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1. Intro Model Equilibrium E¢ 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

2. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. The market for federal funds A market for loans of reserve balances at the Fed.

3. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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

4. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. The market for federal funds

5. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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

9. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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?

10. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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+

11. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Institutional features of the fed funds market Model Fed funds market Search and bargaining

12. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Institutional features of the fed funds market Model Fed funds market Search and bargaining Over-the-counter market

13. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Institutional features of the fed funds market Model Fed funds market Search and bargaining Over-the-counter market [0, T]

14. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Institutional features of the fed funds market Model Fed funds market Search and bargaining Over-the-counter market [0, T] 4:00pm-6:30pm

15. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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

16. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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

17. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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 fUk g

18. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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 fUk g Reserve requirements, interest on reserves...

19. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Bellman equations rV1 (τ) + ˙V1 (τ) = 0 rV0 (τ) + ˙V0 (τ) = αn2 (τ) max fV1 (τ) V0 (τ) ¯R (τ) , 0g rV2 (τ) + ˙V2 (τ) = αn0 (τ) max fV1 (τ) V2 (τ) + ¯R (τ) , 0g Vk (0) = Uk

20. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Bellman equations rV1 (τ) + ˙V1 (τ) = 0 rV0 (τ) + ˙V0 (τ) = αn2 (τ) max fV1 (τ) V0 (τ) ¯R (τ) , 0g rV2 (τ) + ˙V2 (τ) = αn0 (τ) max fV1 (τ) V2 (τ) + ¯R (τ) , 0g Vk (0) = Uk ¯R (τ) = arg max R [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)

21. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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

22. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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)

23. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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.

24. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Analysis ˙S (τ) = δ (τ) S (τ) δ (τ) fr + αφ (τ) [θn2 (τ) + (1 θ) n0 (τ)]g ) S (τ) = e ¯δ(τ) S (0) ¯δ (τ) Z τ 0 δ (x) dx S (τ) 2V1 (τ) V0 (τ) V2 (τ)

25. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Analysis ˙S (τ) = δ (τ) S (τ) δ (τ) fr + αφ (τ) [θn2 (τ) + (1 θ) n0 (τ)]g ) S (τ) = e ¯δ(τ) S (0) ¯δ (τ) Z τ 0 δ (x) dx S (τ) 2V1 (τ) V0 (τ) V2 (τ)

26. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Assumption: S (0) 2U1 U2 U0 > 0 ) S (τ) = e ¯δ(τ) S (0) > 0 for all τ ) φ (τ) = 1 for all τ

29. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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 fr+αθ[n2(T ) n0(T )]gτ n2(T ) e α[n2(T ) n0(T )]T n0(T ) S (0) R (τ) = er∆ fβ (τ) (U2 U1) + [1 β (τ)] (U1 U0)g 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 )

33. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Proposition Suppose 2U1 U2 U0 > 0. Then, the equilibrium supports an e¢ cient allocation of reserve balances.

34. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Intuition for e¢ 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)

35. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Intuition for e¢ 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 Z τ 0 [(1 θ) n2 (z) + θn0 (z)] dz 0

38. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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

44. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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)

47. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Frictionless limit Proposition Let Q ∑2 k=0 knk (T) = 1 + n2 (T) n0 (T). For τ 2 [0, T], as α ! ∞, ρ (τ) ! ρ∞, where 1 + ρ∞ = 8 < : er∆ (U1 U0) if Q < 1 er∆ [θ (U2 U1) + (1 θ) (U1 U0)] if Q = 1 er∆ (U2 U1) if 1 < Q.

48. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Positive implications The theory delivers: (1) Time-varying trade volume (2) Time-varying fed fund rate (3) Intraday convergence of distribution of reserve balances

49. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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) dτ

50. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Fed funds rate The gross rate on a loan at time T τ is: 1 + ρ (τ) = er∆ fβ (τ) (U2 U1) + [1 β (τ)] (U1 U0)g with β (τ) 2 [0, 1] The average daily rate is: ¯ρ = 1 T Z T 0 ρ (τ) dτ

51. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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) n0 (T) Q σ2 (τ) = σ2 (T) 2 [2 + n2 (T) n0 (T)] [n0 (T) n0 (τ)]

52. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. U1 = e r∆ (1 + ir f ) U2 = e r∆ (2 + ir f + ie f ) U0 = e r∆ (iw f ir f + Pw )

53. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. U1 = e r∆ (1 + ir f ) U2 = e r∆ (2 + ir f + ie f ) U0 = e r∆ (iw f ir f + Pw ) Proposition ρ (τ) = β (τ) ie f + [1 β (τ)] (iw f + Pw ) where 1 If n2 (T) = n0 (T), β (τ) = θ 2 If n2 (T) < n0 (T), β (τ) 2 [0, θ], β (0) = θ and β0 (τ) < 0 3 If n0 (T) < n2 (T), β (τ) 2 [θ, 1], β (0) = θ and β0 (τ) > 0.

54. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Intraday interest rate in a balanced market

55. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Intraday interest rate in a market with shortage of reserves

56. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Intraday interest rate in a market with excess reserves

57. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Interest rate and the quantity of reserves

58. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. ¯k = 1 Two scenarios nH 0 (T) , nL 2 (T) nL 0 (T) , nH 2 (T) f0.6, 0.3g f0.3, 0.6g Experiments Bargaining Power (θ) Discount Rate (iw f ) Contact Rate (α) 0.1 0.5 0.9 .0050 360 .0075 360 .0100 360 25 50 100

59. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Bargaining power 16:00 16:30 17:00 17:30 18:00 18:30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x 10 - 5 Surplus Eastern Time θ=0.1 θ=0.5 θ=0.9 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 V 2 -V 1 Eastern Time θ=0.1 θ=0.5 θ=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=0.1 θ=0.5 θ=0.9 16:00 16:30 17:00 17:30 18:00 18:30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x 10 - 5 Surplus Eastern Time θ=0.1 θ=0.5 θ=0.9 16:00 16:30 17:00 17:30 18:00 18:30 1.000007 1.0000075 1.000008 1.0000085 1.000009 V 2 -V 1 Eastern Time θ=0.1 θ=0.5 θ=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=0.1 θ=0.5 θ=0.9

60. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Contact rate 16:00 16:30 17:00 17:30 18:00 18:30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x 10 - 5 Surplus Eastern Time α=25 α=50 α=100 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 V 2 -V 1 Eastern Time α=25 α=50 α=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=25 α=50 α=100 16:00 16:30 17:00 17:30 18:00 18:30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x 10 - 5 Surplus Eastern Time α=25 α=50 α=100 16:00 16:30 17:00 17:30 18:00 18:30 1.000007 1.0000075 1.000008 1.0000085 1.000009 V 2 -V 1 Eastern Time α=25 α=50 α=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=25 α=50 α=100

61. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Discount-Window lending rate 16:00 16:30 17:00 17:30 18:00 18:30 0 0.5 1 1.5 2 2.5 x 10 - 5 Surplus Eastern Time i f w =0.5% i f w =0.75% i f w =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 V 2 -V 1 Eastern Time i f w =0.5% i f w =0.75% i f w =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 i f w =0.5% i f w =0.75% i f w =1% 16:00 16:30 17:00 17:30 18:00 18:30 0 0.5 1 1.5 2 2.5 x 10 - 5 Surplus Eastern Time i f w =0.5% i f w =0.75% i f w =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 V 2 -V 1 Eastern Time i f w =0.5% i f w =0.75% i f w =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 i f w =0.5% i f w =0.75% i f w =1%

62. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. More to be done... Fed funds brokers Banks’portfolio decisions Random “payment shocks” Sequence of trading sessions Ex-ante heterogeneity (αi , θi , Ui k ) Generalized inventories Quantitative work

63. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. The views expressed here are not necessarily re‡ective of views at the Federal Reserve Bank of New York or the Federal Reserve System.

64. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. 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

65. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Price dispersion -.15 -.1 -.05 0 .05 .1 Percent 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

66. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Intermediation: excess funds reallocation 0 50 100 150 BillionsofDollars 2005 2006 2007 2008 2009 2010 Excess Funds Reallocation

67. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Intermediation: proportion of intermediated funds 0 .1 .2 .3 .4 .5 .6 2005 2006 2007 2008 2009 2010 Proportion of Intermediated Funds

68. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Intraday evolution of the distribution of reserve balances -.5 0 .5 1 16:00 16:30 17:00 17:30 18:00 18:30 10th/90th Percentiles 25th/75th Percentiles Median Mean Normalized Balances, 2007

69. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Banks that are “long”...and buy... 0 1,000 2,000 3,000 MillionsofDollars 16:00 16:30 17:00 17:30 18:00 18:30 10th/90th Percentiles 25th/75th Percentiles Median Mean Purchases by Banks with Nonnegative Adjusted Balances, 2007

70. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx. Banks that are “short”...and sell... 0 100 200 300 400 500 MillionsofDollars 16:00 16:30 17:00 17:30 18:00 18:30 10th/90th Percentiles 25th/75th Percentiles Median Mean Sales by Banks with Negative Adjusted Balances, 2007

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