The LOLR Policy and its Signaling Effect in a Time of Crisis

Abstract

When a government implements an LOLR policy during a crisis, creditors can infer a bank’s quality by whether the bank borrows government loans. We establish a formal model to study an LOLR policy in the presence of this signaling effect. We find that three equilibria exist: a separating equilibrium where only low quality banks borrow from the government and two pooling equilibria where both high and low quality banks do and do not borrow from the government. Further, we find that the government’s lending rate serves an important signaling role and that hiding the identity of the banks that borrow government loans tends to encourage banks to do so. We also find two welfare effects of the LOLR policy: the liquidation cost saving and moral hazard. Depending on which effect dominates, the optimal LOLR policy differs.

Appendices

Appendix A: Derivations for r_{M, G} and $\hat {r}_{M}$ in PBB

An equilibrium market rate exists only when: (1) it will give creditors an expected riskless rate of zero; (2) both types of banks are willing to borrow at this rate; and (3) it can actually be paid by H-type banks and L-type banks in the up state. The equilibrium rate derived from Criterion 1 might be too high due to a low h or $\hat {h}$ such that Criteria 2 and 3 are not satisfied. In this case, a market freeze occurs and no equilibrium rate exists.

Based on Criterion 1, an equilibrium market rate r_{M, G} should satisfy:

$$ \begin{array}{@{}rcl@{}} 1 &=& h(1+r_{M,G})+(1-h)\left[p (1+r_{M,G})+(1-p)\frac{AR_{L}}{D_{0}}\right], \end{array} $$

(10)

or equivalently Eq. 3. We impose Condition (1) such that Criteria 2 and 3 hold. For Criterion 2 to hold, an equilibrium market rate must be lower than $\frac {R_{H}}{\gamma }-1$ as proved in Lemma 2. Otherwise, banks will stop borrowing in the market. For Criterion 3 to hold, an equilibrium rate cannot exceed $\frac {AR_{H}-L_{G}(1+r_{G})}{D_{0}-L_{G}}-1$, which is the maximum return rate an H-type bank or an L-type bank in the up state can afford.

Similarly, we find that $\hat {r}_{M}$ takes the same form of r_{M, G} with h being replaced by $\hat {h}$, because it induces a zero expected return rate for creditors with a prior belief of $\hat {h}$. Criterion 2 means that $\hat {r}_{M}$ must be lower than $\frac {R_{H}}{\gamma }-1$. Criterion 3 means that it cannot exceed $\frac {AR_{H}}{D_{0}}-1$, which is the maximum return rate an H-type bank or an L-type bank in the up state can afford when deviating. Thus, we derive Condition (2).

Appendix B: Proof of Proposition 1

An H-type bank’s expected equity when following the equilibrium strategy is

$$ \begin{array}{@{}rcl@{}} E e_{H,G} = q e_{H,G,s_{i}=s_{H}} + (1-q)e_{H,G, s_{i}=\emptyset}, \end{array} $$

(11)

where

$$ \begin{array}{@{}rcl@{}} e_{H,G,s_{i}=s_{H}}&=&AR_{H}-L_{G}(1+r_{G})-(D_{0}-L_{G}), \end{array} $$

(12)

$$ \begin{array}{@{}rcl@{}} e_{H,G, s_{i}=\emptyset} &=& AR_{H} - L_{G}(1+r_{G})-(D_{0}-L_{G})(1+r_{M,G}) \end{array} $$

(13)

are the bank’s equity with and without a signal, respectively. The subscript “G” means that banks borrow the government’s loans, and “s_i = ∅” means no signal.

An L-type bank’s expected equity when following the equilibrium strategy is

$$ \begin{array}{@{}rcl@{}} E e_{L,G} = q p e^{u}_{L,G, s_{i}=s_{L}}+(1-q) p e^{u}_{L,G, s_{i}=\emptyset}, \end{array} $$

(14)

where

$$ \begin{array}{@{}rcl@{}} e^{u}_{L,G, s_{i}=s_{L}} &=& \max\left\{0, \left( A-\frac{D_{0}-L_{G}}{\gamma}\right) R_{H} - L_{G}(1+r_{G})\right\}, \end{array} $$

(15)

$$ \begin{array}{@{}rcl@{}} e^{u}_{L,G, s_{i}=\emptyset}&=&e_{H,G, s_{i}=\emptyset} \end{array} $$

(16)

are its equity in the up state with and without a signal, respectively.

An H-type bank’s expected equity when deviating is

$$ \begin{array}{@{}rcl@{}} E \hat{e}_{H,NG} = q \hat{e}_{H,NG,s_{i}=s_{H}}+ (1-q)\hat{e}_{H,NG,s_{i}=\emptyset}, \end{array} $$

(17)

where

$$ \begin{array}{@{}rcl@{}} \hat{e}_{H,NG,s_{i}=s_{H}}&=& AR_{H} -D_{0}, \end{array} $$

(18)

$$ \begin{array}{@{}rcl@{}} \hat{e}_{H,NG,s_{i}=\emptyset}&=&AR_{H}-D_{0}(1+\hat{r}_{M}) \end{array} $$

(19)

are its equity with and without a signal, respectively. The subscript “NG” means that the banks do not borrow any of the government’s loans.

An L-type bank’s expected equity when deviating is

$$ \begin{array}{@{}rcl@{}} E \hat{e}_{L,NG} = (1-q)p \hat{e}^{u}_{L,NG,s_{i}=\emptyset} , \end{array} $$

(20)

where

$$ \begin{array}{@{}rcl@{}} \hat{e}^{u}_{L,NG,s_{i}=\emptyset}=\hat{e}_{H,NG,s_{i}=\emptyset}=AR_{H}-D_{0}(1+\hat{r}_{M}) \end{array} $$

(21)

is its equity in the up state without a signal. Its equity with a signal is simply zero.

Using Eqs. 11 and 17, we get the no-deviation condition for an H-type bank:

$$ \begin{array}{@{}rcl@{}} E e_{H,G}-E \hat{e}_{H,NG}=(1-q)D_{0}[\hat{r}_{M}-r_{M,G}]+L_{G}[(1-q)r_{M,G}-r_{G}]>0. \end{array} $$

(22)

Thus, we derive Condition (4).³¹

Using Eqs. 14 and 20, we find that

$$ \begin{array}{@{}rcl@{}} E e_{L,G}-E \hat{e}_{L,NG}&=& q p \max\left\{0, \left( A-\frac{D_{0}-L_{G}}{\gamma}\right) R_{H} - L_{G}(1+r_{G})\right\}\\ &&+(1-q)p[D_{0}(\hat{r}_{M}-r_{M,G})+L_{G}(r_{M,G}-r_{G})]. \end{array} $$

(23)

If $E e_{H,G}-E \hat {e}_{H,NG}> 0$, we find that

$$ \begin{array}{@{}rcl@{}} E e_{L,G}-E \hat{e}_{L,NG}&=&p(E e_{H,G}-E \hat{e}_{H,NG})+pqL_{G}r_{G}+ \\ &&q p \max\left\{0, \left( A-\frac{D_{0}-L_{G}}{\gamma}\right) R_{H} - L_{G}(1+r_{G})\right\}>0. \end{array} $$

(24)

Thus, we prove that as long as H-type banks’ no-deviation condition is satisfied, L-type banks’ no-deviation condition must be satisfied too. Thus, we prove Result (1).

When $\hat {h}\ge h$, $\hat {r}_{M}\le r_{M,G}$, and the first term in Eq. 22 is no greater than zero. Thus, the second term must be positive for $Ee_{H,G}-E\hat {e}_{H,NG}$ to be positive. This means r_G < (1 − q)r_{M, G}. Thus, we prove Result (2).

To prove Result (3), we need to find the first order derivatives of $E e_{H,G}-E \hat {e}_{H,NG}$ with respect to L_G, r_G, q, and $\hat {h}$ at $E e_{H,G}-E \hat {e}_{H,NG}=0$. Let $\hat {\Phi }=E e_{H,G}-E \hat {e}_{H,NG}$. We find that:

$$ \begin{array}{@{}rcl@{}} \left.\frac{\partial \hat{\Phi}}{\partial L_{G}}\right\rvert_{\hat{\Phi}=0}=(1-q)r_{M,G}-r_{G}. \end{array} $$

(25)

We have proved that when $\hat {h}\ge h$, the first term in Eq. 22 is no greater than zero. Thus, the second term, (1 − q)r_{M, G} − r_G, must be no less than zero at $\hat {\Phi }=0$. That is, $\left .\frac {\partial \hat {\Phi }}{\partial L_{G}}\right \rvert _{\hat {\Phi }=0}\ge 0$, which means that PBB is more likely to exist with a higher L_G.

Note that $\left .\frac {\partial \hat {\Phi }}{\partial r_{G}}\right \rvert _{\hat {\Phi }=0}=-L_{G}<0$. That is, PBB is more likely to exist with a lower r_G.

In addition, $\left .\frac {\partial \hat {\Phi }}{\partial q}\right \rvert _{\hat {\Phi }=0}=D_{0}(r_{M,G}-\hat {r}_{M})-L_{G} r_{M,G}$. Using Eq. 22, we find that at $\hat {\Phi }=0$, since r_G ≥ 0, $L_{G}r_{M,G}\ge D_{0}(r_{M,G}-\hat {r}_{M})$(the condition holds without equality if r_G > 0). Thus we find that $\left .\frac {\partial \hat {\Phi }}{\partial q}\right \rvert _{\hat {\Phi }=0}\le 0$. That is, PBB is more likely to exist with a lower q.

Finally, $\left .\frac {\partial \hat {\Phi }}{\partial \hat {r}_{M}}\right \rvert _{\hat {\Phi }=0}=(1-q)D_{0}>0$. Note $\frac {\partial \hat {r}_{M}}{\partial \hat {h}}<0$, because a more optimistic belief off the equilibrium path lowers the market rate off the equilibrium path. Thus, $\left .\frac {\partial \hat {\Phi }}{\partial \hat {h}}\right \rvert _{\hat {\Phi }=0}<0$. That is, PBB is more likely to exist with a lower $\hat {h}$.

Appendix C: Proof of Proposition 4

First, we find the threshold level of h_i, $\bar {h}$. If each bank follows a switching strategy of choosing the risky asset if and only if $h_{i}>\bar {h}$, a bank’s expected date 2 equity from choosing the safe asset is given by:

$$ \begin{array}{@{}rcl@{}} Ee^{s}=\pi e^{s,no shock}+(1-\pi)Ee^{s,shock}. \end{array} $$

(26)

Here e^{s, noshock} = AR_H − D₀ denotes the bank’s equity value on date 2 without a crisis. In this case, all the banks are solvent and roll over their debts at the riskless rate of zero. The Ee^{s, shock} denotes the bank’s expected date 2 equity with a crisis. In this case, proportion $\bar {h}$ of the banks is H-type, and proportion $1-\bar {h}$ of the banks is L-type. A bank’s expected date 2 equity from choosing the safe asset is given by:

$$ \begin{array}{@{}rcl@{}} Ee^{s,shock} &= & AR_{H}-L_{G}(1+r_{G})-(D_{0}-L_{G})[(1-q)(1+r_{M,G,\bar{h}})+q], \end{array} $$

(27)

where $r_{M,G,\bar {h}}$ is given by Eq. 8. Again, we confine to the case where $\bar {h}$ is sufficiently high such that an equilibrium market rate exists and there is no market freeze.

A bank’s expected date 2 equity from choosing the risky asset is given by:

$$ \begin{array}{@{}rcl@{}} Ee^{r}=\pi e^{r,no shock}+(1-\pi)Ee^{r,shock}. \end{array} $$

(28)

Here e^{r, noshock} = AR_T − D₀ denotes the bank’s date 2 equity without a crisis. Ee^{r, shock}, the bank’s expected date 2 equity in a crisis, is specified in Eq. 14.³²

The bank with $h_{i}=\bar {h}$ must be indifferent between the two choices. Thus, we have:

$$ \begin{array}{@{}rcl@{}} a_{0}+a_{1} \bar{h}=Ee^{s} - Ee^{r}. \end{array} $$

(29)

Note that:

$$ \begin{array}{@{}rcl@{}} Ee^{s}-Ee^{r}&=&\pi A(R_{H}-R_{T})+ (1-\pi)\left\{\vphantom{A-\frac{D_{0}-L_{G}}{\gamma}}(1-(1-q)p)AR_{H}- \right.\\ && D_{0}[(1-q)(1-p)(1+r_{M,G,\bar{h}})+q]+L_{G}[(1-q)(1-p)r_{M,G,\bar{h}}- \\ && \left.(1-(1-q)p)r_{G}]-qp\max\left\{0, \left( A-\frac{D_{0}-L_{G}}{\gamma}\right) R_{H} - L_{G}(1+r_{G})\right\} \right\}. \end{array} $$

(30)

Thus, we prove Eq. 7.

In the Technical Appendix, we prove that the RHS in the above equation is strictly increasing and concave in h. We also prove that to ensure a stable, unique and interior solution to $\bar {h}\in (0,1)$, we can impose the following conditions: (1) At h = h^f, Ee^s − Ee^r > a₀ + a₁h^f, where h^f is the threshold level of h below which a market freeze occurs.³³ (2) At h = 1, Ee^s − Ee^r < a₀ + a₁h. With these conditions, Ee^s − Ee^r − a₀ − a₁h is strictly decreasing in h around the equilibrium $\bar {h}$.

We confine our attention to symmetric switching strategies where each bank chooses the risky asset if and only if h_i is above a threshold level. Additionally, we assume that parameter values satisfy the above conditions such that a unique interior solution to $\bar {h}$ exists. We find that there exists a symmetric switching strategy perfect Bayesian equilibrium where a bank chooses the risky asset if and only if $h_{i}>\bar {h}$. This switching strategy $\bar {h}$ is optimal for each bank because given that each bank follows this switching strategy, any bank with $h_{i}>\bar {h}$ has the unchanged RHS in Eq. 29 and the higher LHS. This is because both Ee^s and Ee^r do not change in h_i. Thus, it is indeed optimal for the bank to choose the risky asset. Meanwhile, any bank with $h_{i}<\bar {h}$ has the unchanged RHS and the lower LHS. Thus, it is indeed optimal for the bank to choose the safe asset.