Solvency regulation and credit risk transfer

Solvency regulation and credit risk transfer Vittoria Cerasi y Jean-Charles Rochet z This version: May 20, 2008 Abstract This aer analyzes the otimality of credit risk transfer (CRT) in banking. In a model where banks main activity is to monitor loans, we show that a combination of CRT instruments, loan sales and credit derivatives, might be otimal to insure banks against shocks and to otimally redeloy caital when new investment oortunities arise, without imairing incentives. We derive imlications for the otimal design of caital requirements. JEL classi cation: G2 1; G38. Keywords: credit risk transfer; solvency regulation; monitoring. We thank Elena Carletti, Loriana Pelizzon, Gabriella Chiesa and articiants in the Second Conference on anking Regulation, ZEW, Mannheim (October 2007), in the Conference on Interaction of Market and Credit Risk, erlin (December 2007) and in the Unicredit Grou and CSEF Conference on anking and Finance, Nales (December 2007), in seminars at Finance Deartment (Frankfurt University) and Catholic University (Milan) for their useful comments. Financial suort from PRIN2005 and FAR2006 is gratefully acknowledged. y Milano-icocca University, Statistics Deartment, Via icocca degli Arcimboldi 8, 20126 Milano, Italy - hone: +39-02-6448.5821, fax: +39-02-6448.5878, vittoria.cerasi@unimib.it. z Toulouse University (IDEI and GREMAQ), Manufacture des Tabacs, 21 Allée de rienne bat. F, F-31000 Toulouse Cedex, France, rochet@cict.fr 1

1 Introduction In the latest years larger banks have steadily increased their market share in credit risk transfer activities - credit derivatives and loan sales - as extensively documented by EC (2004), IS (2005), Minton et al. (2006) and Du e (2007), among others. When transferring credit risk for risk management uroses, banks reduce their stake in the return from lending, imairing their incentives to monitor loans. If monitoring is imortant for bank credit, then credit risk transfer (CRT, hereafter) may reduce the value of intermediation and increase the risk in the banking sector. The aim of this aer is to exlore the imact of di erent CRT activities on bank monitoring incentives and its imlications for banks solvency regulation. We ut forward an aroach to rudential regulation that di ers markedly from the aroach followed (more or less imlicitly) by banking authorities. Instead of a bu er which aim is to limit the robability of a bank s failure to some redetermined threshold (this is what we call the Value at Risk aroach) we defend the view that bank s caital is needed to rovide bankers with aroriate incentives to monitor borrowers (this is what we call the incentives aroach). These two aroaches have di erent imlications for the rudential treatment of CRT activities. In the VaR aroach, basically all CRT activities justify a reduction in regulatory caital requirements (for a given volume of lending) because they reduce the robability that losses exceed any given threshold. y contrast, in the incentives aroach, CRT activities allow to reduce caital requirements only in so far as they maintain bankers incentives to monitor. Following Holmström s exhaustive statistics aroach, bankers should be, as much as ossible, insured against exogenous risks (such as macroeconomic shocks) but should bear a su cient fraction of all the risks they can in uence by their monitoring activities. In the aer we develo a simle model of rudential regulation where a banker might exert a monitoring e ort in order to reduce entrereneurs oortunism. Deositors cannot observe banker s e ort and condition their funding to the commitment to monitor. ank caital fosters banker s monitoring incentives. Prudential regulation is achieved by setting a minimum caital requirement and a fair deosit insurance remium. To this basic setu we add a solvency shock and new lending oortunities at 2

an interim stage. Once banks have extended loans their ortfolio might be hit by an aggregate shock - corresonding to an economic downturn - negatively a ecting future returns from loans. After the realization of this shock new lending oortunities occur with ositive NPV. Given binding caital adequacy requirements, those oortunities cannot be ursued unless new liquidity is rovided to the bank through access to nancial markets at a fair rice. However since the banker has to be motivated in her monitoring e ort, roviding liquidity is e cient only in uturns. We show that the otimal incentive scheme can be imlemented, in addition to new caital requirements and fair deosit insurance, through a combination of CRT instruments, loan sales to rovide state-contingent liquidity and credit derivatives to insure loan losses. CRT markets resond to di erent functions here: loan sales suly interim liquidity in uturns to undertake, with the roceeds of the sales, the new lending oortunities, while credit derivatives rovide state-contingent insurance. Given that the objective of the rudential regulator is to imrove bank solvency and to avoid sub-otimal under-investment, CRT is art of the otimal incentive scheme. In the model the tension between insurance and incentives is driving the results about otimal solvency regulation and use of CRT: when exected loan losses are dominant the bank must buy rotection (sell credit default swas) to shed risk in downturns, while when the rate of growth of new lending oortunities is large the bank may even take on more risk (buy credit default swas) in downturns to be able to undertake exansion. In conclusion, our simle model of rudential regulation shows that, when taking into account banker s incentives in the di erent states of the economy, caital requirements should not be designed with the unique objective to insure for loan losses in down-turns but also with the concern for under-investment in u-turns. We also confront the otimal solution to the alternative of saving liquidity at the initial stage in order to face latter lending oortunities. In the liquidity hoarding case there is a mis-management of liquidity, because the banker is allowed to exand also in downturns, although it is ex-ante sub-otimal. Therefore, the solution with CRT is referred as it entails a lower caital ratio and greater lending. Several results in our aer are in line with the emirical evidence in Cebenoyan and Strahan (2004), Goderis et al.(2006) and Minton et al. (2006) showing that banks accessing CRT markets tend to increase their lending and hold less caital. 3

In addition, they rovide evidence that larger banks engaging in CRT transactions, articiate in several markets at the same time, showing that loan sales and credit derivatives are comlementary activities. Finally, when looking at articiants in credit derivatives transactions, banks are both buyers and sellers of rotection (as documented by Minton et al., 2006, EC, 2004, and Du e, 2007, among others). The aer is related to the growing literature on CRT (see the survey in Ki et al., 2002) and, in articular, on the imact of CRT instruments on banker s monitoring incentives and bank solvency. The aer builds on Holmstrom and Tirole (1997) alied to banks as delegated monitors where monitoring incentives are rovided through caital regulation, as diversi cation oortunities are scarce. Along this line of research Chiesa and hattacharya (2007) show that CRT insuring for aggregate risks imroves monitoring incentives. Their argument is based on the result that contingent transfers, such as credit derivatives insuring loan ortfolio for common shocks, are otimal mechanisms to achieve maximum e ort when monitoring is more valuable in downturns. In contrast in our model since monitoring e ort enhances ortfolio outcomes in all states, while new interim liquidity is valuable only in uturns, shedding risk in downturns might create under-investment and reduced monitoring incentives. The otimal balance of insurance and incentives is achieved through a combination of di erent CRT instruments, loan sales and credit derivatives. The coexistence of loan sales and credit derivatives in the otimal solution is common to other aers (see for instance Du ee and Zhou, 2001, Thomson, 2006, and Parlour and Winton, 2007). For instance Parlour and Winton (2007) analyze the coexistence of loan sales and credit derivatives for monitoring incentives; however they focus on the imact on loan quality when banks have suerior information comared to investors and disregard rudential regulation. Nicolò and Pelizzon (2006) analyze the imact of caital regulation on the incentives to issue di erent CRT instruments, and show how seci c forms of credit derivatives could emerge as an otimal signaling device for better quality banks in resonse to exogenous caital regulations. Our objective is to analyze the imlications of CRT on monitoring incentives together with otimal caital regulation: therefore we assume that monitoring is exerted after issuing CRT instruments, while dis-regarding the imlications of rivate information 4

in CRT markets. There are other aers centered on the imact of CRT on monitoring incentives. For instance, Aring (2004) argues in favor of credit derivatives since credit rotection fosters the commitment to liquidate and as a result entrereneurial e ort, although the dilution of the claim with investors reduces banker s monitoring incentives. Also Morrison (2001) shows that single-named credit derivatives imact negatively on monitoring when mainly loans nanced by a mix of direct and bank credit are a ected: since banks are risk-averse they bene t from greater insurance on loan losses, but they lose incentives to monitor. We use the idea that loan sales rovide liquidity when other funds are scarce as Gorton and Pennacchi (1995), although we deart by adding credit derivatives and otimal caital regulation. In Parlour and Plantin (2007) loan sales rovide liquidity for new investment oortunities, however, since monitoring is exerted before selling loans, investors cannot distinguish the true motive of the sale, in contrast with our model where information is symmetric, and therefore there could be scarcity of liquidity in the loan sales market. Another strand of the literature analyzes the imact of CRT on the allocation of risks across sectors in the economy. For instance Wagner and Marsh (2006) show that CRT involving transfer of risk from banks to other sectors enhances welfare, when the banking sector is less diversi ed comared to other non-banking sectors, as the greater diversi cation achieved through CRT comensates the reduction in monitoring and is bene cial for nancial stability. Allen and Carletti (2006) show that under some conditions on the distribution of liquidity shocks across banks, transfers of risks from the banking sector to the insurance sector do increase nancial stability. In this aer instead we focus on monitoring incentives at a reresentative bank level, while we do not consider the imact on the allocation of risks across banks or across sectors when banks have access to CRT markets. Wagner (2007) analyses the consequences of credit derivatives for bank solvency. He shows that, although CRT imroves loans liquidity diminishing the likelihood of bank runs, when taking into account ex-ante incentives the greater liquidity induces greater risk-taking by the bank, reversing the ositive e ect on bank stability. Finally, we share with Kashya and Stein (2004) the idea that otimal caital 5

regulation serves both to insure loan losses and to reduce under-investment roblems in the di erent states of the economy, although our conclusions on the otimal caital regulation are a ected by the introduction of CRT instruments. The remainder of the aer is organized as follows. Section 2 describes the model of rudential regulation; we start from a static benchmark model and then extend this model by introducing two additional features, new lending oortunities and a solvency shock. We study the imact of these new features on the otimal caital ratio. In Section 3 we show that this otimal solution can be imlemented by a combination of loan sales and credit derivatives together with a solvency regulation. Section 4 discusses the imlications for liquidity management and caital regulation of ossible alternatives to CRT as for instance liquidity hoarding. In Section 5 we discuss the emirical redictions of the model. Concluding remarks are in Section 6. 2 A model of rudential regulation In this section we model the need for caital regulation in the banking sector. We start by introducing a static benchmark model where minimum caital requirements are justi ed for rudential regulation. In a context where banks, whose main function is to monitor borrowers, have an incentive to exloit their informative advantage to shift ortfolio losses on deositors, minimum caital requirements rovide correct incentives to monitor. We then add to this simle model two ingredients, a negative solvency shock on loan returns and the ossibility to undertake new lending oortunities at a latter stage. These two ingredients add into the benchmark model a roblem of inter-temoral liquidity management which, we claim, can be resolved using credit risk transfer (CRT) instruments. The aim of the model is to analyze in a tractable way the imact of CRT on the monitoring function of banks and the imlications for rudential regulation. 2.1 A static benchmark model Our starting oint is the simle rudential regulation model of Rochet (2004) adated from Holmstrom and Tirole (1997). Consider a two-date economy (t=0,1). At date 0 a bank, with caital E 0, raises deosits D 0 from disersed investors and extends 6

loans L 0 to some entrereneurs. Deositors alternative return er unit invested is 1. Entrereneurs rely on banking nance to undertake a risky roject : each roject requires 1 unit of investment at date 0; and yields a return R > 1 at date 1 with robability 2 [0; 1] and 0 otherwise. The success robability of the loan ortfolio is a ected by the banker s monitoring e ort: un-monitored loans success robability falls to > 0; while the banker saves a rivate cost > 0 er unit lent. We assume constant return to scale for loan returns and rivate bene ts. Loans returns are erfectly correlated as the bank, facing limited oortunities for diversi cation, holds some non-diversi able risk in its ortofolio. 1 Further, we assume that only monitored nance is viable 2 R > 1 > ( )R + (A1) Given that the monitoring e ort is non-observable, the bank is subject to moralhazard. For the banker to monitor the ortfolio of loans the following incentive comatibility condition must be ful lled (RL 0 D 0 ) ( ) (RL 0 D 0 ) + L 0 which can be rewritten as D 0 R L 0 : (1) Given that deositors do not observe the monitoring e ort while the banker derives a rivate bene t from not monitoring, she cannot credibly romise to reay deositors an amount greater than the maximum ledgeable income de ned by the right-hand side in the revious exression. We further assume that a deosit insurance fund (DIF) is in lace: by aying a remium 0 deositors are fully insured against the risk of bank failure at date 1. Date 0 bank s balance sheet is de ned as L 0 + 0 = E 0 + D 0 : (2) 1 There is a literature on the bene ts of diversi cation of loans for banker s incentives to monitor (see Diamond, 1984, and Cerasi and Daltung, 2000, where the result of Diamond is alied to a context similar to the one in this aer). However in that case inside equity fully restores incentives eliminating the need for diversi cation. Holmstrom and Tirole (1997) show indeed that caital - inside equity - strenghten monitoring incentives when oortunities for diversi cation are scarce. In this context the assumtion of erfect correlation is not crucial for the results while it simli es comutations. 2 Given that investors are disersed, they do not have incentives to monitor. Monitored nance is thus rovided by banks. 7

The break-even condition for the DIF is that the exected reayment to deositors, when the banker monitors, must not exceed the remium, that is 0 (1 )D 0 ; and substituting from (2) L 0 E 0 + D 0 : (3) In this simle model we derive the otimal rudential regulation as the contract between the DIF and the banker that maximizes exected social surlus. 3 Proosition 1 The otimal contract between the DIF and the banker can be imlemented by a combination of a fair remium on deosit insurance, 0 = (1 )D 0 ; and a caital adequacy requirement limiting banks lending to a certain multile of their equity, that is where k S 1 R L 0 E 0 (4) k S is the (static) caital ratio: Proof. The otimal contract between the DIF and the banker requires choosing the level of loans L 0 and deosits D 0 that maximize exected social surlus ES = (R 1) L 0 subject to incentive comatibility constraint (1) and break-even condition (3). The otimal solution is obtained by saturating the two constraints. In articular, setting: D 0 = R L 0 h i Substituting into (3), we obtain E 0 1 R L 0: For this result to hold we need to assume that banks need caital, i.e. R < 1: (A2) If (A2) was not ful lled, then banks could be 100% nanced by deosits. y contrast, when (A2) alies, there is a maximum to the amount of deosits that the bank can 3 The idea is that the regulator acts in the interest of deositors (see Rochet, 2004, for a detailed discussion of the otimal rudential regulation). 8

raise: it is given by the maximum ledgeable income in the right hand side of (1). For each unit of loan, the maximum reayment to deositors is (R ) in case of loan success, thus the di erence between the maximum amount of money deositors are willing to suly (R caital E 0. )L 0 at date 0 and loans L 0 ; must be covered with own From Proosition 1 it follows that banks can exand their lending to a maximum of 1=k S of their equity: the static caital ratio k S is increasing in the severity of moral hazard, measured by ; while decreasing in the exected return of the roject, R, as the maximum ledgeable income to deositors decreases accordingly. A greater caital ratio imlies tighter credit conditions. 2.2 The relation with the credit risk literature Our benchmark model is extremely stylized, and makes several irrealistic assumtions for the sake of tractability. In articular, we assume that returns on bank loans are erfectly correlated, which is of course a very strong assumtion. We show in this section that the logic of our model is reserved if we adot seci cations that are closer to those used in the credit risk literature. This will also allow us to clarify the di erence between the VaR aroach and the incentives aroach to rudential regulation. Assume indeed that the return on bank loans has a continuous distribution, derived from a standard credit risk model. Suose for examle that each bank loan returns either R or 0 (zero recovery rate in case of default) but that default is driven by a combination of a common factor ~ f and an idiosyncratic shock ~" i. Default of loan i occurs when ~ f + 1 ~"i s(e) where is a correlation arameter and s(e) is a threshold that deends on the monitoring e ort (e = 0; 1) exerted by the banker, with s(0) > s(1). We assume that conditionally on the common factor ~ f, idiosyncratic shocks ~" i are i.i.d. with a cumulative distribution function. y the law of large numbers, the average loss ~` on the 9

loan ortfolio is comletely determined by the realization of the common factor f: ~ ~` = R Pr ~" i s(e)! f ~ ; 1 ~` = R " # s(e) f ~ : 1 The cumulative distribution function of losses is thus determined by the level of e ort of the banker: F e (`) ~` `je = Pr f ~ s(e) 1 1 ` : R We assume that the c.d.f. of ~ f satis es MLRP. y adating the arguments of Innes (1990), it is easy to see that the otimal contract 4 is similar to that obtained above: The bankers gets a remuneration I(` threshold ` and 0 above this threshold. `) whenever losses ` do not exceed the The bank s default threshold ` is determined by the incentive comatibility condition: Z ` 0 (` `)[df 1 (`) df 0 (`)] = : The minimum caital ratio is equal to the net exected shortfall: Z E I max(`; `)df 1 (`) (R 1): As before, the deosit insurance remium is actuarial: Z = I (` `) + df (`): There are two fundamental di erences with the VaR aroach to rudential regulation. First, the caital requirement is meant to cover not the Value at Risk, but the net exected shortfall. This means that it covers the exected losses above the default threshold `, net of the nominal excess return (R 1) on loans. 4 Like Innes (1990), we restrict attention to contracts such that the marginal remuneration of the banker (as a function of loans returns) is always between 0 and 1. 10

The second di erence is that the default threshold ` is not given by an exogenously determined robability of default " but by the incentive comatibility condition. The default threshold ` is the minimum value that rovides the banker who exerts a monitoring e ort (distribution of losses F 1 ()) with an incremental exected gain (with resect to the case where he shirks, and the distribution of losses F 0 ()) at least equal to the bene t from shirking. This has imortant imlications on the rudential treatment of CRT. In order to cature these di erences we go back to our initial benchmark model (with erfect correlation of loan returns) and extend it to include uncertainty. 2.3 The dynamic model with uncertainty We now add two new ingredients to the simle benchmark model to generate an intertemoral roblem of liquidity management for the bank. The rst ingredient is a negative shock (a credit loss) a ecting the exected return on the ortfolio of loans. In articular we assume that, at date 1=2; an observable shock occurs with robability q 2 [0; 1] and that in this event the loan ortfolio return in case of success is reduced to (R ) er unit lent, instead of R. We assume that 0 < < R. The second ingredient is the occurrence of new lending oortunities after date 1=2; that is after the realization of the shock. In articular, the bank has the ossibility to nance new loans of the same quality of the old ones in roortion to L 0 : loans can be increased u to L 1 = (1 + x)l 0 with x 2 [0; ]: This ingredient catures the idea that new valuable rojects may become available once the bank has already extended loans and is constrained by the caital requirement. Since we assume rigidities in the deosit market, the banker has to raise money from investors to fund these new rojects. This new ingredient requires solving for the otimal amount of new funds, in addition to the otimal lending caacity determined at t = 0. At date 0 the bank raises E 0 +D 0, lends L 0 ; and ays a remium 0 to the DIF as before. At date 1=2 the negative shock occurs with robability q and right afterwards new lending oortunities arise u to of extended loans L 0 : After the realization of the shock and new loans are funded, the banker may monitor loans. Figure 1 might hel to clarify the sequence of events. The uer branch variables (no credit loss) are denoted by a suerscrit +, while the lower branch variables (solvency shock) are 11

denoted by a suerscrit. [Insert Figure 1] For the banker to monitor loans in both states, the following incentive comatibility constraints must hold: R + L+ 1 ; R L 1 ; (5) where R + (resectively, R ) denotes the revenue in case of loan success in the uer (res., lower) branch and L + 1 = (1 + x + )L 0 (res., L 1 = (1 + x )L 0 ) total lending in the uer (res., lower) branch. From an ex-ante ersective, investors are willing to commit to inject new funds at date t = 1=2 if and only if the exected return is greater than the oortunity cost of their caital: (1 q) RL + 1 D 0 R + + q (R )L1 D 0 R (1 q) x + + qx L 0 : (6) The otimal contract between the DIF and the banker is de ned as the vector ( 0 ; L 0 ; x + ; x ) that maximizes exected social surlus ES = (1 q) [R 1] (1 + x + )L 0 + q [(R ) 1] (1 + x )L 0 (7) under constraints (5), (6) and of course x + ; x Proosition. 2 [0; ] : It is derived in the following Proosition 2 The otimal contract between the DIF and the banker is characterized by a fair remium 0 = (1 )D 0 ; and an initial volume of lending limited to L 0 = E 0 k 0 where k 0 = k S [1 + (1 q)] + q denotes the modi ed caital ratio at date t = 0. Moreover x + = ; x = 0: the bank is only allowed to exloit new lending oortunities at date t = 1=2 in state + (boom) but not in state (recession). Proof. See the Aendix. 12

The rudential regulator has two instruments to achieve the otimal solution: the rst is given by the two lending growth rates fx + ; x g allowing to reward the banker for her e ort di erently in the two states; the second is the scale of activity, that is the level of loans L 0 constrained by the maximum ledgeable income to deositors at date 0 for a given caital E 0. oth instruments a ect the reward of the banker in the two states ful lling the incentive conditions (5). The otimal solution requires setting x = 0 while x + = for a given L 0 : In other words the banker is not allowed to grant new loans in state, while she can lend at full caacity in state +, which allows to maximize her incentives to monitor. 5 This is due to the fact that monitoring is more valuable in state +; as its marginal bene t is greatest while its marginal cost is constant. 6 To foster banker s incentives the regulator leaves a greater rent in state + while rewarding e ort the least as ossible in state. However by doing this the maximum ledgeable income to deositors is a ected, as it takes di erent values in the two states. Total deosits, and as a consequence lending, are constrained by the minimum of the ledgeable income across states, leaving an extra-rent to the banker in one of the two states. There is scoe for insuring the maximum ledgeable income to deositors in order to boost lending. As a result, the otimal caital ratio is greater comared to that in the static model, due to the insurance cost, and thus credit conditions are tighter. We analyze the recise measure of this e ect in the next section. 3 Otimal rudential regulation and CRT In this section we show that there is an otimal mix of CRT instruments (a combination of loan sales and credit derivatives) and rudential regulation that imlements the otimal solution characterized above. De ne k 0 to be the caital ratio at date 0, imlying that loans L 0 have to be at most a multile 1=k 0 of the bank s caital E 0 (which is exogenously given by assumtion) (minimum caital requirement). The DIF remium is set, as before, to 0 = (1 the face value of deosits is unchanged. )D 0 ; since the robability of default on At date 1=2 in state + the banker can raise new funds L 0 by selling a fraction y 5 Note that this is true even when the NPV of loans is ositive in state. 6 This is di erent in Chiesa and hattacharya (2007) where instead monitoring is more valuable in state and the MLRP roerty does not aly to the monitoring e ort. 13

of its old loans. Denote the date 1 reayment to investors buying these loans as Y L 0 : The banker s incentive to monitor in state + is reserved whenever (RL 0 Y L 0 ) ( ) (RL 0 Y L 0 ) + L 0 The banker can romise to reay at maximum Y = (R therefore the unit rice of a loan in state + is P = R : ) when loans succeed, This is the maximum rice at which the banker retains incentives to monitor the loans she sells. To raise enough liquidity to extend L 0 new loans, the banker has to sell y such that yp L 0 = L 0 ; therefore y = R : Note that, due to assumtion (A2), P < 1; the bank has to sell more loans than it grants new ones, in order to maintain its incentives, that is y >. If state + revails, the banker receives (1 + ) L 0 at date 1. 7 y contrast, if state revails, the reward to the banker is only L 0, as in the static model. We show that the imlementation of the otimal solution requires, in addition to loan sales at date 1=2 to nance new loans in state +, state-contingent transfers at date 1: In other words it imlies an otimal transfer of funds across the two states through an insurance contract where the bank ays investors S + if state + occurs and S in state. For this insurance to be fairly riced, the contingent transfers S + and S must ful ll the condition qs + (1 q)s + = 0: The maximum ledgeable income to investors in state S = (R )L 0 D 0 L 0; at date 1 is while the maximum ledgeable income to investors in state + at date 1 is S + = RL 0 [1 + y] D 0 L 0(1 + 7 The banker s rivate bene ts in state + are given by (1 + y) L 0 on the loans retained in the ortfolio until their maturity at date 1; in addition the banker earns y L 0 imlicit in the rice P aid by investors at date 1=2: 14 y):

After easy comutations, we nd indeed that the exressions of S simli ed into: and We can state the following result. S = (1 q) S + = q k S L 0 ; k S L 0 : and S + can be Proosition 3 The otimal contract between the regulatory authority and the banker can be imlemented by the following series of state contingent caital ratios: k 0 = (1 q)(1 + )k S + q(k S + ) = (1 + )k S + qw (8) at date 0, and at date 1=2: k S in state + and k S + in state. Given that the bank increases its volume of loans by a fraction in state +, regulatory caital must equal (at least) k S (1 + )L 0 in state + and (k S + )L 0 in state. Regulatory caital at date 0, k 0 L 0 equals the exected value of regulatory caital at date 1=2. New loans are nanced by loan sales at date 1=2 in state +; of a fraction y = R (9) of initial loans. In state ; the banker is not allowed to issue new loans. Finally adjustments in regulatory caital are rovided by state contingent transfers (contracted uon at date 0 and interreted as CDS) with W h i k S. S + = qw L 0 ; S = (1 q)w L 0 (10) In order to get the intuition behind the results in this Proosition, let us consider rst the case without loan losses and without new lending oortunities, that is = = 0: In this case the otimal caital ratio is the static caital ratio k S and there is no role for credit risk transfer, neither loan sales at date 1=2 nor state contingent transfers at date 1 as y = S + = S = 0: 15

This result shows that the two ingredients, solvency shock on the value of loans and interim lending oortunities, are crucial to justify CRT instruments for otimal rudential regulation. When > 0; but = 0 (no new lending oortunity at t = 1=2), the bank does not need to sell loans in state +, in fact y = 0, but it uses state contingent transfers to insure againts its credit losses through a Credit Default Swa (CDS, hereafter). The caital ratio at date 0 is augmented relatively to the static model by q; reresenting the CDS remium aid when the ortofolio of loans succeeds with robability : To understand why there is need for insurance, we comute the maximum amount of deosits without the CDS in the two states. The ledgeable income to deositors in state without CDS is D 0 = R L 0 L 0 ; while in state + without CDS it is D + 0 = R L 0 : Total deosits are given by the minimum of these two ledgeable incomes D 0 = min(d 0 ; D 0 + ) = R L 0 L 0 that is state of deositors in state maximum ledgeable income. Loan losses are 100% on the shoulders ; while the banker earns an extra-rent in state +; since R + = L 0 + L 0 : There is scoe for smoothing income across states: by aying an insurance remium ql 0 to recover the losses L 0 in state ; the maximum ledgeable income would be reduced in state just by the amount of the CDS remium and not by the full 100% of loan losses D 0 = RL 0 L 0 ql 0 : This allows to boost total deosits and increase lending: thus the solution with CDS dominates the one without CDS. In this case the combination of debt at date 0 and a 16

state-contingent insurance contract at date 1 is otimal. 8 In the solution with CRT the banker s reward in state + is reduced to eliminate the extra-rent, in order to increase lending, while reserving monitoring incentives. When ; > 0 things are a little bit more comlicated. To foster banker s incentives new liquidity is injected and loans are extended u to L 0 in state +: Further loan losses occur in state in state + without CDS D + 0 = RL + 1 R + y at date 1: We comute the maximum ledgeable income R L 0 = R L 0 k SL 0 and in state without CDS D 0 = R L 0 L 0 Total deosits are given by the minimum of the two exressions above, that is D 0 = min(d 0 ; D 0 + ) = R L 0 max k S; L 0 (11) Assume that > k S ; the banker s reward in state + is given by the following exression R + = RL+ 1 D 0 y R L 0 and after substituting total deosits from (11) it is easy to derive that R + = (1 + ) L 0 + W L 0 Since in this case W = k S > 0; deosits are determined by state ledgeable income, this leaves an extra-rent W to the banker in state +: There is scoe to smooth income to deositors across the two states by selling a CDS (to buy rotection) which in exchange of a remium qw insures W in state fraction 1 k S (which amounts to insure only a of loan losses). In other words the otimal solution requires a transfer of resources from state + to state : 8 The otimality of state-contingent transfers in combination to initial debt for roviding incentives when information is revealed before the e ort is exerted is similar to Chiesa (1992) where the solution is debt cum warrants. Our model introduces the ossibility to inject new liquidity at an interim stage which comlicates the solution. 17

Assume instead that < k S ; the banker s reward in state following exression is given by the R = RL 1 D 0 L 0 and after substituting total deosits from (11) it easy to derive that R = L 0 W L 0 Since in this case W = k S < 0; deosits are bound by state + maximum ledgeable income, which leaves an extra-rent to the banker of ( W ) > 0 in state : There is scoe to smooth income to deositors across the two states by selling a CDS (to sell rotection) which in exchange of a remium qw romises to ay W in state +: In other words the otimal solution requires a transfer of resources from state to state +: The sign of the term W catures two contrasting e ects. maximum ledgeable income is smaller in state When W > 0 the due to loan losses. The otimal solution requires to redistribute funds from state + (the lucky state) to state unlucky state). (the To achieve this the banker could buy rotection through a CDS insuring for loan losses in the event of the negative shock; for each unit of remium qw > 0 the banker receives a refund of W > 0 in state. When W < 0 instead the maximum ledgeable income is smaller in state + due to funding of new loans by investors. To boost lending caacity in state + the solution requires to redistribute funds from state to state +: To achieve this the banker takes on more risk by selling rotection through a CDS: the banker receives the remium qw in both states and ays W < 0 when state all new lending oortunities. occurs. This maximizes the resources in state + to fund To understand the otimal caital ratio, we comute interim (i.e.date 1 2 ) caital ratios, denoted by k + and k : In state +, the bank is allowed to sell a fraction y of its initial loans, in order to nance a fraction of new loans. As already noted, the banker s incentive to monitor the loans that she sells are only maintained if she kees an equity osition E 1 = (y )L 0 = k S yl 0 in these loans. Moreover the (unconsolidated) balance sheet equation of the bank in state + is L 0 (1 + y) + 0 = E + + S + + D 0 ; 18

which gives after simli cation: E + = k S L 0 (1 + y) On total the bank is required to maintain total caital E + + E 1 = k S (1 + )L 0 ; that is a caital ratio k + = k S (1 + ) in state +: However when de ning the consolidated balance sheet, i.e. the balance sheet of the bank and that of a Secial Purose Vehicle (SPV) in which sold loans and CDS ayments are accounted togheter, the consolidated caital ratio is equal to the static caital ratio k S. Thus there is no change to the static caital ratio, rovided that the bank maintains a su cient equity stake in the loans that have been sold, and that the solvency ratio is also satis ed at the consolidated level. y contrast, in state the caital ratio has to be increased, due to the deterioration of ro tability. Indeed, the balance sheet equation of the bank in state is: L 0 + 0 = E + S + D 0 ; which gives after simli cation E = (k S + )L 0 imlying a caital ratio k = (k S + ) in state : This higher caital ratio revents the bank from increasing its lending in state, which would destroy the banker s incentive to monitors her loans. The otimal caital ratio at date 0 is the exected value of the two interim caital ratios k + and k ; that is k 0 = (1 q)k + + qk = (1 q)k S (1 + ) + q(k S + ) from which the exression of the modi ed caital ratio follows. We can now derive the following results on the e ect of changes in the arameters on the otimal caital ratio at date 0. Proosition 4 The otimal caital ratio at date 0 increases with and. The e ect of an increase in the robability of a shock q on the caital ratio is ositive (res. negative) when W > 0 (res., W < 0). 19

Proof. It is easy to derive from the otimal caital ratio in (8) the following results: @k 0 @ @k 0 @ @k 0 @q = q > 0; = (1 q)k S > 0; = W: Caital requirements must be tighter the larger the loan losses and the greater the rate of growth of new lending oortunities. Finally, the imact of a larger robability of a solvency shock on the otimal caital ratio deends on the relative strength of the two oosite motives catured in the sign of W. To conclude, the mix of CRT instruments together with caital regulation is exlained by the tension between incentives and insurance. The tension is resolved in two di erent ways according to the sign of W: When the solvency shock is dominant (W > 0) insurance hels to restore incentives, while when new lending oortunities dominate (W < 0), there is the usual trade-o between insurance and incentives. More seci cally, in the rst case in the otimal solution buying insurance restores banker s incentives, by reducing the banker s extra-rent in state +: insurance fosters incentives to monitor. In the second case instead, in the otimal solution the bank has to take on more risk by selling insurance on loan losses to transfer funds from state to state +, in order to reduce the banker s extra-rent in state : therefore incentives are restored by reducing insurance. 4 Alternatives to CRT The solution of the model has imlications for liquidity management and caital regulation. In articular in this section we discuss one ossible alternative to the use of CRT instruments which is to hoard liquidity at date 0 and use it at date 1=2 to fund new loans. We show that liquidity hoarding is dominated by the solution where banks access CRT markets. 20

4.1 Liquidity hoarding The intuition is that liquidity hoarding requires the bank to save a xed amount of liquidity before the realization of the shock at date 1=2. At this stage not all information is available. The ex-ante otimal level of liquidity to hold at time t = 0 is therefore di erent from the ex-ost otimal level of liquidity and this imairs banker s incentives. To mitigate this ex-ante incentive roblem, caital ratio adjusts to a higher level, reducing total lending in the rst stage. On the contrary access to CRT markets rovides state-contingent liquidity at t = 1=2, that is when uncertainty about the shock is resolved. Assume that the bank raises E 0 + D 0 and lends L 0 ; ays the remium to the DIF as before and hoards L e 0 as liquidity to be used at date t = 1=2. From date 0 bank s balance sheet, we have: L 0 + 0 + L e 0 = E 0 + D 0 : At date t = 1=2 when new lending oortunities L 0 arise, the banker can invest u to xl 0 of his hoarded liquidity L e 0 : Notice that this amount cannot be made conditional uon the realization of the shock, since there is no credible commitment not to emloy it at time t = 1=2: Regardless of the state of the economy the banker funds new loans u to L 0 in both states as is the otimal growth rate. Since there is a constant level of liquidity hoarded at date 0, that is x + = x ; and given that the exected surlus in (7) is increasing in this constant level of liquidity, the otimal rate of growth is and thus L + 1 = L 1 = (1 + )L 0. Given the (fair) DIF remium, date 0 balance s sheet becomes (1 + ) L 0 = E 0 + D 0 (12) The banker s exected return at t = 1 is thus R + = R(1 + )L 0 D 0 R = (R ) (1 + )L 0 D 0 (13) At date 1=2 for the banker to monitor the following incentive constraints must hold: R + > R L 1 21

from which R = L 1: This sets an uer limit to the amount of deosits the bank can raise, that is state ledgeable income D 0 = R (1 + ) L 0 : (14) Substituting (14) into the balance sheet in (12) we derive the caital adequacy requirement E 0 e k 0 L 0 where e k 0 = (1 + ) [k S + ]. It is easy to check that e k0 > k 0 that is the caital ratio is greater (tighter credit conditions) comared to the caital ratio when CRT is available, to comensate for the soft-budget constraint given by the liquidity hoarded at time 0. We can state the following result: Proosition 5 The solution with liquidity hoarding is sub-otimal comared to the solution with CRT markets. Proof. We can comare the exected surlus in the two cases. From exression (7) substituting the otimal caital ratio and the two otimal levels x + = ; x derive ES = E 0 k 0 f(r 1)(1 + ) qg = 0 we While comuting the exected surlus in the liquidity hoarding solution, we have ES LH = E 0 e k0 f(r 1) (1 + ) q (1 + )g Given that e k 0 > k 0 and that the term in brackets is smaller in the exression below we conclude that ES > ES LH : For a given level of caital the banker will lend less in this case, and therefore liquidity hoarding imlies a sub-otimal solution comared to the case where the bank has access to CRT markets. There are two reasons why this solution is dominated by the solution with CRT. The rst reason is that liquidity hoarding does not comly with the tough incentive scheme of x = 0: as a matter of fact the banker is equally rewarded in the two states, but this leaves her a greater rent reducing the maximum 22

ledgeable income to deositors. As a consequence, the scale of activity of the bank is smaller. The second reason is that liquidity is better rovided through statecontingent nancial contracts allowing to transfer funds across states once information about the shock is revealed. The solution with CRT allows to imlement a better management of the liquidity, by roviding funds in the state in which liquidity is worth more. 4.2 Other ossible alternatives There are other ossible alternatives other than liquidity hoarding to raise liquidity at date 1/2, such as collecting new deosits, resorting to inter-bank lending or issuing outside equity to relax caital requirements. We brie y discuss these alternatives in what follows. Deosits. In the model we have ruled out the ossibility for the bank to raise deosits at date 1/2. To increase the volume of deosits the bank has most likely to oen new branches, as it is documented in the emirical literature showing the imortance of the notion of "distance" in retail banking cometition. Furthermore the decision to oen a new branch is a long-term decision not easy to reverse. In contrast, nancial markets, and in articular CRT markets, rovide exibility for funding at the time when new investment oortunities arise and in the contingencies in which it is desirable. In our model nancial markets are better roviders of the interim liquidity needed only in state + to imlement the otimal solution. If on the contrary banks were able to raise new deosits by oening new branches, this liquidity would be available also in state. ut this solution would be equivalent to the liquidity hoarding case discussed in the revious subsection and we know it is sub-otimal. Inter-bank lending. Other banks could in rincile suly liquidity at date 1/2 through the inter-bank market at the same terms as rivate investors. However other banks should have extra-liquidity when the borrowing bank is short of liquidity. This requires banks to be hit by idiosyncratic shocks, while in the model the solvency shock is a common shock associated with state. Therefore all banks are simultaneously on the same side of the liquidity market and hence the inter-bank market would not be a feasible substitute for CRT. 23

Outside equity. To nance new loans banks could raise equity in nancial markets at date 1/2, relaxing caital requirements and enabling the bank to undertake new loans. However outside equity is costly as any accrued bene ts from monitoring has to be shared with outside shareholders imairing insider s incentives (as shown for instance in Cerasi and Daltung, 2000). While in our context outside equity discourages monitoring e ort, inside equity fully restore incentives as shown by Holmstrom and Tirole (1997). As a matter of fact in our model inside equity is the initial caital which we assume it cannot be increased further at a latter stage. 5 Emirical redictions The model has numerous redictions that can be discussed in the light of the emirical literature. First of all, one of the imlications of the liquidity hoarding case is that banks with access to CRT markets tend to hold less caital and increase their lending comared to other banks. This rediction nds suort in Cebenoyan and Strahan (2004) as they confront the di erent behavior of US banks active in the loan sale market and show that they hold less caital and lend more comared to other banks without access to CRT markets. Also Goderis et al.(2006) nd evidence on a samle of banks worldwide issuing collateralized loan obligations, which they use as a ublic signal of access to CRT markets, exand their lending by 50%, while Minton et al. (2006) rovide evidence of lower caital ratios for US banks who are net buyers of credit rotection. Second, one of the imlications of the model is that CRT instruments are comlement more than substitute, as they resond to di erent needs: while loan sales rovide state-contingent liquidity, CDS serve to insure against loan losses. As a matter of fact in the otimal solution banks use a combination of CRT instruments, loan sales and CDS. Cebenoyan and Strahan (2004) show evidence that banks using derivatives are more likely to sell loans, while Goderis et al. (2006) use a similar argument when using loan sales as a roxy for a more wide access to CRT markets. Also Minton et al. (2006) rovide evidence that banks selling loans tend to access credit derivative markets more likely than others banks. Third, the model redicts that banks with access to CRT markets might be on 24

either sides of the credit derivative markets, both as buyers and sellers of credit rotection, according to their need to insure loan losses or to ursue lending exansion. Several aers rovide gures in suort to the fact that banks are both rotection sellers and rotection buyers (see among others Minton et al., 2006, EC, 2004, Du e, 2007). 6 Conclusions In a model where bank monitoring is imortant but non-observable we have shown that the access to CRT markets imrove incentives, rovided that the caital ratio is adjusted accordingly. The model has imlications for solvency regulation, in articular for caital requirements, and for liquidity management. CRT markets serve as state-contingent roviders of liquidity at future dates when the bank is caital constrained. Loan sales suly interim liquidity, while credit derivatives rovide state-contingent transfers to balance between incentives and insurance for loan losses. We show that banks accessing CRT markets must hold an equity osition in sold loans in order to maintain monitoring incentives. Further, caital ratio should be adjusted in order to let the bank to exand in uturns and forego investment oortunities in down-turns. The model has imlications for the design of caital ratios. According to the literature on risk management in banks, asel II caital ratios are derived from VaR models where the threshold of bank solvency is set at an exogenous level. In our simle model of rudential regulation we have shown that the otimal caital requirement should be state-deendent as it must account for banker s incentives in the di erent states of the economy. Furthermore the model shows that caital ratios should not be designed with the unique objective to insure for loan losses in down-turns but also with the concern for under-investment in u-turns (see Kashya and Stein, 2004, who make a similar oint). In the model we do not discuss the lemon roblem associated with informational asymmetries between CRT sellers and buyers, in articular in the market of loan sales. In our model monitoring takes lace after CRT contracts are issued. This eliminates the roblem of asymmetries of information between banks and investors as the bank does not have suerior information at the time of selling a ortion of the loan or an 25

insurance osition on the loan. Furthermore, this assumtion eliminates any cost due to the coexistence of credit derivatives and loan sales, as exlored in Du ee and Zhou (2001) and Thomson (2006), where the introduction of credit derivatives may cause a break-down in the market of loan sales. However, our model is not t to exlore the imlications of loan sales or credit derivatives on loan quality as done for instance in Parlour and Winton (2007). In the model banks are homogenous with regard to their moral hazard costs and exosure to shocks. We leave for future research the task of exloring otimal solvency regulation in a setting where banks are heterogeneous with regard to the size of loan losses or rivate bene ts. For instance Rochet (2004) exlores otimal closure rules and caital regulation when banks are hit di erently by the same macroeconomic shock. 26

Aendix Proof of Proosition 2 The otimal contract between the DIF and the banker requires choosing the level of loans L 0 ; deosits D 0 and a rate of growth of loans in both states 0 x ; x +, that maximize exected social surlus (7) under the incentive comatibility constraints (5) and the break-even conditions (3) and (6). De ne A + (1 + x + )L 0 and A (1 + x )L 0. The exected surlus is increasing in both A + ; A ; therefore the two incentive comatibility constraints (5) are binding. The otimization roblem amounts to maximize ES = (1 q) [R 1] A + + q [(R ) 1] A given the constraint, once we substitute the break-even condition (3) and the two binding constraints (5), E 0 (1 q) 1 (R ) A + + q 1 (R ) A (15) De ne h the following i arameters h (1 q)(ri 1); q [(R ) 1] ; a (1 q) 1 (R ) ; b q 1 (R ) +. Given our assumtions, when also > 0; all four arameters are ositive. Thus, given that the exected surlus is increasing in both A + ; A the solution requires the constraint (15) to be saturated. One can derive from the constraint the exression for A + (res. A ); substitute it into the ES and comute the derivative w.r.t. A (res. A + ) des da a b = ; des a da = b + a : b It is easy to see that a b = q(1 q) < 0, therefore the solution imlies choosing A as smallest as ossible, while A + as greatest. Note that it would be true even in the case of < 0; namely with a negative net resent value of the monitored roject in state, as the sign of a b would still be negative. Since the otimal solution imlies choosing x = 0; x + = and saturating the other constraints, condition (6) becomes: D 0 (1 q) R (1 + ) + q R L 0 (1 q)l 0 and substituting it into (3), we obtain E 0 [1 + (1 q)] 1 R L 0 + ql 0 : 27