economicsslides

Intro Model Equilibrium Effi ciency Frictionless Implications Policy Examples Conclusion Appx.

The OTC Theory of the Fed Funds Market:A Primer

Gara Afonso Ricardo Lagos

FRB of New York New York University

The market for federal funds

A market for loans of reserve balances at the Fed.

What’s traded?

Unsecured loans (mostly overnight)

How are they traded?

Over the counter

Who trades?

Commercial banks, securities dealers, agencies and branches offoreign banks in the U.S., thrift institutions, federal agencies

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 offset liquidity shocks and manage reserves)

Determines the interest rate on the shortest maturityinstrument in the term structure

Is the “epicenter”of monetary policy implementation

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 aneffi cient reallocation of funds?

The model

A trading session in continuous time, t ∈ [0,T ], τ ≡ T − t

Unit measure of banks hold reserve balances k (τ) ∈ {0, 1, 2}

{nk (τ)} : distribution of balances at time T − τ

Linear payoffs from balances, discount at rate r

Uk : payoff 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 ∆ ∈ R+

Institutional features of the fed funds market

Model Fed funds market

Search and bargaining

Over-the-counter market

[0,T ] 4:00pm-6:30pm

{nk (T )}Distribution of reservebalances at 4:00pm

{Uk} Reserve requirements,interest on reserves...

Search and bargaining Over-the-counter market

[0,T ] 4:00pm-6:30pm

[0,T ]

4:00pm-6:30pm

[0,T ] 4:00pm-6:30pm

[0,T ] 4:00pm-6:30pm

{nk (T )}

Distribution of reservebalances at 4:00pm

[0,T ] 4:00pm-6:30pm

[0,T ] 4:00pm-6:30pm

{Uk}

Reserve requirements,interest on reserves...

[0,T ] 4:00pm-6:30pm

Bellman equations

rV1 (τ) + V1 (τ) = 0

rV0 (τ) + V0 (τ) = αn2 (τ)max {V1 (τ)− V0 (τ)− R (τ) , 0}

rV2 (τ) + V2 (τ) = αn0 (τ)max {V1 (τ)− V2 (τ) + R (τ) , 0}

Vk (0) = Uk

R (τ) = argmaxR[V1 (τ)− R − V0 (τ)]θ [V1 (τ) + R − V2 (τ)]1−θ

= θ [V2 (τ)− V1 (τ)] + (1− θ) [V1 (τ)− V0 (τ)]

R (τ) ≡ e−r (τ+∆)R (τ) (PDV of repayment)

Bellman equations

rV1 (τ) + V1 (τ) = 0

rV0 (τ) + V0 (τ) = αn2 (τ)max {V1 (τ)− V0 (τ)− R (τ) , 0}

rV2 (τ) + V2 (τ) = αn0 (τ)max {V1 (τ)− V2 (τ) + R (τ) , 0}

Vk (0) = Uk

R (τ) = argmaxR[V1 (τ)− R − V0 (τ)]θ [V1 (τ) + R − V2 (τ)]1−θ

= θ [V2 (τ)− V1 (τ)] + (1− θ) [V1 (τ)− V0 (τ)]

R (τ) ≡ e−r (τ+∆)R (τ) (PDV of repayment)

Bellman equations

rV1 (τ) + V1 (τ) = 0

rV0 (τ) + V0 (τ) = αn2 (τ) φ (τ) θS (τ)

rV2 (τ) + V2 (τ) = αn0 (τ) φ (τ) (1− θ) S (τ)

where:

S (τ) ≡ 2V1 (τ)− V0 (τ)− V2 (τ)

φ (τ) =

{1 if 0 < S (τ)0 if S (τ) ≤ 0

Time-path for the distribution of balances

n0 (τ) = αφ (τ) n2 (τ) n0 (τ)

n1 (τ) = −2αφ (τ) n2 (τ) n0 (τ)

n2 (τ) = αφ (τ) n2 (τ) n0 (τ)

DefinitionAn equilibrium is a value function, V, a path for the distribution ofreserve balances, n (τ), and a path for the distribution of tradingprobabilities, φ (τ), such that:

(a) given the value function and the distribution of tradingprobabilities, the distribution of balances evolves according to thelaw of motion; and

(b) given the path for the distribution of balances, the valuefunction and the distribution of trading probabilities satisfyindividual optimization given the bargaining protocol.

Analysis

S (τ) = −δ (τ) S (τ)

δ (τ) ≡ {r + αφ (τ) [θn2 (τ) + (1− θ) n0 (τ)]}

⇒

S (τ) = e−δ(τ)S (0)

δ (τ) ≡∫ τ

0δ (x) dx

S (τ) ≡ 2V1 (τ)− V0 (τ)− V2 (τ)

Analysis

S (τ) = −δ (τ) S (τ)

δ (τ) ≡ {r + αφ (τ) [θn2 (τ) + (1− θ) n0 (τ)]}

⇒

S (τ) = e−δ(τ)S (0)

δ (τ) ≡∫ τ

0δ (x) dx

S (τ) ≡ 2V1 (τ)− V0 (τ)− V2 (τ)

Assumption: S (0) ≡ 2U1 − U2 − U0 > 0

⇒

S (τ) = e−δ(τ)S (0) > 0 for all τ

⇒

φ (τ) = 1 for all τ

Assumption: S (0) ≡ 2U1 − U2 − U0 > 0

⇒

S (τ) = e−δ(τ)S (0) > 0 for all τ

⇒

Assumption: S (0) ≡ 2U1 − U2 − U0 > 0

⇒

S (τ) = e−δ(τ)S (0) > 0 for all τ

⇒

Proposition

Suppose 2U1 − U2 − U0 > 0. Then the unique equilibrium is:

n2 (τ) = n0 (τ) + n2 (T )− n0 (T )n1 (τ) = 1− 2n0 (τ) + n0 (T )− n2 (T )n0 (τ) = [n2(T )−n0(T )]n0(T )

eα[n2 (T )−n0 (T )](T−τ)n2(T )−n0(T )

S (τ) =[n2(T )−e−α[n2 (T )−n0 (T )](T−τ)n0(T )]e−{r+αθ[n2 (T )−n0 (T )]}τ

n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )S (0)

R (τ) = er∆ {β (τ) (U2 − U1) + [1− β (τ)] (U1 − U0)}

β (τ) ≡ θ[n2(T )−e−α[n2 (T )−n0 (T )](T−τ)n0(T )]e−αθ[n2 (T )−n0 (T )]τ

n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )

+[1−e−αθ[n2 (T )−n0 (T )]τ]n2(T )n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )

Proposition

eα[n2 (T )−n0 (T )](T−τ)n2(T )−n0(T )

n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )S (0)

R (τ) = er∆ {β (τ) (U2 − U1) + [1− β (τ)] (U1 − U0)}

n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )

Proposition

eα[n2 (T )−n0 (T )](T−τ)n2(T )−n0(T )

n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )S (0)

R (τ) = er∆ {β (τ) (U2 − U1) + [1− β (τ)] (U1 − U0)}

n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )

Proposition

eα[n2 (T )−n0 (T )](T−τ)n2(T )−n0(T )

n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )S (0)

R (τ) = er∆ {β (τ) (U2 − U1) + [1− β (τ)] (U1 − U0)}

n2(T )−e−α[n2 (T )−n0 (T )]T n0(T )

Proposition

Suppose 2U1 − U2 − U0 > 0. Then, the equilibrium supports aneffi cient allocation of reserve balances.

Intuition for effi ciency result

rV0 (τ) + V0 (τ) = αn2 (τ) θS (τ)

rλ0 (τ) + λ0 (τ) = αn2 (τ) S∗ (τ)

rV1 (τ) + V1 (τ) = 0

rλ1 (τ) + λ1 (τ) = 0

rV2 (τ) + V2 (τ) = αn0 (τ) (1− θ) S (τ)

rλ2 (τ) + λ2 (τ) = αn0 (τ) S∗ (τ)

S (τ) = e−δ(τ)S (0)

S∗ (τ) = e−δ∗(τ)S (0)

δ∗(τ)− δ (τ) = α

∫ τ

0[(1− θ) n2 (z) + θn0 (z)] dz ≥ 0

rV0 (τ) + V0 (τ) = αn2 (τ) θS (τ)

rλ0 (τ) + λ0 (τ) = αn2 (τ) S∗ (τ)

rV1 (τ) + V1 (τ) = 0

rλ1 (τ) + λ1 (τ) = 0

rV2 (τ) + V2 (τ) = αn0 (τ) (1− θ) S (τ)

rλ2 (τ) + λ2 (τ) = αn0 (τ) S∗ (τ)

S (τ) = e−δ(τ)S (0)

S∗ (τ) = e−δ∗(τ)S (0)

δ∗(τ)− δ (τ) = α

∫ τ

0[(1− θ) n2 (z) + θn0 (z)] dz ≥ 0

rV0 (τ) + V0 (τ) = αn2 (τ) θS (τ)

rλ0 (τ) + λ0 (τ) = αn2 (τ) S∗ (τ)

rV1 (τ) + V1 (τ) = 0

rλ1 (τ) + λ1 (τ) = 0

rV2 (τ) + V2 (τ) = αn0 (τ) (1− θ) S (τ)

rλ2 (τ) + λ2 (τ) = αn0 (τ) S∗ (τ)

S (τ) = e−δ(τ)S (0)

S∗ (τ) = e−δ∗(τ)S (0)

δ∗(τ)− δ (τ) = α

∫ τ

0[(1− θ) n2 (z) + θn0 (z)] dz ≥ 0

rV0 (τ) + V0 (τ) = αn2 (τ) θS (τ)

rλ0 (τ) + λ0 (τ) = αn2 (τ) S∗ (τ)

rV1 (τ) + V1 (τ) = 0

rλ1 (τ) + λ1 (τ) = 0

rV2 (τ) + V2 (τ) = αn0 (τ) (1− θ) S (τ)

rλ2 (τ) + λ2 (τ) = αn0 (τ) S∗ (τ)

S (τ) = e−δ(τ)S (0)

S∗ (τ) = e−δ∗(τ)S (0)

δ∗(τ)− δ (τ) = α

∫ τ

0[(1− θ) n2 (z) + θn0 (z)] dz ≥ 0

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 ∈ [0,T ], with “=”only for τ = 0

⇒ The planner “discounts”more heavily than the equilibrium

⇒ S∗ (τ) < S (τ) for all τ ∈ (0, 1]⇒ Social value of loan < joint private value of loan

Equilibrium:

Planner:

Equilibrium:

Planner:

Equilibrium:

Planner:

Equilibrium:

Planner:

Equilibrium:

Planner:

Planner internalizes that searching borrowers and lendersmake it easier for other lenders and borrowers to find partners

These “liquidity provision services” to others receive nocompensation in the equilibrium, so individual agents ignorethem when calculating their equilibrium payoffs

The equilibrium payoff to lenders may be too high or too lowrelative to their shadow price in the planner’s problem:

E.g., too high if (1− θ) S (τ) > S∗ (τ)

Frictionless limit

Proposition

Let Q ≡ ∑2k=0 knk (T ) = 1+ n2 (T )− n0 (T ).

For τ ∈ [0,T ], as α→ ∞, ρ (τ)→ ρ∞, where

1+ ρ∞ =

er∆ (U1 − U0) if Q < 1er∆ [θ (U2 − U1) + (1− θ) (U1 − U0)] if Q = 1er∆ (U2 − U1) if 1 < Q.

Positive implications

The theory delivers:

(1) Time-varying trade volume

(2) Time-varying fed fund rate

(3) Intraday convergence of distribution of reserve balances

Trade volume

Flow volume of trade at time T − τ:

υ (τ) = αn0 (τ) n2 (τ)

Total volume traded during the trading session:

υ =∫ T

0υ (τ) dτ

Fed funds rateThe gross rate on a loan at time T − τ is:

1+ ρ (τ) = er∆ {β (τ) (U2 − U1) + [1− β (τ)] (U1 − U0)}

withβ (τ) ∈ [0, 1]

The average daily rate is:

ρ =1T

∫ T

0ρ (τ) dτ

Cross-sectional distribution of reserve balances

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

µ (τ) = 1+ n2 (T )− n0 (T ) ≡ Q

σ2 (τ) = σ2 (T )− 2 [2+ n2 (T )− n0 (T )] [n0 (T )− n0 (τ)]

U1 = e−r∆(1+ i rf )

U2 = e−r∆(2+ i rf + ief )

U0 = −e−r∆(iwf − i rf + Pw )

Proposition

ρ (τ) = β (τ) ief + [1− β (τ)] (iwf + Pw ) where

1 If n2 (T ) = n0 (T ), β (τ) = θ

2 If n2 (T ) < n0 (T ), β (τ) ∈ [0, θ], β (0) = θ and β′ (τ) < 0

3 If n0 (T ) < n2 (T ), β (τ) ∈ [θ, 1], β (0) = θ and β′ (τ) > 0.

U1 = e−r∆(1+ i rf )

U2 = e−r∆(2+ i rf + ief )

U0 = −e−r∆(iwf − i rf + Pw )

Proposition

ρ (τ) = β (τ) ief + [1− β (τ)] (iwf + Pw ) where

1 If n2 (T ) = n0 (T ), β (τ) = θ

2 If n2 (T ) < n0 (T ), β (τ) ∈ [0, θ], β (0) = θ and β′ (τ) < 0

3 If n0 (T ) < n2 (T ), β (τ) ∈ [θ, 1], β (0) = θ and β′ (τ) > 0.

Intraday interest rate in a balanced market

Intraday interest rate in a market with shortage of reserves

Intraday interest rate in a market with excess reserves

Interest rate and the quantity of reserves

k = 1

Two scenarios{nH0 (T ) , n

L2 (T )

} {nL0 (T ) , n

H2 (T )

}{0.6, 0.3} {0.3, 0.6}

Experiments

Bargaining Power (θ) Discount Rate (iwf ) Contact Rate (α)

0.1 0.5 0.9 .0050360

.0075360

.0100360 25 50 100

Bargaining power

16:00 16:30 17:00 17:30 18:00 18:300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10 5

Sur

plus

Eastern Time

θ=0.1θ=0.5θ=0.9

16:00 16:30 17:00 17:30 18:00 18:301.000006

1.000008

1.00001

1.000012

1.000014

1.000016

1.000018

1.00002

V2V

1

Eastern Time

θ=0.1θ=0.5θ=0.9

16:00 16:30 17:00 17:30 18:00 18:30

0.3

0.4

0.5

0.6

0.7

0.8

ρ (%

)

Eastern Time

θ=0.1θ=0.5θ=0.9

16:00 16:30 17:00 17:30 18:00 18:300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10 5

Sur

plus

Eastern Time

θ=0.1θ=0.5θ=0.9

16:00 16:30 17:00 17:30 18:00 18:30

1.000007

1.0000075

1.000008

1.0000085

1.000009

V2V

1

Eastern Time

θ=0.1θ=0.5θ=0.9

16:00 16:30 17:00 17:30 18:00 18:30

0.3

0.4

0.5

0.6

0.7

0.8

ρ (%

)Eastern Time

θ=0.1θ=0.5θ=0.9

Contact rate

16:00 16:30 17:00 17:30 18:00 18:300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10 5

Sur

plus

Eastern Time

α=25α=50α=100

16:00 16:30 17:00 17:30 18:00 18:301.000006

1.000008

1.00001

1.000012

1.000014

1.000016

1.000018

1.00002

V2V

1

Eastern Time

α=25α=50α=100

16:00 16:30 17:00 17:30 18:00 18:30

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

ρ (%

)

Eastern Time

α=25α=50α=100

16:00 16:30 17:00 17:30 18:00 18:300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10 5

Sur

plus

Eastern Time

α=25α=50α=100

16:00 16:30 17:00 17:30 18:00 18:30

1.000007

1.0000075

1.000008

1.0000085

1.000009

V2V

1

Eastern Time

α=25α=50α=100

16:00 16:30 17:00 17:30 18:00 18:30

0.35

0.4

0.45

0.5

0.55

ρ (%

)Eastern Time

α=25α=50α=100

Discount-Window lending rate

16:00 16:30 17:00 17:30 18:00 18:300

0.5

1

1.5

2

2.5x 10 5

Sur

plus

Eastern Time

ifw=0.5%

ifw=0.75%

ifw=1%

16:00 16:30 17:00 17:30 18:00 18:30

1.000006

1.000008

1.00001

1.000012

1.000014

1.000016

1.000018

1.00002

V2V

1

Eastern Time

ifw=0.5%

ifw=0.75%

ifw=1%

16:00 16:30 17:00 17:30 18:00 18:30

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

ρ (%

)

Eastern Time

ifw=0.5%

ifw=0.75%

ifw=1%

16:00 16:30 17:00 17:30 18:00 18:300

0.5

1

1.5

2

2.5x 10 5

Sur

plus

Eastern Time

ifw=0.5%

ifw=0.75%

ifw=1%

16:00 16:30 17:00 17:30 18:00 18:30

1.000007

1.0000075

1.000008

1.0000085

1.000009

1.0000095

1.00001

1.0000105

V2V

1

Eastern Time

ifw=0.5%

ifw=0.75%

ifw=1%

16:00 16:30 17:00 17:30 18:00 18:30

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

ρ (%

)Eastern Time

ifw=0.5%

ifw=0.75%

ifw=1%

More to be done...

Fed funds brokers

Banks’portfolio decisions

Random “payment shocks”

Sequence of trading sessions

Ex-ante heterogeneity (αi , θi , U ik )

Generalized inventories

Quantitative work

The views expressed here are not necessarily reflective ofviews at the Federal Reserve Bank of New Yorkor the Federal Reserve System.

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 buyThere are banks that are “very short” and sell

Price dispersion

.15

.1

.05

0

.05

.1

Per

cent

16:00 16:30 17:00 17:30 18:00 18:30

10th/90th Percentiles 25th/75th Percentiles

Median Mean

Intraday Distribution of Fed Funds Spreads, 2005

Intermediation: excess funds reallocation

0

50

100

150

Bill

ions

of D

olla

rs

2005 2006 2007 2008 2009 2010

Excess Funds Reallocation

Intermediation: proportion of intermediated funds

0

.1

.2

.3

.4

.5

.6

2005 2006 2007 2008 2009 2010

Proportion of Intermediated Funds

Intraday evolution of the distribution of reserve balances

.5

0

.5

1

16:00 16:30 17:00 17:30 18:00 18:30

Median Mean

Normalized Balances, 2007

Banks that are “long”...and buy...

0

1,000

2,000

3,000

Milli

ons

of D

olla

rs

16:00 16:30 17:00 17:30 18:00 18:30

Median Mean

Purchases by Banks with Nonnegative Adjusted Balances, 2007

Banks that are “short”...and sell...

0

100

200

300

400

500

Milli

ons

of D

olla

rs

16:00 16:30 17:00 17:30 18:00 18:30

Median Mean

Sales by Banks with Negative Adjusted Balances, 2007

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