Designing a banking system to eliminate the potential for catastrophe

Transcription

1 Desigig a bakig system to elimiate the potetial for catastrophe Ketaro Asai Jauary 14, 2015 Abstract This paper ivestigates how bak risk-takig iteracts with deposit market structure, bak trasparecy, deposit isurace, ad the icetive structure of executive bakers. Ulike previous studies, our model edogeizes both the portfolio choice ad the default decisio of baks. As a result, the bakig sector ca attai multiple equilibria, ofte icludig oe that iduces high default risk or risk-shiftig. Although direct asset restrictios or market cocetratio ca elimiate the potetial for a ufavorable equilibrium, these policies harm social welfare. Public disclosure of baks risk ca deregulate these policies; however, the favorable impact of trasparet bakig is offset by deposit isurace. Istead, debt-type maagerial compesatio elimiates risk-shiftig. Moreover, it removes the potetial for high default risk i a ecoomic dowtur by simultaeously providig liquidity to baks durig crises. Surprisigly, this alterative scheme either sacrifices social welfare or the coverage of deposit isurace. Policy packages based o the calibrated model are also proposed. 1

2 1 Itroductio Excessive fiacial risk-takig has played a sigificat role i the global fiacial crisis (GFC). I the aftermath of the crisis, as part of the Dodd Frak Act (DFA), US regulators icreased disclosure ad trasparecy requiremets for the bakig busiess ad ehaced the compesatio oversight of the fiacial idustry. The Capital Requiremets Directive IV (CRD IV), the EU s ew legislative package, aims to establish liquidity requiremets ad revise the icetive structure of executive bakers. Simultaeously, to prevet bak rus, the US raised the threshold o deposit isurace ad evetually covered all oiterest-bearig trasactio accouts. Similarly, the EU raised the threshold o deposit isurace. Although the de facto atitrust exemptio for bakig has bee erodig i the US ad the EU for the last decades 1, both systems took measures agaist competitio durig the crisis: the US arraged mergers betwee Bear Stears ad JP Morga ad betwee Merrill Lych ad Bak of America, whereas the UK arraged a merger betwee Lloyds TBC ad HBOS. To avoid aother fiacial crisis, regulators are required to uderstad the forces that affect the risk-takig of baks ad examie what policies ca efficietly prevet the excessive risk-takig of baks. The purpose of this paper is to aalyze the risk-takig behavior of baks ad idetify a set of policies that improves the soudess of the bakig sector without harmig social welfare. Previous studies have ivestigated the relatioship betwee the soudess of the fiacial sector ad direct asset restrictios 2 [Nicoló et al., 2012, Goodhart et al., 2012], deposit market competitio [Matutes ad Vives, 2000, Alle ad Gale, 2004, Boyd ad Nicoló, 2005, Martiez-Miera ad Repullo, 2010], public disclosure of baks risk [Cordella ad Yeyati, 1998], deposit isurace [Diamod ad Dybvig, 1983], ad compesatio structure [Bolto et al., 2011, DeMarzo et al., 2014]. Accordig to these studies, direct asset restrictios are effective for the reductio of default risk while they sigificatly reduce social welfare. The correlatio betwee risk-takig ad deposit market competitio ca be both positive ad egative. Public disclosure of baks risk reduces default risk but this is offset by deposit isurace, which prevets bak rus. Debt-like compesatio for executives reduces risk for fiacial istitutios. However, few papers have edogeized both the portfolio choice ad the default decisio of baks. Because of the limited strategic optios available to baks, they might have missed the equilibria that potetially could emerge. As a result, they have fewer equilibria tha this paper does. Diamod ad Dybvig [1983] suggested that the soudess of the bakig sector chages whe switchig from oe equilibrium to aother much more tha by the perturbatio withi a sigle equilibrium. As show later, such regime-switchig ca raise the probability of default from egligible to catastrophic. Moreover, it ca chage 1 See OECD [2009]. 2 I practice asset restrictios relate to capital asset ratios, with risk-weighted measures of assets, ad limitatios o large exposures ad the cocetratio of risks [Matutes ad Vives, 2000]. 2

3 the level of risky asset exposure from miimum to maximum. Although the first priority of regulators is to elimiate the potetial for a ufavorable equilibrium to exist, few studies have aalyzed the iteractio of multiple policies i the cotext of multiple equilibria. As a exceptio, Ega et al. [2014] has ivestigated this issue, but these authors did ot edogeize the portfolio choice of baks. Therefore, the implicatio for potetial excessive risky asset exposure (risk-shiftig) still remais ukow. I additio, although previous literature has addressed the iteractio betwee deposit market competitio, bak trasparecy, ad deposit isurace as well as bak trasparecy, deposit isurace, ad compesatio regulatios, it has ot aalyzed the iteractio betwee deposit market competitio ad compesatio regulatios. To the best of our kowledge, this paper is a first attempt to edogeize both the portfolio choice ad the default decisio of baks, takig ito accout strategic iteractios amog stakeholders. This eables us to aalyze multiple equilibria, icludig the oe that iduces high default risk or risk-shiftig, ad propose a scheme that elimiates the potetial for a ufavorable equilibrium. Moreover, our framework allows regulators to simultaeously aalyze the policies from multiple domais, icludig atitrust policies ad compesatio regulatios, i a itegrated way. Our model suggests that direct asset restrictios or market cocetratio ca reduce the potetial for equilibria to exist i either high default risk or risk-shiftig. Nevertheless, these policies harm social welfare. Coversely, disclosig the risks that baks pose to depositors leads to relaxatio of the restrictios o the level of risky loas ad the etries to deposit market market, which is ecessary to prevet catastrophic cosequeces. However, the favorable impact of bak trasparecy is offset by deposit isurace. Istead, debttype maagerial compesatio ca elimiate risk-shiftig without harmig social welfare or shrikig the coverage of deposit isurace. This is because market competitio is complemetary to the implemetatio of debt-type maagerial compesatio. I additio, it ca elimiate the potetial for a high-risk equilibrium to exist i a ecoomic dowtur if regulators simultaeously provide sufficiet liquidity to baks durig crises. Accordig to our calibratio, a catastrophic equilibrium may emerge uder the curret bakig system without regulatios. We propose policy packages based o the calibrated model. The rest of this paper is structured as follows. Sectio 2 describes the bechmark model. Sectio 3 solves the equilibria of the bechmark model. Sectio 4 aalyzes the impact of trasparet bakig. Sectio 5 characterizes the optimal tax o the CDS spread of a bak. Sectio 6 provides quatitative implicatios from the calibrated model. Sectio 7 discusses policy recommedatios. 3

4 2 Model I this sectio, we describe our bechmark model. We assume the same fudig structure of baks, as that i Ega et al. [2014]. Baks are fiaced through deposits, which have to be repaid at the ed of each period. The residual claimats are deep pocket equity holders, as that i Lelad [1994]. If there is a shortfall, the the equity holders ca decide whether to iject eough fuds to repay the deposits or to default. I the case of a default, the equity holders are protected by limited liability. At bakruptcy, the bak is sold ad the proceeds are used to repay the depositors. 2.1 Players There are idetical baks ( 2) that compete with each other to collect fuds from the same pool of depositors. There is o outside optio that the depositors ca avail i the absece of baks. Each bak is ru by a risk-eutral maager, who is hired by shareholders uder a liear icetive cotract 3 that depeds o the bak s stock price. I other words, maagerial icetives are perfectly aliged with the iterests of the shareholders 4. All depositors cosider iterest rates whe they choose their baks. Some depositors are ot isured by the deposit isurace authority; thus, their preferece is also sesitive to the default risk of each bak. O the other had, the isured depositors do ot cosider baks default risk because ay shortfall is compesated by the deposit isurace authority. As isured households observe all the iformatio they eed, they are excluded from this game. Istead, the deposit isurace authority requires a bak to pay a actuarially fair premium; thus its strategy matters to the shareholders of the bak. Cosequetly, the players of this game are (1) baks (maagerial icetives aliged with the iterests of shareholders); (2) the deposit isurace authority; ad (3) the uisured depositors. Each bak simultaeously chooses its risk profile. The deposit isurace authority sets the premium for each bak without observig the riskiess of each bak. The uisured depositors choose their baks without observig the default risk of each bak. We deote the strategy of the maager at bak k by σ k ad the strategies of other players by σ k. 2.2 Timig The timig of the bechmark game is similar to that of Bolto et al. [2011]. 1. For each bak k, the icumbet equity holders hire a maager uder a icetive cotract: 3 For example, a maager receives a fixed salary at the begiig of the year ad a equity-liked bous at the ed of the year. Liear cotracts are commo i practice [Bose et al., 2011]. 4 Later, we will relax this assumptio ad cosider the situatio where maagerial icetives are ot perfectly aliged with the iterests of shareholders. 4

5 W (σ k, σ k ) = W 0 + δ E V (σ k, σ k ), where W 0 is a fixed wage, δ E is the shares of equity (δ E > 0), ad V (σ k, σ k ) is the equity value of the bak. The maager of the bak immediately receives W The deposit isurace authority sets the vector of actuarially fair premiums ξp (σ I ), where P (σ I ) is the vector of the default rates, σ I is the deposit isurace authority s belief i the strategies of all maagers, ad ξ is a fire-sale discout rate (0 < ξ 1). 3. Each bak simultaeously chooses σ k. The bak also determies deposit rates ad aouces them to the depositors. 4. The depositors are iformed about the deposit rates of baks, but they do ot observe the strategies of baks. Istead, they believe σ N for the strategies of baks ad choose their baks. They expect the vector of the baks default risk to be P (σ N ). Each bak acquires deposits from the same pool of isured ad uisured depositors, M I ad M N, respectively (M I + M N = 1, M I 0, ad M N 0). 5. V (σ k, σ k ) is determied by a efficiet stock market. Each bak pays a bous to the maager i accordace with the icetive cotract. 6. The retur o the bak assets, Rk, is realized. After payig the premium to the deposit isurace authority ad iterest to the depositors, the shareholders make a default decisio. I the case of a default, the bak assets ca be sold at a discout. As ξ is the fire-sale discout rate, the depositors recover (1 ξ) of their claims. However, the deposit isurace authority compesates the isured depositors for ay shortfall. 2.3 Baks strategies Each bak k determies (1) portfolio choice; (2) default decisio; (3) isured deposit rate; ad (4) uisured deposit rate. Because (3) ad (4) ca be uiquely determied by (1) ad (2), we ca evetually reduce the strategy space of the bak to that of (1) ad (2). For (1), we allow the maager to ivest i either risky loas or riskless bods. Let q k be the exposure to risky loas (q k > 0). The, the maager ivests 1 q k of the bak assets ito riskless bods. Therefore, we ca represet the retur o the bak assets as R k = (1 q k )µ 0 +q k R, where µ0 is the risk-free rate ad R is the retur o risky loas. We assume that the retur o risky loas follows a ormal distributio, R N[µ, ν], where ν > 0. Without loss of geerality, we ca set µ 0 = 0 by measurig each rate relative to the risk-free rate. Throughout the paper, we deote Φ(.), φ(.), λ(.) as the CDF, PDF, ad iverse Mills Ratio of stadard ormal distributio, respectively. Regardig (2), Hortaçsu et al. [2011] showed that shareholders pla the reservatio rate to make a default decisio. Let the reservatio rate be q k R k > 0, such that the bak cotiues to operate if q k R > qk R k, otherwise, it liquidates its assets. The, the probability of default is Φ( R k µ ν ). Let the ormalized reservatio rate be z k, where z k = R k µ ν. 5

6 We represet the default decisio of the bak by z k because it sufficietly represets the probability of default. Cosequetly, the bak optimizes both the overall risk q k (portfolio choice) ad tail risk z k (default decisio). The, its strategy is represeted by σ k where σ = (q k, z k ). We also deote the strategies of other players by σ k = {{σ k } k k, σ I, σ N }. 2.4 Depositors strategies We model the demad for deposits i a discrete choice framework. As the isured depositors are protected by deposit isurace, they oly care about deposit rates whe choosig a bak for opeig accouts. O the other had, the uisured depositors are ot protected by the deposit isurace; therefore, they also cosider the default risk of baks besides the deposit rates 5. Household j derives idirect utility from holdig isured ad uisured deposits at bak k, where ũ I j (i k ) = αi k + ɛ j,k ũ N j (i k, σ N ) = α(i k ξp k (σ N )) + ɛ j,k. Here i k ad P k (σ N ) represet the deposit rate ad the probability of default associated with bak k, respectively. The parameter α measures depositors effective deposit rate sesitivity, which is the total expected retur o a depositor s claim, takig ito accout the default risk of a bak. For example, if the deposit rate is 10%, the probability of a default is 5%, ad the fire-sale discout rate is 50%, the uisured depositor expects to gai 10 dollars ad lose 5 dollars. Therefore, the total expected retur o the uisured depositor s claim is 5 dollars. The, the effective deposit rate is 5%. Further, ɛ j,k is the cosumer s idiosycratic utility shock which follows a iid Type 1 Extreme Value distributio. Assumig that there are ifiitely may depositors, bak k acquires market shares i isured ad uisured deposit markets, where s I (i k, σ k ) = s N (i k, σ k ) = exp(αi k ) k =1 exp(αi k ) exp(αi k γφ(z N k )) k =1 exp(αi k γφ(zn k )). The uisured depositors become worse off if their belief regardig the strategy of each bak differs from the truth. They might choose the wrog bak, which would ot be best for them. Thus, the optimal strategy of the uisured depositors is to set their belief regardig the strategy of each bak idetical to the actual oe. 5 Ega et al. [2014] empirically rejected that the demad for isured deposits is sesitive to the baks default risks, while the uisured depositors care about them. 6

7 2.5 Deposit isurace authority s strategy As the deposit isurace authority attempts to achieve actuarially fair isurace, it sets the premium equal to the probability of default multiplied by the fire-sale discout ξ for each bak k as follows: ξp k (σ I ) = ξφ(z I k ). We assume that the deposit isurace authority is strictly worse off whe it sets the premium of each bak either strictly above or below the expected paymet to the isured depositors of the bak. Thus, the authority s optimal strategy is to set their belief regardig the strategy of each bak idetical to the actual oe, like the uisured depositors. 3 Equilibrium I this sectio, we solve the equilibrium of the bechmark game ad derive policy implicatios from our results. Our aalysis predicts the potetial for a high-risk or risk-shiftig equilibrium without regulatio, the eed for direct asset restrictios or market cocetratio to prevet the potetial for such equilibrium, ad the egative side effects of these policies o social welfare. 3.1 Symmetric local Nash equilibrium Our goal is to characterize the symmetric local Nash equilibrium. We deote the game described above by Γ. We assume symmetry to obtai the aalytical closed form expressios, which are particularly attractive to regulators because of their simplicity. A local Nash equilibrium is a weaker equilibrium cocept, but it is still robust to the local perturbatio of each player s strategy. Let the strategy space of the bak be Σ = Q R, where Q is the closed iterval [q, q], q > 0, ad q > q. The, the strategy space for all baks is Σ ad the strategy space of the deposit isurace authority is Σ, which is the space for all baks strategies. Similarly, the strategy space of the uisured depositors is Σ. Thus, the strategy space for all players i this game is Σ Σ Σ. Extedig the defiitio by Ratliff et al. [2013] for + 2 players, we defie the symmetric local Nash equilibrium as the followig. Defiitio 1. A strategy {{σ k } k=1, σn, σ I } Σ Σ Σ is a symmetric local Nash equilibrium of Γ if σ k = σ, k = 1,...,, ad either (i) there exist ope rectagles W Σ, k = 1,...,, such that σ W, V (σ, {{σ} k k, σ N, σ I }) V (σ, {{σ} k k, σ N, σ I }), σ W \ σ, σ N = σ I = σ, ad q (q, q), or (ii) there exist half-ope rectagles, i.e. the cartesia products of right (left) half-ope iterval ad ope iterval, W Σ, k = 1,...,, such that 7

8 σ W, V (σ, {{σ} k kσ N, σ I }) V (σ, {{σ} k kσ N, σ I }), σ W \ σ, ad σ N = σ I = σ, ad q = q( q). 3.2 Deposit rates First, we characterize the optimal deposit rates chose by the bak (i I (σ k, σ k ), i N (σ k, σ k ) ). At the optimum, the bak makes the expected markup equal to the iverse price elasticity of the residual demad as follows: q k [µ + νλ(z k )] ξφ(z I k ) ii (σ k, σ k ) = q k [µ + νλ(z k )] i N (σ k, σ k ) = 1 α[1 s I (σ k, σ k )] (1) 1 α[1 s N (σ k, σ k )], (2) where s I (σ k, σ k ) = s I (i I (σ k, σ k ), σ k ), s N (σ k, σ k ) = s N (i N (σ k, σ k ), σ k ). The LHS of (1) ad (2) are strictly decreasig i deposit rate; whereas the RHS of (1) ad (2) are strictly icreasig i it. Moreover, the RHS of (1) ad (2) approach positive ifiity as the deposit rate approaches positive ifiity, whereas they coverge to 1 α as the deposit rate approaches egative ifiity. Furthermore, the LHS of (1) ad (2) approach egative ifiity as the deposit rate approaches positive ifiity; whereas they approach positive ifiity as the deposit rate approaches egative ifiity. These facts suggest that, for a give σ k, optimal deposit rates uiquely exist. Lemma 1. For ay (σ k, σ k ), i I (σ k, σ k ) ad i N (σ k, σ k ) satisfyig (1) ad (2) uiquely exist. 3.3 Valuatio of equity The expected retur ad profit of the bak is characterized as the weighted average of the markups extracted from the isured ad uisured depositors as follows: π(σ k, σ k ) = θ I (σ k, σ k ) α[1 s I (σ k, σ k )] + 1 θi (σ k, σ k ) α[1 s N (σ k, σ k )]. Here θ I (σ k, σ k ) is the weight of isured deposits i the bak liability. I accordace with this, we also characterize the expected profit of the bak as follows: Π(σ k, σ k ) = M I s I (σ k, σ k ) α[1 s I (σ k, σ k )] + (1 M I )s N (σ k, σ k ) α[1 s N. (σ k, σ k )] Give the reservatio rate strategy, we ca characterize the equity value as the expected profit of the bak multiplied by the survival probability discouted by the risk-adjusted 8

9 rate, i.e. the sum of the ormal discout rate, r (0 < r 0.15) 6, ad the default risk of the bak: 3.4 Default decisio V (σ k, σ k ) = (1 Φ(z k))π(σ k, σ k ). r + Φ(z k ) I geeral, the margial value of takig tail risks is 7 as follows: V (σ k, σ k ) z k = { ( M I s I (σ k, σ k ) + (1 M I )s N (σ k, σ k ) ) r + Φ(z k ) ( ) } φ(z k ) 1+r r+φ(z) π(σ k, σ k ) q k ν(λ(z k ) z k ). (3) At equilibrium, this is simplified to the followig: 1 + r qν [λ(z) z] = 0. (4) r + Φ(z) α( 1) Here (4) implies that the goig cocer value of the bak has to be equal to the shortfall eeded for the bak to cotiue busiess whe the reservatio rate is realized. If the goig cocer value of the bak is lower (higher) tha the shortfall at the threshold, the shareholders are uwillig (willig) to iject capital eve if the realized retur is slightly above (below) the threshold. Therefore, at the threshold, the shareholders have to be idifferet betwee stoppig ad cotiuig bak busiess. Rewritig (4), we have α( 1)qν =r + Φ(z) 1 + r [λ(z) z]. (5) The LHS of (5) is the retur o the bak equity divided by the stadard deviatio of the portfolio retur, amely, the Sharpe ratio of the bak equity 8. Moreover, the RHS of (5) is the cost of capital. Note that the retur required by the shareholders is ot a fuctio of exposure to risky loas. I other words, we ca separate the retur required by the shareholders from the optimal portfolio choice. This eables a simple characterizatio of the equilibrium of the game. We deote the Sharpe ratio of the bak equity by g(q, ), which decreases i the exposure to risky loas ad the umber of baks i the deposit market because market 6 Our model requires r to be reasoably low to have meaigful implicatios. 7 Hortaçsu et al. [2011] showed that the optimal reservatio rate is the root of the last term i (3). This is exactly the first-order coditio for the optimal tail risk. 8 Note that every retur is relative to the risk free rate i our model. 9

10 competitio reduces the markup eared by the bak, whereas icreasig the exposure to risky loas raises the stadard deviatio of the portfolio retur. We also deote the RHS of (5) by h(z). As h(z) is ot affected by ay policy parameter, the Sharpe ratio of the bak equity determies the tail risk at equilibrium. As g(q, ) is ot affected by µ, the default risk of each bak is isesitive to µ as log as q is fixed. If a slight decrease i the reservatio rate satisfyig (5) makes the goig cocer value of the bak larger tha the required capital ijectio, the shareholders further decrease the reservatio rate; otherwise, they would irratioally liquidate the bak assets eve if it is higher tha the required capital ijectio. Moreover, if a slight icrease i the reservatio rate satisfyig (5) makes the goig cocer value smaller tha the required capital ijectio, shareholders further icrease the reservatio rate; otherwise, they would irratioally cotiue bak busiess eve if the value of cotiuig busiess is lower tha the shortfall. We fid that the locally stable reservatio rate requires the followig coditio: The, the sufficiet coditio for local stability is λ (z) h (z) 0. (6) 3.5 Fiacial stability λ (z) h (z) >0. (7) Whe the discout rate is modestly low 9, h(z) decreases i z for z < z 1, attais a local miimum at z = z 1, strictly icreases i z for z 1 < z < z 2, attais a local maximum at z = z 2, ad strictly decreases i z for z > z 2. We ca cofirm this from Figure 1. Moreover, we defie z = if{z g(q, ) = h(z)} ad z = sup{z g( q, ) = h(z)}. We categorize the equilibrium based o the performace of fiacial stability as follows. Defiitio 2. A equilibrium strategy σ is high-risk if z z 2, low-risk if z z 1, ad middle-risk if z 1 < z < z 2. We otice that h (z) 0 if z z 1 ad z z 2 while h (z) > 0 if z 1 < z < z 2. The, we claim the followig. Lemma 2. If z z 1 or z z 2, the z satisfies (7). For referece, we documet the critical values for moderately low r i the table give below. The default risk of each bak is at least more tha half for a high-risk equilibrium, whereas it is, at most, 0.04 for a low-risk equilibrium. 9 We verify that this holds for 0 < r

11 r Φ(z 1 ) Φ(z 2 ) Portfolio choice Next, we cosider the optimal portfolio choice. exposure to risky loas is I geeral, the margial value of the V (σ k, σ k ) = 1 Φ(z k) ( M I s I (σ k, σ k ) + (1 M I )s N (σ k, σ k ) ) (µ + νλ(z k )). (8) q k r + Φ(z k ) At equilibrium, this is simplified to 1 Φ(z) r + Φ(z) µ + νλ(z). (9) Note that the margial value of the overall risk is positive if µ > νλ(z), eutral if µ = νλ(z), ad egative if µ < νλ(z). Therefore, if the expected retur o risky loas is greater tha the risk-free rate, the margial value is always positive. Moreover, eve if the expected retur is smaller tha the risk-free rate, the margial value of the exposure to risky loas ca be positive because the bak maager beefits from the upside of the uprofitable gamble without icurrig its dowside because of limited liability. Lemma 3. Suppose σ is a equilibrium strategy. q = q if µ > νλ(z) ad q = q if µ < νλ(z). There is o locally stable q if µ = νλ(z). Proof. This is obvious from (9). While q is arbitrary if µ = νλ(z), it is ot robust to local perturbatio. If q [q, q), a slight icrease i z makes µ > νλ(z + ɛ), ad hece q jumps up to q. Moreover, if q = q, the a slight decrease i z makes µ < νλ(z ɛ), ad hece q jumps dow to q. 3.7 Credit cotrol We also categorize the equilibrium based o the performace of credit cotrol. We assume that a society is risk-eutral because its portfolio is well-diversified. It the wats a bak to udergo the largest exposure to risky loas, as log as the expected retur exceeds the risk-free rate, but the least exposure whe the expected retur is below the risk-free rate. Therefore, we evaluate the equilibrium based o baks credit cotrol as follows. Defiitio 3. A equilibrium strategy σ is uderivestig if q = q ad µ > 0 ad riskshiftig if q = q ad µ < 0. The strategy ca be termed as optimally credit-cotrollig if it is either uderivestig or risk-shiftig. 11

12 3.8 Results Propositio 1. There exists at least oe symmetric local Nash equilibrium uder Γ. If σ is a symmetric local Nash equilibrium strategy of Γ, σ satisfies (5), (6) ad either q = q µ > νλ(z) or q = q µ < νλ(z). Coversely, if σ satisfies (5), (7), ad either q = q µ > νλ(z) or q = q µ < νλ(z), σ is a symmetric local Nash equilibrium strategy of Γ. Proof. If µ νλ(z), the σ = ( q, z) is a equilibrium because z is o the low or high domai, which is locally stable accordig to Lemma 2, ad µ + νλ(z) > µ + νλ(z) 0. If µ < νλ(z), the (q, z) is a equilibrium because z is o the low or high domai, which is locally stable accordig to Lemma 2. The remaiig proof of Propositio 1 is straightforward from the defiitio of a symmetric local Nash equilibrium. Propositio 2. Suppose that q is the exposure to risky loas at equilibrium. If g(q, ) > h(z 2 ), the there exists a uique low-risk equilibrium. If g(q, ) = h(z 2 ), the there exist two equilibria, oe of which is high-risk ad the other low-risk. If h(z 1 ) < g(q, ) < h(z 2 ), the there exist at least two equilibria, oe of which is high-risk ad the other low-risk, ad there may exist a middle-risk equilibrium besides them. If g(q, ) = h(z 1 ), the there exist two equilibria, oe of which is high-risk ad the other low-risk. If g(q, ) < h(z 1 ), the there exists a uique high-risk equilibrium. Proof. Use Defiitio 2 ad Lemma 2. Propositio 3. A symmetric local Nash equilibrium is optimally credit-cotrollig if µ 0 or µ λ( z). It is risk-shiftig if λ(z) µ < 0. It ca be both risk-shiftig ad optimally credit-cotrollig if λ( z) < µ < λ(z). There is o uderivestig equilibrium. Proof. Let z satisfy (6). If µ 0, the µ + λ(z) 0, z so q = q by Lemma 3. From Defiitio 3, it is ot risk-shiftig. If µ λ( z), the µ + λ(z) 0 for all z such that g(q, ) = h(z), where q [q, q]. Therefore, q = q by Lemma 3. From Defiitio 3, it is also ot risk-shiftig. If λ(z) µ < 0, the µ + λ(z) 0, z so q q. Sice µ + λ(z) > µ + λ(z) 0 for all z such that g( q, ) = h(z), q = q. From Defiitio 3, it is risk-shiftig. For the remaiig case, q = q if z = z ad q = q if z = z. As either z or z is o the middle domai, they are both locally stable. Therefore, the bakig sector attais multiple equilibria, oe of which is optimally credit-cotrollig ad the other is risk-shiftig. There is o uderivestig equilibrium because the icetive cotract elimiates the potetial for a uderivestig equilibrium. Our results, however, suggest that there are 12

13 ofte multiple equilibria oe of which ivolves a high default risk or risk-shiftig. Figure 2 ad Table 1 illustrate the cases i which multiple equilibria ca arise. O the oe had, the depositors may expect a low default risk ad allow baks to offer low deposit rates, which icreases the goig cocer value of baks. This lowers reservatio rates. Moreover, this discourages risk-shiftig because the gai from risk-shiftig is proportioal to the payoff of equity holders that is coditioal o bak survival, which is amplified by the reservatio rates of baks. O the other had, the depositors may expect a high default risk ad may require baks to offer high deposit rates, which decreases the cotiuatio value of baks. The, the opposite feedback occurs. Cosequetly, the bakig sector ca take excessive risk without regulatio. 3.9 Implicatio for direct asset restrictios ad atitrust policy Next, we attempt to fid the domai of policy parameters that iduces bakig robustess to the wide rage of mea returs o risky loas. Propositio 4. Let the exposure to risky loas at equilibrium be q. If g(q, ) > h(z 2 ), the there exists a uique low-risk equilibrium. If g(q, ) h(z 1 ), the there exists a low-risk equilibrium. Moreover, g(q,) q < 0 ad g(q,) < 0. Proof. Use Propositio 2. The last statemet is obvious from the defiitio of g(q, ). Propositio 5. The probability of certaily attaiig a risk-shiftig equilibrium is icreasig i ad q. The probability of certaily attaiig a equilibrium that attais optimal credit cotrol is decreasig i ad q. Proof. From Propositio 3, the probability of attaiig oly risk-shiftig equilibria is P [µ 0] + P [µ λ( z)]. I additio, the probability of attaiig oly credit-ehacig equilibria is P [ λ(z) µ < 0]. z z z The our proof completes by: > 0, > 0, q > 0, ad z q > 0. Our fidigs suggest that direct asset restrictios or market cocetratio ca elimiate the potetial for a high-risk equilibrium by raisig the Sharpe ratio of the bak equity. Moreover, these policies ca improve credit cotrol i the fiacial sector by expadig the domai of the mea returs o risky loas i which risk-shiftig ever occurs Welfare criteria for fiacial regulatio Although direct asset restrictios or market cocetratio ca improve the soudess of the bakig sector, these policies may harm the welfare of the depositors. By cotrollig 13

14 the default risk, the depositors are better off with a higher exposure to risky loas because they ca receive a higher iterest. Moreover, market competitio icreases the surplus of the depositors by providig the depositors with more optios for baks while decreasig the market power of baks. For verifyig this claim, we characterize the ex-ate surplus of the depositors via a cotiget valuatio estimate of multiomial logit model [Petri, 2002]. The value of the depositors by percetage is D({σ k } k=1 ) = M I l { k=1 exp(αii k )} + (1 M I ) l { k=1 exp(αin k γφ(z k) } + κ, α (10) where κ is Euler costat (0.5772). At a symmetric equilibrium, with deposit rates i I (σ) ad i N (σ), D(σ) = l() + α(m I i I (σ) + (1 M I )i N (σ) (1 M I )γφ(z) + κ. α At symmetric equilibrium, (1) ad (2) suggest: i I (σ) = i N (σ) ξφ(z) i N (σ) = q(µ + νλ(z)) α( 1). The, we obtai the value of the depositors at a symmetric equilibrium as follows: D(σ) = q(µ + νλ(z)) ξφ(z) + l() + κ α α( 1). (11) The first term represets the gai from credit supply. We fid that the shareholders extra gai due to limited liability is partly trasferred to the depositors because market competitio forces the shareholders to give it to the depositors for collectig more fuds. The secod term represets the expected loss associated with a default. The last two terms are the gai from market competitio. Propositio 6. Suppose σ is a equilibrium strategy. The welfare of the depositors is strictly icreasig i. Moreover, it is icreasig i q ad decreasig i q. Proof. If µ + νλ(z) > 0, the q = q. As the welfare of the depositors is strictly icreasig i q if µ + νλ(z) > 0, it is icreasig i q. If µ + νλ(z) = 0, the there is o equilibrium. 14

15 If µ + νλ(z) < 0, the q = q. As the welfare of the depositors is strictly decreasig i q if µ + νλ(z) < 0, it is decreasig i q. Fially, it is easy to check that the last two terms are strictly icreasig i. As the welfare of the depositors is proportioal to the gai from limited liability (νλ(z)), the depositors partly extract the gai of risk-shiftig from the surplus of equity holders. This suggests the welfare of the depositors should be distict from social welfare whe risk-shiftig takes place. Whe µ > 0, both the depositors ad the society are worse off with a lower q. Whe µ < 0, the depositors are worse off with a lower q, whereas the society is better off with this. O the other had, both the depositors ad society are better off with a lower q. From this isight, we establish the welfare criteria for fiacial regulatio as follows. Defiitio 4. The proposed policy harms social welfare if it either decreases, decreases q whe µ > 0, or icreases q. Thus, both direct asset restrictios ad market cocetratio harm social welfare. I other words, there is a trade off betwee a soud bakig system ad social welfare. For the ext two sectios, we seek to make the bakig sector attai both fiacial stability ad optimal credit cotrol without sacrificig social welfare. 4 Public Disclosure of Baks Risk Previous studies, such as Cordella ad Yeyati [1998], suggest that public disclosure of baks risk improves the soudess of the bakig sector. Followig these studies, we revisit the above game uder the coditio of a greater trasparecy of baks risk. We fid that this coditio mitigates restrictios o bak assets ad etries to the deposit market that regulators require to prevet a high default risk or risk-shiftig. This is because baks iteralize the egative impact of the tail risk o the amout they ca collect from the uisured depositors if they kow that the depositors kow their risks whe choosig their baks. For implemetig this policy, we predict the eed for govermetal itervetio if equity holders caot commit to ijectig capital beyod the goig cocer value of baks. Although public disclosure of baks risk uambiguously improves the soudess of the bakig sector, its favorable effect decreases i the fractio of the isured depositors. This is because the isured depositors oly cosider deposit rates ad are isesitive to the default risk of baks. This suggests that a icrease i the coverage of deposit isurace ruis the favorable effect of trasparet bakig. 4.1 Modified game structure Formally, step 4 of the previous game is altered to the followig. 15

16 The depositors observe both deposit rates ad strategies of all the baks ad choose their baks. Each bak k acquires the deposits from the same pool of isured ad uisured depositors. The, the uisured depositors are excluded from this game because their belief regardig the default risk of each bak is iteralized by the bak maager. We deote the modified game by Γ 1. Defiitio 5. A strategy {{σ k } k=1, σi } Σ Σ is a symmetric local Nash equilibrium of Γ 1 if σ k = σ, k = 1,...,, ad either (i) there exist ope rectagles W Σ, k = 1,...,, such that σ W, V (σ, {{σ} k k, {σ, {σ} k k}, σ I }) V (σ, {{σ} k k, {σ, {σ} k k), σ I }), σ W \ σ, σ I = σ, ad q (q, q), or (ii) there exist half-ope rectagles, i.e. the cartesia products of right (left) half-ope iterval ad ope iterval, W Σ, k = 1,...,, such that σ W, V (σ, {{σ} k k, {σ, {σ} k k}, σ I }) V (σ, {{σ} k k, {σ, {σ} k k}, σ I }), σ W \ σ, ad σ I = σ, ad q = q( q). 4.2 Default decisio Uder the modified game, the margial value of the tail risk is chaged to V (σ k, σ k ) z k = { ( M I s I (σ k, σ k ) + (1 M I )s N (σ k, σ k ) ) r + Φ(z k ) φ(z ( k )(1 Φ(z k ))(1 θ I (σ k, σ k ))ξ ) +φ(z k ) 1+r r+φ(z) π(σ k, σ k ) q k ν(λ(z k ) z k ) }. (12) At symmetric equilibrium, the first-order coditio is modified to 1 + r r + Φ(z) α( 1) qν [λ(z) z] = (1 Φ(z))(1 M I )ξ < 0. (13) Rewritig the equatio that determies the reservatio rate yields α( 1)qν + (r + Φ(z))(1 Φ(z))(1 M I )ξ (1 + r)qν = r + Φ(z) 1 + r [λ(z) z]. (14) Compared with the origial equatio, the LHS uambiguously shifts up. We defie the ew term of the LHS as the cost of losig the uisured deposits c N (q, z, M I ). c N (q, z, M I ) strictly icreases i z for z 1 r 2 ad strictly decreases i z for z 1 r 2. Moreover, c N (q, z, M I ) is strictly decreasig i M I. Furthermore, lim z c N (q, z, M I ) = 0 ad 16

17 lim z c N (q, z, M I ) = (1 M I )ξ qν. Therefore, there exists at least oe root satisfyig (14) for ay q. Ulike the previous game, (13) implies that the goig cocer value of the bak has to be less tha the shortfall eeded for the bak to cotiue busiess whe the reservatio rate is realized. This implies that equity holders have a ex-ate icetive to lower the reservatio rate eve if it is uprofitable ex-post, because they ca collect more moey from the uisured by promisig that they would iject capital beyod the cotiuatio value. Accordigly, coditios required for local stability are modified to λ (z) h (z) φ(z) ξ(1 M I ] ) [αξm I ( 1)2 (r + Φ(z))(1 Φ(z)) (1 + r)qν 2 (1 r 2Φ(z)) (15) λ (z) h (z) > φ(z) ξ(1 M I ] ) [αξm I ( 1)2 (r + Φ(z))(1 Φ(z)) (1 + r)qν 2 (1 r 2Φ(z)). (16) The the stability coditio becomes more restrictive tha before. Eve if z is ot i the middle domai, it may ot satisfy (15). Lemma 4. If M I is sufficietly large, the z o the low ad high domais satisfies (16). If M I is sufficietly small, the z o the low domai satisfies (16). Proof. It is obvious from (16). For the secod statemet, the RHS of (16) becomes egative with sufficietly small M I because 1 r 2Φ(z) > 0, z z 1 for modestly low r. 4.3 Results The choice of the overall risk is determied i the same way as before. We ca characterize the equilibrium of the modified game as follows. Propositio 7. If σ is a symmetric local Nash equilibrium strategy of Γ 1, σ satisfies (14), (15), ad either q = q µ > νλ(z) or q = q µ < νλ(z). Coversely, if σ satisfies (14), (16), ad either q = q µ > νλ(z) or q = q µ < νλ(z), σ is a symmetric local Nash equilibrium strategy of Γ 1. Proof. Similar to the proof of Propositio 1. Propositio 8. Suppose M I is sufficietly small or large. If there exists a uique lowrisk equilibrium uder Γ, the there exists a uique low-risk equilibrium uder Γ 1 as well. 17

18 However, the coverse is ot true. Moreover, if there exists a uique low-risk equilibrium uder Γ 1, the there remais a uique low-risk equilibrium uder Γ 1 whe M I decreases. However, the coverse is ot true. Proof. Whe M I is sufficietly small or large, Lemma 4 suggests that the stability coditio o the low domai of z is automatically satisfied. Let the exposure to risky loas at equilibrium be q. If g(q, ) + c N (q, z, M I ) > h(z 2 ), z z 2, the there exists a uique low-risk equilibrium. If g(q, ) > h(z 2 ), the g(q, ) + c N (q, z, M I ) > h(z 2 ), z z 2. This proves the first part. Moreover, cn (q,z,m I ) < 0. This proves the secod part. For the coverse, you ca see M I our calibratio results described later. There you fid that public disclosure of baks risk makes the bakig sector attai a uique low-risk equilibrium, whereas the bakig sector ca attai a high-risk equilibrium without disclosure. Moreover, you fid the case i which the larger fractio of the isured depositors mitigates the favorable effect of trasparet bakig that elimiates the potetial for a high-risk equilibrium. Propositio 9. Suppose M I is sufficietly small ad g( q, ) h(z 1 ) or M I is sufficietly large. The probability of certaily attaiig a risk-shiftig equilibrium is greater uder Γ tha that uder Γ 1. O the other had, the probability of certaily attaiig a equilibrium that achieves optimal credit cotrol is greater uder Γ 1 tha that uder Γ. Moreover, uder Γ 1, the probability of certaily attaiig a risk-shiftig equilibrium is icreasig i M I, while that of certaily attaiig a equilibrium that achieves optimal credit cotrol is decreasig i M I. Proof. Whe M I is sufficietly large, there exists a either low-risk or high-risk equilibrium uder Γ 1, regardless of q. Whe M I is sufficietly small ad g( q, ) h(z 1 ), there exists at least a low-risk equilibrium uder Γ 1, regardless of q. Defie z 1 = if{z g(q, )+c N (q, z, M I ) = h(z)} ad z 1 = sup{z g( q, )+c N ( q, z, M I ) = h(z)}. It is easy to show z 1 < z 1. If µ 0 µ νλ( z 1 ), the the system certaily attais equilibria with optimal credit cotrol. If νλ(z 1 ) µ < 0, the the system certaily attais risk-shiftig equilibria. c N (q, z, M I ) > 0 implies z > z 1 ad z > z 1. This proves the first statemet of our claim. c N (q,z,m I ) < 0 implies z M I 1 > 0 ad z 1 M I our claim. M I > 0. This proves the secod statemet of Our results suggest that there is a chace of strictly improvig the soudess of fiacial sector by disclosig baks risk to households. Coditioal o the existece of a equilibrium, the public disclosure of baks risk decreases the default risk ad iduces 18

19 the system to attai optimal credit cotrol. Cosequetly, regulators ca relax direct asset restrictios ad ehace market competitio without sacrificig the soudess of the bakig sector. However, the favorable effect of trasparet bakig decreases the fractio of the isured depositors because they do ot have a icetive to moitor the default risk of each bak. Figure 3 illustrates how the trasparet bakig ca improve the soudess of the bakig sector ad how its favorable effect is offset by a greater coverage of deposit isurace. 4.4 Implicatio for the role of govermet durig fiacial crises Public disclosure of baks risk ca improve the soudess of bakig sector, because it remids baks of the moitorig ad threateig roles of the depositors, which is essetial for bakig prudece, accordig to Diamod ad Raja [2001]. However, it is ot clear if shareholders ca commit to ijectig capital beyod the promised amout. Oce the fuds are collected from the depositors, they may ot have a icetive to raise more capital tha the goig cocer value of the bak. I this case, govermets may eed to iject capital o behalf of the icumbet shareholders. If the depositors kow that govermets iject capital o behalf of the shareholders for the amout that caot be committed, the they ca believe that the default risk is lower tha the ex-post optimal rate for the icumbet shareholders. Ulike too-importat-to-fail (TITF) subsidies, the govermet bailout is urelated to the bak size. Therefore, shareholders are uwillig to take excessive exposure to risky loas, raise deposit rates, ad icrease market share. I this sese, the govermet itervetio does ot cause a moral hazard. O the other had, deposit isurace is ot compatible with trasparet bakig. Govermets ofte icrease the coverage for deposit isurace to prevet bak rus i a ecoomic dowtur; however, this ruis the favorable effect of disclosig baks risk. Therefore, if regulators try to achieve fiacial stability by the public disclosure of baks risk, they should limit the coverage of deposit isurace to the miimum level required to prevet bak rus. Thus, if govermets require baks to disclose their strategies to households, they should help iject capital to the level maximizig the ex-ate value of equity while keepig the coverage of deposit isurace to the miimum level required to prevet bak rus. 5 Debt-type Maagerial Compesatio I the previous sectios, maagerial iterests are perfectly aliged with the icetives of shareholders. I this sectio, we lik maagerial compesatio to the tail risk of a bak. This iduces a coflict of iterest betwee a maager ad the shareholders. We show that the agecy problem betwee them rather helps a bak achieve optimal credit cotrol without sacrificig social welfare; moreover, our proposal is uaffected by the fractio of 19

20 the isured depositors. Furthermore, the prevetio of risk-shiftig ca also elimiate the potetial for a high-risk equilibrium whe µ < 0, if regulators cap the miimum exposure to risky loas. While regulators are reluctat to restrict the maximum exposure to risky loas, they are justified to restrict the miimum exposure to risky loas. 5.1 Modified game structure Eve if the maager is able to choose the exposure to risky loas idepedetly, it is difficult for them to madate the shareholders to iject the specified amout of capital. Therefore, we assume that the maager oly determies the exposure to risky loas while the shareholders determie the tail risk. We chage steps 1, 3, ad 5 of Γ as follows. At each bak k, the icumbet equity holders hire a maager uder a icetive cotract as they do i Γ. However, the govermet imposes taxes o the compesatio of the maager, which are liked to the credit default swap (CDS) spread of the bak. Therefore, the maager s compesatio is: W (q k, z k, σ k ) = W 0 + δ E V (q k, z k, σ k ) τ(c(z k )), where W 0 is a fixed wage, δ E is the shares of equity (δ E > 0), ad τ(c(z k )) is the tax associated with the CDS spread of the bak. We also deote it by f(z k ) = τ(c(z k )). The maager of the bak immediately receives W 0. The maager of each bak simultaeously chooses q k ad the shareholders of each bak determies z k after observig q k. After observig both, the maager determies deposit rates ad aouces them to households. V (q k, z k, σ k ) ad C(z k ) are determied by efficiet markets. Each bak pays bouses to the maager followig the icetive cotract ad the govermet collects taxes τ(c(z k )) from the maager. We deote the modified game by Γ 2. Defiitio 6. A strategy ({q k } k=1, {z k} k=1, σn, σ I ) Q R Σ Σ is a symmetric local Nash equilibrium of Γ 2 if q k = q, z k = z, k = 1,...,, ad either (i) there exist ope itervals M Q, k = 1,...,, such that q M, W (q, z, {σ} k k, σ N, σ I ) W (q, z, {σ} k k, σ N, σ I ), q M \ q, σ N = σ I = σ, ad q (q, q), or (ii) there exist right (left) half-ope itervals, M Q, k = 1,...,, such that q M, V (q, z, {σ} k k, σ N, σ I ) V (q, z, {σ} k k, σ N, σ I ), q M \ q, σ N = σ I = σ, ad q = q( q), ad there exist ope itervals E R, k = 1,...,, such that z E, V (q, z, {σ} k k, σ N, σ I ) V (q, z, {σ} k k, σ N, σ I ), z E \ z, σ N = σ I = σ. 20

21 5.2 Portfolio choice The optimality coditio for the tail risk is the same as the oe i Γ, but the margial value of the overall risk is modified to V (σ k, σ k ) f (z k ) dz k q k δ E dq k = 1 Φ(z k) r + Φ(z k ) ( M I s I (σ k, σ k ) + (1 M I )s N (σ k, σ k ) ) (µ + νλ(z k )) f (z k ) δ E dz k dq k. (17) Ulike the previous games, the maager takes ito accout the margial cost of icreasig the default risk determied by the shareholders because of the additioal exposure to risky loas. Eve if the maager is ot able to choose the tail risk, he ca affect it by adjustig the exposure to risky loas. At symmetric equilibrium, the margial value of the overall risk is as follows: 1 Φ(z) r + Φ(z) µ + νλ(z) 5.3 Tax o the tail risk of a bak f (z) δ E α( 1) q[µ+νλ(z)] ( q 2 ν λ (z) h (z) ). (18) Our goal is to elimiate risk-shiftig without sacrificig social welfare. If (18) is egative for all q if µ < 0 ad positive for all q if µ > 0, the q becomes a equilibrium exposure to risky loas wheever µ > 0 while q becomes a equilibrium exposure to risky loas wheever µ < 0. If f (z) 0, we ca iduce the equilibrium by settig f (z) as the followig: f (z) =δ E g 1 (z, ) 2 ν ( λ (z) ) h (z) α( 1) g 1 (z,)νλ(z) 1 Φ(z) νλ(z) r + Φ(z). (19) Here g 1 (z, ) satisfies g(g 1 (z, ), ) = h(z), which is well-defied ad cotiuous. The, we fid that the sufficiet coditio for f (z) 0 is summarized by the followig Lemma. Lemma 5. If α( 1) g 1 ( z,)νλ( z) If is sufficietly large, the > 0, the f (z) 0, where z is a equilibrium tail risk. α( 1) g 1 ( z,)νλ( z) > 0. Proof. As λ (z) h (z) 0 at equilibrium, α( 1) g 1 (z,)νλ(z) > 0 is sufficiet to achieve f (z) 0, where z is a equilibrium tail risk. As z z, α( 1) g 1 ( z,)νλ( z) > 0 α( 1) g 1 (z,)νλ(z) > 0. 21

22 The secod statemet is cofirmed by lim α( 1) g 1 ( z,)νλ( z) = 1 α > 0. Note that market competitio is a complemet to the avoidace of risk-shiftig, whereas it was a substitute to the prevetio of risk-shiftig i our previous results. I fact, the sufficiet coditio for the previous Lemma is satisfied if is sufficietly large. O oe had, a icrease i the exposure to risky loas decreases the Sharpe ratio of the bak equity by icreasig the portfolio retur volatility. O the other had, it icreases the cotiuatio value of the bak as it eables the bak to collect more fuds from the depositors by raisig default rates whe µ + λ(z) > 0. I competitive markets, the former effect domiates the latter effect because the icrease i market share associated with the icrease i deposit rates is smaller. Suppose that α( 1) g 1 ( z,)νλ( z) > 0. Settig the boudary coditio f(z) = 0, the optimal debt-based tax is characterized by the followig: ˆ z f(z) = 1[f (x) 0]f (x)dx. (20) z We ca assig ay value to the margial icrease i tax with respect to tail risk for z that caot be at equilibrium; therefore, we set 0 for z that satisfies λ (z) h (z) < 0 ad hece f (z) < 0. Our goal is to costruct the tax as a fuctio of the CDS spread. A o arbitrary coditio suggests: C(z k ) = ξφ(z k ). The, z ca be expressed as the mootoic trasformatio of the CDS spread ad f(z) ca be expressed as a fuctio of the CDS spread. Fially, we characterize the optimal compesatio as where f(.) satisfies (20). W (q k, z k, σ k ) =W 0 + δ E V (q k, z k, σ k ) f α( 1) g 1 ( z,)νλ( z) ( ( )) Φ 1 C(zk ), (21) ξ Propositio 10. Suppose that satisfies > 0. Every equilibrium is( optimally ( credit-cotrollig )) uder Γ 2 if the govermet assigs the tax: τ(c(z k )) = f Φ 1 C(zk ) ξ, where f(.) satisfies (20). Propositio 10 suggests that regulators ca elimiate the potetial for risk-shiftig without direct asset restrictios or market cocetratio. Moreover, this approach is idepedet of depositor compositio; therefore, it is eutral to the coverage of deposit isurace. Furthermore, this compesatio structure ca be implemeted by the tax liked to the CDS spread of a bak. 22

23 I additio, ote that the prevetio of risk-shiftig ca elimiate the potetial for a high-risk equilibrium i a ecoomic dowtur. If q is small eough to satisfy g(q, ) > h(z 2 ), the we ca elimiate the potetial for a high-risk equilibrium because the maager volutarily reduces exposure to risky loas to the miimum level if µ < 0. Propositio 11. Suppose that satisfies α( 1) g 1 ( z,)νλ( z) > 0 ad q satisfies g(q, ) > h(z 2 ). The bakig sector attais a uique low-risk equilibrium whe µ < 0 uder Γ 2 if ( the govermet assigs the tax τ(c(z k )) = f Φ 1 ( C(zk ) ξ )), where f(.) satisfies (20). Ehaced market competitio lowers the Sharpe ratio of the bak equity. Therefore, regulators may eed to lower q to maitai the Sharpe ratio of the bak equity. I the previous fidigs, restrictios o the maximum exposure to risky loas are ot fully recommeded because they harm credit ehacemet i the society. However, restrictios o the miimum exposure to risky loas do ot harm social welfare. This is because the society is willig to lower exposure to the risky loas that are uprofitable. Although the depositors may be worse off by a lower q, the decrease i their surplus is associated with a reductio o the gai from risk-shiftig that the depositors extract from the surplus of shareholders. Cosequetly, regulators are justified to icrease the Sharpe ratio of the bak equity by providig liquidity to baks ad lowerig q i a ecoomic dowtur, eve though they may reduce the welfare of the depositors. 6 Calibratio I this sectio, we calibrate the model ad desig the policy packages that improve the soudess of the US commercial bakig sector. 6.1 Data We obtai the demad parameters from Ega et al. [2014]. As these authors reported the demad estimates separately for each type of depositors, we use the middle of them for our calibratio. From the demad estimates for the uisured depositors, we ca recover the fire-sale discout rate by dividig the sesitivity to deposit rates with the sesitivity to the default risk. We fid that the correspodig recovery rate is 50%, which is i lie with previous studies, such as Carrizosa ad Rya [2013]. We use the data from the Federal Reserve H8 to obtai the exposure to risky loas. We compute cash as well as Treasury ad agecy securities as a proportio of total assets ad subtract it from 1. As Ega et al. [2014] focused o large US baks, we use the data for large domestically chartered commercial baks. As regulators are iterested i whether the curret policy is robust eough, we estimate the maximum exposure to the risky loas by usig the data as of November 12, Note that the expected retur o risky loas is likely to be above the risk-free rate i The, baks are likely to ivest i risky loas 23