Treasury Bills on its balance sheet uring the perio. These assets earn the security-market interest rate of. When the capital constraint is bining the bank s balance-sheet constraint is given by B = D K = K q Equation emonstrates the bank s operations management in lening since the total assets in the left-han sie are finance by emanable eposits an equity capital in the right-han sie. In erton s 974 moel the I: Issue 3 olume 8 arch 9

4 equity of a firm is viewe as a call option on the firm s assets. The reason is that equity holers are resiual claimants on the firm s assets after all other obligations have been met. The strike price of the call option is the book value of the firm s liabilities. When the value of the firm s assets is less than the strike price the value of equity is zero. We assume that the bank is willing to sell the amount of toxic loans < to the government what the TAP is meant to target. 3 The market value of the bank s unerlying assets follows a geometric Brownian motion of the form: = μ t W σ =. 3 U.. commercial banks hol 3.8 trillion in resiential an commercial mortgages accoring to Feeral eserve ata plus $. trillion in mortgage-relate securities. That represents more than 5% of banks total financial assets of $9 trillion which means that they are highly expose to the real estate market Economist 9c. Creit-efault swap is an insurance policy against a asset efault. American International Group AIG sol fistfuls on mortgage-relate securities that have collapse in value. Wang 9 points out that AIG takes the worl insurance an the financial service leaer also has pai the huge price. is the value of the bank s risky loans with an instantaneous rift μ an an instantaneous volatility σ. W is a stanar Wiener process. With government help when the carrying value of the bank s mortgage loan books is far above market price it is reasonable to assume that the bining value of toxic loans sol to the government is set to equal its book value. If the TAP falls flat it is more likely to be because of a lack of sellers. 4 Even with government help bis may not be high enough to tempt banks to eal since any price below the carrying value will force banks to take a write-own an eplete precious capital. The market value of equity will then be given by the Black an choles 973 formula for call options: = Ze 3 4 Banks are still holing assets particularly whole loans at values far above their market price because uner accrual accounting losses can be booke over several years. I: Issue 3 olume 8 arch 9

5 K Z = D [ K ] q q = ln δ σ σ Z = σ δ is the sprea rate between an D is the cumulative ensity function of the stanar normal istribution. We note that the amount of toxic loans sol is replace by its book value of e in the option-pricing valuation. In aition the efault probability of the bank s equity in equation 3 is the probability when is less than Z. Using information in equation 3 we apply assalou an Xing 4 an efine the efault probability as: P ef = 3 4 = ln μ σ. σ Z 3 3 represents how many stanars the log of Z nees to eviate from its mean in orer for efault to occur. 3. olutions an esults The first orer conitions are given by: = Z Ze e = = Z e 5 Ze = 6 = Ze = Ze Z = < q = [ D K ] < q I: Issue 3 olume 8 arch 9

6 ] [ < = D q q K Z = ] [ ] [ <. We require that the secon orer conitions be satisfie there are an. To ensure a unique market equilibrium obtaine we further assume < < Δ. > Both the optimal loan rates an are etermine when the marginal risk-ajuste value of risky-asset repayments equals to the marginal risk-ajuste value of net-obligation payments. We further substitute these two optimal loan rates to obtain the efault probability in equation 4 staying on the equity maximization. We consier next the impacts on both the bank s optimal loan rates an thus on the bank s margins from changes in the amount of toxic loans sol to the government. Implicit ifferentions of equations 5 an 6 with respect to yiel: = Δ 7 = Δ 8 = = e Z δ I: Issue 3 olume 8 arch 9

7 = σ Z >. Z Before proceeing with the analyses of the results we efine the term as the risk-ajuste factor elasticity effect. This effect reflects the risk state face by the bank in the option-base optimization. The sign of this effect can be equivalent to the sign of the ifference. The former represents the reciprocal risk-ajuste factor elasticity of risky-asset repayments while the latter represents that elasticity of net-obligation payments. When the ifference is negative the effect inicates that the bank is operating uner greater risk since the former is more sensitive than the latter. When the ifference is positive the bank has a ecreasing risk magnitue for its equity return. The term captures only the variance effect on from a change in the term associate with. This is because term loans is not the TAP s target. There is > when the bank is operating uner greater risk. The first two terms of represents the mean profit effect on from a change in while the thir term represents the variance effect. The mean effect is positive in sign. The variance effect is governe by the risk-ajuste factor elasticity effect. There is > when the bank is operating uner greater risk. The results of equations 7 an 8 are state as follows. With strategic complements in Bulow Geankopolos an Klemperer s 985 sense an increase in the toxic loans sol to the government ecreases the loan portfolio at an increase in equation 7 an in equation 8 when the bank encounters greater risk. If either loan eman is relatively rate-elastic a smaller loan portfolio is possible at an increase margin. Another way to unerstan the sell of the toxic loans is to see its impact on efault risk in equity return since a bank s efault probability is relate to its volatility which is a key input in the Black-choles 973 option-pricing formula. Implicit ifferentiation of equation 4 evaluate at both the I: Issue 3 olume 8 arch 9

8 optimal loan rates with respect to yiels: P ef P ef P ef P ef Pef Pef = 3 P ef 9 Z = < σ Z = σ = σ Z 3 Z Z < 3 Z <. The first term on the right han sie of equation 9 represents the irect eff ect while the secon term through an the thir term through represent the inirect effects. Both the irect an inirect effects are negative since there are > an > known as in equations 7 an 8 respectively. Equation 9 emonstrates that an increase in the toxic loans sol to the government ecreases the bank s efault probability in equity return. 4. Conclusion We evelop an option-pricing moel to etermine the market value of bank equity an its efault probability in equity return explicitly incorporating the TAP. Within the setting we show that with strategic complements an increase in the toxic loans sol to the government by the bank increases the bank s margin an ecreases the bank s efault probability in equity return when the bank is operating uner greater risk. With the TAP s help banks will be easier to entice as long as bis are high enough to tempt them to eal. etting up the TAP is a goo move for banks. Of course whether the authority has picke the effective approach is another matter. The alternative chosen by Britain to leave the assets in place an insure them for a fee has many proponents Economist 9a. uch concerns are beyon the scope of this paper an so are not aresse here. What this paper oes emonstrate however is the important role planne by the TAP in affecting the operating ecisions of banks. Acknowlegement : The first author s research has been supporte partially by a research grant from the ational cience Council I: Issue 3 olume 8 arch 9

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