Amplification Mechanisms in Liquidity Crises

Arvind KrishnamurthyNorthwestern University

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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

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Amplification Mechanisms

• I am going to describe two financial mechanisms that have played an important role in the crisis1. Balance sheet amplification2. Uncertainty amplification

• I omit …– Subprime was the trigger for a real estate

bubble bursting– Aggregate demand effects

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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:– φA , φB

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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:– Ps = 1 – (1 – L/2) φA φB

– Liquidity discount = (1 – L/2) φA φB

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Balance Sheet Considerations• Define the “equity net worth” of an investor as

W = Pt – Ds

• Suppose date t holdings are subject to a capital/collateral constraint

m Θt < W• 1 – Θt is amount liquidated if constraint binds:

1 – Θt = 1 - (Pt – Ds) /m

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Pt = 1

Lt

Lt = 1 – (Pt - Ds)/2m

E1

E2

E3

Consider states (A) or (B)

• P = 1 is equilibrium if L is small

• If Ds 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

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Pt = 1

Lt

Lt = 1 – (Pt - ds)/2m

E1

E2

E3

Policy Response: Add liquidity (increase L)

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Pt = 1

Lt

Lt = 1 – (Pt - ds)/2m

E1

E2

E3

Policy Response: Discount loans at m* < m

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Pt = 1

Lt

Lt = 1 – (Pt - ds)/2m

E1

E2

E3

Policy Response: Buy distressed assets

Crisis Policy

1. Liquidity injection2. Buying troubled assets3. Discount lending4. Equity injections …

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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.

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Recap

• So far, liquidation model

• Next, Uncertainty and Crises

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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

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Modeling:

• Standard expected utility– max{c} EP u(c)

– P refers to the agent’s subjective probability distribution

• Modeling ambiguity/uncertainty/robustness:– max{c} min{Q ϵ Q } EQ u(c)

– Q is the set of probability distributions that the agent entertains

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Uncertainty in the baseline model• Recall, agents may receive liquidity

shocks that makes them sell assets at date t• Shock probabilities are φA , φB

• 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 } EQ 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:• Ps = 1 – (1 – L/2) φ• Liquidity discount = (1 – L/2) φ

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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:– Ps = 1 – (1 – L/2) φA φB

– Liquidity discount = (1 – L/2) φA φB

• Uncertainty magnifies the importance of the liquidation event: order(φ) versus order(φ2)

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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 φA φB

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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

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Ex-ante Policy

• Policing new innovations, these are the trouble spots– Regulations slow new innovations

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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

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