# Amplification Mechanisms in Liquidity Crises Arvind Krishnamurthy Northwestern University 1.

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Amplification Mechanisms in Liquidity Crises Arvind Krishnamurthy Northwestern University 1

Amplification Losses on Subprime Mortgages (Fall 07 est.) – At most \$500 bn Decline in world stock market (Sep 08 to Oct 08) – Close to \$26,000 bn Expected output losses (IMF forecast) – \$4,700 bn 2

Amplification Mechanisms I am going to describe two financial mechanisms that have played an important role in the crisis 1.Balance sheet amplification 2.Uncertainty amplification I omit … – Subprime was the trigger for a real estate bubble bursting – Aggregate demand effects 3

Liquidity model Investors (continuum) A and B own one unit of an asset at date s Intermediary (bank/market-maker/trading desk) provides price support at date t>s: – Promises to provide liquidity to sellers at P=1 – But, Bank has only 2 > L > 1 units of liquidity Investors may receive shocks that require them to liquidate: – φ A, φ B 4

Fundamental equilibrium at date t One of four states – No shocks: P = 1 – A shock: P = 1 – B shock:P = 1 – A and B shocks:P = L/2 Date s price: – P s = 1 – (1 – L/2) φ A φ B – Liquidity discount = (1 – L/2) φ A φ B 5

Balance Sheet Considerations Define the “equity net worth” of an investor as W = P t – D s Suppose date t holdings are subject to a capital/collateral constraint m Θ t < W 1 – Θ t is amount liquidated if constraint binds: 1 – Θ t = 1 - (P t – D s ) /m 6

7 P t = 1 LtLt L t = 1 – (P t - D s )/2m E1 E2 E3 Consider states (A) or (B) P = 1 is equilibrium if L is small If D s is large, liquidation curve shifts up and right Or, larger fundamental liquidity shock, liquidation curve shifts up and right Or, m increases, twists liquidation function All cases, multiple equilibria

8 P t = 1 LtLt L t = 1 – (P t - d s )/2m E1 E2 E3 Policy Response: Add liquidity (increase L)

9 P t = 1 LtLt L t = 1 – (P t - d s )/2m E1 E2 E3 Policy Response: Discount loans at m * < m

10 P t = 1 LtLt L t = 1 – (P t - d s )/2m E1 E2 E3 Policy Response: Buy distressed assets

Crisis Policy 1.Liquidity injection 2.Buying troubled assets 3.Discount lending 4.Equity injections … 11

Ex-ante Policy If we push the model further (I wont here), there is another policy that pops up: – Ex-post externalities that agents don’t internalize ex-ante Over-leveraging in the financial sector Ex-ante leverage limitation. 12

Recap So far, liquidation model Next, Uncertainty and Crises 13

Uncertainty Subprime crisis: – Complex CDO products, splitting cash flows in unfamiliar ways – Substantial uncertainty about where the losses lie – But less uncertainty about the direct aggregate loss (small) Knightian uncertainty, ambiguity aversion, uncertainty aversion, robustness preferences 14

Modeling: Standard expected utility – max {c} E P u(c) – P refers to the agent’s subjective probability distribution Modeling ambiguity/uncertainty/robustness: – max {c} min {Q ϵ Q } E Q u(c) – Q is the set of probability distributions that the agent entertains 15

Uncertainty in the baseline model Recall, agents may receive liquidity shocks that makes them sell assets at date t Shock probabilities are φ A, φ B Suppose agents are uncertain about the correlation between their liquidity shocks of A and B. ρ (A,B) ϵ [0, 1] 16

Worst-case decision rules max {c} min {Q ϵ Q } E Q u(c) Worst-cases for A (and B) is ρ (A,B) = 1 Agents subjective probs only consider two states No shocks: P = 1 A and B shocks together:P = L/2 Date s price: P s = 1 – (1 – L/2) φ Liquidity discount = (1 – L/2) φ 17

Compare to baseline case One of four states – No shocks: P = 1 – A shock: P = 1 – B shock:P = 1 – A and B shocks:P = L/2 Date s price: – P s = 1 – (1 – L/2) φ A φ B – Liquidity discount = (1 – L/2) φ A φ B Uncertainty magnifies the importance of the liquidation event: order(φ) versus order(φ 2 ) 18

Crisis Policy LLR policy again Inject liquidity into bank in the event that both shocks hit. – Liquidity discount = (1 – L/2) φ – Larger effect on agent’s uncertainty, but CB delivers only with probability φ A φ B 19

Ex-ante Policy In liquidity externality model, it was to reduce date s leverage More generally, this is about incentivizing better ex-ante risk management But does the central bank really know better? – Especially when it comes to new financial products – History … everyone is blindsided in the same way 20

Ex-ante Policy Policing new innovations, these are the trouble spots – Regulations slow new innovations 21

Summary Two financial amplification mechanisms – Interactions Crisis policies are similar Ex-ante policies are different – Regulate leverage of financial sector – Regulate growth in particular of financial innovation 22

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