# Uncertainty and Liquidity Crises Arvind Krishnamurthy Northwestern University

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Uncertainty and Liquidity Crises Arvind Krishnamurthy Northwestern University 1

Crisis theory • Liquidation models – Diamond-Dybvig bank runs – Balance sheet/asset price feedback effects – Lender of last resort policy • Uncertainty and crises – Current subprime crisis – Unknowns and immeasurable risk 2

The two issues for any theory • Why is there “insufficient” liquidity? – – Insufficient in the sense that asset prices reflect liquidity premia Or, Difficult to trade assets Too few market participants Market participants have too few resources to trade • What are appropriate policy responses? – Why intervene? – Channels for policies. Optimal policy. 3

Outline • Liquidation model – Bank runs – Financial markets/capital constraints • Uncertainty and crises – Motivation – Theoretical framework – Policy insights 4

Liquidity model • Investors (continuum) A and B own one unit of an asset at date 0 • Intermediary (bank/market-maker/trading desk) provides price support at date 1: – 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: – q. A , q. B 5

Fundamental equilibrium at date 1 • One of four states – No shocks: – A shock: – B shock: – A and B shocks: P=1 P=1 P = L/2 • Date 0 price: – P 0 = 1 – (1 – L/2) q. A q. B – Liquidity premium = (1 – L/2) q. A q. B 6

Bank run a la Diamond-Dybvig • Bank imposes sequential service constraint: – If all agents liquidate, first liquidators get 1, until all of L is used up – Agents after L get 0 if liquidate, or P<1 if they hold on past date 1 • Now, liquidation equilibrium is possible even with no fundamental liquidity shocks – If an investor conjectures that everyone else will liquidate, he will try to get in line first … • LLR policy: Inject (2 – L) into bank if everyone liquidates 7

Financial market version • Define the “equity net worth” of an investor as W = P – D 0 • Suppose date 1 holdings are subject to a capital/collateral constraint X 1 P < m W • Define Y = 1 – X 1 as amount liquidated • If constraint binds: Y = 1 – m + D 0 / P • Lower price induces more liquidation … lowers price further … 8

• P = 1 is equilibrium if Y is small P = L / 2 Y or Y = L/2 P • If D 0 is large, liquidation curve shifts right • If fundamental liquidity shock affects either agent, liquidation curve shifts Y • Either case, multiple equilibria Y = 1 – m + D 0 / P P=1 P 9

Liquidation externality • Familiar liquidity spiral: lower prices induces liquidations, which lowers prices further • Amplification mechanism (volatility) • Multiple equilibria • Policy response: Increase L in the event of large liquidations – Shifts out supply, dampens the liquidity spiral 10

Crisis Policy • Rule out bad equilibrium via LLR • Reduce balance sheet pressures by easing credit 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 – Ex-ante agents will “underinsure” against crises • In this context, agents will choose too much D 0 – Policy aims to correct the underinsurance problems • Ex-ante leverage limitation in the model. • See for example, Caballero-Krishnamurthy on Emerging Markets 12

Literature • Bank Run Models: Diamond-Dybvig (JPE 1983), Allen-Gale (MANY), Morris-Shin (AER 1998), Rochet-Vives (JEEA 2004) • Financial Markets: Allen-Gale (AER 1994), Kiyotaki-Moore (JPE 1997), Morris-Shin (ROF 2004), Gromb-Vayanos (JFE 2001), Brunnermeier-Pedersen (RFS 2007) • Ex-ante Liquidity/Risk Management: Holmstrom-Tirole (JPE 1998), Caballero-Krishnamurthy (JME 2001), Diamond. Rajan (JF 2005) • Dynamic GE Models: Xiong (JFE 2001), Vayanos (2007), He. Krishnamurthy (2007) 13

He-Krishnamurthy • Asset market: d. D(t) = g dt + d. Z(t) D(t) • Asset market investors/intermediaries with CRRA utility: Can raise outside funds < m w(t) • Households: lenders, savers, etc. (fill in the gaps) • Calibrate m to hedge fund data; match average intermediary leverage; household labor income; g and to stock market. 14

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Crisis Recovery Experiment Start model at 20% risk premium state and simulate until economy reaches: Transit to 15% 12. 5% 10% 7. 5% 6% Time (yrs) 0. 23 0. 46 1. 02 2. 62 5. 91 Increment (yrs) 0. 23 0. 56 1. 60 3. 29 16

Recap • So far, liquidation model • Next, Uncertainty and Crises 17

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 18

Ellsberg Paradox • Urn A: 50 red balls and 50 green balls • Urn B: unknown proportions of red and green balls. • Subject is offered a gamble: – Choose either urn A or urn B, and choose a color to bet on. • If subjective prob(red, B) < ½, then subject should choose B and green. Or, vice-versa – But subjects always strictly prefer urn A • Agents act to reduce ambiguity 19

Ellsberg Paradox (2) • Urn with 30 red balls and 60 others, which are either green or black balls • Gamble A: Choose red or green. – If P(green|green or black)< ½, choose red • Gamble B: Choose [red or black] or [green or black] – If chose red on gamble A, should choose red again • Subjects choose red in A and [green or black] in B • Agents act to reduce ambiguity 20

Modeling: • Standard expected utility – max{c} EQ 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 21

Financial markets context • Uncertainty kicks-in when Q is large – Agents don’t know, and have diffuse priors – Current bout of CDO uncertainty • The “min” in the objective function emphasizes the tails of the distribution – Worst-case decision rules. VAR as an important input into decisions • Conjecture: Principal-agent issues become magnified when there is uncertainty 22

Uncertainty in the baseline model • Recall, agents may receive liquidity shocks that makes them sell assets at date 1 – For example, they realize losses on subprime investments and have to reoptimize portfolios Shock probabilities are q. A , q. B • Suppose agents are uncertain about the correlation between their liquidity shocks of A and B. • ρ (A, B) ϵ [0, 1] n 23

Examples • Counterparty risk: Will bank have enough liquidity to deliver on price support? • How widespread are the subprime losses? Where are the losses buried? • Even if A can assess his own risks (q. A), what about other risks (i. e. q. B ), that end up affecting agent A? 24

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 0 price: – P 0 = 1 – (1 – L/2) q – Liquidity premium = (1 – L/2) q 25

Compare to baseline case • One of four states – No shocks: – A shock: – B shock: – A and B shocks: P=1 P=1 P = L/2 • Date 0 price: – P 0 = 1 – (1 – L/2) q. A q. B – Liquidity premium = (1 – L/2) q. A q. B • Uncertainty magnifies the importance of the liquidation event: order(q) versus order(q 2) 26

Quantity interpretation • Suppose the asset was a project that required an initial outlay from investors and purchase of liquidity insurance from the bank – Total cost = 1 • We can interpret the equilibrium as follows: – Since each agent effectively thinks the world is either (no shock) or (A and B shocks) – A requires the bank to put aside L/2 to fully back A liquidity events. Vice-versa for B – Overcollateralization eliminates counterparty risk. 27

Quantity interpretation • Suppose the asset was a project that required an initial outlay from investors and purchase of liquidity insurance from the bank – Total cost = 1 • Each group of agents will require the bank to put aside L/2 to fully back their own liquidity event. • Agents undertake only L/2 of the project • Bank liquidity goes underused in 1 shock states 28

Policy • LLR policy again • Inject liquidity into bank in the event that both shocks hit. – Liquidity premium = (1 – L/2) q – Larger effect on agent’s uncertainty, but CB delivers only with probability q. A q. B – Or, ex-ante quantity response proportional to injection. 29

Uncertainty and crises literature • Caballero-Krishnamurthy (JF 2007) – Collective bias: Individual worst cases cannot simultaneously occur; yet those are the subjective probabilities underlying each investors’ decisions – Flight to quality, liquidity hoarding • Routledge-Zin (working paper): “Model Uncertainty and Liquidity” – Uncertainty leads to trading halt, widening of bid-ask spread • Equity premium puzzle (aggregates versus idiosyncratic): – Ambiguity/ Uncertainty/ Robustness: Hansen-Sargent, Epstein. Schneider, Maenhout, Kleshchelski … – Extreme events: Barro, Weitzman 30

Other Crises • Claim: This is more general than subprime – Most important crises have uncertainty at their core 1. 1970 Penn Central Default 2. 1993/1994 MBS market meltdown 3. 1998 LTCM episode 31

Penn Central Default • 1970 Penn Central default on $82 m of prime rated commercial paper – CP market not as mature as today • “Ratings” were not fine tuned • Back-up liquidity facilities (standard practice today) did not exist • Default spooked money-market investors – Re-evaluate credit models, don’t trust ratings – Investors stopped buying CP completely • Fed stepped in to encourage banks to buy CP 32

Mercury Finance in 1997 • Mercury Finance defaults on $500 m of CP – Much larger in real terms than Penn Central • Surprise at the time, but it quickly became clear that this was a case of fraudulent accounting in Mercury Finance – No effects on the CP market 33

MBS Market in 1993 • MBS securitization in late 1980 s, as banks shift mortgages off balance sheet 1600 MBS Outstanding ($bn) 1400 1200 1000 800 600 400 200 0 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 • Valuations dependent on assumed consumer prepayment model – But, relatively short time series to calibrate 34

Prepayment surprise • 1993: Large wave of consumer prepayments – Not predicted by the historical models – Model uncertainty, searching for right model • “Worst-case” prepayment models and pricing in the interim 35

1998 Hedge Fund Crisis 70 Total Hedge Funds AUM ($bn) 60 50 40 30 20 10 0 1991 1992 1993 1994 1995 1996 1997 • Hedge funds still relatively new in 1998 • Substantial growth over the 1990 s 36

Correlations in 1998 • High crisis correlations – Even sophisticated investors such as LTCM had not anticipated this in their risk management • We now know that hedge funds had similar strategies and had filled up a similar asset space – Same marginal investors across many asset markets – Liquidations simultaneous across many markets • But at the time, hedge funds did not know this and certainly creditor banks did not understand this point. 37

Amaranth in 2007 • $5 bn loss (>> LTCM loss) • Liquidations … • But clearly the result of a specific trading mistake • Almost no reaction across the hedge fund asset space 38

Other uncertainty events • 1987 Stock market crash – What was equilibrium with portfolio insurance? • 9/11 – Was there more? How much would financial markets be affected 39

Financial Innovation is about the New • Financial innovations are complex even to sophisticated market participants • Risk management of an unknown product – Learning – Model risk – Interpreting and acting on outcomes that the model does not expect • Mistakes and uncertainty seem inevitable 40

Ex-ante Policy • In liquidity externality model, it was to reduce date 0 leverage • More generally, this is about incentivizing better ex-ante risk management • But how does the central bank know what better risk management is for a new innovation? 41

1998 Hedge Fund Crisis • Pre-crisis risk management: Stress testing based on historical correlations • Crisis: Liquidity shortages cause unusual comovement. • Post-crisis risk management: Stress testing scenarios include liquidity events 42

1987 Stock Market Crash • Insurance Strategies: Synthetic Puts/ Portfolio Insurance • Pre-1987: Compare implied volatilities on outof-the-money put options to at-the money options (Bates 2000) – Out-of-the moneys have 3% higher implied vols • Post-1987: Same comparison – Out-of-the moneys have 10% higher implied vols • The crisis led agents to better understand the true costs of portfolio insurance 43

Ex-ante Policy: Some conjectures • Policing new innovations, these are the trouble spots • Put in place systems that allow for quick information revelation 44

Summary • The Uncertainty element – Can help to understand aspects of crises – Rationalize LLR, but now due to benefit of uncertainty reduction – Need to rethink crisis prevention strategies 45

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