Bank Stability and Market Discipline: Debt-for- Equity Swap versus Subordinated Notes

Transcription

1 ank Stabiity and Market Discipine: Debt-for- Equity Swap versus Subordinated Notes Aon Raviv Abstract Severa studies have recommended reiance on subordinated debt as a too for monitoring banks by investors and for enhancing depositors protection. However, subordinated debenture increases the eve of everage and thus the probabiity of costy faiure. We propose a nove financia instrument, Debt-for-Equity Swap contract (DES, that pays to its hoder a fixed income uness the vaue of the bank s assets fas beow a predetermined threshod. In such an event, the debt obigation is automaticay converted to the bank s common equities. y using a contingent caims vauation approach we present cosed-form soutions for the vauation of iabiities, the cost of deposit insurance and the vaue of bankruptcy costs of a bank that incudes DES or aternativey subordinated debt in its capita structure. We compare and evauate quantitativey the effects of DES contract versus subordinated debt on bank stabiity, depositor protection, incentives for risk taking, the abiity to provide market discipine and the vaue of bankruptcy costs. The impications of the paper highight the fact that the DES contract has saient advantages over subordinated debt as an efficient too for enhancing market stabiity and bank efficiency, since it reduces the vaue of bankruptcy costs. The advantage of the DES over subordinated debt as a provider of depositors protection depends on the eve of mandatory intervention, assets vaue and voatiity as we as on the ratio of bankruptcy costs. The mode iustrates the pros and cons of each of the two capita instruments as a too for enhancing market discipine. Whie the vaue of subordinated debt increases with the vaue of assets, its disadvantage as a monitoring too derives from its ow sensitivity to changes in assets voatiity when the eve of reguatory intervention is reativey high in terms of capita adequacy and the rate of bankruptcy costs is reativey ow. The DES contract is beneficia as a too for monitoring due to its negative sensitivity to increase in assets risk. However, when the conversion ratio is reativey high its price might increase as the everage ratio increases. J.E.L. Cassification Codes: G1, G13, G1, G8, G38, E58. Keywords: bank, financia stabiity, market discipine, deposit insurance, options pricing, subordinated debt, Debt for Equity Swap. The Hebrew University usiness Schoo. Corresponding address: The Hebrew University usiness schoo, Mount Scopus, Jerusaem, Israe. Emais: araviv@stern.nyu.edu. I woud ike to acknowedge Dan Gaai, and Zvi Wiener for their hepfu comments and discussion. This paper is schedued for presentation in the Annua Meeting of the FMA and in the Annua Meeting of the German Finance Association. 1

2 Introduction Unike firms in non-financia, unreguated industries, the primary creditors in a bank, the depositors, do not have sufficient incentives to monitor the bank, because of the impicit or expicit guarantees that are provided to commercia banks by nationa governments. As a resut, governments, as the depositors insurers, have a strong incentive to monitor banks in order to avoid insovency. Moreover, the motivation of governments stemmed from the fact that bank insovency may spiover to other sectors of the financia system and, through it, to the rea economy (See Aen and Herring (001 and Fannery (00. Over the past decades, as the size and compexity of financia firms have enormousy increased, government supervisors have found it more difficut and chaenging to monitor and contro banks in a timey manner by using traditiona supervisory techniques such as minimum capita requirements and reguar review of a bank s risk management procedures. Accordingy, considerabe efforts have been made by banks and their supervisors to incude capita instruments that enhance market discipine. 1 Recent studies have recommended subordinated notes and debentures (hereafter SND as a preferred too to discipine banks and poicymakers are activey considering requiring banks to issue SND. It is argued that the expected negative effect on subordinated debt prices to excessive risk-taking encourage their hoders to monitor the bank cosey on an on-going basis, in a way which is aigned with those of the deposits' insurers. Moreover, it is asserted that the SND can impose discipine indirecty by providing risk signas to other market participants and reguators who can then discipine the bank (See Evanoff and Wa (001.

3 Whie heping to monitor a bank s activities and increasing the size of the financia cushion for the deposit insurer, the subordinated debenture aso increases its everage and the probabiity of defaut by the issuing financia institution. According to the cassica structura approach, in the event of firm faiure equity investors simpy surrender the firm to the bond investors who proceed to operate the firm in the most efficient manner avaiabe. 3 In reaity, the costs of distress and bankruptcy are substantia, especiay in the banking industry, in which a weak credit rating imits a bank's abiity to trade foreign exchange and OTC derivatives and to extend ines of credit to borrowing customers. To overcome the disadvantages of the SND mechanism, Fannery (00 presents quaitativey a potentia new capita instrument: Reverse Convertibe Debentures (RCD, which is converted into common stocks if the issuing firm's capita ratio fas beow some pre-specified eve. The RCD conversion is triggered automaticay: neither the issuer nor the investor wi have an option regarding this conversion and the debentures convert at the current share price. When the firm s share price is depressed, part of the outstanding debt is forgiven and thus the incidence of costy faiures is reduced. Reying on the market price of the iabiities as the trigger for conversion may ead to price puzzing, since it is uncear under which conditions conversion woud take pace. The market price of each corporate iabiity is a function of the ex-post possibe payoffs that are contingent on the vaue of the underying asset of the corporation. However, by determining that the payoff of a caim coud be repaced by another payoff at some future time, if its vaue ex-ante (market vaue has touched some predetermined eve, may ead to an undesired equiibrium in which conversion may not occur even if the corporation is in a bad standing and vice versa. 3

4 In this paper we eaborate the conversion mechanism by suggesting a capita instrument: Debt-for-Equity-Swap contract (hereafter DES, which has a fixed payoff upon maturity, uness the vaue of assets fas beow some pre-specified conversion threshod at any time unti debt s maturity. In such event, the debt contract is converted automaticay according to a predetermined conversion ratio into the bank s common stocks. Under the presented mechanism, when the vaue of the bank s assets is depressed, the outstanding amount of the DES is forgiven and thus the incidence of costy faiures is reduced. For the evauation and quantitative comparison of the effects of issuing DES contract versus subordinated debt, we adopt a contingent caim framework, à a ack and Cox (1976, hereafter C and Ericsson and Reneby (1998, where each of the bank s iabiities, under each of the two capita structures, is mimicked by a portfoio of barrier options. y using this moduar approach we derive cosed-form soutions for the iabiities of a bank, the cost of deposit insurance and the vaue of bankruptcy costs. Reying on the derived cosed-form soutions, we compare the effects of the DES and SND contracts on severa crucia poicy targets as bank stabiity, depositor protection, minimizing the vaue of bankruptcy costs and enhancement of market discipine. We anayze how the eves of voatiity, everage ratio, bankruptcy costs and the mandatory supervisory intervention affect these goas. The vauation of a bank's iabiities requires setting a trigger point for mandatory supervisory intervention. Previous modes for pricing SND contracts (See Levonian (001 and Nivorozhkin (001 assume based on the ack-schoes-merton mode that defaut can occur ony at debt maturity if the vaue of assets fas beow the debt's face vaue. We, ike C, assume that bank faiure can occur at the first time when the vaue of assets fas beow a predetermined threshod. In such event, the remaining assets of the 4

5 bank are distributed among the caimhoders according to their seniority. However, in our mode, conversey to C, iquidation may be costy. Moreover, the trigger point in our mode is based on the assumption that the sovency of a bank is determined by its reguator according to a minimum adequate capita eve, which is expressed as a percentage of the book vaue of equity. Thus, a mandatory supervisory intervention occurs usuay when the vaue of assets is we above the bank s outstanding debt. Aware of this fact, the iquidation threshod in our mode is set at east equa to the vaue of the bank s outstanding debt. 4 We prove that the difference between the defaut probabiities of a bank with SND and a bank with DES contract, hoding everything ese constant, is aways positive. The power of DES as provider of depositors protection is more questionabe. We prove that the ratio between the cost of insurance of a deposit in a bank that incudes DES contract in its capita structure, and a simiar insurance of a deposit in a bank which incudes instead SND may be ower, equa or arger than one. We show that the ratio between the costs of insurance of a deposit in a bank with SND contract and an identica deposit in a bank with DES contract is a decreasing function of assets voatiity, everage ratio and the eve of mandatory supervisory intervention, which is expressed as the ratio between the vaue of assets and the tota face vaue of debts. We demonstrate that for a reativey ow ratio of bankruptcy costs, the cost of deposit insurance is ower for a bank with SND contract, whie the conversey occurs when bankruptcy costs are reativey high. However, as the eve of supervisory intervention increases (in the form of reativey high iquidation threshod the costs of deposits insurance are equaized at a higher ratio of bankruptcy costs. It is argued that the SND contract can impose direct discipine on banks by charging high funding costs once excessive risk-taking activities are detected. The resuts 5

6 of empirica studies that examine whether risky debt, issued by banks and bank hoding companies, faciitates market monitoring and the contro of risk taking have been mixed. Using our theoretica framework, we demonstrate that unike the DES contract, the SND contract coud be amost insensitive to changes in assets voatiity if the bank is highy reguated and the rate of bankruptcy costs is reativey ow. We find that the effect of the reguatory intervention poicy on the sensitivity of the SND's price to increase in assets voatiity is not obvious, and depends on the everage ratio. It is shown that whie the sensitivity of the SND to changes in asset price is aways positive, the sensitivity of the DES contract coud be negative under reativey high conversion ratio, and its efficiency as a too for market monitoring is therefore questionabe. Jensen and Mecking (1976 assert that the interests of bondhoders and sharehoders in a everage firm strongy diverge regarding the risk that can accompany higher firm profits, since the increase in assets voatiity resuts in a transfer of vaue from the debthoders of a firm to its equityhoders. However, as demonstrated by Reisz and Perich (004, if the iquidation threshod is higher than the sum of the tota iabiity, the probabiity of going bankrupt becomes too arge and sharehoders wi shy away from any risky project. In this case we might observe a risk-avoidance probem à a John and rito (000. In consistency with this approach, Fannery (00 asserts that the introduction of subordinated debt into a bank s capita structure increases the eve of everage and thus exacerbates the probem of risk-avoidance and might even hurt the competitive nature of a bank and its task as iquidity provider. He suggests that the incusion of Reverse convertibe debentures (RCD instead of subordinated debt, can foresta financia distress without distorting the stockhoders risk-taking incentives. In this paper, it is shown that the incusion of DES contract in a bank s capita structure may not avoid the distortion of the sharehoders risk taking incentive. This effect depends 6

7 among other factors on the eve of the conversion threshod and on the conversion ratio. Moreover, the sharehoders risk-taking incentive in a bank with DES contract may be ower than the incentive of sharehoders in bank that is identica in a other respects except for repacing the DES contract with contract that does not incude the conversion feature (i.e.: SND contract. The rest of the paper is organized as foows. Section presents the basic assumptions for the vauation of corporate iabiities of a bank with DES contract in its capita structure. Section 3 anayzes in a simiar manner a bank with SND contract in its capita structure. Section 4 shows the vauation of a bank s caims under each of the two capita structures via options repication. Section 5 compares quantitative y the effects of DES contract versus SND on bank s stabiity, depositor protection, the enhancement of market discipine and the vaue of bankruptcy costs. Section 6 concudes. A proofs are contained in the appendices. 7

8 . Capita structure with DES cont ract In this section we discuss the basic economic setting on which we base our mode for pricing corporate iabiities of a bank with capita structure that consists of senior deposit, DES contract and equity. We show the vauation of these iabiities, and derive cosed form soutions for the vaues of bankruptcy costs and for the cost of deposit insurance under the presented capita structure..1 Setup and Key Assumptions Consider a hypothetica bank with assets ω that are continuo usy traded in an arbitragefree and compete market with riskess borrowing or ending at a constant rate of r. The vaue of the bank s assets is independent of its capita structure, and is we described under the risk neutra probabiity by the foowing stochastic differentia equation: dω = ( r δ ωdt + σωdw (1 where W is a standard rownian motion, δ is the institution s payout ratio and σ is the instantaneous constant standard deviation of the rate of return of the bank. 5 To finance its assets, the bank issues three types of caims: a singe zero-coupon deposit, a singe DES contract and a residua equity caim with market vaue denoted by S. The zero-coupon deposit matures at time T, has principa vaue of F and market vaue of. The depositor is the most senior security hoder, and thus has prior ity over a casses of securities in a way that wi be specified ater on. It is assumed that the government, which supervises the bank through a speciaized reguator, woud force 8

9 iquidation or reorganization at any time t [0, T] if the vaue of assets has reaches an exogenous ower threshod K, where this threshod is defined as: K = λ F where λ 1 ω F. ( In contrast to the convention in most structura modes in our mode, the threshod eve is at east equa to the face vaue of debt. Previous modes assume, based on the nature of existing safety covenants or common practice and aw, that the debthoders or reguators have the power to enforce iquidation or reorganization ony if the vaue of assets has reached the debt s principa vaue. 6 However, reguators usuay determine the sovency of a bank according to a minimum adequate capita eve, which is expressed as a percentage of the book vaue of equity, and thus the reorganization of a bank occurs we above the bank s debt face vaue. The time of defaut, where iquidation is decared, is denoted by τ and is defined formay by: τ { t > 0 K } = inf ω (3 t If defaut has occurred, so T τ, a fraction 0 γ 1 of vaue wi be ost due to bankruptcy costs. 7 The costs of bankruptcy consist of, for exampe, osses due to suspended deiveries by cautious suppiers or the ex-post costs of over or underinvestment incentives. In the banking industry, bankruptcy costs coud aso incude the cutting down of trades in financia derivative, where counterparties aware of the bank s 9

10 condition woud reduce derivative transactions in which they might have credit exposure to the distressed bank. Simiar to the deposit, the DES contract matures at time T and has a principa vaue of D F and market vaue of D. The DES is converted automaticay into α ( 0 α 1 common stocks if the vaue of assets fas beow some pre-specified conversion threshod, denoted by c K, at any time prior to debt s maturity. Whie the eve of the iquidation threshod is determined exogenousy by the behavior of the reguator, the conversion threshod is set in the contract terms. Since the main goa of the DES issue is to reduce the incidence of costy faiures, a natura choice is to set the conversion threshod, ike the iquidation threshod, at a eve which is at east equa to c D the sum of the principa vaue of the deposit and the DES, and thus K ( F + F. This threshod eve ensures that the event of bank insovency may not occur before the time of an enforced conversion, and thus the DES hoder has no ega support to force eary iquidation. However, the reguator may have incentive to iquidate the bank before the event of conversion if the DES principa amount is imited and the conversion threshod is reativey cose to the eve of the deposit face vaue. To ensure the efficiency of the conversion mechanism, the threshod eve shoud be set at a sufficient eve above the tota sum of a caims face vaue. The time of conversion is denoted by τ c and is defined formay by: τ c c { t > 0 K } = inf ω (4 t 10

11 .. Vauation of the ank iabiities with DES contract The market vaue of the bank,v, is equa to the sum of the vaue of its securities, i.e.: V t = S + + D. In the presence of bankruptcy costs and the deductibiity of tax and interest payments, this vaue is not in genera equa to the vaue of assets, ω. 8 We assume neither tax nor interest deductibiity, since we want to concentrate on the effects of the DES contract on the vaue of the bank s iabiities and on the vaue of bankruptcy costs. Under this assumption, the vaue of assets are at east equa to the bank s market vaue, such that ω V. As presented by C and simiary by Ericsson and Reneby (1998, the vaue of zero-coupon corporate security can be decomposed into two sources of vaue: first, its vaue at maturity, assuming the bank is not prematurey iquidated, and second, its vaue if the bank is iquidated before debt maturity, T. 9 Athough these two components are mutuay excusive, they are both possibe outcomes and accordingy each contributes to the present vaue of both equity and debt. Since the issuing of the DES contract invoves the introduction of a conversion threshod, which is efficienty ocated above the iquidation threshod and beow the vaue of the bank s assets, the vaues of the stock and the DES shoud be decomposed to three mutua excusive sources of vaue, and can be expressed as: S e D ( ω { } T F F 1 τ c > T rτ S + e Φ 1 { τ T} ( ( + 1α ω F 1 { } Q rt = E T τ T > τ c (5 D = E Q rt D e F 1 { } τ + c> T rt rτ D α e ( ω { } + Φ { } T F 1 τ T > τ e 1 c τ T (6 11

12 where Q E denotes the conditiona expectation under a risk neutra measure Q given a avaiabe information at time zero, and 1ψ is the indicator function of the event ψ. The symbos S Φ and D Φ are the payoff functions upon iquidation of the stock and the DES contract respectivey, which depend on the ratio of bankruptcy costs, and can be expressed as: Φ S 0 = ( 1 α [ K ( 1-γ -F ] F γ 1- ω esewhere (7 Φ D 0 = α [ K ( 1- γ -F ] F γ 1 - ω esewhere (8 The first term on the right hand side of each of equations (5 and (6 accounts for the cash fows that are generated if the vaue of assets has not touched the conversion threshod unti maturity( τ > T c, as depicted in Figure 1.A. In such cases, the debthoder is fuy paid and the equityhoder receives the residua amount. The second term on the right hand side of each equation accounts for the cash fows that are generated if conversion has occurred but no iquidation process has happened unti debt maturity, i.e. τ > T τ c, as depicted in Figure 1.. The stockhoder woud deiver a portion of α common stock to the DES hoder, whie the DES hoder woud waive the debt obigation. The payoff to the DES hoder at maturity in this state is equa to α units of the difference between the vaue of assets and the deposit face vaue, F. The initia stockhoder woud receive 1

13 ( 1 α units of the same payoff. Upon iquidation, where T τ, as depicted in Figure 1.C, the bank incurs costs that represent a portion γ of the vaue of its assets. If the proceeding assets are distributed according to absoute priority, then the depositor, as the most senior security hoder, woud receive the minimum between the debt face vaue and the remaining assets ( 1 γ K. In the extreme case, when the senior depositor has been fuy paid off, the initia equityhoder and the DES hoder woud receive the residua as expressed through equations (7 and (8 respectivey. The current vaue of the senior deposit, provided neither conversion nor iquidation have occurred unti the current time, is expressed by: : = E Q rt rτ [ e F ] 1 { τ > T} + e Φ 1 { T τ } (9 where Φ is the payoff function upon iquidation, which depends on the ratio of bankruptcy costs, and can be written as: Φ K ( 1-? = S F F γ 1 K esewhere (10 The current vaue of bankruptcy costs, denoted by C, refects the market vaue of a payoff of γ K shoud iquidation occur. The vaue of the bank, V, is equa to the vaue of assets minus the vaue of bankruptcy costs. Therefore, we can write its current vaue as: 13

14 V = ω C = ω E Q rτ Q rt rτ [ e K γ ] = E [ e ω 1 + e K (1 γ ] 1{ τ } { } 1 T T T< τ { τ T } (11 The payoffs to the caimhoders for various reaized asset vaues and the aternative ratio of bankruptcy cost are summarized in Tabe 1..3 The Introduction of Deposit Insurance in the Presence of DES Contract. Two of the primary objectives of the reguator, who supervises the banking system, are to attain financia stabiity by avoiding systematic risk and to protect investors by ensuring that the financia services firm wi be abe to honor its iabiities to its depositors. Whie the two measures capita adequac y and deposit insurance serve these two objectives, by imposing minimum capita adequacy the reguator tries to prevent ex-ante bank s run, where by introducing expicit or impicit deposit insurance an ex-post measure, which compensates the depositor upon defaut, is empoyed. 10 In our mode, the deposit insurance is assumed to absorb a risk for the senior debt and none for the DES contract and thus the deposit is fuy insured or guaranteed. 11 In that case, the iabiity of the provider of the deposit insurance (the guarantor s iabiity can be expressed as: G = E Q rτ { e ( F K 1 γ 1 { T τ } } ( (1 14

15 For a bank of given size, this iabiity is affected by four factors: the degree of deposit everage which is refected in the reationship between ω and F ; the rate of bankruptcy costs, the riskiness of the bank s assets, which is refected in the bank s choice of assets voatiity and the reguator mandatory intervention poicy, which is refected in the distance between the iquidation threshod and the deposit s principa amount. The higher the spread between the iquidation threshod and the deposit's principa amount the smaer is the oss of the depositor upon iquidation. Securing the face vaue of the deposit at any possibe state can be achieved by both measures, via deposit insurance and/or by setting a reativey high eve of minimum capita adequacy in the form of a reativey high iquidation threshod that causes the equity hoder to carry a the burden of bankruptcy costs. According to Equation (1, deposit insurance and minimum capita adequacy are substitutive measures for eiminating the potentia osses of the depositor upon iquidation. The reguator can increase the iquidation threshod, K, in a way which nuifies the payoff of the insurer upon iquidation. However, in our mode, conversey to the minimum capita adequacy measure, the presence of deposit insurance has no infuence on the bank's defaut probabiity Capita structure with Subordinated Notes In this section we present a mode for vauation of bank iabiities with traditiona capita structure that incudes senior debt, SND contract and equity. We present the added assumptions to those presented in Section.1 and then derive vauation equations for 15

16 each of the bank s securities under each possibe state. In a simiar manner, the cost of deposit insurance and the vaue of bankruptcy costs are derived. Since we assume that the vaue of bank s assets is independent of its capita structure a the assumptions about the economy that were presented in Section.1 are hod ing. As in the previous presented capita structure, the bank is financed with equity and a singe zero-coupon deposit with principa amount of F and market vaue of. However, instead of issuing a DES contract, the bank has issued a singe zero-coupon subordinated debt that matures at time T, has a principa vaue of J F and a market vaue of J. As in Section.1, the reguator poicy is to force iquidation or reorganization if the vaue of assets fas beow an exogenous threshod: J ω K = λ ( F + F where λ 1 J ( F + F (13 where the iquidation threshod is denoted by K and the star superscript denotes a iquidation threshod of a bank with capita structure that incudes SND. The iquidation threshod is at east equa to the sum of the principa vaues of the senior deposit and the SND. The time of defaut, where iquidation is decared, is denoted by τ and is defined formay by: { t > 0 K } τ = inf ω (14 t 16

17 If iquidation has not occurred unti debt s maturity( τ >T, the debthoders woud receive its debt s face vaue, whie the stockhoder woud receive the residua assets (See Figure.A. Upon iquidation, the residua assets of the bank, after bankruptcy costs have been incurred, woud be distributed among the caimhoders according to their seniority. The vaue of equity, provided that iquidation has not occurred by the current time can be expressed as: S = Ε Q rt J rτ [ e ( ω F + F 1 (15 T S { } 1 { } ] + e Φ τ > T τ T t where S t is the vaue of a stock issued by a bank with capita structure that incudes SND, and S Φ is the stock s payoff function upon iquidation, which depends on the ratio of bankruptcy costs and can be written as: Φ S 0 = K (1 γ ( F + F J K ( F γ K esewhere + F J (16 If the remaining assets after bankruptcy costs are not sufficient to competey pay off the deposit then the SND s hoder receives nothing. Esewhere, the hoder of the SND receives the minimum between its debt s face vaue and the remaining assets. The SND can be vaued as: J [ Q rt J = Ε e F τ 1 rτ J { } 1 { } ] + e Φ (17 τ > T T 17

18 18 where the function J Φ is the SND s payoff function upon iquidation, which equas: + + > > = Φ ( ( (1 0 γ γ γ γ J J J J K F F K F K F F K K F K F K K F K (18 The vaue of the deposit can be written as: { } { } ] 1 1 [ T r T rt Q e F e E τ τ τ > Φ + = (19 where Φ is the deposit s payoff function in the event of iquidation, which is equa to: = Φ esewhere 1 F K F K -? ( K γ (0. The vaue of a bank with capita structure that incudes SND, denoted by V, is equa to the vaue of assets minus the vaue of bankruptcy costs, denoted by C. Therefore we can write the bank vaue as: [ ] [ ] } { } { } { t 1 (1 1 1 T r T T rt Q T r Q t t t t K e e E K e E C V < + = = = τ τ τ τ τ γ ω γ ω ω τ (1

19 The need for deposit insurance under the presumed capita structure arises ony in states in which the size of bankruptcy costs is arger than the difference between the eve of the iquidation threshod and the deposit s face vaue. The cost of a deposit insurance that fuy compensates the depositor upon iquidation is denoted by G and expressed by: G { } Q rτ t = E e [ F K (1 γ ]1 { T τ } ( t The payoffs to the caimhoders for various reaized asset vaues and the aternative ratio of bankruptcy cost are summarized in Tabe. 4. Pricing the ank s Caims by Repicating Payoffs The foowing vauation method utiizes the fact that each of the bank s securities can be expressed as a combination of four buiding bocks: down-and-out ca and down-and-in ca options, down-and-out and down-and-in Heaviside ca options. Assuming no arbitrage, caims with identica payoff function must have equivaent vaue. Hence, in order to price the different iabiities we simpy need to mimic each security by using combinations of these four basic caims. In this section we define the payoff function and the vaue of each option that serves as one of the buiding bocks for pricing the bank s iabiities, and then we show how to repicate the bank s different caims under each of the two capita structures. 19

20 4.1 Definitions of the asic Caims Definition 1: If the vaue of assets has (has not hit a ower barrier K unti maturity then the hoder of a down-and-in (out ca option receives at maturity,t, the maximum between zero and the difference between the vaue of assets, ω T and an exercise price of F. Lemma 1: The price of a down-and-in ca option and down-and-out ca option (with payoff given by definition 1 is: + [( ω F ] di rt Q C ( T, F, K = e E T 1 { τ < T} C + [( ω F ] do rt Q ( T, F, K = e E T 1 { τ T} where C di ( T, F, K and C do ( T, F, K are the current vaues of down-and-in ca and down-and-out ca, with expiry at time T. The strike price of the options is equa to F and the barrier is equa to K. The superscript di indicates a down-and-in type contract and the superscript do indicates a down-and-out type contract.τ is the first time that the vaue of assets has touched the ower barrier K. Definition : If the vaue of assets has hit a ower barrier K unti maturity, T, the hoder of a down-and-in Heaviside ca (at hit woud receive a unit of 1$ at the hitting time τ. Lemma : The price of a down-and-in Heaviside ca (with payoff given by definition is: H rτ [ e ] di Q ( T, K = E 1 { T τ} 0

21 where H di ( T, K is the current vaue of a down-and-in Heaviside ca with expiry at time T and a barrier eve of K. Definition 3: If the vaue of assets has not hit a ower barrier K unti maturity, T, the hoder of a down-and-out Heaviside ca woud receive a unit of 1$ at maturity. Lemma 3: The price of a down-and-out Heaviside ca (with payoff given by definition 3 is: H [ ] do rt Q ( T, K = e Et 1 { T < τ} where H do ( T, K is the vaue of a down-and-out Heaviside ca with expiry at time T and barrier eve K. The reader wi find in Appendix-A a reminder of the pricing formuas for a four basic barrier options that are needed. 4. Repicating Corporate Securities that Incude DES Contract The foowing options portfoios mimic the payoffs of the stock and the DES contract respectivey: S = C do ( T, F (1α C + F di D, K ( T, F c + (1α C, K di + (1α Φ ( T, F S H di, K c ( T, K (3 D = F D H do αc ( T, K di c ( T, F + αc, K di ( T, F + αφ D, K H c do ( T, K c (4 1

22 The ong down-and-out ca position represents the stock payoff if neither iquidation nor conversion ha ve occurred. In this state, the hoder of the DES contract has D F units of down-and-out Heaviside ca option. The ong down-and-in ca position account for the combined effects that occur upon reaching the conversion threshod: the diution of the initia stockhoder and the reduction of the eve of debt to F. To account for eary iquidation and for the transfer of contro to the senior depositor, a short down-and-in ca position is introduced. This option, with barrier K, offset exacty upon iquidation the ong down-and-in payoff with barrier eve c K, eaving the stockhoder with zero payoffs. In the extreme case, in which the senior S depositor has been fuy paid, and thus K ( 1γ F > 0, the initia equityhoder and the DES hoder woud receive the residua assets. The senior deposit is not infuenced in any case by the conversion activity and thus its payoff shoud be mimicked ony by options with barrier eve that equa to the iquidation threshod: do di = F H ( T, K + Φ H ( T, K (5 The current vaue of bankruptcy costs is equa to its magnitude times the present vaue of 1$ conditiona on future defaut. Thus its vaue is isomorphic to down-and-in Heaviside ca option with payoff of minus the vaue of bankruptcy costs and can be expressed as: γ K. The vaue of a bank, V, refects its assets vaue di V = ω C = ω γk H ( T, K (6

23 The cost of deposit insurance can be expressed as a down-and-out Heaviside ca option, with payoff upon iquidation which is equa to the difference between the face vaue of the deposit and the remaining assets of the bank: [ F K (1 γ ] di G = H ( T, K (7 4.3 Repicating Corporate Securities that Incude SND The vaue of each of the bank iabiities is identica to the vaue of a portfoio that consists of two types of options. The first are down-and-out options that mimic the payoff if eary iquidation has not occurred, and the second are down-and-in options that mimic the payoff in the event of iquidation. The vaues of the stock, the SND contract and the senior deposit can be mimicked by the foowing options: S do J S di = C ( T, F + F, K + Φ H ( T, K (8 J do J di J = F H ( T, K + Φ H ( T, K (9 do di = F H ( T, K + Φ H ( T, K (30 As for a bank with capita structure that incudes DES contract, the vaue of the bank, V, is equa to the vaue of its assets minus a down-and-in Heaviside ca option with payoff equa to the size of bankruptcy costs: V di = ω C = ω γk H ( T, K (31 3

24 where C is the current vaue of bankruptcy costs of a bank that incudes SND in its capita structure. The cost of deposit insurance, denoted by G, can be mimicked by a down-and-out ca option with iquidation threshod K, which can be expressed as: G di = [ F K (1 γ ] H ( T, K (3 5. Debt-for-Equity Swap Contract versus Subordinated Notes Equipped with cosed form soutions for the vauation of a bank s iabiities, which incudes a DES or aternativey an SND contract as part of its capita structure, we can compare these two capita structures with respect to the fundamenta tasks of preventing costy faiures in banking, reducing the cost of deposits insurance, enhancing market discipine and reducing the current vaue of bankruptcy costs. The effects of each contract on promoting the presented goas are comp ared in the foowing sections. As a base case for our anaysis, we assume a bank with capita structure that is composed of a singe zero-coupon deposit with F = 95, a singe DES D contract with F = 5, and a stock. oth iabiities mature in one year. The vaue of the bank s assets equas 130, the risk free interest rate is r = 3%, the voatiity of the bank's assets is equa to 1% and no payout is expected ( δ = 0. ankruptcy costs, as percentage of the vaue of the bank s assets upon iquidation, are equa to 0% and the parameter c λ is equa to 1.05, which means that conversion woud occur if the vaue of assets is 4

25 equa to 105% of the tota sum of the principa amounts (deposit pus DES. At the first time that the vaue of assets reaches the conversion threshod, the DES hoder woud receive automaticay 0.5 of common stock issued by the bank in exchange for waiving the debt contractua obigation. The iquidation threshod is equa to 96.9 and therefore iquidation occurs when the vaue of assets is equa to 10% of the tota outstanding debt, which consists by that time ony of the senior deposit. The compared bank is identica in a respects except for its having an SND instead of DES contract. The SND contract has identica terms as the DES contract except conversion. We assume that the reguator has a consistent mandatory intervention poicy in the form of a iquidation threshod that equa to a constant parameter ( λ = λ mutipied by the tota debt s principa amount. As a resut, the iquidation threshod of a bank with SND in its capita structure, exampe. Tabe 3 summarizes the input for the base case. K, equas 10 in our 5.1 Reducing the Incidence of Costy faiures. In this section we compare the risk neutra probabiity of defaut of the two capita structures and derive anaytic expression for the difference between the two probabiities. The risk neutra probabiity that the vaue of assets woud touch a ower threshod unti debt maturity is equivaent to the probabiity that the running minimum of the og-asset vaue at maturity, T, woud be beow the adjusted defaut threshod n( K ω. As presented in Giesecke (003, empoying the fact that the distribution of the minimum is inverse Gaussian and setting m = ( r δ σ, we can write this probabiity as: 5

26 n P( τ < T = N ( K ω σ mt T + e mn σ ( K ω ( K ω n + mt N σ T where N is the standard norma distribution functio n. Lemma 4: If the mandatory supervisory intervention poicy is set as a constant fraction of the tota debt s principa amount for a banks, then the difference between the risk neutra defaut probabiities of a bank with SND contract in its capita structure and a bank with DES contract, which are identica in a other respects, can be cacuated as: P( τ T τ n = N n N S D D m n( λ ( F + F ω D ( λ ( F + F ω mt n ( λ ( F + F ω σ T S m n( λ F ωt ( λ F ω mt n ( λ F ω σ T e + e σ + mt σ N σ T + mt N σ T (33 It is noteworthy that the cosed-form expression reported in Lemma 4 for the difference between the defaut probabiities is an increasing function of the foowing factors: voatiity of assets, assets vaue, eve of capita adequacy, which is measured in the form of the ratio between the iquidation threshod, and the tota face vaue of debt. To iustrate, Tabe 4 presents the difference between the defaut probabiities for various eves of voatiity, vaue of assets and iquidation threshod. When ω = 130, λ = λ = 1. 0 and σ = 1% the difference is equa to.0%. However, when the vaue of assets changes to 10 or to 140, the difference is equa to 8.16% and 0.38% respectivey. A decrease in the eve of the minimum capita adequacy, in the form of ower iquidation threshod, where λ = λ = 1. 0, decreases the difference between the defaut probabiities to 1.34%. 6

27 However, a ower iquidation threshod coud increase the cost of deposit insurance, as wi be described at the foowing section. 5. Depositors Protection: The Cost of Deposits Insurance Whie the DES contract definitey enhances financia stabiity under any market conditions compared to the SND contract, its advantage as a measure for reducing the cost of deposits insurance and enhancing depositors protection is more ambiguous and depends on the ratio of bankruptcy costs, the vaue of assets (or the everage ratio, the voatiity of assets and the eve of the iquidation threshod. Lemma 5: If the mandatory supervisory intervention poicy is set as a constant fraction of the tota debt s principa amount for a banks, then the difference between the costs of deposit insurance of a bank with SND contract and a bank with DES contract, that are a ese identica, can be positive, negative or equa to zero. Proof: See Appendix. Tabe 5 iustrates that the ratio between the cost of insurance of a deposit in a bank with SND in its capita structure and the cost of insurance of a deposit in a bank with DES contract decreases with the eve of iquidation threshod. When ω =130, λ = λ = 1.0 and σ = 18%, the difference between the risk neutra probabiities of defaut is equa to 5.83% (See Tabe 4, and the ratio between the cost of deposits insurance is equa to 143.9%. However, when the iquidation threshod increases, such that λ = λ = 1. 1, the difference between the risk neutra probabiities of defaut increases to 1.38% whie the ratio between the costs of deposits insurance 7

28 decreases to 98.8%, meaning that the cost of insurance of a deposit in a bank with DES contract becomes more expensive than an identica insurance of a deposit in a bank with SND contract in its capita structure. As the vaue of the bank s assets decreases (the everage is increased, issuing SND contract increases the probabiity of costy defaut and decreases the cost of deposit insurance compared to the DES contract. A decrease in the vaue of the bank s assets to ω =10 increases the difference between the defaut probabiities from 5.83% to 11.18% whie decreasing the ratio between the costs of deposit insurance from 143.9% to 19.5%. The behavior of the cost of deposit insurance with respect to the ratio of bankruptcy costs is shown in Figure 3. When the iquidation threshod is set at a reativey ow eve, such that λ = λ = 1. 0, we find that the cost of insurance of a deposit in a bank with DES contract is higher than a deposit in a bank with SND contract unti the ratios of bankruptcy costs are approximatey equa to 7%. If the iquidation threshod is increased (in our exampe: λ = λ = 1. 1, the costs of deposits insurance are equaized when the ratios of bankruptcy costs are approximatey equa to 16%. Intuitivey, we coud think of it in the foowing way. Under a ow ratio of bankruptcy costs the depositor in a bank with SND is fuy paid in the event of iquidation if the eve of iquidation threshod is reativey high, since the ayers of equity and SND absorb a osses. The depositor in a bank with DES contract is not protected by an extra ayer of debt, since the event of conversion woud aways precede the iquidation event and thus the depositor begins to suffer osses at a ower ratio of bankruptcy costs. When the ratio of bankruptcy costs increases both depositors woud not be fuy paid upon iquidation. 8

29 However, the probabiity of a iquidation event is consideraby higher for a bank with SND contract in its capita structure. 5.3 Efficiency Enhancement: The Vaue of ankruptcy Costs Athough the deadweight vaue in the event of a bank faiure is usuay not imited to the coapsing bank, it is interesting to compare the vaue of bankruptcy costs of the two capita structures as a signa for efficiency. Lemma.6: If the mandatory supervisory intervention poicy is set as a constant fraction of the tota debt s principa amount for a banks, then the difference between the vaues of bankruptcy costs of a bank with SND contract and a bank with DES contract, that are identica in a other respects, is aways positive. Proof: See Appendix. When the vaue of assets is reativey high (ow everage ratio, the difference between the vaues of bankruptcy costs (as % of assets vaue of a bank with SND in its capita structure and a bank with DES contract, that are identica in a other respects, increases with assets voatiity (See Figure 4. For a reativey high eve of assets voatiity iquidation woud certainy occur under each of the two capita structures, and thus the difference between the two vaues of bankruptcy costs woud converge to γ ( K K ω, which in the presented exampe equa to 0.93%. Conversey, when the vaue of assets is reativey ow (high everage ratio, the difference between the vaues of bankruptcy costs is hump shaped with respect to assets voatiity, and as the vaue of assets increases this difference is maximized at a higher eve of assets voatiity. 9

30 5.4 Risk-Contro and Market Monitoring In this section, we evauate and compare the effects of the DES and the SND contracts on enhancing market monitoring and controing the risk taking by the sharehoders. A contract can enhance market monitoring due to its negative sensitivity to increased everage or due to its negative sensitivity to an increase in assets voatiity. The infuence of a contract on the sharehoders risk taking is measured by the sensitivity of the stock to changes in assets' voatiity. a. Market Monitoring The presence of most subordinated debt instruments within a bank s capita structure is justified by their protective nature and their abiity to enhance market monitoring by charging high funding costs once excessive risk-taking activities are detected. Levonian (001 presents a theoretica mode for pricing SND, which is based on the ack-schoes-merton contingent caims anaysis, and finds that an increase in the risk of assets wi decrease the vaue of subordinated debt for sovent banks The resuts of empirica studies that examine whether risky debt issued by banks and bank hoding companies enhances risk monitoring have been mixed. Studies done prior to 199 faied to find a significant reationship between firm risk and yieds on subordinated debt. 13 More recent studies do indicate that risk is being appropriatey priced (See Fannery and Sorescu,1996, and Jagtiani, Kaufman and Lemieux, In a recent study, Krishnan, Ritchken, and Thomson (003 examine whether changes in credit spreads refect changes in firm specific risks. After controing for changes in market-wide and iquidity factors, the authors do not find any consistent evidence for 30

31 connection between the two. Moreover, the fact that banking firms are highy reguated coud not expain the insensitivity of the subordinated debt spreads. In order to expore and to compare the sensitivity of the SND and the DES prices to changes in assets voatiity, we present in figures 5 and 6 the vega of the two contracts against the ratio of bankruptcy costs for different eves of iquidation threshod and everage. 15 When the ratio of iquidation costs is reativey ow and banking firms are highy reguated, in the form of a reativey high iquidation threshod ( λ =1.06, the oss given defaut of the SND is zero. As a resut, the SND price is insensitive to the eve of voatiity, and the SND contract is inefficient as a too for providing market monitoring (See Figure 5. However, when the ratio of bankruptcy costs increases the SND hoder woud not be fuy paid upon defaut, and therefore the contract price has negative vega. The effect of the mandatory reguatory intervention poicy on the contract price is not obvious and depends heaviy on the everage ratio. When the everage ratio is reativey J ow [( F + F ω = ], the SND's vega decreases with the eve of the iquidation threshod. Conversey, when the everage is reativey high ( ω = 110 the SND's vega increases with the eve of the iquidation threshod. The vega of the DES contract receives negative vaues since a mandatory conversion woud aways precede the iquidation event (See Figure 6. The efficiency of the SND over the DES contract as a too for providing market monitoring exists in regions in which the ratio of bankruptcy costs is reativey high. A reativey high ratio of bankruptcy costs decreases the recovery rate of both contracts upon iquidation, however, the potentia compensation of the DES hoder upon conversion, in the form of common stocks, reduces its vega compared to the SND. 31

32 Figure 7 highights the imitedness of the DES contract as a too for providing market monitoring by potting the vaue of the SND and the DES contract against the vaue of assets. Whie the vaue of the SND contract increases with assets vaue, the behavior of the DES contract depends on its conversion ratio (α. An increase in the vaue of assets produces two opposite effects: increasing the probabiity that the DES hoder woud be fuy paid upon maturity and decreasing the probabiity of eary mandatory conversion. As the conversion ratio increases (decreases the ater effect becomes more dominant (minor and the vaue of the DES decreases (increases with the vaue of assets. b. Risk-Avoidance or Risk-Transfer? In their semina work, ack and Schoes (1973 and Merton (1974 offer the insight that equity vaue is identica to the price of a standard European ca option on the tota market vaue of the firm s assets with an exercise price equa to the promised payment of corporate iabiities. However, as Gaai and Masuis (1976 first pointed out, this option anaogy suffers from the fact that the vaue of a standard ca option is stricty increasing with assets voatiity of the underying assets. Hence, a sharehoder-aigned manager, who is faced with a choice between two different projects woud invest in the project of higher variance. Moreover, it is even possibe that a more profitabe investment project wi be rejected in favor of a project with a higher variance of percentage returns, thereby transferring weath from bondhoders to sharehoder s. This asset-substitution probem was much deveoped in the agency iterature, starting with Jensen and Mecking (1976. According to this approach, the interests of bondhoders and sharehoders in a everage firm strongy diverge regarding the risk that can accompany higher firm profits 3

33 since the increase in assets voatiity resuts in a transfer of vaue from the debthoders of a firm to its equityhoders. The traditiona methods empoyed by debthoders to dea with asset-substitution probems incude increasing the required rate of return on their financia caims, design of safety covenants that imit a firm s abiity to shift risk, and simpe termination of the reationships with the companies. Athough the market is abe to constrain the behavior of non-bank firms, commercia banks debthoders have weak incentives to protect the vaue of their caims due to the effects of impicit or expicit guarantees provided to commercia banks by nationa governments. However, by setting a minimum eve of mandatory intervention, government s reguators have a strong impact on equityhoders abiity to transfer risk. As anayzed by Reisz and Perich (004, if the iquidation threshod is higher than the sum of the tota iabiity, the derivative of the stock pr ice with respect to assets voatiity (vega of the option is negative for a σ : any risky investment makes the probabiity of going bankrupt too arge and sharehoders wi shy away from any risky project. In this case, far from witnessing an asset substitution probem à a Jensen and Mecking (1976, we might observe a risk-avoidance probem à a John and rito (000: a sharehoder-aigned manager, afraid of osing growth options privy ony to her, can shy away from risk and undertake projects with suboptima risk eves. Reisz and Perich (004 support this finding numericay and find that the vaue of equity decreases for ow everage ratio. Consistent with this attitude, Fannery (00 asserts that the introduction of subordinated debt into a bank s capita structure increases the eve of everage and thus derives a higher iquidation threshod that exacerbates the probem of risk avoidance and might even hurt the competitive nature of bank and its task as iquidity provider. 33