Trade Dynamics in the Market for Federal Funds

Transcription

1 Trade Dynamics in the Market for Federal Funds Gara Afonso FRB of New York Ricardo Lagos New York University

2 The market for federal funds A market for loans of reserve balances at the Fed.

3 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 The market for federal funds

5 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 maturity instrument in the term structure Is the epicenter of monetary policy implementation

6 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 maturity instrument in the term structure Is the epicenter of monetary policy implementation

7 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 maturity instrument in the term structure Is the epicenter of monetary policy implementation

8 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 maturity instrument in the term structure Is the epicenter of monetary policy implementation

9 In this paper we... (1) Develop a model of trade in the fed funds market that explicitly accounts for the two key OTC frictions: Search for counterparties Bilateral negotiations

10 In this paper we... (2) Use the theory to address some elementary questions: Positive: What are the determinants of the fed funds rate? How does the market reallocate funds? Normative: Is the OTC market structure able to achieve an effi cient reallocation of funds?

11 In this paper we... (3) Calibrate the model and use it to: Assess the ability of the theory to account for empirical regularities of the fed funds market: Intraday evolution of reserve balances Dispersion in fed funds rates and loan sizes Skewed distribution of number of transactions Skewed distribution of proportion of intermediated funds

12 In this paper we... (3) Calibrate the model and use it to: Assess the ability of the theory to account for empirical regularities of the fed funds market: Intraday evolution of reserve balances Dispersion in fed funds rates and loan sizes Skewed distribution of number of transactions Skewed distribution of proportion of intermediated funds Conduct policy experiments: What is the effect on the fed funds rate of a 25 bps increase in the interest rate that the Fed pays on reserves?

13 The model A trading session in continuous time, t [, T ], τ T t Unit measure of banks hold reserve balances k (τ) K = {, 1,..., K } {n k (τ)} k K : distribution of balances at time T τ Linear payoffs from balances, discount at rate r Fed policy: U k : payoff from holding k balances at the end of the session u k : flow payoff from holding k balances during 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 +

14 Institutional features of the fed funds market Model Search and bargaining Fed funds market

15 Institutional features of the fed funds market Model Search and bargaining Fed funds market Over-the-counter market

16 Institutional features of the fed funds market Model Search and bargaining Fed funds market Over-the-counter market [, T ]

17 Institutional features of the fed funds market Model Search and bargaining Fed funds market Over-the-counter market [, T ] 4:pm-6:3pm

18 Institutional features of the fed funds market Model Search and bargaining Fed funds market Over-the-counter market [, T ] 4:pm-6:3pm {n k (T )} k K

19 Institutional features of the fed funds market Model Search and bargaining Fed funds market Over-the-counter market [, T ] 4:pm-6:3pm {n k (T )} k K Distribution of reserve balances at 4:pm

20 Institutional features of the fed funds market Model Search and bargaining Fed funds market Over-the-counter market [, T ] 4:pm-6:3pm {n k (T )} k K Distribution of reserve balances at 4:pm {u k, U k } k K

21 Institutional features of the fed funds market Model Search and bargaining Fed funds market Over-the-counter market [, T ] 4:pm-6:3pm {n k (T )} k K Distribution of reserve balances at 4:pm {u k, U k } k K Reserve requirements, interest on reserves...

22 Bank with balance k contacts bank with balance k at time T τ

23 Bank with balance k contacts bank with balance k at time T τ The set of feasible post-trade balances is: Π ( k, k ) = {( k + k y, y ) K K : y {, 1,..., k + k }}

24 Bank with balance k contacts bank with balance k at time T τ The set of feasible post-trade balances is: Π ( k, k ) = {( k + k y, y ) K K : y {, 1,..., k + k }} The set of feasible loan sizes is: Γ ( k, k ) = { b { K,...,,..., K } : ( k b, k + b ) Π ( k, k )}

25 Bank with balance k contacts bank with balance k at time T τ The set of feasible post-trade balances is: Π ( k, k ) = {( k + k y, y ) K K : y {, 1,..., k + k }} The set of feasible loan sizes is: Γ ( k, k ) = { b { K,...,,..., K } : ( k b, k + b ) Π ( k, k )} V k (τ) : value of a bank with balance k at time T τ

26 Bargaining Bank with balance k contacts bank with balance k at time T τ. The loan size b, and the repayment R maximize: [ ] V k b (τ) + e r (τ+ ) 1 2 [ ] R V k (τ) V k +b (τ) e r (τ+ ) 1 2 R V k (τ) s.t. b Γ ( k, k ), R R

27 Bargaining Bank with balance k contacts bank with balance k at time T τ. The loan size b, and the repayment R maximize: [ ] V k b (τ) + e r (τ+ ) 1 2 [ ] R V k (τ) V k +b (τ) e r (τ+ ) 1 2 R V k (τ) s.t. b Γ ( k, k ), R R b arg max b Γ(k,k ) [V k +b (τ) + V k b (τ) V k (τ) V k (τ)] e r (τ+ ) R = 1 2 [V k +b (τ) V k (τ)] [V k (τ) V k b (τ)]

28 Value function rv i (τ) + V i (τ) = = u i + α 2 n j (τ) φ ks ij (τ) [V k (τ) + V s (τ) V i (τ) V j (τ)] j,k,s K

29 Value function rv i (τ) + V i (τ) = = u i + α 2 n j (τ) φ ks ij (τ) [V k (τ) + V s (τ) V i (τ) V j (τ)] j,k,s K with V i () = U i, and φ ks ij (τ) = { φ ks ij (τ) if (k, s) Ω ij [V (τ)] if (k, s) / Ω ij [V (τ)]

30 Value function rv i (τ) + V i (τ) = = u i + α 2 n j (τ) φ ks ij (τ) [V k (τ) + V s (τ) V i (τ) V j (τ)] j,k,s K with V i () = U i, and with φ ks ij (τ) = { φ ks ij (τ) if (k, s) Ω ij [V (τ)] if (k, s) / Ω ij [V (τ)] Ω ij [V (τ)] arg max [V (k,s k (τ) + V s (τ) V i (τ) V j (τ)] ) Π(i,j) where φ ks ij (τ) and k K s K φ ks ij (τ) = 1

31 Time-path for the distribution of balances For all k K, ṅ k (τ) = αn k (τ) α i K j K s K i K j K s K n i (τ) φ sj ki (τ) n i (τ) n j (τ) φ ks ij (τ)

32 Definition 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.

33 Assumption A. For any i, j K, and all (k, s) Π (i, j), the payoff functions satisfy: u i+j 2 + u i+j 2 u k + u s U i+j 2 + U i+j 2 U k + U s, > unless k { i+j 2, } i+j 2 where for any x R, x max {k Z : k x} x min {k Z : x k}

34 Proposition Let the payoff functions satisfy Assumption A. Then: (i) An equilibrium exists. The paths V (τ) and n (τ) are unique. (ii) The equilibrium path for φ (τ) = {φ ks ij φ ks ij (τ) = (τ)} i,j,k,s K is { φ ks ij (τ) if (k, s) Ω ij if (k, s) / Ω ij where φ ks ij (τ) and φ ks (k,s) Ωij ij (τ) = 1, with Ωij = {( i+j 2, i+j {( 2 i+j 2, )} ) i+j 2, ( i+j 2, )} i+j 2 if i + j even if i + j odd.

35 Proposition Let the payoff functions satisfy Assumption A. Then, the equilibrium supports an effi cient allocation of reserve balances.

36 Positive implications The theory delivers: (1) Time-varying distribution of trade sizes, trade volume (2) Time-varying distribution of fed fund rates (3) Endogenous intermediation

37 Trade volume Flow volume of trade at time T τ: ῡ (τ) = υ ks ij i K j K k K s K (τ) where υ ks ij (τ) αn i (τ) n j (τ) φ ks ij (τ) k i

38 Trade volume Flow volume of trade at time T τ: ῡ (τ) = υ ks ij i K j K k K s K (τ) where υ ks ij (τ) αn i (τ) n j (τ) φ ks ij (τ) k i Total volume traded during the trading session: ῡ = T ῡ (τ) dτ

39 Fed funds rate If a bank with i borrows k i = j s from bank with j at time T τ, the interest rate on the loan is: [ ] [ ] Rij ln ks (τ) V ρ ks k i ln j (τ) V s (τ) j s S ij ks (τ) j s ij (τ) = = r + τ + τ +

40 Fed funds rate If a bank with i borrows k i = j s from bank with j at time T τ, the interest rate on the loan is: [ ] [ ] Rij ln ks (τ) V ρ ks k i ln j (τ) V s (τ) j s S ij ks (τ) j s ij (τ) = = r + τ + τ + The daily average (value-weighted) fed funds rate is: where ρ = 1 T ρ (τ) T ρ (τ) dτ ωij ks i K j K k K s K ω ks ij (τ) υ ks ij (τ) /ῡ (τ) (τ) ρ ks ij (τ)

41 Endogenous intermediation Cumulative purchases: O p = N max {k n k n 1, } n=1 Cumulative sales: O s = N min {k n k n 1, } n=1

42 Endogenous intermediation Cumulative purchases: O p = N max {k n k n 1, } n=1 Cumulative sales: O s = N min {k n k n 1, } n=1 Bank-level measures of intermediation Excess funds reallocation: X = O p + O s O p O s

43 Endogenous intermediation Cumulative purchases: O p = N max {k n k n 1, } n=1 Cumulative sales: O s = N min {k n k n 1, } n=1 Bank-level measures of intermediation Excess funds reallocation: X = O p + O s O p O s Proportion of intermediated funds: ι = X O p + O s

44 Analytics for special case with K = {, 1, 2} Intuition for effi ciency result Frictionless limit Analytical figures Small-scale simulations: K = {, 1, 2}

45 Payoff functions u k = (k ) 1 ɛ i d + if k (k ) 1+ɛ i d if k < with ɛ e r f U k = k + i r f k + i e f (k k) if k k k + i r f k i w f ( k k ) if k < k where k k k

46 Payoff functions u k = (k ) 1 ɛ i d + if k (k ) 1+ɛ i d if k < with ɛ e r f U k = k + i r f k + i e f (k k) if k k k + i r f k i w f ( k k ) if k < k where k k k k is a shifter

47 Distribution of balances at 16: for typical day in 27 Sample: N = 142 commercial banks that traded fed funds at least once during 27 Q2 ˆk i : bank i s average balance at 16: over a given two-week maintenance period during 27 Q2, divided by bank i s daily average required operating balance over the same period

48 Distribution of balances at 16: for typical day in 27 Sample: N = 142 commercial banks that traded fed funds at least once during 27 Q2 ˆk i : bank i s average balance at 16: over a given two-week maintenance period during 27 Q2, divided by bank i s daily average required operating balance over the same period K = {,..., 25}, k = 1, K K k, k = 5

49 Distribution of balances at 16: for typical day in 27 Sample: N = 142 commercial banks that traded fed funds at least once during 27 Q2 ˆk i : bank i s average balance at 16: over a given two-week maintenance period during 27 Q2, divided by bank i s daily average required operating balance over the same period K = {,..., 25}, k = 1, K K k, k = 5 n k (T ) = 1 N N I { ˆk i [k k,k k +1)} i=1

50 Distribution of balances at 16: for typical day in 27 Sample: N = 142 commercial banks that traded fed funds at least once during 27 Q2 ˆk i : bank i s average balance at 16: over a given two-week maintenance period during 27 Q2, divided by bank i s daily average required operating balance over the same period K = {,..., 25}, k = 1, K K k, k = 5 n k (T ) = 1 N Q = N I { ˆk i [k k,k k +1)} i=1 25 (k k ) n k (T ) 1.4 k=

51 Policy parameters as in 27 T f i d + i d i r f i e f i w f i c f i o f r

52 Calibrated parameters, 27 targets (α, P w ) = ( 1,.525 ) 36 Model Data fed funds rate * std. dev. of balances at 6:3 pm * median number of counterparties 7 2 mean number of counterparties intermediation index.65.43

53 Fed funds rate (%) Proportion of banks Proportion of loans Balances Spreads to market rates (%) Standard deviation of balances Reserve balances and fed funds rates (27).45.4 Opening balances End of day balances Opening and end of day balances 15 16: 16: 15 16: 3 16: 45 17: 17: 15 17: 3 17: 45 18: 18: 15 18: 3 Eastern Time 16: 16:3 17: 17:3 18: 18:3 Eastern Time : 16:3 17: 17:3 18: 18:3 Eastern Time Loan rates (% ) 6. 16:16:1516:316:4517:17:1517:317:4518:18:1518:3 Eastern Time

54 Proportion of funds traded Proportion of banks Proportion of loans Proportion of banks Proportion of banks Loan sizes Distributions of loan sizes and trading activity (27) Proportion of amount of funds traded Proportion of number of loans traded 16: 16:3 17: 17:3 18: 18:3 Eastern Time Loan sizes : 16: 15 16: 3 16: 45 17: 17: 15 17: 3 17: 45 18: 18: 15 18: 3 Eastern Time Number of counterparties Number of counterparties banks lend to Number of counterparties banks borrow from

55 Proportion of banks Intermediation (27) Proportion of intermediated f unds

56 Policy parameters as in 211 i r f i e f i w f i c f i o f

57 Calibrated parameters, 211 targets (α, P w ) = (1, ) Model Data fed funds rate * std. dev. of balances at 6:3 pm * median number of counterparties 2 mean number of counterparties intermediation index.2.35

58 Policy experiments (baseline policy as in 211) i f Q/ k =.5 Q/ k = 1. Q/ k = if w Q/ k =.5 Q/ k = 1. Q/ k =

59 Corridor system (i f, if w ) Q/ k =.5 Q/ k = 1 Q/ k =

60 Corridor system 2 Fed funds rate (%) 2 Fed funds rate (%) 2 Fed funds rate (%) i f w i f w i f w 1 1 i f.8.8 i f i f Fed funds rate (%) 2 Fed funds rate (%) 2 Fed funds rate (%) i f w.8 i f w i f 1.8 i f w.6 i f i f

61 IOR Policy intuition from the analytical example Proposition If r, ρ f (τ) β (τ) i e f + [1 β (τ)] i w f where 1 If n 2 (T ) = n (T ), β (τ) = θ 2 If n 2 (T ) < n (T ), β (τ) [, θ], β () = θ and β (τ) < 3 If n (T ) < n 2 (T ), β (τ) [θ, 1], β () = θ and β (τ) >. Figures

62 Ex-ante heterogeneity We also extend the model to allow for: 1 Heterogeneity in contact rates 2 Heterogeneity in bargaining powers 3 Heterogeneity in target balances (or non-bank participants, e.g., GSEs)

63 More to be done... Fed funds brokers Banks portfolio decisions Random payment shocks Sequence of trading sessions Quantiative work with ex-ante heterogeneity

64 The views expressed here are not necessarily reflective of views at the Federal Reserve Bank of New York or the Federal Reserve System.

65 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

66 Percent Price dispersion.1 Intraday Distribution of Fed Funds Spreads, : 16:3 17: 17:3 18: 18:3 1th/9th Percentiles Median 25th/75th Percentiles Mean

67 Billions of Dollars Intermediation: excess funds reallocation 15 Excess Funds Reallocation

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

69 Intraday evolution of the distribution of reserve balances 1 Normalized Balances, : 16:3 17: 17:3 18: 18:3 1th/9th Percentiles Median 25th/75th Percentiles Mean

70 Millions of Dollars Banks that are long...and buy... Purchases by Banks with Nonnegative Adjusted Balances, 27 3, 2, 1, 16: 16:3 17: 17:3 18: 18:3 1th/9th Percentiles Median 25th/75th Percentiles Mean

71 Millions of Dollars Banks that are short...and sell... Sales by Banks with Negative Adjusted Balances, : 16:3 17: 17:3 18: 18:3 1th/9th Percentiles Median 25th/75th Percentiles Mean

72 Billions of Dollars Number of Loans Daily volume 2 Daily Amount of Federal Funds Loans Daily Number of Federal Funds Loans 1,

73 Millions of Dollars Daily Number of Loans Daily volume (size distribution) 1, Daily Distribution of Loan Sizes 8 Distribution of Daily Number of Loans th Percentile Median 25th Percentile Median 9th Percentile Mean 75th Percentile Mean

74 Daily Number of Borrowers per Bank Daily Number of Counterparties Daily Number of Lenders per Bank Daily distribution of the number of counterparties 15 Distribution of Daily Number of C ounterparties th P ercentile 75th P ercentile Median Mean Distribution of Daily Number of C ounterparties B anks Lend to 15 Distribution of Daily Number of C ounterparties B anks B orrow from th P ercentile Median 25th P ercentile Median 75th P ercentile Mean 75th P ercentile Mean

75 Millions of Dollars Millions of Dollars Millions of Dollars Millions of Dollars Millions of Dollars Millions of Dollars Intraday volume (dollar amount) 3, Intraday Total Amount of Loans, 25 2, 5 Intraday Total Amount of Loans, 26 3, Intraday Total Amount of Loans, 27 2, 2, 2, 1, 5 1, 1, 1, 5 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean 3, Intraday Total Amount of Loans, 28 2, 5 Intraday Total Amount of Loans, 29 2, Intraday Total Amount of Loans, 21 2, 1, 5 2, 1, 5 1, 1, 1, : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean

76 Number of Loans Number of Loans Number of Loans Number of Loans Number of Loans Number of Loans Intraday volume (number of loans) 15 Intraday Number of Loans, by Bank, Intraday Number of Loans, by Bank, Intraday Number of Loans, by Bank, : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean 15 Intraday Number of Loans, by Bank, 28 8 Intraday Number of Loans, by Bank, 29 1 Intraday Number of Loans, by Bank, : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean

77 Millions of Dollars Millions of Dollars Millions of Dollars Millions of Dollars Millions of Dollars Millions of Dollars Intraday size distribution of loans Intraday Distribution of Loan Sizes, 25 5 Intraday Distribution of Loan Sizes, 26 5 Intraday Distribution of Loan Sizes, : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean 5 Intraday Distribution of Loan Sizes, 28 1, 5 Intraday Distribution of Loan Sizes, 29 8 Intraday Distribution of Loan Sizes, , : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean

78 Millions of Dollars Millions of Dollars Intraday size distribution of loans 1,5 Intraday 9th Percentile of Loan Sizes (over time) 25 Intraday 1th Percentile of Loan Sizes (over time) 2 1, th Percentile, 15:45 16:15 9th Percentile, 17:45 18:15 1th Percentile, 15:45 16:15 1th Percentile, 17:45 18:15

79 Trading activity by time-of-day Proportion of Total Daily Amount of Loans by Time of Day 1 1 Proportion of Total Number of Loans by Time of Day Loans by 16:15 Loans by 17:15 Loans by 18:15 Loans by 16:15 Loans by 17:15 Loans by 18:15

80 1 Normalized Balances, 29 Intraday evolution of the distribution of reserve balances 1 Normalized Balances, 25 1 Normalized Balances, 26 1 Normalized Balances, : 16: 3 17: 17: 3 18: 18: : 16: 3 17: 17: 3 18: 18: : 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean 1.5 Normalized Balances, 28 1 Normalized Balances, Normalized Balances, : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean

81 Intraday evolution of the distribution of reserve balances 1 S tandard Deviation of Normalized B alances S tandard deviation at 16:15 S tandard deviation at 17:15 S tandard deviation at 18:15 Intraday 1th P ercentile of Normalized B alances (over time).2 15 Intraday 9th P ercentile of Normalized B alances (over time) th P ercentile, 16:15 1th P ercentile, 18:15 9th P ercentile, 16:15 9th P ercentile, 18:15

82 Percent Percent Daily fed funds rate vs. FOMC target 6 Daily Fed Funds Rate and FOMC Target 1 Deviations between Fed Funds Rate and FOMC Target FOMC Target Fed Funds Rate

83 Percent Percent Daily effective fed funds rate vs. FOMC target 6 Daily Effective Fed Funds Rate and FOMC Target Deviations between Effective Fed Funds Rate and FOMC Target FOMC Target Effective Fed Funds Rate

84 Percent Daily fed funds rate dispersion 6 Daily Dispersion of Fed Funds Rate FOMC Target 1th Percentile of Fed Funds Rate 9th Percentile of Fed Funds Rate

85 Percent Fed funds rate vs. effective fed funds rate.6 Daily Spread between Fed Funds and Effective Rate

86 P er cent P er cent P er cent P er cent P er cent P er cent Intraday distribution of fed funds spreads.1 Intraday Distribution of Fed Funds Spreads, 25.2 Intraday Distribution of Fed Funds Spreads, 26.2 Intraday Distribution of Fed Funds Spreads, : 16: 3 17: 17: 3 18: 18: : 16: 3 17: 17: 3 18: 18: : 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean.6 Intraday Distribution of Fed Funds Spreads, Intraday Distribution of Fed Funds Spreads, 29.1 Intraday Distribution of Fed Funds Spreads, : 16: 3 17: 17: 3 18: 18: : 16: 3 17: 17: 3 18: 18: : 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean

87 Percent Intraday distribution of fed funds spreads (over time) Intraday Distribution of Fed Funds Spreads (over time) th Percentile, 16:15 9th Percentile, 16:15 1th Percentile, 18:15 9th Percentile, 18:15

88 P er cent P er cent P er cent P er cent P er cent P er cent Intraday distribution of fed funds/fomc target spreads.2 Intraday Distribution of Spreads to FOMC Target, 25.5 Intraday Distribution of Spreads to FOMC Target, 26 Intraday Distribution of Spreads to FOMC Target, : 16: 3 17: 17: 3 18: 18: : 16: 3 17: 17: 3 18: 18: : 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean.5 Intraday Distribution of Spreads to FOMC Target, 28.2 Intraday Distribution of Spreads to FOMC Target, 29.1 Intraday Distribution of Spreads to FOMC Target, : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean

89 Billions of Dollars Millions of Dollars Daily intermediation 15 E xcess Funds Reallocation.6 P roportion of Intermediated Funds Daily Distribution of E xcess Funds Reallocation.8 Daily Distribution of P roportion of Intermediated Funds th P ercentile Median 1th P ercentile Median 9th P ercentile Mean 9th P ercentile Mean

90 Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Banks that are long...and buy... Purchases by Banks with Nonnegative Adjusted Balances, 25 1, Purchases by Banks with Nonnegative Adjusted Balances, 26 1, 5 Purchases by Banks with Nonnegative Adjusted Balances, 27 3, 8 6 1, 2, 4 5 1, 2 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean Purchases by Banks with Nonnegative Adjusted Balances, 28 6, Purchases by Banks with Nonnegative Adjusted Balances, 29 5 Purchases by Banks with Nonnegative Adjusted Balances, , 3 3 2, : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean

91 Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Mi l l i ons of Dol l ars Banks that are short...and sell... 1, Sales by Banks with Negative Adjusted Balances, 25 5 Sales by Banks with Negative Adjusted Balances, 26 5 Sales by Banks with Negative Adjusted Balances, : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean 2, Sales by Banks with Negative Adjusted Balances, 28 5 Sales by Banks with Negative Adjusted Balances, 21 5 Sales by Banks with Negative Adjusted Balances, 29 1, , : 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 16: 16: 3 17: 17: 3 18: 18: 3 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles 1t h/ 9t h Per cent iles 25t h/ 75t h Per cent iles Median Mean Median Mean Median Mean

92 Percent Daily fed funds rate vs. IOR.5 Daily Dispersion of Fed Funds Rate and IOR th Percentile of Fed Funds Rate 9th Percentile of Fed Funds Rate IOR

93 Percent Percent Daily FFR and daily effective FFR vs. IOR: a puzzle.25 Daily Fed Funds Rate and IOR.25 Daily Effective Fed Funds Rate and IOR IOR Fed Funds Rate IOR Effective Fed Funds Rate

94 Value function (derivation) { min(τα,τ) ( ) J k (x, τ) = E e rz u k dz + I {τα >τ}e r τ U k + e r x + I {τα τ}e r τ α J k bss (τ τ α ) (x + R s s (τ τ α ), τ τ α ) µ ( ds ) }, τ τ α

95 Value function (derivation) { min(τα,τ) ( ) J k (x, τ) = E e rz u k dz + I {τα >τ}e r τ U k + e r x + I {τα τ}e r τ α J k bss (τ τ α ) (x + R s s (τ τ α ), τ τ α ) µ ( ds, τ τ α ) } τ α : time until next trading opportunity b ss (τ) : balance that bank s = (k, x) lends to bank s = (k, x ) at time T τ R s s (τ) : repayment negotiated at time T τ (due at T + ) µ (, τ) : prob. measure over individual states, s = (k, x )

96 Bargaining Bank with s = (k, x) meets bank s = (k, x ) at T τ. The loan size b and the repayment R maximize: [J k b (x + R, τ) J k (x, τ)] 1 2 [ Jk +b ( x R, τ ) ( J k x, τ )] 2 1 s.t. b Γ ( k, k ) R R

97 Value function (derivation) J k (x, τ) = V k (τ) + e r (τ+ ) x where { min(τα,τ) V k (τ) = E e rz u k dz + I {τα >τ}e r τ U k + I {τα τ}e r τ α n k (τ τ α ) k K [ ] } V k bkk (τ τ α ) (τ τ α ) + e r (τ+ τα) R k k (τ τ α )

98 Value function (derivation) J k (x, τ) = V k (τ) + e r (τ+ ) x where { min(τα,τ) V k (τ) = E e rz u k dz + I {τα >τ}e r τ U k + I {τα τ}e r τ α n k (τ τ α ) k K [ ] } V k bkk (τ τ α ) (τ τ α ) + e r (τ+ τα) R k k (τ τ α ) b kk (τ) arg max [V b Γ(k,k k +b (τ) + V k b (τ) V k (τ) V k (τ)] )

99 Value function (derivation) J k (x, τ) = V k (τ) + e r (τ+ ) x where { min(τα,τ) V k (τ) = E e rz u k dz + I {τα >τ}e r τ U k + I {τα τ}e r τ α n k (τ τ α ) k K [ ] } V k bkk (τ τ α ) (τ τ α ) + e r (τ+ τα) R k k (τ τ α ) b kk (τ) arg max [V b Γ(k,k k +b (τ) + V k b (τ) V k (τ) V k (τ)] ) e r (τ+ ) R k k (τ) = 1 [ ] V 2 k +b kk (τ) (τ) V k (τ) + 1 [ ] V k (τ) V 2 k bkk (τ) (τ)

100 Special case with K = {, 1, 2} Bank with i = 2 is a lender, bank with j =, a borrower θ [, 1] : bargaining power of the borrower Only potentially profitable trade is between i = and j = 2 S (τ) 2V 1 (τ) V 2 (τ) V (τ) Conjecture S (τ) > for all τ [, T ] (to be verified later)

101 Special case with K = {, 1, 2} Bank with i = 2 is a lender, bank with j =, a borrower θ [, 1] : bargaining power of the borrower Only potentially profitable trade is between i = and j = 2 S (τ) 2V 1 (τ) V 2 (τ) V (τ) Conjecture S (τ) > for all τ [, T ] (to be verified later) Assumption: 2u 1 u 2 u and 2U 1 U 2 U >

102 Special case with K = {, 1, 2} Bank with i = 2 is a lender, bank with j =, a borrower θ [, 1] : bargaining power of the borrower Only potentially profitable trade is between i = and j = 2 S (τ) 2V 1 (τ) V 2 (τ) V (τ) Conjecture S (τ) > for all τ [, T ] (to be verified later) Assumption: 2u 1 u 2 u and 2U 1 U 2 U >

103 Special case with K = {, 1, 2} Bank with i = 2 is a lender, bank with j =, a borrower θ [, 1] : bargaining power of the borrower Only potentially profitable trade is between i = and j = 2 S (τ) 2V 1 (τ) V 2 (τ) V (τ) Conjecture S (τ) > for all τ [, T ] (to be verified later) Assumption: 2u 1 u 2 u and 2U 1 U 2 U > Given {n k (T )}, the distribution of balances follows: ṅ (τ) = αn 2 (τ) n (τ) ṅ 2 (τ) = αn 2 (τ) n (τ)

104 Time-path for the distribution of balances n 2 (τ) = n 2 (T ) [n (T ) n (τ)] n 1 (τ) = 1 n (τ) n 2 (τ) n (τ) = [n 2 (T ) n (T )] n (T ) n 2 (T ) e α[n 2(T ) n (T )](T τ) n (T )

105 Bargaining The repayment R solves: [ ] θ [ ] 1 θ max V 1 (τ) V (τ) e r (τ+ ) R V 1 (τ) V 2 (τ) + e r (τ+ ) R R e r (τ+ ) R (τ) = θ [V 2 (τ) V 1 (τ)] + (1 θ) [V 1 (τ) V (τ)]

106 Value function rv (τ) + V (τ) = u + αn 2 (τ) θs (τ) rv 1 (τ) + V 1 (τ) = u 1 rv 2 (τ) + V 2 (τ) = u 2 + αn (τ) (1 θ) S (τ) V i () = U i for i =, 1, 2

107 Value function rv (τ) + V (τ) = u + αn 2 (τ) θs (τ) rv 1 (τ) + V 1 (τ) = u 1 rv 2 (τ) + V 2 (τ) = u 2 + αn (τ) (1 θ) S (τ) V i () = U i for i =, 1, 2 Ṡ (τ) + δ (τ) S (τ) = 2u 1 u 2 u δ (τ) {r + α [θn 2 (τ) + (1 θ) n (τ)]}

108 Surplus ( τ S (τ) = e [ δ(τ) δ(z)] ) dz ū + e δ(τ) S () ū 2u 1 u 2 u S () = 2U 1 U 2 U δ (τ) τ δ (x) dx δ (τ) {r + α [θn 2 (τ) + (1 θ) n (τ)]}

109 Fed funds rate R (τ) = e ρ(τ+ ) 1

110 Fed funds rate R (τ) = e ρ(τ+ ) 1 ρ (τ) = ln R (τ) τ + = r + ln [V 2 (τ) V 1 (τ) + (1 θ) S (τ)] τ +

111 Intuition for effi ciency result rv (τ) + V (τ) = u + αn 2 (τ) θs (τ) rλ (τ) + λ (τ) = u + αn 2 (τ) S (τ) rv 1 (τ) + V 1 (τ) = u 1 rλ 1 (τ) + λ 1 (τ) = u 1 rv 2 (τ) + V 2 (τ) = u 2 + αn (τ) (1 θ) S (τ) rλ 2 (τ) + λ 2 (τ) = u 2 + αn (τ) S (τ) τ S (τ) = ū S (τ) = ū τ e [ δ(τ) δ(z)] dz + e δ(τ) S () e [ δ (τ) δ (z)] dz + e δ (τ) S ()

112 Intuition for effi ciency result rv (τ) + V (τ) = u + αn 2 (τ) θs (τ) rλ (τ) + λ (τ) = u + αn 2 (τ) S (τ) rv 1 (τ) + V 1 (τ) = u 1 rλ 1 (τ) + λ 1 (τ) = u 1 rv 2 (τ) + V 2 (τ) = u 2 + αn (τ) (1 θ) S (τ) rλ 2 (τ) + λ 2 (τ) = u 2 + αn (τ) S (τ) τ S (τ) = ū S (τ) = ū τ e [ δ(τ) δ(z)] dz + e δ(τ) S () e [ δ (τ) δ (z)] dz + e δ (τ) S () τ δ (τ) δ (τ) = α [(1 θ) n 2 (z) + θn (z)] dz

113 Intuition for effi ciency result rv (τ) + V (τ) = u + αn 2 (τ) θs (τ) rλ (τ) + λ (τ) = u + αn 2 (τ) S (τ) rv 1 (τ) + V 1 (τ) = u 1 rλ 1 (τ) + λ 1 (τ) = u 1 rv 2 (τ) + V 2 (τ) = u 2 + αn (τ) (1 θ) S (τ) rλ 2 (τ) + λ 2 (τ) = u 2 + αn (τ) S (τ) τ S (τ) = ū S (τ) = ū τ e [ δ(τ) δ(z)] dz + e δ(τ) S () e [ δ (τ) δ (z)] dz + e δ (τ) S () τ δ (τ) δ (τ) = α [(1 θ) n 2 (z) + θn (z)] dz

114 Intuition for effi ciency result rv (τ) + V (τ) = u + αn 2 (τ) θs (τ) rλ (τ) + λ (τ) = u + αn 2 (τ) S (τ) rv 1 (τ) + V 1 (τ) = u 1 rλ 1 (τ) + λ 1 (τ) = u 1 rv 2 (τ) + V 2 (τ) = u 2 + αn (τ) (1 θ) S (τ) rλ 2 (τ) + λ 2 (τ) = u 2 + αn (τ) S (τ) τ S (τ) = ū S (τ) = ū τ e [ δ(τ) δ(z)] dz + e δ(τ) S () e [ δ (τ) δ (z)] dz + e δ (τ) S () τ δ (τ) δ (τ) = α [(1 θ) n 2 (z) + θn (z)] dz

115 Intuition for effi 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 [, T ], with = only for τ = The planner discounts more heavily than the equilibrium S (τ) < S (τ) for all τ (, 1] Social value of loan < joint private value of loan

116 Intuition for effi 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 [, T ], with = only for τ = The planner discounts more heavily than the equilibrium S (τ) < S (τ) for all τ (, 1] Social value of loan < joint private value of loan

117 Intuition for effi 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 [, T ], with = only for τ = The planner discounts more heavily than the equilibrium S (τ) < S (τ) for all τ (, 1] Social value of loan < joint private value of loan

118 Intuition for effi 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 [, T ], with = only for τ = The planner discounts more heavily than the equilibrium S (τ) < S (τ) for all τ (, 1] Social value of loan < joint private value of loan

119 Intuition for effi 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 [, T ], with = only for τ = The planner discounts more heavily than the equilibrium S (τ) < S (τ) for all τ (, 1] Social value of loan < joint private value of loan

120 Intuition for effi 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 [, T ], with = only for τ = The planner discounts more heavily than the equilibrium S (τ) < S (τ) for all τ (, 1] Social value of loan < joint private value of loan

121 Intuition for effi ciency result Planner internalizes that searching borrowers and lenders make it easier for other lenders and borrowers to find partners These liquidity provision services to others receive no compensation in the equilibrium, so individual agents ignore them when calculating their equilibrium payoffs The equilibrium payoff 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)

122 Intuition for effi ciency result Planner internalizes that searching borrowers and lenders make it easier for other lenders and borrowers to find partners These liquidity provision services to others receive no compensation in the equilibrium, so individual agents ignore them when calculating their equilibrium payoffs The equilibrium payoff 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)

123 Intuition for effi ciency result Planner internalizes that searching borrowers and lenders make it easier for other lenders and borrowers to find partners These liquidity provision services to others receive no compensation in the equilibrium, so individual agents ignore them when calculating their equilibrium payoffs The equilibrium payoff 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)

124 Frictionless limit Proposition Let Q K k=1 kn k (T ) = 1 + n 2 (T ) n (T ). For τ [, T ], ρ (τ) = r + ln[(1 e r τ ) u 1 u r + ln r +e r τ (U 1 U )] [ ] (1 e r τ ) u 1 u θū r +e r τ (U 1 U θs ()) r + ln[(1 e r τ ) u 2 u 1 τ+ if Q < 1 τ+ if Q = 1 r +e r τ (U 2 U 1 )] τ+ if 1 < Q.

125 IOR Policy: intuition from the analytical example n 2( T ) = n ( T ) e f r ( )( i + i P c ) θ i + 1 θ + f c f ρ f ( T t) ρ f T t

126 IOR Policy: intuition from the analytical example n 2( T ) < n( T ) i + i + P r f c f c ρ f e f r ( )( i + i P c ) θ i + 1 θ + f c f ρ f ( T t) T t

127 IOR Policy: intuition from the analytical example n ( T ) < n2( T ) e f r ( )( i + i P c ) θ i + 1 θ + f c f ρ f ( T t) e i f ρ f T t

128 IOR Policy: intuition from the analytical example n ( T ) = n2( T ) i + i + P r f c f c n 2( T ) < n( T ) ρ f n ( T ) < n2( T ) e f r ( )( i + i P c ) θ i + 1 θ + f c f ρ f ( T t) e i f ρ f 1 Q

129 Small-scale simulations: K = {, 1, 2} k = 1 Two scenarios { n H (T ), n L 2 (T )} { n L (T ), nh 2 (T )} {.6,.3} {.3,.6} Experiments Bargaining Power (θ) Discount Rate (i w f ) Contact Rate (α)

130 Surplus Surplus ρ (%) ρ (%) Bargaining power 1.6 x 1 5 θ=.1 θ= θ= θ=.1 θ=.5 θ=.9.8 θ=.1 θ=.5 θ= V 2 V : 16:3 17: 17:3 18: 18:3 Eastern Time : 16:3 17: 17:3 18: 18:3 Eastern Time : 16:3 17: 17:3 18: 18:3 Eastern Time 1.6 x 1 5 θ=.1 θ= θ= θ=.1 θ=.5 θ=.9.8 θ=.1 θ=.5 θ= V 2 V : 16:3 17: 17:3 18: 18:3 Eastern Time 16: 16:3 17: 17:3 18: 18:3 Eastern Time 16: 16:3 17: 17:3 18: 18:3 Eastern Time

131 Surplus Surplus ρ (%) ρ (%) Contact rate 1.6 x 1 5 α=25 α=5 1.4 α= α=25 α=5 α= α=25 α=5 α= V 2 V : 16:3 17: 17:3 18: 18:3 Eastern Time : 16:3 17: 17:3 18: 18:3 Eastern Time 16: 16:3 17: 17:3 18: 18:3 Eastern Time 1.6 x 1 5 α=25 α=5 1.4 α=1 1.9 α=25 α=5 α=1.55 α=25 α=5 α= V 2 V : 16:3 17: 17:3 18: 18:3 Eastern Time 16: 16:3 17: 17:3 18: 18:3 Eastern Time 16: 16:3 17: 17:3 18: 18:3 Eastern Time

132 Surplus Surplus ρ (%) ρ (%) Discount-Window lending rate 2.5 x w i =.5% f w i =.75% f w i =1% f i f w =.5% i f w =.75% i f w =1%.8 i f w =.5% i f w =.75% i f w =1% V 2 V : 16:3 17: 17:3 18: 18:3 Eastern Time 16: 16:3 17: 17:3 18: 18:3 Eastern Time 16: 16:3 17: 17:3 18: 18:3 Eastern Time 2.5 x 1 5 w i =.5% f 1.15 i f w =.5% i f w =.5% 2 w i =.75% f w i =1% f i f w =.75% i f w =1%.7.65 i f w =.75% i f w =1% 1.5 V 2 V : 16:3 17: 17:3 18: 18:3 Eastern Time : 16:3 17: 17:3 18: 18:3 Eastern Time : 16:3 17: 17:3 18: 18:3 Eastern Time