# RDP 2019-10: Emergency Liquidity Injections 6. Model Generalisations and Extensions

One of the simplifying model features maintained in previous sections is that there are no interbank transactions in equilibrium. Interbank transactions are not ruled out by the model structure, but do not occur in symmetric equilibria because the liquidity shock is homogeneous. This is not important for the results. To generate asymmetry in ex ante securities holdings, some banks could expect securities to pay zero net returns in the no-shock outcome, i.e. rs = 0. These banks would hold only cash at date 0, but at date 1 profitably purchase securities from other banks if m* > 0. It is straightforward to show that, for the other banks, the previous results are unchanged – the effect is similar to an upward pivot of LS, which is nested in the baseline model. Banks' liquidity positions affect each other only through the date 1 market return on liquidity, without implications for how distressed banks deal with the authority, aside from potentially changing the amount of intervention required.

In the baseline model, the impact of the shock is endogenous to banks' choices, but the shock itself is not. It is feasible that creditors might withdraw more at date 1 from a bank that has taken more liquidity risk at date 0. This could be incorporated by assuming that the liquidity shock is drawn from f (b, si), where ${s}_{i}^{1}<{s}_{i}^{2}$ implies $f\left(b,{s}_{i}^{2}\right)$ stochastically dominates $f\left(b,{s}_{i}^{1}\right)$ . This adds terms to the marginal return on securities that make existence proofs difficult. However, there is little reason to expect conclusions about the differences between liquidity injection policies to change. The generalisation is likely to simply lower the incentives for liquidity risk-taking under the lending policies, without direct implications for how collateral constraints offset fire sale externalities.

## 6.1 Capital Injection Policies and Policy Combinations

To generate insights about a capital injection policy, the model can be examined from an individiual bank's perspective, holding the behaviour of other banks constant. The policy involves the authority providing banks liquidity in return for a stake in their equity. To incorporate equity into the model, at date 0 each bank is endowed with long-term non-marketable assets that cannot be sold at date 1, and that have positive value at date 2. Denote the date 2 value as ai, which may vary across i. At date 1 the authority provides bank i liquidity in return for a share $\varphi$ of bank i's date 2 value, where $\varphi \left(0\right)=0,{\varphi }^{\prime }>0\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\varphi$ is bound below one.

Assume that the authority only provides bank i liquidity that it cannot raise itself. This removes emphasis from the specific functional form of $\varphi$ , because banks do not try to balance the marginal costs across date 1 liquidity sources. It contrasts with the lending policies in Sections 3 and 4, into which banks self-select. Indeed, in reality capital injections have often been targeted at specific banks, whereas lending policies have been offered on a broader basis, through auctions, standing facilities, or voluntary take-up of government guarantees (see Appendix A for examples).

Under the capital injection policy C, bank i's pay-off is

(25) $Π i C ( s i , s −i )= a i +( 1+λ )( l+ s i r s ) +λ( ∫ 0 b ¯ ( s i , s −i ) ( l− s i m*( b, s −i )−b ) 1 1−m*( b, s −i ) f( b )db+ ∫ b ¯ ( s i , s −i ) l −ϕ( b+ s i m*( b, s −i )−l ) a i f( b )db )$

Proposition 7. Under the capital injection policy, any interior equilibrium is characterised by liquidity risk si being weakly decreasing in long-term assets ai.

Corollary 8. The capital injection policy cannot leave a bank with a negative date 2 pay-off whereas lending policies can.

Proposition 7 and Corollary 8 reflect the same intuition – when the authority buys a stake in banks' equity, banks with low equity have little to repay. The policy is unable to disincentivise their liquidity risk-taking, and, despite being close to insolvency, does not leave them insolvent ex post. Avoiding ex post insolvency has costs and benefits. Work on gambling for resurrection argues that low-profitability banks tend to take excessive risks (e.g. Freixas and Rochet 2008, chapter 9). On the other hand, if external creditors' behaviour is endogenous to banks' ex post pay-offs, heightened concerns about ex post solvency could amplify the date 1 liquidity distress (e.g. Rochet and Vives 2004; Morris and Shin 2016). In this case, by Corollary 8, a capital injection may help to stave off further runs. He and Krishnamurthy (2013) also argue that a capital injection can improve banks' access to credit.

To acknowledge this potential benefit, the model could be generalised to permit the authority to combine the capital injection policy with either an unsecured or secured lending policy. If banks are allowed to choose the proportion of their liquidity deficit that is funded by the capital injection, an immediate implication is that no banks would be insolvent at date 2. That is, banks' date 2 profit is positive if they fund the deficit completely through the capital injection (Corollary 8), so date 2 pay-off under their optimal choice must be at least as high. The policy combination would preserve the benefits of secured lending over unsecured lending if collateral constraints still bind for high liquidity shocks. If this would not occur with h = 0, it could be induced with h > 0. Formal treatment is left for future work.

## 6.2 Penalty Rates and Heterogeneous Securities

The conclusions in this paper relate to lending policies that charge penalty rates, a topic with a small literature of its own. Bagehot (1920) highlights the benefit of deterring excessive expansion of the authority's balance sheet; Fischer (1999) explains that penalty rates disincentivise liquidity risk-taking; further reading includes Freixas, Rochet and Parigi (2004), Rochet and Vives (2004) and Castiglionesi and Wagner (2012). As Rochet and Vives (2004) demonstrate, penalty rates can have the undesired effect of discouraging private creditors by lowering banks' expected future profits. This topic is briefly discussed in Section 6.1.

Following Bagehot (1920), penalty rates in this paper are defined relative to outside funding options; that is, relative to securities market liquidity. It is interesting to think about the case in which rP is penalising relative to some securities but not others. Consider a heterogeneous range of securities ranked by liquidity risk and indexed by j, such that j = 1, …, J. Specifically, for equal LD > 0 in each securities market, market illiquidity is ranked ${m}_{1}^{*}>{m}_{2}^{*}>...>{m}_{J}^{*}$ . Fix b, s and the lending policy interest rate rP, defining mP such that rP = mP/(1 –mp) and 0 < mP < 1. For securities with ${m}_{j}^{*}<{m}_{P}$ , rates are penalising and banks prefer selling them over emergency borrowing. For securities with ${m}_{j}^{*}\ge {m}_{P}$ , banks prefer borrowing from the authority, which binds these securities' market illiquidity above at mP.

Assume a bank is liquidity deficient, so it cannot raise enough liquidity by selling all its securities. Under an unsecured lending policy, it sells all its securities with ${m}_{j}^{*}<{m}_{P}$ . It is indifferent between selling securities with ${m}_{j}^{*}={m}_{P}$ and borrowing, so its borrowing ei is somewhere between its remaining liquidity needs when selling all these securities and when selling none of them. Under a secured lending policy it must provide collateral to the authority. It starts by providing securities with ${m}_{j}^{*}={m}_{P}$ , but if these are not sufficient, additionally provides securities whose sales generate the largest losses. In this case there is some $\overline{m}\le {m}_{P}$ for which it sells all securities with ${m}_{j}^{*}<\overline{m}$ and provides the rest as collateral. The implication is that, relative to an unsecured lending policy, the collateral constraints reduce market illiquidity for the most illiquid securities that are accepted as collateral.