# RDP 2019-10: Emergency Liquidity Injections 3. Liquidity Injection through Unsecured Lending

Under an unsecured lending policy, bank i's pay-off function is

(9) $Π i u =( 1−λ )( l+ s i r s )+λ ∫ 0 l [ ( l−b )− s im ( b )m*( b )− e i ( b )( 1+ r u ) ] f( b )db$

subject to the liquidity constraint

(10) $s im ( 1−m* )+ e i ≥b+ s i −l$

and the feasibility constraint

(11) $s im ≤ s i$

The first term on the right-hand side of Equation (9) captures the value of liquid assets after no liquidity shock (l + sirs). The second term represents the liquidity shock outcome, and has three components: the liquidity endowment minus what is lost to the shock (lb); the losses (gains) from any securities sales (purchases) in an illiquid market (simm*); and any date 2 repayments to the authority (ei + eiru). The constraint in Equation (10) ensures that bank i's securities liquidation revenue plus its borrowing is at least its cash shortage. The constraint in Equation (11) reflects that bank i cannot sell more securities than it owns.

Securities sales (sim) and borrowing from the authority (ei) are choices made at date 1 that can be expressed as functions of predetermined variables (b,si,s–i,ru). Whenever m* > 0, banks raise only the minimum liquidity required and do so via the cheapest sources available. Raising one unit of liquidity through the securities market requires selling 1/(1 – m*) securities, which would have each had value 1 at date 2. Therefore, borrowing from the authority is more expensive than securities sales if and only if 1 + ru >1/(1 – m*).

Accordingly, if ru is low enough, the lending policy places an upper bound on market liquidity:

(12) $m*≤ r u 1+ r u$

The securities market cannot be in equilibrium at higher m* because banks would reduce selling and instead source funds from the authority. If there is some liquidity shock b at which Equation (12) binds, it also binds at all higher b, because market illiquidity m* is weakly increasing in b (see Lemma 9 in Appendix B). Denote by b the lowest b at which Equation (12) binds, so $\underset{_}{b}\left({s}_{-i},{r}_{u}\right)$ satisfies

(13)

It is possible that ru is high enough such that Equation (12) does not bind at any b, and $\underset{_}{b}$ is not defined by the first case in Equation (13). In this case set $\underset{_}{b}$ at b = l for notational convenience. An illustration of how ru can bind $m*\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}b\ge \underset{_}{b}$ is presented by the red dashed lines in Figure 4.

If $b\ge \underset{_}{b}$ , banks are indifferent between selling securities and borrowing. Each individual bank's selling is indeterminate, but total securities selling is precisely the quantity that, given b and ru, ensures that Equation (12) binds. Accordingly, bank i's liquidity raised through securities has three cases:

(14) $s im ( 1−m* ){ =b+ s i −l if b≤min{ b _ , b ¯ i } = s i ( 1−m* ) if b ¯ i

Note that if ${\overline{b}}_{i}$ defined in Equation (4) is above $\underset{_}{b}$ , then ${\overline{b}}_{i}$ does not represent a threshold at which bank i is forced to borrow, and has no particular relevance for bank i.

Bank i meets its remaining liquidity deficit by borrowing from the authority:

(15) $e i =max{ 0,b−( l− s i )− s im ( 1−m* ) }$

Combining Equations (9), (14) and (15), bank i's pay-off, taking ru as given, can be expressed as

(16) $Π i u ( s i , s −i )=( 1−λ )( l+ s i r s ) +λ( ∫ 0 min{ b _ ( s −i ), b ¯ i ( s i , s −i ) } l− s i m*( b, s −i )−b 1 1−m*( b, s −i ) f( b )db + ∫ min{ b _ ( s −i ), b ¯ i ( s i s −i ) } l ( l− s i m*( b, s −i )−b )( 1− r u )f( b )db )$

The expression lsim* – b captures bank i's spare liquidity, or liquidity deficit if negative. The expression it is multiplied by – either 1/1(1 – m*) or 1 + ru – is the return on liquidity. This is what would be saved (spent) if bank i held one additional (less) unit of cash. If $b<\mathrm{min}\left\{\underset{_}{b},{\overline{b}}_{i}\right\}$ , banks would raise additional liquidity through the securities market, and the return on liquidity is the date 2 value of the securities that must be sold to raise one unit of cash. If $b>{\overline{b}}_{i},\text{\hspace{0.17em}}\text{so}\text{\hspace{0.17em}}l-{s}_{i}m*-b<0,$ bank i is forced to borrow from the authority, and the return on liquidity is the cost of borrowing 1 + ru. If $b>\underset{_}{b}$, then the return on liquidity is 1 + ru = 1/(1 – m*) whether bank i is borrowing out of necessity or choice.

Proposition 1. Among symmetric choice sets s, there is a unique equilibrium ${S}_{u}^{*}$ . If rs is low enough relative to ru for ${S}_{u}^{*} , then ${S}_{u}^{*}$ is strictly decreasing in ru, bound below only at zero.

Each bank chooses si by comparing the returns that securities provide in the normal state (rs) against the expected costs given a liquidity shock. These costs depend on how many securities other banks hold, because higher S tends to generate more selling pressure, and more market illiquidity whenever m* > 0. Higher market illiquidity lowers the expected marginal return to securities, because depressed prices raise the cost of selling securities and similarly the value of spare cash. Accordingly, as S increases, securities' expected returns decrease, generating a fixed point where S equals the optimal si given S, so that acting in line with other banks is the best response to their choices. This fixed point is decreasing in ru, which enters negatively into banks' marginal return to securities.

Lemma 2. Under an unsecured lending policy, if the level of liquidity risk S that maximises collective bank pay-offs is interior, banks' equilibrium liquidity risk is higher.

Banks' excessive liquidity risk-taking is a form of fire sale externality, modelled the same way as Stein (2012). Market illiquidity is determined by banks' collective liquidity risk-taking, but individual banks' contributions are small enough that they do not factor this cost into their pay-off maximisation. Chernenko and Sunderam (forthcoming) provide empirical evidence for this type of externality in US securities markets. In this model the collective optimum is more informative than a social planner solution. This is because the authority is assumed to not care about its own profit; its benefit from high interest rates is that they deter risk-taking. A social planner would circumvent this deterrent, resulting in a meaningless optimum of S = 0 with infinite funds transferred from the authority to banks.

As mentioned earlier in this section, if ru is high enough, there is no b such that Equation (12) binds. Denote by rpen the lowest ru that induces an equilibrium with this outcome. The condition ru > rpen aligns with the interest rates on emergency lending that Bagehot (1920) recommends, which have since been termed ‘penalty rates’. When ru > rpen, sourcing liquidity from the authority is necessarily more costly than through the securities market, so banks limit their emergency borrowing to the minimum required to avoid failure. The condition also implies strict concavity of ${\Pi }_{i}$ , because raising si lowers ${\overline{b}}_{i}$ , which raises the probability of needing to borrow from the authority and paying penalty rates. Therefore, ru >rpen implies both (i) a declining marginal return to si, and (ii) that banks minimise ei to b + sim* – l.

## Footnotes

The model abstracts from parts of a bank's balance sheet that are not directly relevant to the liquidity dynamics being examined. Accordingly, the fact that the liquidity shock b reduces a bank's asset value, as well as its funding liquidity, is of little concern. This could simply be offset by returning b to bank i at date 2. 

This assumes all banks have access to the liquidity injection policy. If not, market illiquidity could be pushed above ru /(1 + ru) by banks without access. In this case the banks with access would not sell any securities and would rely completely on borrowing from the authority. 

The condition also rules out asymmetric equilibria by making banks' pay-offs strictly concave.