Policy Uncertainty and Bank Bailouts Frank N. Caliendo Utah State University [email protected]

Jason M. Smithy

Nick L. Guo

U Wisconsin, Whitewater Utah State University [email protected]

[email protected]

July 25, 2017

Abstract We model the e¤ect of government bailouts on portfolio choices and welfare. Banks sell bonds to leverage investment in risky projects and households buy bonds under rational expectations about default risk. Bailouts induce greater leverage but reduce equilibrium interest rates. The interest rate e¤ect dominates the leverage e¤ect and bailouts lead to fewer bank failures. Bailouts are e¢ cient but not Pareto optimal: bailouts increase social welfare by mitigating uninsurable risk, which helps banks but hurts households since the insurance gains are not worth the price households must pay to …nance the bailout. Capital requirements are neither e¢ cient nor Pareto optimal. Key words: Portfolio Choice, Welfare, Bailouts, Policy Uncertainty.

We would especially like to thank an anonymous referee for providing suggestions that have greatly improved the paper. We also thank Ben Blau, Aspen Gorry, Bill Shughart and seminar participants at Utah State University and the BYU-USU macro workshop for helpful comments. y Corresponding author: Department of Economics and Finance, Utah State University, Logan, UT 84322-3565.

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1. Introduction The widespread bailout of banks in the US during the Great Recession has caused concern about the moral hazard problem. Right from the start, the bailout architects themselves worried about moral hazard. In a CNN interview on March 16, 2008, Treasury Secretary Henry Paulson said: “I’m as aware as anyone is of moral hazard.” And on May 29, 2008, Chairman Bernanke echoed Paulson’s remarks in a speech at the Federal Reserve Bank of Atlanta: “A central bank that is too quick to act as liquidity provider of last resort risks inducing moral hazard.” In this paper we construct a theoretical model with a …nancial market in order to study the e¤ect of government bailouts on the portfolio choices and welfare of banks and households. Our …nancial market has two participants, a bank and a household. The bank has access to a risky investment project and invests its own capital, together with the proceeds from the sale of bonds to the household, in the risky project. The household also has some initial wealth to invest and likewise has access to the same risky investment project. The household allocates its wealth between direct investment in the risky project (stock) and loans to the bank (bonds). The bank and household meet in a …nancial market where bonds are exchanged. Bank failure occurs when the return on the risky project leaves the bank insolvent. The bank and household are risk averse, each making portfolio decisions to maximize mean-variance utility over random pro…ts. Total surplus— also a random variable— is the sum of the pro…ts of the bank and the pro…ts of the household, and ex ante social welfare is mean-variance utility over total surplus. We de…ne an e¢ cient policy as one that maximizes ex ante social welfare. We compare two equilibria. In the …rst equilibrium (Laissez Faire), there is no government to bail out bank losses from unlucky returns in the risky project. The equilibrium interest rate on bank debt is a Nash equilibrium in the sense that both sides of the …nancial market buy and sell the optimal quantity of bank debt given the risks that they face, and default risk on bank debt is endogenously priced into the interest rate on the debt contract. In the second equilibrium, the government runs a self-…nanced bailout program in which it provides the bank with just enough funds to meet its debt obligation to the household in the event the bank’s investments leave it insolvent. In other words, the government bails out the bank’s creditors but does not rescue the bank’s own assets or shareholders. The household pays an ex ante tax that is su¢ cient to cover the expected future bailout. The equilibrium interest rate on bank debt in this regime is again a Nash equilibrium in the sense that both sides of the …nancial market 2

exchange the optimal quantity of bank debt given the risks that they face, but now the bank is e¤ectively selling risk-free treasury bonds to the household and the equilibrium interest rate re‡ects the absence of default risk. We derive four main results from the model. The …rst two results deal with the positive and normative implications of bailouts. The third result deals with the positive and normative implications of capital requirements. The fourth result relates to the case where the government’s future bailout policy is unpredictable. 1. While bailouts certainly induce greater bank leverage in equilibrium, the probability of bank failure is not necessarily higher in a world with bailouts. Bailouts cause a decrease in the equilibrium interest rate and an increase in the quantity of bank debt. The increase in bank debt is the classic moral hazard problem: banks become more leveraged when their losses are insured and other things equal, this increases the likelihood of bank failure. However, the reduction in the equilibrium interest rate has a countervailing e¤ect on the likelihood of bank failure because a lower interest rate makes it easier for the bank to repay its debts. Which e¤ect dominates is a quantitative question, but in our calibration the second e¤ect dominates and the bank is less likely to fail in a world with bailouts. 2. Bailouts are e¢ cient but not Pareto optimal. Bailouts are e¢ cient because they provide insurance to mitigate the consequences of uninsurable risk. Relative to Laissez Faire, the bank achieves higher expected utility in the bailout equilibrium because the bank can borrow at a lower rate. But the household achieves lower utility in the bailout equilibrium. The household would rather live in Laissez Faire with two risky assets (risk stocky and risky bonds) than in a world with one risky and one safe asset, because in the latter world the household has to pay a steep price for the additional security. It must accept a low interest rate on bank debt and it must pay a tax to …nance the bailout. Quantitatively, the reduction in risk is not worth the price the household has to pay, and therefore the bailout hurts the very group it is intended to help. This result is reminiscent of the First Welfare Theorem in economics— Laissez Faire in the …nancial market produces a Pareto optimal outcome. 3. We also study a di¤erent form of government intervention, a limit on a bank’s debt level, and we compare this policy to the bailout policy. While the bailout policy is socially e¢ cient, the debt limit has no e¤ect on social welfare, and neither policy is a Pareto move. The debt limit leaves social welfare unchanged because it generates zero-sum transfers of utility across market participants 3

without providing any insurance value. Finally, the e¤ect of the debt limit on the probability of bank failure is non-trivial: the lower level of debt makes it easier for the bank to repay its debts, but the debt limit can produce an increase in the equilibrium interest rate which in turn makes it harder for the bank to repay its debts. The …rst e¤ect dominates in our model, and therefore a policy to limit bank debt reduces the likelihood of bank failure. 4. Finally, we expand our analysis to incorporate policy uncertainty. We assume that the current government cannot commit the future government to any particular course of action. Instead, the bank, the household, and the current government all face uncertainty about whether the future government will bail out the bank upon failure. Hence there are two layers of uncertainty in the model: the bank and household do not know the return on the risky investment project, nor do they know if the government will bail out creditors if the bank becomes insolvent. The conclusions from this extension are similar to those outlined in points 1 and 2 above: the bank prefers the policy uncertainty equilibrium to Laissez Faire because the partial insurance protection provided by a potential future bailout drives down their cost of borrowing somewhat; however, the household prefers Laissez Faire to the policy uncertainty equilibrium because the latter equilibrium bears a low interest rate and an ex ante bailout tax on household wealth that erode any welfare gains from having a safe asset in its portfolio.1 Of course, these conclusions are suggestive rather than de…nitive because they are speci…c to the abstract model setting that we have designed. But the main lesson from the paper is that incorporating a …nancial market into the analysis— with an interest rate that appropriately prices default risk— is a powerful feature that can shape the conclusions in rich and interesting ways. Turning to the related literature, Farhi and Tirole (2012) study optimal interest rate policy, bailouts, 1 Our paper …ts into a rapidly growing literature on policy uncertainty that is too big to cite here in a comprehensive way. Brie‡y, research on this topic …ts into two categories. Papers in the …rst category seek to measure policy uncertainty and to understand changes in policy uncertainty over time. Prominent examples include Baker, Bloom, and Davis (2013), Baker, Bloom, Canes-Wrone, Davis, and Rodden (2014), and Fernández-Villaverde, Guerrón-Quintana, Kuester, and RubioRamírez (2013). Papers in the second category study the e¤ects of speci…c types of policy uncertainty on economic outcomes. For example Fernández-Villaverde, Guerrón-Quintana, Kuester, and Rubio-Ramírez (2013) …nd that idiosyncratic volatility shocks to tax rates depress economic activity and can be a stand-alone cause of a recession. Davig and Foerster (2014), Stokey (2014), and Caliendo, Gorry, and Slavov (2015) study one-time …scal reforms that feature uncertainty about both the timing and structure of reform. Davig and Foerster show that uncertainty about the timing and structure of future tax policy can create a recession even if no change in …scal policy actually materializes. Stokey shows that uncertainty about the timing and structure of tax policy creates periods of inaction, in which investors adopt a “wait-and-see” attitude. Caliendo, Gorry, and Slavov show that the welfare loss from uncertainty about Social Security reform depends on how well households are solving an optimal hedging problem. Our paper falls into the second category of analyzing the impact of a speci…c type of policy uncertainty on economic outcomes. Our speci…c focus on uncertainty about bailouts distinguishes our paper from the others in this literature.

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and regulations when banks invest in long-term projects that may require follow-up investments. This underlying maturity mismatch issue, combined with government polices, creates potential moral hazard and government commitment to policy plays an important role. While both their study and ours feature a …nancial market with an endogenous interest rate, our study is di¤erent in that we allow for risk aversion in both the household and bank, unlike the risk neutrality assumption in Farhi and Tirole. Also, the bailout policy that we study is a tax and transfer program that directly rescues bondholders of insolvent banks, whereas they mainly study an interest rate policy (reduced form monetary policy) that provides cheap credit to struggling banks. Chari and Kehoe (2016) recognize that a benevolent government lacks commitment and will pursue bank bailouts in order to avoid costly bankruptcies. The source of the moral hazard in their paper is free riding: …rms (managers) tend to increase bank size, and losses are thus greater when hazardous outcomes are realized and hence bailouts are more likely. Naturally, their policy recommendation features a tax on size as well as a limit on debt-to-value ratios. They also assume risk neutral agents, investors and managers alike. Acharya and Yorulmazer (2007) highlight the cross sectional behavior among banks when the government lacks commitment. Having “too-big-to-fail”in mind, banks, especially small banks, may choose portfolio strategies emulating (herding) the others. The correlation of bank failures is thus endogenized and becomes the source of moral hazard: by herding together with other institutions, small banks rely on a government bailout when big losses occur in multiple banks. The optimal government policy is thus aimed at reducing the correlations of bank investment. Acharya and Thakor (2016) focus on bank regulation when bank liquidity has both positive and negative externalities. The provision of bank liquidity on the one hand facilitates the rest of economy and thus exerts positive externalities (bright side); on the other hand, excessive liquidity increases the risk of bank failure and hence costly government bailouts are more likely (dark side). Their paper studies optimal regulatory intervention in this setting. In the model, there is information asymmetry between the regulator and banks. When an aggregate shock hits the economy, the regulator cannot observe the bank’s performance during the buildup of its …nal outcome. They discuss optimal capital requirements when regulators commit to no bailouts as well as when regulators bail out banks unconditionally, in which case the optimal capital requirement implies zero leverage. Finally, the paper characterizes the optimal capital requirement with a conditional bailout policy when the regulator can bridge the information gap at a cost. Thakor (2014) o¤ers a literature review on the role of bank capital and …nancial stability. He demon5

strates both the theoretical support and empirical evidence that higher capital is positively correlated with …nancial stability, while the optimal level of capital, however, remains an open question. Likewise, in our paper capital requirements promote …nancial stability, but they do not improve e¢ ciency and are not a Pareto improvement over Laissez Faire. Merton and Thakor (2015) distinguish two di¤erent types of agents who use …nancial intermediaries: customers who use banks without bearing any risks and merely make deposits and buy insurance, while investors, on the other hand, are less risk averse and require a smaller risk premium. In their model, customers bear no intermediary-speci…c risk, while investors do for reasons outside the discussion. Pertinent to our paper, the authors use the customer-investor framework to provide a rationale for bailing out big and complex banks. Since customers are very sensitive to their contracts with intermediaries, their reactions will be more drastic during an economic downtown. Moreover, large and complex banks have disproportionately more customers than small banks. Since the government cannot easily distinguish customers from investors, they thus choose to bail out big banks in order to save more customers. In all of these papers, our paper stands out in at least the following two ways. First, we provide a welfare analysis of both banks and investors (households). We can o¤er a clear-cut answer when it comes to who wins and who loses from a bank bailout. Second, our policy uncertainty discussion features the empirically relevant case where the bailout is a random event, re‡ecting the inability of current decision makers to constrain or predict what a future government might do.2

2. Laissez Faire Benchmark There is one bank, one household, and no government. While both the bank and household are utility maximizers, their incentives are not compatible: the bank wants to borrow at a low rate and the household wants to lend at a high rate. The two actors reach an agreement in a …nancial market where the interest rate is determined. We begin with a Laissez Faire setting to establish a benchmark for understanding the e¤ects of bailouts. 2

We have only just scratched the surface of papers that study moral hazard and government intervention. A few other relevant papers include the following. DeYoung, Kowalik, and Reidhill (2013) model a bank’s incentive to create complexity. In their model, a government resolution authority struggles to close complex banks without creating spillover cost on the broader economy. In the model, the moral hazard is a higher degree of complexity since banks can’t choose to be risky or safe. Mailath and Mester (1994) consider bank closure explicitly. In their paper, the regulator must consider and optimize the additional risk that the bank is willing to take due to the structure of the policy in its current and future portfolio decisions against the actual costs of closing the bank. This allows a focus on the credibility of the regulator and opportunity costs of closure. Interestingly, they …nd examples of reduced welfare merely by the existence of the option to close a bank. Black and Hazelwood (2012) empirically investigate the e¤ect of TARP on risk taking by banks. They estimate the degree to which banks used TARP funds ex post to turn around and make risky loans. Finally, Kocherlakota (2010) proposes taxing the risky behavior of banks to internalize the spillover consequences of bank failures.

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The bank has access to a risky investment project. The bank invests its own capital and the proceeds from the sale of bank debt (bonds) in the risky project. On the other side of the …nancial market, the household makes portfolio decisions by purchasing stock directly in the same risky project and by making loans to the bank. Essentially, the bank facilitates household investment in the risky project indirectly through the bond market or the household can invest directly in the risky project and avoid the bank. Consider a two period model. At t = 0 the bank and household meet in a …nancial market where debt is exchanged. The equilibrium interest rate on bank debt is determined endogenously as the rate that equalizes the household’s supply of loanable funds with the bank’s demand for loanable funds. In other words, the bank and household make rational portfolio decisions in the face of uncertainty, and the equilibrium interest rate is a Nash equilibrium in the sense that both sides of the …nancial market behave optimally given the risks that they face. One period later at t = 1 the return on the risky investment project is realized and the bank pays the household what the bank has promised to repay (if the bank is solvent), or the bank repays the household everything that it has (if the bank is insolvent). With this understanding of the timing of the model, we will omit time subscripts from all variables in order to reduce clutter. The rate of return on the risky investment project is a continuous random variable r with support [ 1; R] and p.d.f. f (r). Notice that the bank and household face limited liability in the sense that the worst outcome (r =

1) is to lose everything invested in stock (but not more than everything invested)

and the best the bank can do (r = R) is …nite for simplicity and without loss of generality. The bank’s initial asset is A. It borrows D from the household with a commitment to repay D(1 + i) one period later conditional on bank solvency where i is the equilibrium interest on debt (to be determined below). If the bank is insolvent, it will default on its debt payment to the household and instead return a smaller payment to the household. The household’s initial wealth is W , which it divides between stock S and debt D, hence W = S + D. When the household buys stock directly in the risky project, it receives a stochastic payment S(1 + r) one period later and is not shielded from losses. When the household purchases debt from the bank, the household is partially insulated from the riskiness of the investment project (but not fully insulated because there still is default risk). The combined investment in the risky project is x

A + S + D. The bank is solvent if the risky

project provides a return that is su¢ cient to meet its debt obligation, (A + D)(1 + r)

7

D(1 + i);

or in other words Di A A+D

r

rC (D; i);

where rC (D; i) is the “cut-o¤” rate of return above which the bank is solvent and below which it must default on its debt and give all of its remaining assets to bondholders. Note that the cut-o¤ rate is a function of the level of debt of the bank, D, and the interest rate on that debt, i. Hence, the probability of bank failure P (insolvency) is P =

Z

rC (D;i)

f (r)dr: 1

2.1. Bank Behavior The bank’s pro…t , de…ned as the di¤erence between the …nal and initial value of its assets, is a random variable that depends on the realization of the risky return r, 8 < Ar + D(r i) if r rC (D; i) = : A if r < rC (D; i):

The moments of the random variable E( ) =

Var( ) =

Z

Z

are

rC (D;i)

f (r)( A)dr +

Z

R

f (r)[Ar + D(r

i)]dr;

f (r)[Ar + D(r

i)

rC (D;i)

1

rC (D;i)

f (r)[ A

E( )]2 dr +

Z

R

E( )]2 dr:

rC (D;i)

1

The bank has mean-variance utility with risk aversion .3 For a given interest rate i, the bank chooses optimal debt according to

n max E( )

2

D

o Var( ) :

We denote the solution to this problem as n D (i) = arg max E( )

2

o Var( ) :

Note that D (i) is the demand for loanable funds, which we have derived from the pro…t maximizing portfolio decisions of the bank. 3

Risk aversion is an important assumption. Without it, there is no internal solution to the portfolio choice problem because banks choose as much leverage as possible as long as the mean return on the risky project exceeds the interest rate on borrowed funds.

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2.2. Household Behavior If the bank is solvent, the household collects an equity payment and a debt payment S(1 + r) + D(1 + i); but if the bank is insolvent the household collects a lesser amount x(1 + r): The household chooses a portfolio of debt D and stock S = W

D. We calibrate the model so that the

individual wishes to hold positive amounts of both stock and debt. Re-using notation, household pro…t

=

with moments E( ) =

Var( ) =

Z

Z

is the di¤erence between its …nal and initial wealth

8 <

Sr + Di if r

rC (D; i)

: A(1 + r) + (S + D)r if r < r (D; i) C

rC (D;i)

f (r)[A(1 + r) + (S + D)r]dr +

Z

R

f (r)(Sr + Di)dr;

rC (D;i)

1

rC (D;i)

E( )]2 dr +

f (r)[A(1 + r) + (S + D)r

Z

R

f (r)[Sr + Di

E( )]2 dr:

rC (D;i)

1

The household has mean-variance utility with risk aversion . For a given interest rate i, the household chooses optimal debt (and therefore optimal stock purchases) according to n max E( )

2

D

o Var( ) :

We denote the solution to this problem as n D (i) = arg max E( ) Note that D

2

o Var( ) :

is the supply of loanable funds, which we have derived from the pro…t maximizing portfolio

decisions of the household.

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2.3. Equilibrium in the Financial Market In the Nash equilibrium between the bank and the household, risky debt is priced according to default risk. De…nition 1. Laissez Faire Equilibrium in the Financial Market. For a given p.d.f. f (r) over the rate of return on the risky project, the equilibrium in the …nancial market consists of an interest rate ie and a quantity of bank debt De such that (i) the bank issues the optimal quantity of bank debt D (ie ) given the interest rate ie , (ii) the household purchases the optimal quantity of bank debt D (ie ) (and also the optimal quantity of bank stock) given the interest rate ie , and (iii) bank debt De equals the quantity of debt sold by the bank as well as the quantity of debt purchased by the household, De = D (ie ) = D (ie ). The equilibrium interest rate must be estimated numerically, since the supply and demand functions themselves must be constructed numerically. 2.4. Numerical Examples The model inputs to be calibrated are: the p.d.f. f (r) over the rate of return on the risky project, the risk aversion parameter , the bank’s initial assets A, and the household’s initial wealth W . We calibrate these inputs to replicate three targets: a mean return E(r) = 10% to re‡ect stock market returns on diversi…ed portfolios with variance Var(r) = 0:04 which is approximately equal to the historical variance on stock returns in the U.S. economy, together with an interest rate i = 6% to approximate coupon rates on uninsured, high quality corporate bonds in recent years in the U.S. economy. We are able to match all of these targets by assuming the risky return r follows the beta distribution with p.d.f. f (r) = R R

(r + 1)

1 (r + 1)

with parameters

= 13:0615,

1 (R 1 (R

r) r)

1 1 dr

;

= 10:6866, and R = 1, together with a risk aversion

= 0:50 and initial

bank and household assets A = 1 and W = 13:185. Notice the household begins with signi…cantly more wealth than the bank, but the bank has access to the risky projects so the two can bene…t from each other and they meet in the …nancial market. In equilibrium, the interest rate is ie = 6:00% and the household divides its wealth almost evenly between debt and stock, De = 6:67 and S e = 6:52. The equilibrium cut-o¤ interest rate is rC (De ; ie ) = 7:83% which implies an equilibrium probability of bank failure P = 19:07%. We interpret the equi10

librium probability of bank failure to fall reasonably between the share of bank failures that we observe in good and bad states of the economy. That is, while we might expect far less than 19% of banks to fail each year during a boom period, 42% of all publicly traded banks in the U.S. received TARP bailout funds during the Financial Crisis (Blau, Brough, and Thomas (2013)). Table 1 summarizes the parameterization of the model. Figure 1 shows the determination of the equilibrium in the …nancial market as the intersection of the supply of loanable funds D (i) and the demand for loanable funds D (i). Figure 2 plots bank and household pro…t realizations as a function of the return on the risky project. Notice the ‡oor on bank losses because the worst the bank can do is to lose all of its initial assets. For any return below the cut-o¤ return the bank is insolvent and it cannot repay its debts, so it liquidates its assets to pay debtholders and defaults on the gap between what it owes and what it can pay. On the upside, the household has a natural advantage over the bank because its initial wealth endowment is large and therefore it tends to earn larger pro…ts than the bank for a wide range of positive returns; but, for very high returns the bank’s ability to leverage its initial assets becomes an advantage and it can do better than the household for this reason.

3. Bailouts We now introduce the government into the model. The purpose of the government is to bail out the bank’s creditor (household) in the event the bank is unable to repay its debts in full. The government provides a bailout to banks B that is just su¢ cient to keep the bank solvent. In other words, when the bank is unable to meet its debt obligation to the household, the government intervenes and provides a bailout that is just su¢ cient to allow the bank to use all of its remaining asset base together with the bailout to meet its debt obligation, leaving the bank with nothing. Therefore, the government bails out the bank’s debt obligations but does not bail out stock losses and the bank’s own asset base still gets wiped out after the bailout. The cut-o¤ interest rate is de…ned as before rC (D; i) =

Di A ; A+D

where a bailout is triggered if the random rate of return on the risky project r fall below rC (D; i). The

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bailout provided by the government is a random variable

B=

with mean

8 <

0 if r

: D(i

E(B) =

Z

r)

rC (D; i)

A(1 + r) if r < rC (D; i)

rC (D;i)

f (r)[D(i

r)

A(1 + r)]dr:

1

In order to …nance the bailout, the government charges the household an ex ante tax on its wealth at rate

that is su¢ cient to cover the expected bailout, W = E(B):

3.1. Bank Behavior For a given interest rate i, a given realization of the rate of return on the risky project r, and a given amount invested in the risky asset x, the bank’s realized pro…t is the same in the bailout regime as in Laissez Faire. This is because return realizations that leave the bank solvent (r

rC (D; i)) would confer

the same pro…t as Laissez Faire, while return realizations that leave the bank insolvent (r < rC (D; i)) would also confer the same (negative) pro…t as Laissez Faire because the bank losses all of its initial asset base and the bailout is used to meet the bank’s debt obligation rather than to bailout the bank’s asset base. This does not mean that the bank will make the same portfolio decision in the two worlds, because the interest rate will be di¤erent in the bailout regime and this will trigger di¤erent portfolio choices by the bank. However, the bank’s demand for loanable funds as a function of the interest rate, D (i), is the same as in Laissez Faire. 3.2. Household Behavior Because household debt is what gets bailed out, the household’s willingness to lend to banks will be much di¤erent than in Laissez Faire. The household’s after-tax wealth is divided between stocks and debt, W (1

) = S + D:

Whether the bank is solvent or not, the household collects the same post-return payment (because of the 12

bank bailout) S(1 + r) + D(1 + i); and household pro…t therefore is S+D 1

= S(1 + r) + D(1 + i) = S r with moments E( ) =

Var( ) =

Z

Z

+D i

1

1

R

f (r) S r 1

+D i

1

1

dr; 2

R

f (r) S r 1

+D i

1

1

E( )

dr:

For a given interest rate i, the household solves n max E( )

2

D

We denote the solution as

o Var( ) :

n D (i; ) = arg max E( )

2

o Var( ) ;

where D (i; ) is the supply of loanable funds. 3.3. Equilibrium in the Financial Market De…nition 2. Bailout Regime Equilibrium in the Financial Market. For a given p.d.f. f (r) over the rate of return on the risky project, the equilibrium in the …nancial market consists of an interest rate ie , a quantity of bank debt De , and a tax rate

e

such that (i) the bank issues the optimal

quantity of bank debt D (ie ) given the interest rate ie , (ii) the household purchases the optimal quantity of bank debt D (ie ; rate

e

e

) (and also the optimal quantity of bank stock) given the interest rate ie and tax

, (iii) bank debt De equals the quantity of debt sold by the bank as well as the quantity of debt

purchased by the household, De = D (ie ) = D (ie ; expectation

e

e

), and (iv) the bailout is self …nanced in

W = E(B), where E(B) is the expected bailout which is a function of the interest rate ie

and quantity of bank debt De .

13

3.4. Numerical Examples We hold all model inputs the same as in Laissez Faire in order to draw a fair comparison of the e¤ect of government bailouts on bank and household interactions through the …nancial market. The presence of the bailout triggers an increase in the household’s willingness to lend to the bank since, after all, it is ultimately the household whose bank loans are insured. This causes the equilibrium interest rate to fall from its Laissez Faire value ie = 6:00% to ie = 1:56%; and the equilibrium quantity of bank debt to increase from its Laissez Faire value De = 6:67 to De = 8:77 while the equilibrium quantity of bank stock drops from S e = 6:52 to S e = 4:22. This seems to be a reasonable movement in the equilibrium interest rate since the bank is essentially issuing government-backed treasury bonds and the interest rate on such instruments is often in the neighborhood of 1 to 2 percentage points. The equilibrium tax on household wealth that is ex ante su¢ cient to fund the bailout is

e

= 1:43%.

Figure 3 compares and contrasts equilibrium pro…t realizations of the bank and household across the Laissez Faire and bailout regimes. Notice the household no longer faces the risk of losing all of its wealth. Instead, the worst the household can do is to lose all of its stock investment, and hence low realizations of the risky return led to higher equilibrium household pro…t in the bailout regime than in Laissez Faire. On the other hand, while high realizations of the risky return are great, they are less pro…table to the household in the bailout regime than in Laissez Faire because the household holds less stock in the former equilibrium than in the latter. Meanwhile, in either regime the bank’s pro…ts are limited on the downside to the loss of all of its initial assets, while on the upside the bank fares better in the bailout regime than Laissez Faire for two reasons. First, the bank is more highly leveraged in the bailout regime and hence has a larger stake in the risky project, and second the interest rate on bank debt is low in the bailout regime and this expands the bank’s pro…t margin on its debt. Perhaps surprisingly, the equilibrium cut-o¤ interest rate actually falls from its Laissez Faire value rC (De ; ie ) =

7:83% to rC (De ; ie ) =

8:84%, implying that bank failure is less likely in the bailout

regime than in Laissez Faire. Indeed, the equilibrium probability of bank failure in the bailout regime is P = 17:77%, down from the Laissez Faire value P = 19:07%. Table 2 provides a summary of the parameterization of the model. To see the intuition, consider a …rst-order expansion of rC (D; i) rC (D; i)

D iA + A di + dD: A+D (A + D)2

14

An increase in D and i both cause an increase in rC (D; i) and hence an increase in the probability of bank failure. However, bailouts cause the equilibrium in the …nancial market to shift to a lower interest rate with an increase in bank debt. The increase in bank debt is the classic moral hazard problem: banks become more leveraged when their losses are insured. However, the reduction in the equilibrium interest rate has a countervailing e¤ect on the cut-o¤ interest rate. So it is a quantitative question whether or not the presence of government bailouts would actually increase or decrease the probability of bank failure. Our numerical examples predict a decrease. Like any model, ours is a simpli…ed representation of reality, but it does point to the possibility that, once the model is generalized to include a …nancial market with the equilibrium determination of interest rates that depend on default risk, bailouts may have the opposite e¤ect on bank failures than standard intuition suggests.

4. E¢ ciency The purpose of this section is to study the e¤ect of bailouts on social welfare. The sum of the surplus of the entities in the model is

= where

B

and

H

B

+

H;

are bank and household pro…ts, with moments E( ) = E(

+ E(

H)

E( ))2 ]

Var( ) = E[( = Var(

B)

B)

+ Var(

H)

+ 2Cov(

B;

H ):

The social welfare function is de…ned as mean-variance utility over the sum of the total surplus U

E( )

2

Var( )

= UB + UH where UB

E(

B)

2 Var( B )

and UH

E(

H)

Cov(

B;

2 Var( H ).

H)

Bailouts are e¢ cient if they improve

social welfare. To compute social welfare in Laissez Faire, we can use a shortcut. Note that 15

= xr for any realization

2 (A

of r and hence U = (A + W )E(r)

+ W )2 Var(r). Notice that the socially optimal level of bank

leverage (D) is indeterminate in our simpli…ed model. That is, social welfare does not depend on bank leverage because in the end, all of the initial wealth in the economy (of both the bank and the household) gets invested in the risky project one way of the other, either through the stock channel or the debt channel. And we do not model systemic risk so there isn’t a negative externality tied to the level of bank debt. To compute social welfare in the bailout regime we can make use of our equilibrium calculations of UB and UH from the previous section of the paper and compute the covariance of pro…t Cov(

B;

H)

=

Z +

rC (D;i)

f (r)[ A

E(

B )]

S r

Z

+D i

1

1

E(

1

H)

dr

R

f (r)[Ar + D(r

i)

E(

B )]

S r

rC (D;i)

where E(

B) =

E(

Z

rC (D;i)

H) =

f (r)( A)dr +

1

E(

H)

dr

R

f (r)[Ar + D(r

i)]dr

rC (D;i)

1

Z

Z

+D i

1

R

f (r) S r 1

1

+D i

1

dr:

Table 3 summarizes the utility of the bank and household as well as social welfare. In our calibration, the bailout is socially e¢ cient, but it is not a Pareto e¢ cient policy because there are winners and losers. Relative to Laissez Faire, the bank achieves higher expected utility in the bailout equilibrium but the household achieves lower utility in the bailout equilibrium. Relative to Laissez Faire, the household’s portfolio in the bailout equilibrium is more heavily tilted toward debt and away from stock. The debt held by the household bears a much lower interest rate in the bailout regime than in Laissez Faire, and the household self …nances its own bailout through the payment of taxes on its wealth. Factoring in these equilibrium …nancial market e¤ects, the household would rather live in a Laissez Faire world with two risky assets (risk stocky and risky bonds) than in a world with one risky and one safe asset, because in the latter world the household has to pay a steep price for the additional security: it must except a very low interest rate on bank debt and it must pay a tax to …nance the bailout. Quantitatively, the reduction in risk is not worth the price the household has to pay, and therefore the bailout hurts the very group it is intended to help. To clarify the intuition, we compute a third equilibrium that is just like the bailout equilibrium except that the government provides the bailout free of charge. We recompute the equilibrium interest rate to

16

ensure that the bond market clears, but we don’t require the government’s budget constraint to hold. In other words, the household doesn’t have to …nance the bailout; instead, the government magically provides the necessary funds to bail out debt holdings upon bank failure. Not surprisingly, household welfare in this third equilibrium dominates household welfare in the equilibrium in which bailouts are self …nanced by households. Less obvious, however, is the …nding that household welfare in this third equilibrium dominates household welfare in Laissez Faire. This exercise teaches us that, other things equal, the household would indeed prefer to have a safe bond asset with relatively low interest rate instead of a risky bond asset with a higher expected return. But the household is not willing to pay the price (the tax) required to buy down this risk. Finally, it is worth mentioning that risk taking itself is socially ine¢ cient, at least under the current de…nition of e¢ ciency. Imagine a world in which the bank and household cannot partake in the risky project. Instead, each is consigned to simply eat their initial wealth, which confers zero pro…t with certainty to each. Social welfare would be zero, which dominates social welfare in the Laissez Faire equilibrium with risk taking. This is because the pro…ts of the bank and household are correlated and there is a penalty in social welfare for positive covariance. But this conclusion is sensitive to the way we de…ne social welfare. If we instead de…ne social welfare as the sum of the utilities of the bank and household— rather than mean-variance utility over total surplus, which includes a covariance penalty— then we would conclude that risk taking is socially e¢ cient because both the bank and the household prefer the equilibrium with risk taking to autarky.

5. Capital Requirements Rather than a bailout, suppose the government intervenes by setting a maximum on bank debt, D 2 (0; De ), where De is the Laissez Faire equilibrium interest rate. In this case, the interest rate is indeterminate and can take on any value in the set I, where I = fi : D (i)

D and D (i)

Dg.

But for any i 2 I, social welfare is invariant to the debt limit because the total surplus

= xr is invariant

to the household’s allocation of its wealth between stock and debt. Intuitively, all of the household’s wealth (and all of the bank’s initial assets) ultimately get invested in the risky project, whether directly through the stock channel or indirectly through the debt channel. Hence, limits on bank debt (capital requirements) are neutral in an e¢ ciency sense. Of course, there will be winners and losers but the 17

indeterminacy of the interest rate prevents us determining whether it is the bank or the household that bene…ts from the intervention. For example, suppose we start from the Laissez Faire calibration with no bailouts. Instead of bailouts, the government places a limit on bank debt at D = 6:0 (which is below the Laissez Faire equilibrium level of bank debt De = 6:67). The bank’s demand for loanable funds will exceed the debt limit if i < 6:74%, and the household’s supply of loanable funds will exceed the debt limit if i > 3:33%, so we con…ne ourselves to interest rates in the range of 3:33% to 6:74%. The interest rate is indeterminate within this range and we would need to make some additional assumptions about the bargaining powers of the players in the …nancial market in order to determine how the interest rate responds to the debt limit. Instead of doing that, we consider both of the boundary interest rates from the set I. Whether the interest rate is 3:33% or 6:74%, our model suggests that the probability of bank failure will be below its Laissez Faire value. This may seem obvious at …rst because we have constrained bank debt to be below its equilibrium level, but this is not a trivial result. While a reduction in bank debt reduces the probability of bank failure other things equal, a high interest rate increases the probability of bank failure. The debt e¤ect overpowers the interest rate e¤ect and the end result of a debt limit is to reduce the probability of bank failure. The debt limit has no bearing on social welfare. But it does distort the welfare gains to the bank and household in a non-Pareto-improving way. If the debt limit causes the interest rate to rise, then the household prefers the debt limit to Laissez Faire and the bank does not. The reverse is true if the debt limit causes the interest rate to fall. Why is the bailout policy socially e¢ cient while the capital requirement is neutral? Because with bailouts the government is providing insurance at a fair price to the participants of the …nancial market. In very bad times the government steps in and absorbs large losses but only charges a tax that is ex ante self …nancing. The capital requirement of course provides no such insurance so it does not a¤ect social welfare.

6. Policy Uncertainty To this point in our analysis, we have assumed that both the bank and the household know with certainty whether they are living in a Laissez Faire world or in a bailout regime. Of course, reality is more complicated. Here we assume that the government cannot commit itself to any future policy. Regulated industries such as banking continue to face increasing policy uncertainty in the U.S. Baker, Bloom, Canes-Wrone, Davis, and Rodden (2014) estimate that policy uncertainty is 4 times higher 18

now than it was a half-century ago. This trend is related to a near-doubling in the size of government expenditure as a share of GDP and a 6-fold increase in the size of the Code of Federal Regulations. Growth in policy uncertainty is also accompanied by increased polarization in American politics. The views of the Democratic and Republican parties have increasingly diverged in recent years, creating additional policy uncertainty around political elections (Baker, Bloom, and Davis (2013)). In short, in the words of Sargent (2005), “What we do not know today is how subsequent political deliberations from shifting majority coalitions will render U.S. …scal policy coherent.” More speci…cally, banks and other …nancial institutions, particularly large ones, don’t know in advance whether they will be rescued if their investments fail. For example, in 2008 the U.S. government bailed out Bear Stearns but then let Lehman Brothers and Washington Mutual fail a few months later. The policy uncertainty index created by Baker, Bloom, and Davis (2013) hit a peak after Lehman Brothers failed and President George W. Bush signed the Troubled Asset Relief Program (TARP) into law, suggesting a general feeling of uncertainty about the government’s intentions. Even though 42% of all publicly traded banks in the U.S. received TARP bailout funds (Blau, Brough, and Thomas (2013)), hundreds of banks were left to fail during the Financial Crisis and in the years since (e.g., see the FDIC’s list of failed banks, updated every Monday).4 At t = 0 when the bank and household interact in the …nancial market, they face uncertainty about whether the government will bail out bank debt at t = 1 when random pro…ts are realized. Uncertainty is resolved at t = 1 and the government either bails out bank debt or they do not. It is common knowledge that the ex ante likelihood of a bailout is probability of a future bailout at t = 1 is

2 [0; 1]. The government at t = 0 also knows that the but it cannot commit the future government to any speci…c

4

Moreover, recent legislation does not resolve bailout uncertainty. The Dodd-Frank Wall Street Reform and Consumer Protection Act, signed into law on July 21, 2010, was explicitly meant to put an end to bailouts. In particular, it created the Financial Stability Oversight Council, charged with the duty to “promote market discipline, by eliminating expectations...that the government will shield [banks and other …nancial institutions] from losses in the event of failure.” However, leading regulators still worry about the potential for future bailouts, despite the promises made by current policy makers. For example, a few months after Dodd-Frank, Federal Reserve Chairman Ben Bernanke testi…ed before the Financial Crisis Inquiry Commission that “Simple declarations that the government will not assist …rms in the future, or restrictions that make providing assistance more di¢ cult, will not be credible on their own. Few governments will accept devastating economic costs if a rescue can be conducted at a lesser cost; even if one Administration refrained from rescuing a large, complex …rm, market participants would believe that others might not refrain in the future.” And while Congress was in the process of drafting the Dodd-Frank Act, the President of the Federal Reserve Bank of Minneapolis, Narayana Kocherlakota (2010), warned that “no legislation can completely eliminate bailouts [and] any new …nancial regulatory structure must keep this reality in mind.” See also Chari and Kehoe (2016) for similar remarks. Also see the “Bailout Barometer” calculated by the Federal Reserve Bank of Richmond. In recent years, this barometer predicts that over half of the …nancial sector would be bailed out in the future, notwithstanding recent political e¤orts (like Dodd-Frank) to end bailouts.

19

course of action. For notational convenience, we de…ne the lottery

L=

8 <

1 with probability

: 0 with probability 1

:

If at t = 1 the government bails out the bank, then the government provides a bailout B that is just su¢ cient to keep the bank solvent. In other words, just as before, when the bank is unable to meet its debt obligation to the household, the government intervenes and provides a bailout that is just su¢ cient to allow the bank to use all of its remaining asset base together with the bailout funds to meet its debt obligation, leaving the bank with nothing. The government bails out the bank’s debt obligations but does not bail out stock losses and the bank’s own asset base gets wiped out. The bailout probability

is subjective. We do not model the expectation formation process. Instead,

we simply assume the bank and the household share the same subjective expectations about a future bailout, and we will calibrate this expectation to re‡ect the share of banks that we bailed out during TARP. Technically speaking, in the model both banks and households face uncertainty but they do not face ambiguity because they know the probability of a bailout. Essentially, both parties have full information about the risks that they face, whereas in reality it may be the case that neither party has any information about the probability of a bailout or information may be asymmetric (if, for instance, the bank has a better understanding of the bailout probability). Also, whether the government ultimately chooses to bail out banks in reality has a lot to do with the ex post circumstances of the economy, such as the size and systemic importance of the bank in question as well as the magnitude of its losses— whereas here we simplify matters considerably by assuming the bailout probability is exogenous. We abstract from these complications in order to test the robustness of our conclusions above, which deal with the standard case in the literature where bailouts are certain ( = 1). The cut-o¤ interest rate is de…ned as before rC (D; i) =

Di A ; A+D

where a bailout is triggered if the random rate of return on the risky project r fall below rC (D; i). The bailout provided by the government is a random variable 8 < D(i B= :

r)

A(1 + r) if r < rC (D; i) and L = 1 0 otherwise

20

with mean E(B) =

Z

rC (D;i)

f (r)[D(i

r)

A(1 + r)]dr:

1

In order to …nance the bailout, the government charges the household an ex ante tax on its wealth at rate

that is su¢ cient to cover the expected bailout, W = E(B):

6.1. Bank Behavior The bank’s demand for loanable funds as a function of the interest rate, D (i), is the same as in Laissez Faire. While uncertainty about a potential future bailout will certainly a¤ect the equilibrium in the …nancial market, it does not a¤ect the demand side of the market for loanable functions because the policy uncertainty is relevant only to the household because it is the household that ultimately bene…ts from a bailout (a poor return realization that leaves the bank insolvent causes a total loss of the bank’s initial assets, irrespective of whether the government steps into to cover the household’s debtholdings). 6.2. Household Behavior If the bank is solvent after the realization of the risky return, or if the bank is insolvent and L = 1, then the household collects S(1 + r) + D(1 + i): If the bank is insolvent and L = 0 then the household collects a lesser amount x(1 + r): Therefore, household pro…t is

= where

8 < :

if r < rC (D; i) and L = 0 otherwise

= A(1 + r) + (S + D) r =S r

+D i

1

21

1 1

with moments E( ) =

Var( ) =

Z

Z

rC (D;i)

f (r)[(1

) +

]dr +

R

f (r) dr;

rC (D;i)

1

rC (D;i)

f (r)f(1

Z

E( )]2 + [

)[

1

E( )]2 gdr +

Z

R

f (r)[

E( )]2 dr:

rC (D;i)

For a given interest rate i, the household solves n max E( )

2

D

We denote the solution as

o Var( ) :

n D (i; ) = arg max E( )

2

o Var( ) ;

where D (i; ) is the supply of loanable funds. 6.3. Equilibrium in the Financial Market De…nition 3. Policy Uncertainty Equilibrium in the Financial Market. For a given p.d.f. f (r) over the rate of return on the risky project and a given probability

that the government will bail out

the bank upon failure, the equilibrium in the …nancial market consists of an interest rate ie , a quantity of bank debt De , and a tax rate

e

such that (i) the bank issues the optimal quantity of bank debt D (ie )

given the interest rate ie , (ii) the household purchases the optimal quantity of bank debt D (ie ; also the optimal quantity of bank stock) given the interest rate ie and tax rate

e

e

) (and

, (iii) bank debt De

equals the quantity of debt sold by the bank as well as the quantity of debt purchased by the household, De = D (ie ) = D (ie ;

e

), and (iv) the bailout is self …nanced in expectation

e

W = E(B), where

E(B) is the expected bailout which is a function of the interest rate ie , the quantity of bank debt De , and the probability of being bailed out upon failure

.

6.4. Numerical Examples We leave all parameters the same as everywhere else throughout the paper (as in Tables 1 and 2). The probability of a bank bailout in the future,

, is a di¢ cult parameter to calibrate; it represents the

common perception among the bank, household, and current government that the future government will engage in a bailout, and such perceptions are not easy to quantify. Using the recent past as a guide to how expectations about the future could be formed, we set

= 42% because this is the share of all

publicly traded banks in the U.S. that received TARP bailout funds (Blau, Brough, and Thomas (2013)).

22

The equilibrium interest rate and tax rate are ie = 3:74% and

e

= 0:60%, while the equilibrium

quantity of bank debt is De = 7:90 and the probability of bank failure is P = 18:93%. Notice that the presence of policy uncertainty causes these equilibrium objects to fall between the Laissez Faire and bailout regime equilibrium values. The potential for a future bailout does increase the household’s willingness to lend to the bank, which drives down the equilibrium interest rate below its Laissez Faire level, but the uncertainty about the bailout keeps the interest rate signi…cantly above its level in the bailout regime equilibrium. Relative to Laissez Faire, the increased risk taking by the bank (in the form of higher bank debt) in an equilibrium with policy uncertainty is more or less neutralized by the lower interest rate. As a result, the probability of bank failure remains pretty much the same. This contrasts with a bailout regime in which perfect knowledge of a future bailout causes a large enough reduction in the interest rate that the probability of bank failure is lower than in Laissez Faire. Concerning welfare, the bank prefers the policy uncertainty equilibrium to Laissez Faire because the partial insurance protection provided by the potential for a bailout drives down their cost of borrowing somewhat. However, the household prefers Laissez Faire to the policy uncertainty equilibrium because the latter equilibrium bears a low interest rate and an ex ante bailout tax on household wealth that erode any welfare gains from having a safe asset in its portfolio.

7. Conclusion and Future Work In this paper, we construct a theoretical model with a …nancial market to evaluate the e¤ect of government bailouts on bank and household portfolio choices and welfare. In the model, the equilibrium interest rate clears the bond market. The bonds that banks issue are subject to bank failure risk. Government bailouts cause higher leverage (moral hazard) as well as lower equilibrium interest rates. In terms of welfare, bailouts increase social welfare by mitigating uninsurable risk but are not a Pareto improvement over Laissez Faire since bailouts reduce household welfare. Capital requirements reduce the likelihood of bank failure but are not a Pareto improvement over Laissez Faire either. We have focused speci…cally on the interaction between the bank and household in the …nancial market while abstracting from the broader macroeconomic context that would include …rm production. This has allowed us to concentrate on the fundamental questions that we are interested in— how do bailouts a¤ect portfolio choices and welfare— without the extra complexity of a full blown general equilibrium model. But a full blown model would surely allow for deeper insights into these questions. In addition, we have abstracted from a number of complexities in both the bank and the household portfolio choice problem. More realistic portfolio choices would include a menu of risky assets. Our level of abstraction keeps the 23

analysis clean and tractable but leaves room for additional analysis.

References [1] Acharya, Viral and Anjan Thakor (2016), The Dark Side of Liquidity Creation: LeverageInduced Systemic Risk and Implications of the Lender of Last Resort. Journal of Financial Intermediation, October, 4-21. [2] Acharya, Viral and Tanju Yorulmazer (2007), Too Many to Fail— An Analysis of TimeInconsistency in Bank Closure Policies. Journal of Financial Intermediation 16(1), 1-31. [3] Baker, Scott R., Nicholas Bloom, Brandice Canes-Wrone, Steven J. Davis, and Jonathan Rodden (2014), Why Has US Policy Uncertainty Risen Since 1960? American Economic Review: Papers and Proceedings 104(5): 56-60. [4] Baker, Scott R., Nicholas Bloom, and Steven J. Davis (2013), Measuring Economic Policy Uncertainty. Working Paper, Stanford University. [5] Black, Lamont and Lieu Hazelwood (2012), The E¤ect of TARP on Bank Risk-Taking. Working Paper, Board of Governors of the Federal Reserve System. [6] Blau, Ben, Tyler Brough, and Diana Thomas (2013), Corporate Lobbying, Political Connections, and the Bailout of Banks. Journal of Banking and Finance 37, 3007-3017. [7] Caliendo, Frank N., Aspen Gorry, and Sita Slavov (2015), The Cost of Uncertainty about the Timing of Social Security Reform. Working Paper, Utah State University. [8] Chari, V.V. and Patrick J. Kehoe (2016), Bailouts, Time Inconsistency and Optimal Regulation: A Macroeconomic View. American Economic Review 106(9), 2458-2493. [9] Davig, Troy and Andrew Foerster (2014), Uncertainty and Fiscal Cli¤s. Working Paper, Federal Reserve Bank of Kansas City. [10] DeYoung, Robert, Michal Kowalik, and Jack Reidhill (2013), A Theory of Failed Bank Resolution: Technological Change and Political Economics. Journal of Financial Stability 9(4), 612– 627.

24

[11] Farhi, Emmanuel and Jean Tirole (2012), Collective Moral Hazard, Maturity Mismatch and Systemic Bailouts. American Economic Review 102, 60-93. [12] Fernández-Villaverde, Jesús, Pablo Guerrón-Quintana, Keith Kuester, and Juan RubioRamírez (2013), Fiscal Volatility Shocks and Economic Activity. Working Paper, University of Pennsylvanian. [13] Kocherlakota, Narayana (2010), Taxing Risk and the Optimal Regulation of Financial Institutions. Economic Policy Paper, Federal Reserve Bank of Minneapolis. [14] Mailath, George J. and Loretta J. Mester (1994), A Positive Analysis of Bank Closure. Journal of Financial Intermediation 3(3), 272-299. [15] Merton, Robert and Richard Thakor (2015), Customers and Investors: A Framework for Understanding Financial Institutions. NBER Working Paper 21258. [16] Sargent, Thomas J. (2005), Ambiguity in American Monetary and Fiscal Policy. Working Paper, NYU. [17] Stokey, Nancy (2014), Wait-and-See: Investment Options under Policy Uncertainty. Working Paper, University of Chicago. [18] Thakor, Anjan (2014), Bank Capital and Financial Stability: An Economic Trade-o¤ or a Faustian Bargain? Annual Review of Financial Economics 6, 185-223.

25

Table 1. Summary of Laissez Faire Calibration and Equilibrium in the Financial Market

Parameters: r

Random return on risky bank projects

f (r) = R=1

(r+1) RR 1 (r+1)

1 (R

r) 1 (R r)

1 1 dr

Beta p.d.f. over project returns Upper bound on support

= 13:0615

Parameter in beta p.d.f.

= 10:6866

Parameter in beta p.d.f.

E(r) = 10:00%

Mean return on risky projects

Var(r) = 0:0400

Variance of return on risky projects

= 0:50

Risk aversion parameter in mean-variance utility

A=1

Initial bank assets

W = 13:185

Initial household wealth

Equilibrium values: ie = 6:00%

Interest rate on bank debt

De = 6:67

Quantity of bank debt

rC (De ; ie ) =

7:83%

Cut-o¤ return on risky project

S e = 6:52

Quantity of bank stock

P = 19:07%

Probability of bank failure

Note: Laissez Faire is a setting in which bank insolvency causes it to default on household debt obligations. Bank assets are wiped out in attempt to make (reduced) debt payments to households.

26

Table 2. Summary of Bailout Regime Calibration and Equilibrium in the Financial Market

Parameters: r

Random return on risky bank projects

f (r) = R=1

(r+1) RR 1 (r+1)

1 (R

r) 1 (R r)

1 1 dr

Beta p.d.f. over project returns Upper bound on support

= 13:0615

Parameter in beta p.d.f.

= 10:6866

Parameter in beta p.d.f.

E(r) = 10:00%

Mean return on risky projects

Var(r) = 0:0400

Variance of return on risky projects

= 0:50

Risk aversion parameter in mean-variance utility

A=1

Initial bank assets

W = 13:185

Initial household wealth

Equilibrium values: ie = 1:56% e

Interest rate on bank debt

= 1:43%

Bailout tax on household wealth

De = 8:77 rC (De ; ie ) =

Quantity of bank debt 8:84%

Cut-o¤ return on risky project

S e = 4:22

Quantity of bank stock

P = 17:77%

Probability of bank failure

Note: In the bailout regime, if the bank is insolvent then the government pays debtholders the di¤erence between what the bank owes and what the bank is able to pay.

27

Table 3. Winners, Lossers and E¢ ciency of Bank Bailouts

Laissez Faire Equilibrium:

Bailout Equilibrium:

ie = 6:00%

ie = 1:56%

Interest rate on bank debt

De = 6:67

De = 8:77

Quantity of bank debt

S e = 6:52

S e = 4:22

Quantity of bank stock

E(

E(

Mean of bank pro…t

B)

Var(

= 0:5290

B)

= 1:6512

B)

Var(

= 1:0296

B)

= 2:7555

Variance of bank pro…t

UB = 0:1162

UB = 0:3407

Mean-variance bank utility

E(

E(

Mean of household pro…t

H)

Var(

= 0:8892

H)

= 2:6706

H)

Var(

= 0:3701

H)

= 0:7132

Variance of household pro…t

UH = 0:2216

UH = 0:1918

Mean-variance household utility

U=

U=

Social welfare

0:5934

0:1454

Note: All model inputs (parameters) are held constant across settings to draw a fair comparison.

28

Figure 1. Laissez Faire Equilibrium in the Financial Market 7.2

demand for loans, D∗ (i)

quantity of debt, D

7 6.8 6.6 6.4 supply of loans, D∗∗ (i) 6.2 6 5.8 5.6 0.05

0.055

0.06

interest rate, i

0.065

0.07

Figure 2. Laissez Faire Equilibrium Profit Realizations

profit π and density f(r)

5

f(r) 0

bank π −5

cut-off rC −10

household π −15 −1

−0.5

0

random return, r

0.5

1

Figure 3. Bailout Regime Equilibrium Profit Realizations Bailout bank π LF bank π

profit π

5

0

−5

LF household π −10

−15 −1

Bailout household π

−0.5

0

random return, r

0.5

1