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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)
β’ A spot sale of a security and a simultaneous forward agreement torepurchase at a later date (π - repo rate, β - haircut).
β’ Securities used: Treasuries, Bonds, MBS, ABS, other (AAA-rated).
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Repo market
β’ Hedge funds β a typical borrower.
β’ Money market funds β a typical cash investor.
β’ Dealer banks (Goldman, Citigroup, Merrill Lynch, Barclays, PNPParibas, etc.) are the intermediaries.
β’ The Federal reserve uses repos to implement monetary policy.
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Primary dealers' net repo financing
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Related literature
β’ Financial networks: Allen & Gale (2000), Eisenberg & Noe (2001),Acemoglu, Ozdaglar & Tahbaz-Selehi (2013), Glasserman & Young(2014) and many others.
β’ Secured lending: Dang, Gorton & Holmstrom (2013), Zhang (2013),Lee (2013), Eren (2014), Martin, Skeie & von Thadden (2014).
β’ Instability of credit: Hawtrey (1923), Hawtrey (1934), Schumpeter(1934), Minsky (1957), Minsky (1967) and others.β One bank's cash outflow is another's inflow.β Spending of one bank induces spending at another, and so on ...
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Modeling & analysis overview
β’ A dynamic model of the repo market.β Perspective: modeler has the same information set as the market,
β π½ = {β±π }π β₯0 - information filtration observed by the market.
β’ Write SDEs tomodel repo events occuring on a network.
SDEExpectation
βΉ ODEJacobian
βΉ Stability
β’ Analysis: derive spectra of ODE system Jacobian:β out-of-equilibrium system trajectories,
β equilibria: stability, chaotic behaviour, etc.
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Model construction (purchase only)
β’ Network of π β₯ 1 dealer banks. At time π β₯ 0,
π ππ - units of collateral posted by π in repo,
πΌπ (π) - name of π 's counterparty,
πΌππ - size of π's collateral pool.
β’ SDE model takes the form
ΞπΌππ = βΞπ π
π +π
βπ=1
π{πΌπ (π)=π}Ξπ ππ (1)
β’ Take expectation to obtain ODEs.
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Probabilistic modeling assumptions(Intuitive general principles which are empiricallya supported.)
β’ Borrow in proportion to collateral pool size.
π πβ β β«
β
0πΌπ
π ππ is a martingale. (2)
β’ Lend in proportion to cash available. On {Ξπ ππ = 1},
π (πΌπ (π ) = π | β±π β) β πππ β π£ππ (3)
where πππ is the cash balance of dealer π at time π β₯ 0.
(π£ππ - probability both parties agree to contact.)
a(Kirk, McAndrews, Sastry & Weed 2014)
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Deterministic laws
β’ Integrate SDE, take expectation, apply assumptions and differentiate:
[ππ
ππ ]=
[β1 π΄/πΆ1 βπ΄/πΆ ] [
ππ
ππ ](4)
for π = 1, β¦ , π where at time π β₯ 0
ππ(π ) = π[πΌππ ]
ππ(π ) = π[πππ ]
π΄ = βππ=1 πΌπ (total system assets)
πΆ = βππ=1 ππ (total system cash)
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Purchase market stabilityπΆ - total cash, π΄ - total assets.
Theorem. Trajectories {(ππ(π ), ππ(π ))}π β₯0 converge to a globally stableequilibrium (πβ
π , πβπ ) = limπ ββ(ππ(π ), ππ(π )) where
πβπ = π΄/πΆ
1 + π΄/πΆ ππ (0) (5)
πβπ = (πΆ/π΄) πβ
π (6)
on (β+, β+)\{(0, 0), (π΄, πΆ)}.
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Repurchase & Rehypothecation
β’ Rehypothecation is the re-use of collateral.
β’ Unlimited rehypothecation implies (no repurchase) system debt
π·(π ) β₯ π΄π > πΆ (system cash) (7)
(equals when π = 0) where π΄ is the total system assets.
β’ Sum over all dealer banks to get system rates.
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Rehypothecation by primary dealers
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Model assumptions (repurchase)
β’ Dealers repurchase only when cash is available, i.e.
πΎ πβ β β«
β
0ππ
π π{ππ>0,πΏππ >0} ππ is a martingale. (8)
where πΏπ = π π β πΎ π has not been repurchased and
πΎ ππ - units of security repurchased by π,
π ππ - units of collateral posted by π in repo.
πππ - cash holdings of dealer π.
β’ Also assume repo interest is paid daily.
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Deterministic lawsβ’ Integrate SDE, take expectation, apply assumptions and differentiate:
[ππ
ππ ]=
[β1 1 + π΄/πΆ
1 + β + π β(1 + π ) β (1 β β)π΄/πΆ ] [ππ
ππ ]+ β¦
for π = 1, β¦ , π where at time π β₯ 0
ππ(π ) = π[πΌππ ]
ππ(π ) = π[πππ ]
π΄ = βππ=1 πΌπ (total system assets)
πΆ = βππ=1 ππ (total system cash)
π = repo rate
β = haircut
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Repo market stabilityπΆ - total cash, π΄ - total assets, ππ β (0, 1) - demand to collect repointerest, β - haircut, π - repo rate.
Theorem. Trajectories {(ππ(π ), ππ(π ))}π β₯0 converge to a globally stableequilibrium (πβ
π , πβπ ) = limπ ββ(ππ(π ), ππ(π )) where
πβπ = (1 + π΄/πΆ )πβ
π β πππΆ (9)
πβπ = (β/π )πππΆ β ππ (0)
β/π β π΄/πΆ (10)
on (β+, β+) if and only if
1 β€ π΄/πΆ < β/π (bounded leverage) (11)
or π΄/πΆ < 1 and π β€ 0.
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Infinite-debt equilibriumπΆ - total cash, π΄ - total assets, β - haircut, π - repo rate.
β’ Bounded leverage condition
1 β€ π΄/πΆ < β/π (bounded leverage) (12)
β’ Netted inter-dealer debt is finite but system debt β β.
β’ Sum over all dealer banks to obtain system rates.
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Finite-debt equilibria (intuition)
β’ Suppose there is only a single security in the market.
β’ The security performs a random walk over the network moving fromdealer π to dealer π at event time ππ with probability
ππππβπΆ (13)
β’ On each move it leaves a loan on dealer π's balance sheet.
β’ Repurchase at rate proportional the number of loans.
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Model assumptions (repurchase)
β’ Dealers repay in proportion to loans held, i.e.
πΎ πβ β β«
β
0πΏπ
π ππ is a martingale. (14)
where πΏπ = π π β πΎ π has not been repurchased and
πΎ ππ - units of security repurchased by π,
π ππ - units of collateral posted by π in repo.
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Deterministic lawsβ’ Integrate SDE, take expectation, apply assumptions and differentiate:
[ππ
ππ ]=
[β1 β π§ π΄/πΆ
1 + π§(1 + π ) βπ΄/πΆ ] [ππ
ππ ]β π§ππ (0)
[β1
(1 + π ) ]
for π = 1, β¦ , π where at time π β₯ 0
ππ(π ) = π[πΌππ ]
ππ(π ) = π[πππ ]
π΄ = βππ=1 πΌπ (total system assets)
πΆ = βππ=1 ππ (total system cash)
π = repo rate
π§ = fraction in repo
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Repo market instabilityπΆ - total cash, π΄ - total assets, π§ β (0, 1] - fraction in repo, π - repo rate.
Theorem. Trajectories {(ππ(π ), ππ(π ))}π β₯0 converge to a globally stableequilibrium (πβ
π , πβπ ) = limπ ββ(ππ(π ), ππ(π )) where
πβπ = ππ (0) (15)
πβπ = (πΆ/π΄) πβ
π (16)
on (β+, β+) only if π β€ 0. If π > 0, equilibrium (15)-(16) is unstable.
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Remedies for instability
β’ Solution 1: Allow for (Fed) open market operations.β Sell assets to soak up excess reserves in parts of network.β Purchase assets to inject liquidity in rest of network.
β’ Solution 2: Ensure return π > 0 on portfolio with π = π β π satisfiesπ§
1 β π§ < 1 + π 1 + π . (17)
β’ The equilibrium associated with (17) is a finite-debt equilibrium.
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Nonlinear phenomena
β’ Suppose only one dealer demands a haircut β. For constant π§ > 0.
π = βπ + π΄πΆ π β π + ππΆ (18)
π = π β (1 β β) π΄πΆ π + βπ π
πΆ β (1 + π )(π β π β π) β ππΆ (19)
β = π§ β ππ (20)
β’ Two (non-trivial) equilibria corresponding to low and high regimes
Β±(βπ§π΄/πΆ, βπ§πΆ/π΄, 0) (21)
β’ 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
β’ Variance and asymptotic analysis,
β’ Dealer defaults,
β’ Restricted network topology (e.g. central clearing),
β’ Heterogeneous repo contracts,
β’ Some simple emperical evaluation.
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Some conclusions
β’ Model of cash and collateral flows with survival contraints.
β’ The framework provides an elegant way analyze the behaviour of avery complex system.
β’ The model is highly sensitive without amplification through shocks.
β’ Aggregate variables (e.g. π΄/πΆ) 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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Eisenberg, Larry & Thomas Noe (2001), `Systemicrisk in financial systems', Management Science47(2), 236--249.
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Minsky, Hyman (1957), Central Banking and MoneyMarket Changes, Quarterly Journal of Eco-nomics.
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