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:
– φA , φB
4
Fundamental equilibrium at date t
• One of four states
– No shocks:
– A shock:
– B shock:
– A and B shocks:
P=1
P=1
P=1
P = L/2
• Date s price:
– Ps = 1 – (1 – L/2) φA φB
– Liquidity discount = (1 – L/2) φA φB
5
Balance Sheet Considerations
• Define the “equity net worth” of an investor as
W = P t – 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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Consider states (A) or (B)
• P = 1 is equilibrium if L is small
• If Ds is large, liquidation curve shifts up
and right
Pt = 1
• Or, larger fundamental liquidity shock,
liquidation curve shifts up and right
E1
• Or, m increases, twists liquidation function
E2
• All cases, multiple equilibria
E3
Lt = 1 – (Pt - Ds)/2m
Lt
7
Policy Response: Add liquidity (increase L)
Pt = 1
E1
E2
Lt = 1 – (Pt - ds)/2m
E3
Lt
8
Policy Response: Discount loans at m* < m
Pt = 1
E1
E2
Lt = 1 – (Pt - ds)/2m
E3
Lt
9
Policy Response: Buy distressed assets
Pt = 1
E1
E2
Lt = 1 – (Pt - ds)/2m
E3
Lt
10
Crisis Policy
1.
2.
3.
4.
Liquidity injection
Buying troubled assets
Discount lending
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} 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
15
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
17
Compare to baseline case
• One of four states
– No shocks:
– A shock:
– B shock:
– A and B shocks:
P=1
P=1
P=1
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)
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 φA φB
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
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