Partial State-Owned Bank Interest Margin, Default Risk, and Structural Breaks: A Model of Financial Engineering

Transcription

1 Partial State-Owne Bank Interest Margin, Default Risk, an Structural Breaks: A Moel of Financial Engineering JYH-HORNG IN,CHING-HUI CHANG * AND ROSEMARY JOU Grauate Institute of International Business Tamkang University lin905@mail.tku.eu.tw Grauate Department of Applie Statistics an Information Science, Ming Chuan University chchang@mcu.eu.tw Grauate Institute of Management Sciences Tamkang University TAIWAN @s9.tku.eu.tw Abstract: Recent reports emonstrate that the grip of nationalism is tightest in banking (Economist, 009a, an in France an Britain, politicians powering taxpayers money into ailing banks are emaning that the cash be lent at home (Economist, 009b. This paper proposes a call option moel for bank interest margin an efault risk valuation base on an event-epenent, structural break framework uner the return of banking nationalization. The primary feature is the existence of the iscontinuity an fat tails of asset returns an risks cause by structural changes. We show that an increase in the structural breaks in mean return iscontinuity an volatility fluctuation cause by the increasing egree of the bank s partial nationalization increases the bank s interest margin an efault risk. We conclue that nationalization may be a high return-volatility structure break. Key-wors: Partial Nationalization, Structural Change, Retail Banking, Interest Margin Introuction * Wiesprea partial nationalization in has generate large reports concerning the effect of ownership on Corresponing Author. ISSN: Issue, olume 9, January 00

2 bank performance. For example, the grip of nationalism is tightest in banking (Economist, 009a. In France an Britain, politicians powering taxpayers money into ailing banks are emaning that the cash be lent at home (Economist, 009b. Northern Rock, a British bank which was nationalize in early 008 an was originally tol to shut its oors to new borrowers an shrink into its book, abruptly change course in February. It now aims to len an extra 5 billion in mortgages in 009, an up to aitional 9 billion in 00 (Economist, 009c. These reports emonstrate the prouct of both market forces an political interference on banks to change their lening courses. From a financial engineering perspective, the return of bank nationalization raises the nee for conceptual frameworks able to link the event-epenent, structural changes to bank operations management an avance our unerstaning of the associate asset prices an efault risks. Moreover, the challenge to better unerstan bank interest margin One of the ba bank solutions is partial nationalization, the government accepts a majority ownership share in a bank in exchange for the ba assets (Manel, 009. The three largest U.S. banks, JPMorgan chase, Citigroup, an Bank of America, rely increasingly on a special class of stock, much of its owne by the government, that reuires big ivien payouts (Henry, Golstein, an Farza, 009. ecisions to efault has been recently invigorate by regulators guest to embe moels of creit risk into partial state-owne bank capital structures an bank-capital reuirements (Basel Committee on Bank Supervision, 00. In this paper, we investigate the optimal behavior of a partial state-owne bank allowe to efault an stuy the bank margin pricing implications in the presence of this event-epenent creit risk relate to bank euity capital structure. Much of the literature follows Merton s (974 moel by explicitly linking the risk of a firm s efault process to the variability in the firm s asset price an viewing the market value of firm s euity as a stanar call option on the market value of the firm s asset with strike price euity the promise payment of corporate liabilities. Stock an Watson (00 an Kholoilin an Yao (006, for example, recognize one possible weakness of the approach that efault only occurs at the normal istribution or continuity of asset returns. There is ample evience on the presence of structural breaks in financial time series, which contribute to the fat tails or iscontinuity of financial returns (see, for example, Aggarwal, Incl an an enl, ISSN: Issue, olume 9, January 00

3 999, an Stock an Watson, 00. The principal avantage of this structure-change option approach is the explicit treatment of break events. This approach, however, omits a key aspect of bank behavior that structural breaks occur not only in moeling return an volatility but also operations management recognize as the case of the return of bank nationalization. As a result, we argue that these may inee be significant synergies between structural breaks in return/volatility an in bank operations management. The return of partial nationalization is of theoretical interest because of the insight it offers into the long-staning ebate over whether state-owne banks perform poorly. Unerstaning the impact of bank partial nationalization is important because political interference relate to bank nationalization on profits an risks may be through the presence of state-owne share changes in bank euity capital structure an the presence of fat tails an iscontinuity of asset returns an their associate efault risks Both theoretical an empirical stuies have trie to ientify the proper statistical moel for common stocks. Blattberg an Gonees (974 provie a comprehensive review of various statistical properties of stock return. Alternatively, to avoi nationalizing its major banks, the government will insure their riskiest ebts. See a recent Britain s bank rescue plan in in, Chang, an in (009a an the Trouble Asset Relief Program of the Unite States in in, Chang, an in (009b. cause by structural changes in lening activities. As we iscuss further below, the former can be motivate base on a bank interest margin etermination in the spirit of Zarruk an Maura (99 an Wong (997, while the latter can be motivate base on structural changes in financial time series in the spirit of Hansen (00 an Stock an Watson (00. In light of previous work, the purpose of this paper is to evelop a moel of banking firm behavior that integrates the structural changes cause by banking nationalism of the portfolio-theoretic approach with government-owne share conitions an loan rate-setting behavioral moe of the firm-theoretic approach. We show that an increase in the structural breaks in mean an volatility with partial banking nationalization increases the bank s interest margin an efault risk. By ignoring the existence of structural breaks, the conventional viewpoint of political interferences unervalues levere euity an efault risk by an amount eual to the fat-tail form of the call option valuation. This paper is organize as follows. Section evelops the basic structure of the moel. Section erives the solution of the moel an the comparative static analysis. The final section conclues. ISSN: Issue, olume 9, January 00

4 The Moel We consier a single-perio ( 0 t, fat-tail istribute call option moel of a banking firm. At bank accepts D t 0, the ollars of eposits an provies epositors with a rate of return eual to the risk-free market rate. R D A partial nationalization is possible the government has a majority ownership share an private investors have a minority one in the bank. The amounts of euity capital hel by the government an the private investors are mk an K, respectively, m >. Euity capital ( + mk hel by the bank is tie by regulation to be a fixe proportion (+ m K D capital-to-eposits ratio of the bank s eposits,. The reuire is assume to be an increasing function of the amount of the loans hel by the bank at t 0, / > 0 (see Zarruk an Maura, 99. The bank is assume to hol two types of earning assets: open market securities B an loans. The bank acts as a price taker in the open market so that the interest rate on open market securities R is given. The bank faces a ownwar sloping eman curve for its loans, expresse by ( R an / < 0, an chooses profits. This assumption implies the bank exercises some monopoly power in its loan market. Empirical evience by Hancock (986 supports the presence of rate-setting behavior in loan markets. The value of the bank s euity return at t is the resiual value of the bank after meeting all of the obligations: S max[ 0, (+ R + (+ R B ( + R s.t. D D] ( + B D + ( + m K ( + m K( + In euation (, the bank s total costs are only the eposit payment costs. The constraint is the balance sheet constraint which captures the bank s operations management in lening since the total assets on the left-han sie are finance by emanable eposits an euity capital on the right-han sie. Specifying the objective of euation (, we apply Black an Scholes (97 call options on unerlying assets with structural changes an further set maximize the market value of euity return. 4 In particular, the unerlying assets is the risky-loan repayments, ( + R, an the ate of structural to the loan rate R, so as to maximize 4 in, in, an Jou (009a, b, for example, also use the similar conceptual framework. ISSN: Issue, olume 9, January 00

5 breaks is known such as the implementation ate of partial nationalization of the bank by increasing state-own shares, Δm > 0, in our moel. To o this, we view the market value of the bank s euity as the call option on the market value of with strike price eual to the promise eposit payments net of the liui-asset repayments, Z ( + RD D ( + RB, respectively. We further propose to incorporate structural breaks that efault occurs at maturity of the ebt. The rational is that as the breaks cause by partial nationalization, the urge to keep the bank operating, it is reasonable to recognize an exclusive case that efault occurs prior to the maturity. et structural breaks exist in the mean an volatility of, which is assume to follow a one-imensional Brownian motion. The ynamics of follows: M [ μ + α( m M ] t + [ σ + β ( m M ] W, if τ t 0, otherwise ( instantaneous rift an volatility of, respectively, an W is a Wiener process. α (m an β (m are the ifference between with an without the bank s structural changes in the mean an volatility of, respectively. There are three possible cases euation ( can be simplifie. First, when the structural breaks o not occur ( M 0, or α ( m 0 an β ( m 0, motion. follows a geometric Brownian Secon, α ( m < 0 an β (m < 0 emonstrate the structural breaks in the mean an volatility make events. be lower, respectively, meaning ba Thir, α ( m > 0 an β ( m > 0 stan for goo events which can enlarge. A possible structural change may be expecte to influence the fat tails of loan repayments, α ( m > 0 an β ( m > 0. In this case, istributions of loan repayments have flatter tails an higher centers when compare to a normal istribution, meaning that more of the outcomes are locate in the tails rather than towar the center of the istribution. 5 Further, α / > 0 an β / > 0 represent structural In euation (, M is a ummy variable, τ is the ate of the known structural breaks, μ an σ are 5 It is recognize that fat tails are influence by customer (borrower acceptance (see Asosheha, Bagherpour, an Yahyapour, 008. The etails of what rive loan eman are unimportant for our purposes, so this abstraction is sufficient. ISSN: Issue, olume 9, January 00

6 changes in the mean an volatility make ln ~ φ( ln + ( δ ( σ + β ( m, σ loan repayments be higher, respectively, (5 meaning goo events owning to the φ ( enotes a normal partial nationalization. Contrary, α / < 0 an β / < 0 stan for ba events which shrink loan repayments. These two alternatives are use in a later section when the comparative static results are erive. Given conition (, the expecte euity value of the call option with the structural changes in the risk-neutral istribution with fat tails. This moel can tractably isclose the impact of the structural changes cause by the partial nationalization on contingent claim pricing. Specifically, the market value of the bank s euity S is expresse by the Black an Scholes (97 formula for call options: state of euation ( can be rewritten as: δ S N( Ze N( (6 Ŝ Ŝ(max[ 0, Z] ( ( + R Z D ( + m K ( + R[( + m K( + ] [ln + δ + ( σ + β ( m σ + β ( m Z ( σ + β ( m N( the cumulative ensity function ] From the neutral-state valuation argument, the call option price S is the value of this iscounte at the risk-free net-obligation sprea rate that is, δ R R, S e δ Ŝ (4 In a risk-neutral state, ln has the probability istribution in the following form: D In aition, using information in objective (6, we can apply assalou an Xing (004 an efine the efault probability as follows. The efault probability of the bank s euity return is the probability when the value of the loan repayments in less than the value of the net-obligation payments enote by P ef prob( Z prob(ln 0 ln Z 0 (7 ISSN: Issue, olume 9, January 00

7 Given the conitions of euations (5 an (7, we can specify the efault probability as follows: prob(ln lnz + ( δ ( σ + β( m + ( σ + β( m 0 ln + δ ( σ + β ( m prob( Z (8 σ + β ( m ε ( t P ef ε ( t W ( t W (0, an ε ( t ~ N(0, Accoringly, we can express the istance to efault as [ln ( ( m ( ( m + μ + α σ + β ] σ +β( m Z (9 Default risk occurs when the ratio of / Z is less than, or its log is negative. tells us by how many stanar eviations with structural changes the log of this ratio nees to eviate from its mean in orer for efault to occur. Notice that although the value of the call option in euation (6 oes not epen on is because μ + α(m, oes. This epens on the future value of loan repayments which is given in an in euation (6. We use the theoretical istribution implie by Merton s (974 moel, which is the normal istribution with fat tails. In that case, the theoretical probability of efault is given by: P ef N( ln + ( μ + α( m ( σ + β( m N( Z σ + β( m (0 Probability (0 states a nonlinear stochastic one which mimics the efault probability ensity function of the bank s euity returns with structural breaks. The effect that efault risk may have on euity returns is not obvious since euity holers are resiual claimants on the bank s cash flows with structural breaks an there is no promise normal return in bank euities. Solution an Results Partially ifferentiating euation (6 with respect to conition is given by: S N( N( + Z δ e N( δ N( Ze, the first-orer ( 0 ISSN: Issue, olume 9, January 00

8 A process in simplifying euilibrium conition ( is in calculating the cumulative normal istribution with fat tails N ( euation (, we can have. In N ( ( (ln + δ Z e π N( Ze Accoringly, we have δ (4 + ( σ + β ( m (ln Z ( σ + β ( m + δ ( N Ze N( ( δ (5 Further, we follow Hull (99 an use the numerical proceures to irectly calculate N(. One such approximation is 0 N( ( a k + a k N ( + ak ( Imposing conition (5 on the euilibrium conition of euation (, we have the simplifie form in the following: k /( , a 0.468, a , a N( ( / e π > Using euations ( an (, we rewrite the term N( / in euation ( as: 0 S Z 0 Z δ N( e N( + ( + R < 0 ( R RD ( + m K [ + ( + R] The term / can (6 < 0 be rewritten as ( + / η, ISSN: Issue, olume 9, January 00

9 η ( /( + R ( ( + R /. η is efine as the interest rate elasticity of loan eman. There is η < since loan eman face by the bank is ( R / < 0. We have + / η < 0 an hence / R < 0. The term Z / is also negative in sign. Inspection of euation (0 reveals that a necessary conition for the optimal loan rate is that the risk-ajuste value for marginal loan repayments of loan rate euals the risk-ajuste value for marginal net-obligation payments. A sufficient conition for the optimum is that S / < 0. Consier next the impact on the bank s loan rate (an thus on the bank s margin from changes in the egree of the partial nationalization relate to structural breaks in return an volatility of lening. This result is use later when the structure-break event on efault risk probability is analyze. To show this result, we ifferentiate from euation (6 with respect to m : S S / (7 S Z δ e N( N ( + ( N( δ Ze N( + Z ( R R N( N ( N( D K' < 0 β Z β ( σ + β σ + β Z Z (+ R of D S / change in S / K ( + R K( + < 0 The first term on the right-han sie the mean profit effect on a change in can be interprete as S / m, the secon term can be from explaine as the variance or risk effect, an the thir term can be specifie as the structural volatility-break effect. The mean profit effect captures the, evaluate at the optimal loan rate, ue to an increase in m, ceteris paribus. It is unambiguously positive because an increase in m provies a return to a larger euity base. One way the bank may attempt to augment its total returns is by shifting its investments to the liui assets an away from its loan portfolio at an increase loan rate. If loan eman is relatively rate-elastic, a less loan portfolio is ISSN: Issue, olume 9, January 00

10 possible at an increase margin. Before proceeing with the analysis of the variance effect, we explain the term / associate with right-han sie of as follows. The term β / on the / can be interprete as the structural break effect, while the term associate with Z / can be interprete as the incremental capital effect. The former can be motivate base on a structure-break argument in the spirit of the portfolio-theoretic approach, while the latter can be motivate base on a liuiity-management argument in the firm-theoretic approach. Base on rather general assumptions within the bank interest margin etermination framework, it is reasonable to believe that the portfolio-theoretic effect is either reinforce or insufficient to offset the firm-theoretic effect. Thus, we have the result of / > 0. The variance effect arises because an increase in increases in the bank s profit by its lening evaluate at the optimal loan rate in every possible state. As usual, the sign of this variance effect is ineterminate. However, euation (7 provies as with a hunch that the variance effect is positive since / > 0 m. Intuitively, an incremental state-owne capital implies a higher risk-state of the bank an as a result, the bank raises its loan rate to cut risky lening. The positive structural volatility-break effect emonstrates the positive impact on the bank s loan rate from a change in m when the break is recognize as a ba event ( β / < 0 which shrinks loan repayments at a ecrease margin. Since both the variance effect an the volatility-break effect reinforce the mean profit effect to give an overall positive response of the optimal loan rate to an increase in m, we establish the following proposition. Proposition : An increase in the egree of partial nationalization increases in the bank s interest margin when the structural break in volatility is recognize as a ba event. One immeiate application of this proposition is to evaluate the changes in the egree of partial nationalization of lening arrangements propose as alternatives for future loans. One freuent suggestion is that partial nationalization is possible the government increasingly accepts a majority ownership share in the bank, for example, in exchange for the ba loans. Uner the suggestion, it is reasonable to believe that the bank s lening ecreases ISSN: Issue, olume 9, January 00

11 at an increase interest margin. Accoringly, when structural break in volatility is recognize as a ba event, we can argue that an increase in the partial privatization or a ecrease in the partial nationalization increases the bank s lening amount at a ecrease margin. Total ifferentiating euation (0 evaluate at the optimal loan rate, we can obtain the following expression for how the partial nationalization relate at the structural breaks preicts the efault probability in the bank s euity return: P ef m P ef N ( N( ( / e π [ σ + β Z P ef N( Pef Pef + (8 ( α + β R > 0 Z α + ( α + β + R ( R Z β ] Z The first term on the right-han sie of euation (8 can be interprete as the irect effect on the efault probability from a change in m, while the secon term can be interprete as the inirect effect through the optimal bank interest margin etermination. The irect effect captures the change in the efault probability ue to an increase in margin constant. increases with m, holing the optimal sign of the term / Note that N(. Conseuently, the irect effect is governe by the relationship between an m. The is etermine by the incremental capital effect ( Z / an the structure-break effects ( α / an β /. As state earlier, there are Z / < 0 an β / < 0. The sign of the term α / is negative since there is β / < 0. If both the incremental capital effect an the structural-break volatility effect is insufficient to offset the structure-break mean effect, we have / m < 0. The irect impact can be unerstoo by recalling that / Z m etermine by the ifference between the log of the in ecreases an conseuently is an its mean value in orer for efault to occur. An increase increases the efault probability when the optimal loan rate remains constant. The inirect impact arises when every possible state of the optimal loan ISSN: Issue, olume 9, January 00

12 rate is variant. This inirect effect is positive in sign since P / > 0 an. ecreases at an increase optimal loan rate by shifting the bank s investments to the liui assets an away from its risky loans. Since the inirect effect reinforces the irect effect to give an overall positive response of m to an increase in ef / > 0 shown in Proposition P ef, we establish the following proposition. Proposition : An increase in the egree of partial nationalization increases the bank s efault risk in euity return when the structural breaks are recognize as ba events. 4 Conclusion This paper proposes a framework for bank interest margin an efault risk valuation base on event-epenent, structure-break options uner the return of banking nationalism an argues that event epenency is an intrinsic characteristic of bank euity an efault risk. The stanar structure-break option approach ignores the possibility of event-epenent bank euity capital structure an implicitly values bank euity as break-inepenent processes in the portfolio-theoretic approach. In essence, we can argue that little attention has been pai to the effects of efault risk on euity return. Actual bank interest margins an thus bank euities, in contrast, follow event-epenent processes because their payoffs epen on the particular event followe by the unerlying asset when structural breaks are realize. The results of the moel show that an increase in the structural breaks in mean return iscontinuity an volatility cause by the increasing egree of the bank s partial nationalization reflecte by its euity capital structure changes increases the bank s interest margin an its associate with efault risk. However, one issue that has not been aresse is the expecte isavantages to having government-owne rivals. The obvious one is unfair competition. Of course, in a loan market without such strict expectations reuirements, other factors woul affect lening courses. For example, strategic consierations may play a very important role, as woul more extreme problems of price variation conjectures. Such concerns are beyon the scope of this paper an so are not aresse here. What this paper oes emonstrate, however, is the important role play by political interference in affecting bank lening courses. ISSN: Issue, olume 9, January 00

13 Acknowlegement:The secon author s research has been supporte partially by a research grant from the National Science Council (98-8-M-0-00 References []. Aggarwal, R., C. Incl an, an R, eal (999, olatility in Emerging Stock Markets, Journal of Financial an Quantitative Analysis, 4,, []. Asosheha, A., S. Bagherpour, an N. Yahyapour (008, Extene Acceptance Moels for Recommener System Aaption, Case of Retail an Banking Service in Iran, WSEAS Transactions on Business an Economics, 5, 5, []. Basel Committee on Banking Supervision (00, The New Basel Capital Accor, Basel: Bank for International Settlements. [4]. Black, F., an M. Scholes (97, The Pricing of Options an Corporate iabilities, Journal of Political Economy, 8,, [5]. Blattberg, R. C., an N. J. Gonees (974, A Comparison of the Stable an Stuent Distributions as Statistical Moels for Stock Prices, Journal of Business, 47,, [6]. Economist (009a, The Return of Economic Nationalism, Economist, February 7 th, 9-0. [7]. Economist (009b, Homewar Boun, Economist, February 7 th, 6-6. [8]. Economist (009c, Exit Right: The Contract between Society an Banks Will Get Stricter, Economist, A Special Report on International Banking, May 6 th, 5-9. [9]. Hancock, D. (986, A Moel of the Financial Firm with Imperfect Asset an Deposit Elasticities, Journal of Banking an Finance, 0,, [0]. Hansen, B. E. (00, The New Econometrics of Structural Change: Dating Breaks in U.S. abor Prouctivity, Journal of Economic Perspectives, 5, 4, 7-8. []. Henry, D., M. Golstein, an R. Farza (009, The Fix Has Not Fixe BofA, BusinessWeek, February 9, 4-6. []. Hull, J. C. (99 Options, Futures, an Other Derivative Securities, n e., onon: Prentice-Hall International, Inc. []. Kholoilin, K. A., an. W. Yao (006, Moeling the Structural Break in olatility, Applie Economics etters,,, [4]. in, J. J., C. H. Chang, an J. H. in ISSN: Issue, olume 9, January 00

14 (009a, Rescue Plan, Bank Interest Margin an Future Promise ening: An Option-Pricing Moel, WSEAS Transactions on Information Science an Applications, 6, 6, [5]. in, J. J., C. H. Chang, an J. H. in (009b, Trouble Asset Relief Program, Bank Interest Margin an Default Risk in Euity Return: An Option-Pricing Moel, WSEAS Transactions on Mathematics, 8,, 7-6. [6]. in, J. H., J. J. in, an R. Jou (009a, Global Diversification, Hegin Diversification, an Default Risk in Bank Euity: An Option-Pricing Moel, WSEAS Transactions on Mathematics,, 8, [7]. in, J. J., J. H. in, an R. Jou (009b, The Effects of Sunshine-Inuce Moo on Bank ening Decisions an Default Risk: An Option-Pricing Moel, WSEAS Transactions on Information Science an Applications, 6, 6, [8]. Manel, M. (009, The Ba Bank Solution, BusinessWeek, February 9, 7. [9]. Merton, R. C. (974, On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance, 9,, [0]. Stock, J. H., an M. W. Watson (00, Has the Business Cycle Change an Why? NBER Working Paper, No. 97. []. Wong, K. P. (997, On the Determinants of Bank Interest Margins uner Creit an Interest Rate Risks, Journal of Banking an Finance,,, 5-7. []. assalou, M., an Y. Xing (004, Default Risk in Euity Returns, Journal of Finance, 59,, []. Zarruk, E. R., an J. Maura (99, Optimal Bank Interest Margin uner Capital Regulation an Deposit Insurance, Journal of Financial an Quantitative Analysis, 7,, ISSN: Issue, olume 9, January 00