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A Deviation to the safe strategy
In the first case, the bank has sufficient net worth to satisfy the first order condition (26) while remaining within the deposit threshold
Second, the inward shift in the deposit threshold
required for this case to be valid.
Note also that the deviating bank’s expected payoff is given by the expression
which differs from (39) only in terms of ls|g.66 This reflects that a shift to bad sentiments has no impact on the bank’s ability to borrow when its net worth lies above nr|g.
In the second case, bank net worth falls short of nr|g such that it is not possible to satisfy (26) without breaching the deposit threshold
Using the budget constraint, the price of deposits can also be written as
and the deviating bank’s expected payoff is given by the expression
which is lower than (79) due to the increase in bank funding costs.
The solvency constraint binds in the third case. The quantity of loans is determined implicitly by the expression
attained by using (47) and (81) to substitute for q (γg,ds|g) and
and this case is valid when net worth is below the boundary
B Household’s recursive problem
Households supply labour inelastically to firms and have risk averse preferences with their flow utility u (c) given by a standard CRRA specification. The representative household’s problem can be written as
where Γ(.) is the law of motion for the aggregate state variables and υh(.) represents the household’s continuation value under sovereign default. Lemma 6 provides an expression for υh(.).
Lemma 6 The continuation value for households in the steady state S is
where w is given by
Proof. Provided in the Technical Appendix. ■
Observe that consumption c in the steady state is positively related to household wealth after sovereign default, which is increasing in the recovery rate θ of domestic deposits. Using the above expressions, the first order conditions for risk-free assets D* and domestic bank deposits D can be written as
where uc(.) is marginal utility.
As in section 3.1.5, the recovery rate anticipated by households depends on household expectations about the bank’s domestic sovereign bond exposure
The deposit demand schedule is attained by combining this expression with the household’s first order conditions.
C Liquidity Provision
In periods t ≤ T, the model is characterized by
I am indebted to Chryssi Giannitsarou, Giancarlo Corsetti, Vasco Carvalho and Luca Dedola for invaluable advice. I am grateful for comments and suggestions from Luigi Bocola, Charles Brendon, Fernando Broner, Giovanni Dell’Ariccia, Filippo De Marco, Nicola Gennaioli, Marcus Hagedorn, Peter Karadi, Igor Livshits, Alberto Martin, Maria Soledad Martinez Peria, Monica Petrescu, Jun Uno, Alexandros Vardoulakis, Jaume Ventura as well as anonymous referees and seminar participants at Cambridge University, Oxford University, ECB, IMF, University of Vienna, Bank of England, University of Bristol, Federal Reserve Bank of Atlanta, Southern Methodist University, George Washington University, Bank of Canada, Danmarks Nationalbank, Toulouse School of Economics, Bocconi University, the XX Workshop on Dynamic Macroeconomics, the 3rd Macro Banking Finance Workshop, EDGE Jamboree 2015, the ECB workshop on non-standard monetary policy measures, SED 2016, RIEF 2016, EEA 2016, SBM 2017, ESEM European Meeting 2017 and EFA 2017. I thank Oliver Shand for superb research assistance. I gratefully acknowledge financial support from the Royal Economic Society, the Keynes Fund and the Cambridge-INET Institute for financial support. All errors are my own.
Non-contractibility of portfolio exposures may arise due to (sufficiently) costly enforcement on behalf of depositors or information frictions such as opacity in bank balance sheets preventing depositors from observing bank portfolios in detail.
Deposit insurance schemes typically guarantee deposits only up to a limit (Demirguc-Kunt et al., 2008). In real terms, depositor losses can take the form of a suspension of convertibility and a currency re-denomination as well as an explicit bail-in.
I rely on expected rents from imperfect competition to moderate banks’ risk-taking incentives. When there is perfect competition in the banking sector, the combination of limited liability and non-contractibility of portfolio exposures always leads to a gambling equilibum.
When banks’ portfolio exposures are contractible, it is always optimal for banks to commit to a safe portfolio as by doing so they may reduce their funding costs to the risk-free rate.
The multiplicity mechanism considered here differs from bank-runs à la Diamond and Dybvig (1983) in that it pertains to banks’ ex-ante risk-taking decisions rather than ex-post withdrawals. Farhi and Tirole (2012) and Acharya et al. (2016) also propose models with multiplicity in bank risk-taking. In these studies, multiple equilibria arise due to strategic complementarities across banks as correlation in bank exposures makes it ex-post optimal for the government to provide support. This paper instead focuses on strategic complementarities between banks and depositors.
Like Bocola (2016), I treat sovereign default risk as driven by some exogenous latent factor. Abstracting from the government’s default decision allows me to focus sharply on the properties of the novel mechanism my model is about.
See also Fact 1 in the next section.
See Acharya and Steffen (2015) for an empirical analysis. They reach the same conclusion with a regression that controls for bank and country characteristics.
While under-capitalized banks in Germany have a higher exposure to domestic sovereign bonds than their well-capitalized counterparts, their holdings do not increase over the crisis. Since German government bonds were widely considered as a safe asset throughout the sovereign debt crisis, these holdings may be due to their use as collateral or regulatory requirements.
The patterns in Figure 2 are compatible with moral suasion under the condition that risky governments can exert greater pressure on under-capitalized banks to purchase domestic sovereign debt. Note, however, that the gambling and moral suasion channels are not mutually exclusive. In fact, gambling relies on moral suasion in the sense that the government neglects to regulate against the domestic sovereign exposure of local banks.
For further empirical evidence on the effects of the sovereign debt crisis on credit to the private sector, see Acharya et al. (2014b), Becker and Ivashina (2014), De Marco (2017) and Popov and Van Horen (2015).
Acharya et al. (2014a) show that changes in sovereign CDS explain changes in bank CDS even after controlling for aggregate and bank-level determinants of credit spreads.
The absence of risk-free assets among banks’ investment opportunities serves only to simplify the exposition. Their inclusion would be completely inconsequential in this set up as purchasing a safe asset is either equivalent to or less profitable than a reduction in deposits by the same amount.
In other words, the contracting space between households and banks is limited to time deposits.
This helps simplify the exposition without any actual impact on the model mechanisms.
This is the reduced-form outcome of a re-negotiation game between firms and banks after loans become non-performing. As firms are perfectly competitive and banks have market power, the latter extracts all of the remaining revenues after salary payments. Implicitly, this relies on the absence of information asymmetries, which can be motivated by relationship banking. This also makes it prohibitively costly for households and foreign entities to lend directly to firms. The domestic banking sector thus acts as a financial intermediary that channels funds to firms. Note that the outcome here is equivalent to the issuance of state-contingent debt by firms.
The assumption of risk neutrality only serves to attain a tractable expression for the deposit demand schedule. The results presented below retain their validity under risk aversion, which is introduced in section 4.
D* can be interpreted as deposits in a safe foreign bank or simply as a safe real asset. As there is a unit continuum of homogenous households, individual households’ deposits are identical to the aggregate quantities. I abuse notation by using the aggregate terms (D, D*) to describe the household’s problem.
There is no deposit insurance or bailot guarantees in the baseline model. These are evaluated as policy interventions in section 6.
This can also be interpreted as a leverage threshold
The Technical Appendix is available online at https://sites.google.com/site/anlari/files/Technical_Appendix
(19) differs slightly from (3) as it is from the perspective of an individual bank. L represents aggregate bank lending which is taken as given by the representative bank.
Observe that there is no deposit market mark-up in the safe region of the deposit demand schedule. This is because banks face a horizontal deposit demand schedule in this region as their deposits become perfectly substitutable with safe assets.
Implicitly, this is a complementary slackness condition for an occassionally binding non-negativity constraint b > 0. This constraint never binds under the gambling strategy due to the higher valuation of domestic sovereign bonds. An equivalent constraint for lending (l ≥ 0) is also slack at all times since ql declines in response to a fall in l.
I abstain from mixed equilibria, as this would complicate the model solution significantly without yielding any interesting insights in addition to those provided by analyzing symmetric equilibria. Note also that the candidate equilibria described here, and the conditions under which they are valid, would remain unchanged even when mixed equilibria are taken into account.
Under risk neutrality, bank deposits are priced at their expected value and the curvature of the deposit demand schedule is such that the mark-up
Observe that the rate of change in the deposit supply schedule changes direction. This occurs at
In the definiton for
The parameter regions under which the safe equilibrium with the selected bs value exists fully encompasses that of safe equilibria with lower bs values. In other words, whenever the safe equilibria with lower bs values exist, so does the selected equilibrium, which is identical to them in all other aspects.
This mechanism becomes even stronger when the solvency constraint binds, since the downward pivot in the deposit demand schedule under bad sentiments leads to a tightening of the solvency constraint as shown in the third panel of Figure 12.
The number of banks, and the bankers that manage them are constant over time. Insolvent banks are replaced with a new bank that has zero net worth. Bankers that exit from solvent banks are replaced with new bankers which do not contribute to net worth.
The consumption of portion (1 - ψ) of profits and overhead costs ω serve to prevent the accumulation of infinite net worth by banks in the steady state after sovereign default. The former aspect is standard in dynamic financial models while the latter is necessitated by the excess profits banks make due to imperfect competition. Overhead costs are waived when π < ω so as to ensure that they never drive the bank into insolvency or affect the recovery rate θ on deposits.
Global games constitutes an alternative approach to sunspots in resolving multiple equilibria that creates an endogenous relationship between economic fundamentals and equilibrium selection. This approach, however, is not implementable in the context of the multiplicity considered in this paper since the strategic complementary is between banks and households rather, and takes place through a market mechanism that is capable of aggregating diverse beliefs. To see this, consider the introduction of a private signal to households about
The immediate recovery in productivity only serves to simplify the exposition. This can be replaced with any continuation path for productivity as long as there is perfect foresight about it.
There is no need take a stance on when and whether the government returns to sovereign bond markets as long as there is no further default risk. If the government is able to issue bonds, they are priced at qb = q* and banks are indifferent to holding them.
Solving the household’s problem when q* differs from the discount factor β is trivial but leads to a balanced growth path for consumption rather than a steady state value. I abstain from this since it leads to additional complication without yielding any insights of interest.
In the small open economy setting, the markets for goods and sovereign bonds are cleared through trade with foreign agents. Therefore, there is no need to explicitly include the clearing conditions for these markets in the equilibrium definition.
I use bonds with a remaining maturity of 3 months due to the quarterly calibration of the model. While the standard benchmark for measuring sovereign default risk is the yield/CDS spreads on 10 year bonds, it is not possible to extract quarter-on-quarter default probabilities from these measures without imposing additional restrictions on the yield curve.
This implies a relatively high output cost of default compared to the previous literature. It is worth noting, however, that the calibration for θl can be reconciled with lower output costs with the introduction of bankruptcy costs or real frictions that limit the ability of firms to decrease salary costs following sovereign default. Note also that, under the baseline calibration, the parameter restrictions in (15) are satisfied for a wide range of recovery rates θl ∈ [0:59;0:99]. The qualitative results presented throughout the paper, including the non-emptiness of the multiple equilibria region, remain valid at all points within this range.
The relationship between the mark-up and the steady state price of loans is given by (71). I match this with pre-crisis interest rates in order to isolate the excess return due to market power.
This stems from the lack of other types of aggregate risk within the model environment. It can, however, be interpreted as the reduced form outcome of a richer environment with capital regulation based on risk-weighted assets. In this environment, capital requirements faced by a bank depend on the risk-weight attached to its portfolio. For assets with non-sovereign risk, positive risk weights align the bank’s incentives towards following a safe strategy. If sovereign bonds have zero risk-weight, Sovereign bonds, on the other hand, have a zero risk-weight, then gambling is only possible in the presence of sovereign default risk. The preferential treatment for sovereign bonds described here approximately reflects the regulatory framework in the Euro area (Bank for International Settlements, 2013).
To be precise, the payoff is independent of P (S) when the solvency constraint is binding, which is the case at the boundary of net worth
Recall that the economy immediately moves to the steady state following sovereign default. The impulse responses in Figure 15 correspond to a timeline where, in each successive period, it is revealed that the government remains solvent.
Recall that banks take household sentiments as given when deciding on their optimal strategies.
Household and bank values are higher under the safe equilibrium at all times despite the sharp decline in output. With regard to bank values, this finding is due to the increase in net worth under the safe equilibrium. Households place a higher value on the safe equilibrium due to risk aversion. Although the deleveraging by banks causes an initial decline in wages, it precludes a haircut on deposits in the event of sovereign default. Since consumption is already low following sovereign default due to the decline in productivity, risk averse households place a high value on avoiding the haircut. Note that this is true regardless of bank net worth as the size of a haircut on deposits under the gambling equilibrium is proportionate to the decline in bank lending in the safe equilibrium. Furthermore, due to the slow increase in net worth under the gambling equilibrium, this calculus is not affected by the extent of sovereign risk in the current period, but rather the cumulative probability of default until exit from the multiplicity region, which is significantly higher.
Although the latter containts non-depository liabilities which are not directly present in the model, the nature of deposits as a choice variable captures the optimal leverage decision of banks.
I abstain from collateral requirements on debt issued to the central bank. In practice, collateral requirements do not preclude the form of gambling considered here as long as risky domestic sovereign debt is eligible as collateral. This is the case with LTROs since the ECB’s decision to suspend collateral eligibility requirements for sovereign debt issued by distressed Euro area countries (European Central Bank, 2012). In this context, placing a haircut on sovereign debt pledged as collateral is equivalent to a reduction in
This is true unless the liquidity provided by the central bank exceeds total bank revenues under sovereign default. The restriction
Figure 18 provides an example of this where the solvency constraint remains slack. It is also possible for the solvency constraint to become binding as shown in the third panel of Figure 12. In this case, liquidity provision leads to a relaxation of the solvency constraint.
The changes in the deposit demand schedule and the bank’s problem are similar to the two period model. I relegate the relevant expressions to Appendix C in the interest of brevity.
The equilibrium allocation in the steady state after sovereign default is independent of
The LTROs had a 3 year maturity with an early repayment option after 1 year (European Central Bank, 2011). In the context of the model, exercising the early repayment option is equivalent to choosing
When the policy expires at T + 1, the extended model becomes identical to the baseline model. Therefore, future expectations at T for
I calibrate T = 12 in line with LTROs and set
The impulse responses under good sentiments, and those for loan interest rates are excluded as they remain identical to the baseline case in Figure 15.
This does not necessarily need to take the form of an explicit arrangement where depositors have greater seniority. When the central bank has priority in debt repayments, providing the liquidity schedule above under bad sentiments completely crowds out deposit funding. Without deposits to act as a buffer, bank insolvency results in losses for the central bank.
Note that targeted liquidity provision differs from the targeted longer-term refinancing operations (TLTROs) implemented by the ECB in that the latter provide liquidity conditional on bank lending. In the setting here, liquidity provision conditional on l does not affect incentives to gamble since banks have the ability to further increase their leverage to purchase sovereign bonds after satisfying the lending conditionality. Therefore, it is largely similar to non-targeted liquidity provision, with the addition that it may lead to a rise in bank lending in the gambling equilibrium when sufficient liquidity is provided along with a risk transfer.
If participation in the deposit insurance and macroprudential regulation scheme is non-voluntary, the failure of the policy may lead to the use of deposit insurance funds in equilibrium. In the region with a unique gambling equilibrium, banks respond to a non-voluntary scheme by following a gambling strategy despite satisfying the regulatory constraint.
As with the safe equilibrium, domestic sovereign bond purchases bs|g and deposits ds|g are indeterminate in this case but have no impact on expected payoff.
The discontinuous jump in μd(γg,d) as deposits ds|g cross the threshold d(μg) leads to the possibility of a fourth case. In this case, net worth is below nr|g but the first order condition (80) associated with the second case leads the bank to select a level of deposits within the threshold ds|g ≤ d γg). The optimal behaviour of the deviating bank, and the associated net worth boundaries can be then be determined by treating the deposit threshold as a binding constraint. I relegate this case to the Technical Appendix, as it does not have an impact on the mechanism or the outcome.