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Systemic risk in the repo market.
Alexander ShkolnikUC [email protected]
IPAM. Systemic Risk and Financial Networks. March 25, 2015.Joint work with Robert Anderson, Kay Giesecke and Lisa Goldberg.
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The repurchase agreement (repo)
โข A spot sale of a security and a simultaneous forward agreement torepurchase at a later date (๐ - repo rate, โ - haircut).
โข Securities used: Treasuries, Bonds, MBS, ABS, other (AAA-rated).
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Repo market
โข Hedge funds โ a typical borrower.
โข Money market funds โ a typical cash investor.
โข Dealer banks (Goldman, Citigroup, Merrill Lynch, Barclays, PNPParibas, etc.) are the intermediaries.
โข The Federal reserve uses repos to implement monetary policy.
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Primary dealers' net repo financing
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Related literature
โข Financial networks: Allen & Gale (2000), Eisenberg & Noe (2001),Acemoglu, Ozdaglar & Tahbaz-Selehi (2013), Glasserman & Young(2014) and many others.
โข Secured lending: Dang, Gorton & Holmstrom (2013), Zhang (2013),Lee (2013), Eren (2014), Martin, Skeie & von Thadden (2014).
โข Instability of credit: Hawtrey (1923), Hawtrey (1934), Schumpeter(1934), Minsky (1957), Minsky (1967) and others.โ One bank's cash outflow is another's inflow.โ Spending of one bank induces spending at another, and so on ...
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Modeling & analysis overview
โข A dynamic model of the repo market.โ Perspective: modeler has the same information set as the market,
โ ๐ฝ = {โฑ๐ }๐ โฅ0 - information filtration observed by the market.
โข Write SDEs tomodel repo events occuring on a network.
SDEExpectation
โน ODEJacobian
โน Stability
โข Analysis: derive spectra of ODE system Jacobian:โ out-of-equilibrium system trajectories,
โ equilibria: stability, chaotic behaviour, etc.
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Model construction (purchase only)
โข Network of ๐ โฅ 1 dealer banks. At time ๐ โฅ 0,
๐ ๐๐ - units of collateral posted by ๐ in repo,
๐ผ๐ (๐) - name of ๐ 's counterparty,
๐ผ๐๐ - size of ๐'s collateral pool.
โข SDE model takes the form
ฮ๐ผ๐๐ = โฮ๐ ๐
๐ +๐
โ๐=1
๐{๐ผ๐ (๐)=๐}ฮ๐ ๐๐ (1)
โข Take expectation to obtain ODEs.
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Probabilistic modeling assumptions(Intuitive general principles which are empiricallya supported.)
โข Borrow in proportion to collateral pool size.
๐ ๐โ โ โซ
โ
0๐ผ๐
๐ ๐๐ is a martingale. (2)
โข Lend in proportion to cash available. On {ฮ๐ ๐๐ = 1},
๐ (๐ผ๐ (๐ ) = ๐ | โฑ๐ โ) โ ๐๐๐ โ ๐ฃ๐๐ (3)
where ๐๐๐ is the cash balance of dealer ๐ at time ๐ โฅ 0.
(๐ฃ๐๐ - probability both parties agree to contact.)
a(Kirk, McAndrews, Sastry & Weed 2014)
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Deterministic laws
โข Integrate SDE, take expectation, apply assumptions and differentiate:
[๐๐
๐๐ ]=
[โ1 ๐ด/๐ถ1 โ๐ด/๐ถ ] [
๐๐
๐๐ ](4)
for ๐ = 1, โฆ , ๐ where at time ๐ โฅ 0
๐๐(๐ ) = ๐[๐ผ๐๐ ]
๐๐(๐ ) = ๐[๐๐๐ ]
๐ด = โ๐๐=1 ๐ผ๐ (total system assets)
๐ถ = โ๐๐=1 ๐๐ (total system cash)
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Purchase market stability๐ถ - total cash, ๐ด - total assets.
Theorem. Trajectories {(๐๐(๐ ), ๐๐(๐ ))}๐ โฅ0 converge to a globally stableequilibrium (๐โ
๐ , ๐โ๐ ) = lim๐ โโ(๐๐(๐ ), ๐๐(๐ )) where
๐โ๐ = ๐ด/๐ถ
1 + ๐ด/๐ถ ๐๐ (0) (5)
๐โ๐ = (๐ถ/๐ด) ๐โ
๐ (6)
on (โ+, โ+)\{(0, 0), (๐ด, ๐ถ)}.
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Repurchase & Rehypothecation
โข Rehypothecation is the re-use of collateral.
โข Unlimited rehypothecation implies (no repurchase) system debt
๐ท(๐ ) โฅ ๐ด๐ > ๐ถ (system cash) (7)
(equals when ๐ = 0) where ๐ด is the total system assets.
โข Sum over all dealer banks to get system rates.
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Rehypothecation by primary dealers
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Model assumptions (repurchase)
โข Dealers repurchase only when cash is available, i.e.
๐พ ๐โ โ โซ
โ
0๐๐
๐ ๐{๐๐>0,๐ฟ๐๐ >0} ๐๐ is a martingale. (8)
where ๐ฟ๐ = ๐ ๐ โ ๐พ ๐ has not been repurchased and
๐พ ๐๐ - units of security repurchased by ๐,
๐ ๐๐ - units of collateral posted by ๐ in repo.
๐๐๐ - cash holdings of dealer ๐.
โข Also assume repo interest is paid daily.
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Deterministic lawsโข Integrate SDE, take expectation, apply assumptions and differentiate:
[๐๐
๐๐ ]=
[โ1 1 + ๐ด/๐ถ
1 + โ + ๐ โ(1 + ๐ ) โ (1 โ โ)๐ด/๐ถ ] [๐๐
๐๐ ]+ โฆ
for ๐ = 1, โฆ , ๐ where at time ๐ โฅ 0
๐๐(๐ ) = ๐[๐ผ๐๐ ]
๐๐(๐ ) = ๐[๐๐๐ ]
๐ด = โ๐๐=1 ๐ผ๐ (total system assets)
๐ถ = โ๐๐=1 ๐๐ (total system cash)
๐ = repo rate
โ = haircut
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Repo market stability๐ถ - total cash, ๐ด - total assets, ๐๐ โ (0, 1) - demand to collect repointerest, โ - haircut, ๐ - repo rate.
Theorem. Trajectories {(๐๐(๐ ), ๐๐(๐ ))}๐ โฅ0 converge to a globally stableequilibrium (๐โ
๐ , ๐โ๐ ) = lim๐ โโ(๐๐(๐ ), ๐๐(๐ )) where
๐โ๐ = (1 + ๐ด/๐ถ )๐โ
๐ โ ๐๐๐ถ (9)
๐โ๐ = (โ/๐ )๐๐๐ถ โ ๐๐ (0)
โ/๐ โ ๐ด/๐ถ (10)
on (โ+, โ+) if and only if
1 โค ๐ด/๐ถ < โ/๐ (bounded leverage) (11)
or ๐ด/๐ถ < 1 and ๐ โค 0.
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Infinite-debt equilibrium๐ถ - total cash, ๐ด - total assets, โ - haircut, ๐ - repo rate.
โข Bounded leverage condition
1 โค ๐ด/๐ถ < โ/๐ (bounded leverage) (12)
โข Netted inter-dealer debt is finite but system debt โ โ.
โข Sum over all dealer banks to obtain system rates.
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Finite-debt equilibria (intuition)
โข Suppose there is only a single security in the market.
โข The security performs a random walk over the network moving fromdealer ๐ to dealer ๐ at event time ๐๐ with probability
๐๐๐๐โ๐ถ (13)
โข On each move it leaves a loan on dealer ๐'s balance sheet.
โข Repurchase at rate proportional the number of loans.
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Model assumptions (repurchase)
โข Dealers repay in proportion to loans held, i.e.
๐พ ๐โ โ โซ
โ
0๐ฟ๐
๐ ๐๐ is a martingale. (14)
where ๐ฟ๐ = ๐ ๐ โ ๐พ ๐ has not been repurchased and
๐พ ๐๐ - units of security repurchased by ๐,
๐ ๐๐ - units of collateral posted by ๐ in repo.
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Deterministic lawsโข Integrate SDE, take expectation, apply assumptions and differentiate:
[๐๐
๐๐ ]=
[โ1 โ ๐ง ๐ด/๐ถ
1 + ๐ง(1 + ๐ ) โ๐ด/๐ถ ] [๐๐
๐๐ ]โ ๐ง๐๐ (0)
[โ1
(1 + ๐ ) ]
for ๐ = 1, โฆ , ๐ where at time ๐ โฅ 0
๐๐(๐ ) = ๐[๐ผ๐๐ ]
๐๐(๐ ) = ๐[๐๐๐ ]
๐ด = โ๐๐=1 ๐ผ๐ (total system assets)
๐ถ = โ๐๐=1 ๐๐ (total system cash)
๐ = repo rate
๐ง = fraction in repo
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Repo market instability๐ถ - total cash, ๐ด - total assets, ๐ง โ (0, 1] - fraction in repo, ๐ - repo rate.
Theorem. Trajectories {(๐๐(๐ ), ๐๐(๐ ))}๐ โฅ0 converge to a globally stableequilibrium (๐โ
๐ , ๐โ๐ ) = lim๐ โโ(๐๐(๐ ), ๐๐(๐ )) where
๐โ๐ = ๐๐ (0) (15)
๐โ๐ = (๐ถ/๐ด) ๐โ
๐ (16)
on (โ+, โ+) only if ๐ โค 0. If ๐ > 0, equilibrium (15)-(16) is unstable.
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Remedies for instability
โข Solution 1: Allow for (Fed) open market operations.โ Sell assets to soak up excess reserves in parts of network.โ Purchase assets to inject liquidity in rest of network.
โข Solution 2: Ensure return ๐ > 0 on portfolio with ๐ = ๐ โ ๐ satisfies๐ง
1 โ ๐ง < 1 + ๐ 1 + ๐ . (17)
โข The equilibrium associated with (17) is a finite-debt equilibrium.
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Nonlinear phenomena
โข Suppose only one dealer demands a haircut โ. For constant ๐ง > 0.
๐ = โ๐ + ๐ด๐ถ ๐ โ ๐ + ๐๐ถ (18)
๐ = ๐ โ (1 โ โ) ๐ด๐ถ ๐ + โ๐ ๐
๐ถ โ (1 + ๐ )(๐ โ ๐ โ ๐) โ ๐๐ถ (19)
โ = ๐ง โ ๐๐ (20)
โข Two (non-trivial) equilibria corresponding to low and high regimes
ยฑ(โ๐ง๐ด/๐ถ, โ๐ง๐ถ/๐ด, 0) (21)
โข Model exhibits complex and chaotic behavior.
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Stable orbit about high regime
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Stable orbit converges to high regime
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Stable orbit leaves high regime
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Oscillations between the two regimes
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Extensions
โข Variance and asymptotic analysis,
โข Dealer defaults,
โข Restricted network topology (e.g. central clearing),
โข Heterogeneous repo contracts,
โข Some simple emperical evaluation.
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Some conclusions
โข Model of cash and collateral flows with survival contraints.
โข The framework provides an elegant way analyze the behaviour of avery complex system.
โข The model is highly sensitive without amplification through shocks.
โข Aggregate variables (e.g. ๐ด/๐ถ) directly related to market instability.
Questions
Systemic risk in the repo market. March 25, 2015. 28-1ReferencesAcemoglu, Daron, Asuman Ozdaglar & Alireza
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Systemic risk in the repo market. March 25, 2015. 28-2Glasserman, Paul & Peyton Young (2014), How likely
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