Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

The OTC Theory of the Fed Funds Market: A Primer

Gara Afonso

Ricardo Lagos

FRB of New York

New York University

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

The market for federal funds

A market for loans of reserve balances at the Fed.

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

The market for federal funds What’s traded? Unsecured loans (mostly overnight) How are they traded? Over the counter Who trades? Commercial banks, securities dealers, agencies and branches of foreign banks in the U.S., thrift institutions, federal agencies

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

The market for federal funds

Implications

Policy

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Why is the fed funds market interesting? It is an interesting example of an OTC market (Unusually good data is available) Reallocates reserves among banks (Banks use it to o¤set liquidity shocks and manage reserves) Determines the interest rate on the shortest maturity instrument in the term structure Is the “epicenter” of monetary policy implementation

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Why is the fed funds market interesting? It is an interesting example of an OTC market (Unusually good data is available) Reallocates reserves among banks (Banks use it to o¤set liquidity shocks and manage reserves) Determines the interest rate on the shortest maturity instrument in the term structure Is the “epicenter” of monetary policy implementation

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Why is the fed funds market interesting? It is an interesting example of an OTC market (Unusually good data is available) Reallocates reserves among banks (Banks use it to o¤set liquidity shocks and manage reserves) Determines the interest rate on the shortest maturity instrument in the term structure Is the “epicenter” of monetary policy implementation

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Why is the fed funds market interesting? It is an interesting example of an OTC market (Unusually good data is available) Reallocates reserves among banks (Banks use it to o¤set liquidity shocks and manage reserves) Determines the interest rate on the shortest maturity instrument in the term structure Is the “epicenter” of monetary policy implementation

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

In this paper we ...

(1)

Propose an OTC model of trade in the fed funds market

(2)

Use the theory to address some elementary questions, e.g., Positive: How is the fed funds rate determined? Normative: Is the OTC market structure able to achieve an e¢ cient reallocation of funds?

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

The model A trading session in continuous time, t 2 [0, T ], τ

T

t

Unit measure of banks hold reserve balances k (τ ) 2 f0, 1, 2g

fnk (τ )g : distribution of balances at time T

τ

Linear payo¤s from balances, discount at rate r Uk : payo¤ from holding balance k at the end of the session Trade opportunities are bilateral and random (Poisson rate α) Loan and repayment amounts determined by Nash bargaining Assume all loans repaid at time T + ∆, where ∆ 2 R+

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Institutional features of the fed funds market

Model Search and bargaining

Fed funds market

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Institutional features of the fed funds market

Model Search and bargaining

Fed funds market Over-the-counter market

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Institutional features of the fed funds market

Model Search and bargaining

[0, T ]

Fed funds market Over-the-counter market

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Institutional features of the fed funds market

Model

Fed funds market

Search and bargaining

Over-the-counter market

[0, T ]

4:00pm-6:30pm

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Institutional features of the fed funds market

Model

Fed funds market

Search and bargaining

Over-the-counter market

[0, T ]

4:00pm-6:30pm

fnk (T )g

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Institutional features of the fed funds market

Model

Fed funds market

Search and bargaining

Over-the-counter market

[0, T ]

4:00pm-6:30pm

fnk (T )g

Distribution of reserve balances at 4:00pm

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Institutional features of the fed funds market

Model

Fed funds market

Search and bargaining

Over-the-counter market

[0, T ]

4:00pm-6:30pm

fnk (T )g

Distribution of reserve balances at 4:00pm

f Uk g

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Institutional features of the fed funds market

Model

Fed funds market

Search and bargaining

Over-the-counter market

[0, T ]

4:00pm-6:30pm

fnk (T )g

Distribution of reserve balances at 4:00pm

f Uk g

Reserve requirements, interest on reserves...

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Bellman equations rV1 (τ ) + V˙ 1 (τ ) = 0 rV0 (τ ) + V˙ 0 (τ ) = αn2 (τ ) max fV1 (τ )

V0 ( τ )

rV2 (τ ) + V˙ 2 (τ ) = αn0 (τ ) max fV1 (τ )

V2 (τ ) + R¯ (τ ) , 0g

Vk ( 0 ) = U k

R¯ (τ ) , 0g

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Appx.

Bellman equations rV1 (τ ) + V˙ 1 (τ ) = 0 rV0 (τ ) + V˙ 0 (τ ) = αn2 (τ ) max fV1 (τ )

V0 ( τ )

R¯ (τ ) , 0g

rV2 (τ ) + V˙ 2 (τ ) = αn0 (τ ) max fV1 (τ )

V2 (τ ) + R¯ (τ ) , 0g

Vk ( 0 ) = U k R¯ (τ ) = arg max [V1 (τ ) R

= θ [ V2 ( τ ) R¯ (τ )

e

r (τ +∆)

R

V0 (τ )]θ [V1 (τ ) + R

V1 (τ )] + (1

θ ) [ V1 ( τ )

R (τ ) (PDV of repayment)

V2 (τ )]1

V0 (τ )]

θ

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Bellman equations rV1 (τ ) + V˙ 1 (τ ) = 0 rV0 (τ ) + V˙ 0 (τ ) = αn2 (τ ) φ (τ ) θS (τ ) rV2 (τ ) + V˙ 2 (τ ) = αn0 (τ ) φ (τ ) (1

θ ) S (τ )

where: S (τ ) φ (τ ) =

2V1 (τ )

V0 ( τ )

1 if 0 < S (τ ) 0 if S (τ ) 0

V2 ( τ )

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Time-path for the distribution of balances

n˙ 0 (τ ) = αφ (τ ) n2 (τ ) n0 (τ ) n˙ 1 (τ ) =

2αφ (τ ) n2 (τ ) n0 (τ )

n˙ 2 (τ ) = αφ (τ ) n2 (τ ) n0 (τ )

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

De…nition An equilibrium is a value function, V, a path for the distribution of reserve balances, n (τ ), and a path for the distribution of trading probabilities, φ (τ ), such that: (a) given the value function and the distribution of trading probabilities, the distribution of balances evolves according to the law of motion; and (b) given the path for the distribution of balances, the value function and the distribution of trading probabilities satisfy individual optimization given the bargaining protocol.

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Analysis

S˙ (τ ) = δ (τ )

δ (τ ) S (τ )

fr + αφ (τ ) [θn2 (τ ) + (1

θ ) n0 (τ )]g

) S (τ )

δ¯ (τ )

= e

S (0)

δ¯ (τ )

Z τ

S (τ )

2V1 (τ )

0

δ (x ) dx V0 ( τ )

V2 ( τ )

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Analysis

S˙ (τ ) = δ (τ )

δ (τ ) S (τ )

fr + αφ (τ ) [θn2 (τ ) + (1

θ ) n0 (τ )]g

) S (τ )

δ¯ (τ )

= e

S (0)

δ¯ (τ )

Z τ

S (τ )

2V1 (τ )

0

δ (x ) dx V0 ( τ )

V2 ( τ )

Conclusion

Appx.

Intro

Model

Equilibrium

Assumption:

E¢ ciency

Frictionless

S (0)

2U1

Implications

U2

Policy

U0 > 0

) S (τ ) = e

δ¯ (τ )

S (0) > 0 for all τ

) φ (τ ) = 1 for all τ

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

Assumption:

E¢ ciency

Frictionless

S (0)

2U1

Implications

U2

Policy

U0 > 0

) S (τ ) = e

δ¯ (τ )

S (0) > 0 for all τ

) φ (τ ) = 1 for all τ

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

Assumption:

E¢ ciency

Frictionless

S (0)

2U1

Implications

U2

Policy

U0 > 0

) S (τ ) = e

δ¯ (τ )

S (0) > 0 for all τ

) φ (τ ) = 1 for all τ

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

Proposition Suppose 2U1

E¢ ciency

U2

Frictionless

Implications

Policy

n1 ( τ ) n0 ( τ )

=

S (τ )

=

R (τ )

= e r ∆ f β ( τ ) ( U2

β (τ )

Conclusion

U0 > 0. Then the unique equilibrium is:

= n0 ( τ ) + n2 ( T ) n0 ( T ) = 1 2n0 (τ ) + n0 (T ) n2 (T )

n2 ( τ )

Examples

[n 2 (T ) n 0 (T )]n 0 (T ) e α[n2 (T ) n0 (T )](T τ ) n 2 (T ) n 0 (T )

[ n 2 (T )

e

τ ) n (T ) e fr +αθ [n 2 (T ) n 0 (T )]gτ 0 α[n 2 (T ) n 0 (T )]T n (T ) 0

α[n 2 (T ) n 0 (T )](T

n 2 (T ) e

θ [ n 2 (T ) e

U1 ) + [ 1

[1

e

β (τ )] (U1

α[n 2 (T ) n 0 (T )](T

n 2 (T ) e

+

]

U0 )g

τ ) n (T ) e αθ [n 2 (T ) n 0 (T )]τ 0 α[n 2 (T ) n 0 (T )]T n (T ) 0

αθ [n 2 (T ) n 0 (T )]τ

n 2 (T ) e

S (0)

]

n 2 (T ) α[n 2 (T ) n 0 (T )]T n (T ) 0

]

Appx.

Intro

Model

Equilibrium

Proposition Suppose 2U1

E¢ ciency

U2

Frictionless

Implications

Policy

n1 ( τ ) n0 ( τ )

=

S (τ )

=

R (τ )

= e r ∆ f β ( τ ) ( U2

β (τ )

Conclusion

U0 > 0. Then the unique equilibrium is:

= n0 ( τ ) + n2 ( T ) n0 ( T ) = 1 2n0 (τ ) + n0 (T ) n2 (T )

n2 ( τ )

Examples

[n 2 (T ) n 0 (T )]n 0 (T ) e α[n2 (T ) n0 (T )](T τ ) n 2 (T ) n 0 (T )

[ n 2 (T )

e

τ ) n (T ) e fr +αθ [n 2 (T ) n 0 (T )]gτ 0 α[n 2 (T ) n 0 (T )]T n (T ) 0

α[n 2 (T ) n 0 (T )](T

n 2 (T ) e

θ [ n 2 (T ) e

U1 ) + [ 1

[1

e

β (τ )] (U1

α[n 2 (T ) n 0 (T )](T

n 2 (T ) e

+

]

U0 )g

τ ) n (T ) e αθ [n 2 (T ) n 0 (T )]τ 0 α[n 2 (T ) n 0 (T )]T n (T ) 0

αθ [n 2 (T ) n 0 (T )]τ

n 2 (T ) e

S (0)

]

n 2 (T ) α[n 2 (T ) n 0 (T )]T n (T ) 0

]

Appx.

Intro

Model

Equilibrium

Proposition Suppose 2U1

E¢ ciency

U2

Frictionless

Implications

Policy

n1 ( τ ) n0 ( τ )

=

S (τ )

=

R (τ )

= e r ∆ f β ( τ ) ( U2

β (τ )

Conclusion

U0 > 0. Then the unique equilibrium is:

= n0 ( τ ) + n2 ( T ) n0 ( T ) = 1 2n0 (τ ) + n0 (T ) n2 (T )

n2 ( τ )

Examples

[n 2 (T ) n 0 (T )]n 0 (T ) e α[n2 (T ) n0 (T )](T τ ) n 2 (T ) n 0 (T )

[ n 2 (T )

e

τ ) n (T ) e fr +αθ [n 2 (T ) n 0 (T )]gτ 0 α[n 2 (T ) n 0 (T )]T n (T ) 0

α[n 2 (T ) n 0 (T )](T

n 2 (T ) e

θ [ n 2 (T ) e

U1 ) + [ 1

[1

e

β (τ )] (U1

α[n 2 (T ) n 0 (T )](T

n 2 (T ) e

+

]

U0 )g

τ ) n (T ) e αθ [n 2 (T ) n 0 (T )]τ 0 α[n 2 (T ) n 0 (T )]T n (T ) 0

αθ [n 2 (T ) n 0 (T )]τ

n 2 (T ) e

S (0)

]

n 2 (T ) α[n 2 (T ) n 0 (T )]T n (T ) 0

]

Appx.

Intro

Model

Equilibrium

Proposition Suppose 2U1

E¢ ciency

U2

Frictionless

Implications

Policy

n1 ( τ ) n0 ( τ )

=

S (τ )

=

R (τ )

= e r ∆ f β ( τ ) ( U2

β (τ )

Conclusion

U0 > 0. Then the unique equilibrium is:

= n0 ( τ ) + n2 ( T ) n0 ( T ) = 1 2n0 (τ ) + n0 (T ) n2 (T )

n2 ( τ )

Examples

[n 2 (T ) n 0 (T )]n 0 (T ) e α[n2 (T ) n0 (T )](T τ ) n 2 (T ) n 0 (T )

[ n 2 (T )

e

τ ) n (T ) e fr +αθ [n 2 (T ) n 0 (T )]gτ 0 α[n 2 (T ) n 0 (T )]T n (T ) 0

α[n 2 (T ) n 0 (T )](T

n 2 (T ) e

θ [ n 2 (T ) e

U1 ) + [ 1

[1

e

β (τ )] (U1

α[n 2 (T ) n 0 (T )](T

n 2 (T ) e

+

]

U0 )g

τ ) n (T ) e αθ [n 2 (T ) n 0 (T )]τ 0 α[n 2 (T ) n 0 (T )]T n (T ) 0

αθ [n 2 (T ) n 0 (T )]τ

n 2 (T ) e

S (0)

]

n 2 (T ) α[n 2 (T ) n 0 (T )]T n (T ) 0

]

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Proposition Suppose 2U1 U2 U0 > 0. Then, the equilibrium supports an e¢ cient allocation of reserve balances.

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Intuition for e¢ ciency result rV0 (τ ) + V˙ 0 (τ ) = αn2 (τ ) θS (τ ) r λ0 (τ ) + λ˙ 0 (τ ) = αn2 (τ ) S (τ ) rV1 (τ ) + V˙ 1 (τ ) = 0 r λ1 (τ ) + λ˙ 1 (τ ) = 0 rV2 (τ ) + V˙ 2 (τ ) = αn0 (τ ) (1 θ ) S (τ ) r λ2 (τ ) + λ˙ 2 (τ ) = αn0 (τ ) S (τ )

S (τ ) S (τ )

= e = e

δ¯ (τ )

S (0)

δ¯ (τ )

S (0)

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result rV0 (τ ) + V˙ 0 (τ ) = αn2 (τ ) θS (τ ) r λ0 (τ ) + λ˙ 0 (τ ) = αn2 (τ ) S (τ ) rV1 (τ ) + V˙ 1 (τ ) = 0 r λ1 (τ ) + λ˙ 1 (τ ) = 0 rV2 (τ ) + V˙ 2 (τ ) = αn0 (τ ) (1 θ ) S (τ ) r λ2 (τ ) + λ˙ 2 (τ ) = αn0 (τ ) S (τ )

S (τ ) S (τ ) δ¯ (τ )

δ¯ (τ ) = α

Z τ 0

= e = e

[(1

δ¯ (τ )

S (0)

δ¯ (τ )

S (0)

θ ) n2 (z ) + θn0 (z )] dz

0

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result rV0 (τ ) + V˙ 0 (τ ) = αn2 (τ ) θS (τ ) r λ0 (τ ) + λ˙ 0 (τ ) = αn2 (τ ) S (τ ) rV1 (τ ) + V˙ 1 (τ ) = 0 r λ1 (τ ) + λ˙ 1 (τ ) = 0 rV2 (τ ) + V˙ 2 (τ ) = αn0 (τ ) (1 θ ) S (τ ) r λ2 (τ ) + λ˙ 2 (τ ) = αn0 (τ ) S (τ )

S (τ ) S (τ ) δ¯ (τ )

δ¯ (τ ) = α

Z τ 0

= e = e

[(1

δ¯ (τ )

S (0)

δ¯ (τ )

S (0)

θ ) n2 (z ) + θn0 (z )] dz

0

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result rV0 (τ ) + V˙ 0 (τ ) = αn2 (τ ) θS (τ ) r λ0 (τ ) + λ˙ 0 (τ ) = αn2 (τ ) S (τ ) rV1 (τ ) + V˙ 1 (τ ) = 0 r λ1 (τ ) + λ˙ 1 (τ ) = 0 rV2 (τ ) + V˙ 2 (τ ) = αn0 (τ ) (1 θ ) S (τ ) r λ2 (τ ) + λ˙ 2 (τ ) = αn0 (τ ) S (τ )

S (τ ) S (τ ) δ¯ (τ )

δ¯ (τ ) = α

Z τ 0

= e = e

[(1

δ¯ (τ )

S (0)

δ¯ (τ )

S (0)

θ ) n2 (z ) + θn0 (z )] dz

0

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result Equilibrium: Gain from trade as perceived by borrower: θS (τ ) Gain from trade as perceived by lender: (1

θ ) S (τ )

Planner: Each of their marginal contributions equals S (τ ) δ (τ )

δ (τ ) for all τ 2 [0, T ], with “=” only for τ = 0

) The planner “discounts” more heavily than the equilibrium ) S (τ ) < S (τ ) for all τ 2 (0, 1] ) Social value of loan < joint private value of loan

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result Equilibrium: Gain from trade as perceived by borrower: θS (τ ) Gain from trade as perceived by lender: (1

θ ) S (τ )

Planner: Each of their marginal contributions equals S (τ ) δ (τ )

δ (τ ) for all τ 2 [0, T ], with “=” only for τ = 0

) The planner “discounts” more heavily than the equilibrium ) S (τ ) < S (τ ) for all τ 2 (0, 1] ) Social value of loan < joint private value of loan

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result Equilibrium: Gain from trade as perceived by borrower: θS (τ ) Gain from trade as perceived by lender: (1

θ ) S (τ )

Planner: Each of their marginal contributions equals S (τ ) δ (τ )

δ (τ ) for all τ 2 [0, T ], with “=” only for τ = 0

) The planner “discounts” more heavily than the equilibrium ) S (τ ) < S (τ ) for all τ 2 (0, 1] ) Social value of loan < joint private value of loan

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result Equilibrium: Gain from trade as perceived by borrower: θS (τ ) Gain from trade as perceived by lender: (1

θ ) S (τ )

Planner: Each of their marginal contributions equals S (τ ) δ (τ )

δ (τ ) for all τ 2 [0, T ], with “=” only for τ = 0

) The planner “discounts” more heavily than the equilibrium ) S (τ ) < S (τ ) for all τ 2 (0, 1] ) Social value of loan < joint private value of loan

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result Equilibrium: Gain from trade as perceived by borrower: θS (τ ) Gain from trade as perceived by lender: (1

θ ) S (τ )

Planner: Each of their marginal contributions equals S (τ ) δ (τ )

δ (τ ) for all τ 2 [0, T ], with “=” only for τ = 0

) The planner “discounts” more heavily than the equilibrium ) S (τ ) < S (τ ) for all τ 2 (0, 1] ) Social value of loan < joint private value of loan

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result Equilibrium: Gain from trade as perceived by borrower: θS (τ ) Gain from trade as perceived by lender: (1

θ ) S (τ )

Planner: Each of their marginal contributions equals S (τ ) δ (τ )

δ (τ ) for all τ 2 [0, T ], with “=” only for τ = 0

) The planner “discounts” more heavily than the equilibrium ) S (τ ) < S (τ ) for all τ 2 (0, 1] ) Social value of loan < joint private value of loan

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result

Planner internalizes that searching borrowers and lenders make it easier for other lenders and borrowers to …nd partners These “liquidity provision services” to others receive no compensation in the equilibrium, so individual agents ignore them when calculating their equilibrium payo¤s The equilibrium payo¤ to lenders may be too high or too low relative to their shadow price in the planner’s problem: E.g., too high if (1

θ ) S (τ ) > S (τ )

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result

Planner internalizes that searching borrowers and lenders make it easier for other lenders and borrowers to …nd partners These “liquidity provision services” to others receive no compensation in the equilibrium, so individual agents ignore them when calculating their equilibrium payo¤s The equilibrium payo¤ to lenders may be too high or too low relative to their shadow price in the planner’s problem: E.g., too high if (1

θ ) S (τ ) > S (τ )

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intuition for e¢ ciency result

Planner internalizes that searching borrowers and lenders make it easier for other lenders and borrowers to …nd partners These “liquidity provision services” to others receive no compensation in the equilibrium, so individual agents ignore them when calculating their equilibrium payo¤s The equilibrium payo¤ to lenders may be too high or too low relative to their shadow price in the planner’s problem: E.g., too high if (1

θ ) S (τ ) > S (τ )

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Frictionless limit

Proposition Let Q

∑2k =0 knk (T ) = 1 + n2 (T )

n0 ( T ) .

For τ 2 [0, T ], as α ! ∞, ρ (τ ) ! ρ∞ , where 8 r∆ < e ( U1 U 0 ) ∞ 1+ρ = e r ∆ [ θ ( U2 U1 ) + ( 1 : r∆ e ( U2 U1 )

θ ) ( U1

if Q < 1 U0 )] if Q = 1 if 1 < Q.

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Positive implications

The theory delivers:

(1)

Time-varying trade volume

(2)

Time-varying fed fund rate

(3)

Intraday convergence of distribution of reserve balances

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Trade volume

Flow volume of trade at time T

τ:

υ (τ ) = αn0 (τ ) n2 (τ )

Total volume traded during the trading session: υ¯ =

Z T 0

υ (τ ) d τ

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Fed funds rate The gross rate on a loan at time T 1 + ρ ( τ ) = e r ∆ f β ( τ ) ( U2

τ is:

U1 ) + [ 1

with β (τ ) 2 [0, 1] The average daily rate is:

ρ¯ =

1 T

Z T 0

ρ (τ ) d τ

β (τ )] (U1

U0 )g

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Cross-sectional distribution of reserve balances

Let µ (τ ) and σ2 (τ ) denote the mean and variance of the cross-sectional distribution of reserve balances at t = T τ

µ ( τ ) = 1 + n2 ( T )

σ 2 ( τ ) = σ 2 (T )

n0 ( T )

2 [ 2 + n2 ( T )

Q

n0 (T )] [n0 (T )

n0 (τ )]

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

U1 = e U2 = e U0 =

Implications

r∆

(1 + ifr )

r∆

(2 + ifr + ife )

e

r∆

(ifw

Policy

ifr + P w )

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

U1 = e U2 = e U0 =

Implications

r∆

(1 + ifr )

r∆

(2 + ifr + ife )

e

r∆

(ifw

Policy

Examples

Conclusion

ifr + P w )

Proposition ρ (τ ) = β (τ ) ife + [1 1

2

3

β (τ )] (ifw + P w )

where

If n2 (T ) = n0 (T ), β (τ ) = θ If n2 (T ) < n0 (T ), β (τ ) 2 [0, θ ], β (0) = θ and β0 (τ ) < 0 If n0 (T ) < n2 (T ), β (τ ) 2 [θ, 1], β (0) = θ and β0 (τ ) > 0.

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Intraday interest rate in a balanced market

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Appx.

Intraday interest rate in a market with shortage of reserves

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intraday interest rate in a market with excess reserves

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Interest rate and the quantity of reserves

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

k¯ = 1 Two scenarios n0H (T ) , n2L (T )

n0L (T ) , n2H (T )

f0.6, 0.3g

f0.3, 0.6g

Experiments Bargaining Power (θ)

0.1

0.5

0.9

Discount Rate (ifw ) .0050 360

.0075 360

.0100 360

Contact Rate (α)

25

50

100

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Appx.

Bargaining power -5

1.6

x 10

1.00002

θ=0.1 θ=0.5 θ=0.9

1.4

θ=0.1 θ=0.5 θ=0.9

1.000018

1.2

0.8

1.000016

θ=0.1 θ=0.5 θ=0.9

0.7

1.000014

2

0.8

ρ (%)

1

V -V

Surplus

1

1.000012

0.6

0.6

0.5 1.00001

0.4

0.4 1.000008

0.2

0.3 0 16:00

16:30

17:00

17:30

18:00

1.000006 16:00

18:30

16:30

Eastern Time

17:00

17:30

18:00

18:30

16:00

16:30

17:00

17:30

18:00

18:30

18:00

18:30

Eastern Time

Eastern Time

-5

1.6

x 10

1.4

θ=0.1 θ=0.5 θ=0.9

θ=0.1 θ=0.5 θ=0.9

1.000009

0.8

θ=0.1 θ=0.5 θ=0.9

1.2

0.7

1.0000085

2

0.8

ρ (%)

1

V -V

Surplus

1

1.000008

0.6 0.5

0.6

1.0000075

0.4

0.4

1.000007 0.2 0 16:00

0.3

16:30

17:00

17:30

Eastern Time

18:00

18:30

16:00

16:30

17:00

17:30

Eastern Time

18:00

18:30

16:00

16:30

17:00

17:30

Eastern Time

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Appx.

Contact rate -5

1.6

x 10

1.00002

α=25 α=50 α=100

1.4

α=25 α=50 α=100

1.000018

1.2

0.66

1.000016 1

2

0.8

1.000014

ρ (%)

1

0.64

V -V

Surplus

α=25 α=50 α=100

0.7 0.68

1.000012

0.58

1.00001

0.4

0.62 0.6

0.6

0.56

1.000008

0.2

0.54 0 16:00

16:30

17:00

17:30

18:00

1.000006 16:00

18:30

16:30

Eastern Time

17:00

17:30

18:00

18:30

16:00

16:30

17:00

17:30

18:00

18:30

18:00

18:30

Eastern Time

Eastern Time

-5

1.6

x 10

1.4

α=25 α=50 α=100

α=25 α=50 α=100

1.000009

0.55

α=25 α=50 α=100

1.2 0.5

1.0000085

2

0.8

ρ (%)

1

V -V

Surplus

1

1.000008

0.6

0.45

1.0000075

0.4

1.000007

0.35

0.4 0.2 0 16:00

16:30

17:00

17:30

Eastern Time

18:00

18:30

16:00

16:30

17:00

17:30

Eastern Time

18:00

18:30

16:00

16:30

17:00

17:30

Eastern Time

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Appx.

Discount-Window lending rate -5

2.5

x 10

w f w f w f

i =0.5%

w f w f w f

iw=0.5%

i =0.5%

f w f w f

1.00002

i =0.75% 2

i =0.75%

i =0.75% 1.000018

i =1%

0.8

i =1%

i =1%

0.75

1

ρ (%)

0.7

1.000014

2

V -V

Surplus

1.000016 1.5

1

1.000012

0.65 0.6 0.55

1.00001 0.5 0.5

1.000008

0.45 0.4

1.000006 0 16:00

16:30

17:00

17:30

18:00

18:30

16:00

16:30

Eastern Time

17:00

17:30

18:00

18:30

16:00

16:30

17:00

17:30

18:00

18:30

18:00

18:30

Eastern Time

Eastern Time

-5

2.5

x 10

1.0000105 w f w f w i =1% f

w f w f iw=1% f

i =0.5%

i =0.5%

1.00001

i =0.75% 2

1.0000095

w f w f w i =1% f

i =0.5% 0.7

i =0.75%

0.65

i =0.75%

1

ρ (%)

1 2

V -V

Surplus

0.6

1.000009

1.5

1.0000085 1.000008

0.55 0.5 0.45 0.4

1.0000075 0.5

0.35

1.000007

0.3 0 16:00

16:30

17:00

17:30

Eastern Time

18:00

18:30

16:00

16:30

17:00

17:30

Eastern Time

18:00

18:30

16:00

16:30

17:00

17:30

Eastern Time

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

More to be done... Fed funds brokers Banks’portfolio decisions Random “payment shocks” Sequence of trading sessions Ex-ante heterogeneity (αi , θ i , Uki ) Generalized inventories Quantitative work

Policy

Examples

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

The views expressed here are not necessarily re‡ective of views at the Federal Reserve Bank of New York or the Federal Reserve System.

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Evidence of OTC frictions in the fed funds market

Price dispersion Intermediation Intraday evolution of the distribution of reserve balances There are banks that are “very long” and buy There are banks that are “very short” and sell

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Price dispersion Intraday Distribution of Fed Funds Spreads, 2005 .1

.05

Percent

0

-.05

-.1

-.15 16:00

16:30

17:00

17:30

18:00

10th/90th Percentiles

25th/75th Percentiles

Median

Mean

18:30

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Intermediation: excess funds reallocation Excess Funds Reallocation

150

Billions of Dollars

100

50

0

2005

2006

2007

2008

2009

2010

Conclusion

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intermediation: proportion of intermediated funds Proportion of Intermediated Funds .6

.5

.4

.3

.2

.1

0

2005

2006

2007

2008

2009

2010

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Intraday evolution of the distribution of reserve balances Normalized Balances, 2007 1

.5

0

-.5 16:00

16:30

17:00

17:30

18:00

10th/90th Percentiles

25th/75th Percentiles

Median

Mean

18:30

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Banks that are “long”...and buy... Purchases by Banks with Nonnegative Adjusted Balances, 2007 3,000

Millions of Dollars

2,000

1,000

0 16:00

16:30

17:00

17:30

18:00

10th/90th Percentiles

25th/75th Percentiles

Median

Mean

18:30

Appx.

Intro

Model

Equilibrium

E¢ ciency

Frictionless

Implications

Policy

Examples

Conclusion

Banks that are “short”...and sell... Sales by Banks with Negative Adjusted Balances, 2007 500

Millions of Dollars

400

300

200

100

0 16:00

16:30

17:00

17:30

18:00

10th/90th Percentiles

25th/75th Percentiles

Median

Mean

18:30

Appx.