Trade Dynamics in the Market for Federal Funds

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

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

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

The market for federal funds

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

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

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?

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

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 +

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

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

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

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

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

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

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

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

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 }}

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 )}

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)] + 1 2 [V k (τ) V k b (τ)]

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

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)]

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

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)

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.

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}

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.

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

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

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

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

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 + 1 2 S ij ks (τ) j s ij (τ) = = r + τ + τ +

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 + 1 2 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)

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

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

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

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

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

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

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 2.5 24 2.5 24 22 24 1 7 36.36.625 36 36.725 36.925 36.1 36

Calibrated parameters, 27 targets (α, P w ) = ( 1,.525 ) 36 Model Data fed funds rate.527.525 * std. dev. of balances at 6:3 pm 1.2 1.15 * median number of counterparties 7 2 mean number of counterparties 7 4.5 intermediation index.65.43

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 15 16.35 1 14.3 5 12.25 1.2 8.15 5 6.1 4.5 1 2 5 5 1 15 2 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 5.8 5.7 5.6 5.5 5.4 5.3 5.2.51.43.36.29.22.14.7 6. 4. 2.. 2. 4. 5.1 16: 16:3 17: 17:3 18: 18:3 Eastern Time 2 4 6 8 1 12 Loan rates (% ) 6. 16:16:1516:316:4517:17:1517:317:4518:18:1518:3 Eastern Time

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) 1.9.8.7.6.5.4.3.2.1 Proportion of amount of funds traded Proportion of number of loans traded 16: 16:3 17: 17:3 18: 18:3 Eastern Time.65.58.51.43.36.29.22.14.7 1 11 21 31 41 51 61 71 Loan sizes 3 25 2 15 1 5 16: 16: 15 16: 3 16: 45 17: 17: 15 17: 3 17: 45 18: 18: 15 18: 3 Eastern Time.18.25.25.16.14.2.2.12.1.15.15.8.1.1.6.4.5.5.2 2 4 6 8 1 12 14 16 18 2 22 24 Number of counterparties 2 4 6 8 1 12 14 16 18 2 22 24 Number of counterparties banks lend to 2 4 6 8 1 12 14 16 18 2 22 24 Number of counterparties banks borrow from

Proportion of banks Intermediation (27).12.1.8.6.4.2.2.4.6.8 1 Proportion of intermediated f unds

Policy parameters as in 211 i r f i e f i w f i c f i o f.25 36.25 36.75 36.175 36.415 36

Calibrated parameters, 211 targets (α, P w ) = (1, ) Model Data fed funds rate.29.25 * std. dev. of balances at 6:3 pm 31 31 * median number of counterparties 2 mean number of counterparties.1 2.9 intermediation index.2.35

Policy experiments (baseline policy as in 211) i f Q/ k =.5 Q/ k = 1. Q/ k = 3 38 36 1 25 51 5 26 5 64 63 51 if w Q/ k =.5 Q/ k = 1. Q/ k = 3 5 39 38 26 75 51 5 26 1 63 61 26

Corridor system (i f, if w ) Q/ k =.5 Q/ k = 1 Q/ k = 3 25 14 13 1 25 5 39 38 26 5 75 64 63 51 75 1 89 88 76 1 125 114 113 11

Corridor system 2 Fed funds rate (%) 2 Fed funds rate (%) 2 Fed funds rate (%) 1.8 1.8 1.8 1.6 1.6 1.6 i f w 1.4 1.4 i f w 1.4 1.2 1.2 1.2 1 i f w 1 1 i f.8.8 i f.8.6.4 i f.6.6.4.4.2.2.2 5 1 15 2 25 3 35 4 45 5 5 1 15 2 25 3 35 4 45 5 5 1 15 2 25 3 35 4 45 5 2 Fed funds rate (%) 2 Fed funds rate (%) 2 Fed funds rate (%) 1.8 1.8 1.8 1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 1 1.8 i f w.8 i f w.6.6.4.4 i f 1.8 i f w.6 i f.4.2.2.2 5 1 15 2 25 3 35 4 45 5 i f 5 1 15 2 25 3 35 4 45 5 5 1 15 2 25 3 35 4 45 5

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

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)

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

The views expressed here are not necessarily reflective of views at the Federal Reserve Bank of New York or 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 buy There are banks that are very short and sell

Percent Price dispersion.1 Intraday Distribution of Fed Funds Spreads, 25.5.5.1.15 16: 16:3 17: 17:3 18: 18:3 1th/9th Percentiles Median 25th/75th Percentiles Mean

Billions of Dollars Intermediation: excess funds reallocation 15 Excess Funds Reallocation 1 5 25 26 27 28 29 21

Intermediation: proportion of intermediated funds.6 Proportion of Intermediated Funds.5.4.3.2.1 25 26 27 28 29 21

Intraday evolution of the distribution of reserve balances 1 Normalized Balances, 27.5.5 16: 16:3 17: 17:3 18: 18:3 1th/9th Percentiles Median 25th/75th Percentiles Mean

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

Millions of Dollars Banks that are short...and sell... Sales by Banks with Negative Adjusted Balances, 27 5 4 3 2 1 16: 16:3 17: 17:3 18: 18:3 1th/9th Percentiles Median 25th/75th Percentiles Mean

Billions of Dollars Number of Loans Daily volume 2 Daily Amount of Federal Funds Loans Daily Number of Federal Funds Loans 1, 15 1 5 5 25 26 27 28 29 21 25 26 27 28 29 21

Millions of Dollars Daily Number of Loans Daily volume (size distribution) 1, Daily Distribution of Loan Sizes 8 Distribution of Daily Number of Loans 8 6 6 4 4 2 2 25 26 27 28 29 21 25 26 27 28 29 21 1th Percentile Median 25th Percentile Median 9th Percentile Mean 75th Percentile Mean

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 12 9 6 3 25 26 27 28 29 21 25th 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 15 12 12 9 9 6 6 3 3 25 26 27 28 29 21 25 26 27 28 29 21 25th P ercentile Median 25th P ercentile Median 75th P ercentile Mean 75th P ercentile Mean

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, 5 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

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, 25 15 Intraday Number of Loans, by Bank, 26 15 Intraday Number of Loans, by Bank, 27 1 1 1 5 5 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 15 Intraday Number of Loans, by Bank, 28 8 Intraday Number of Loans, by Bank, 29 1 Intraday Number of Loans, by Bank, 21 8 6 1 6 4 4 5 2 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

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, 27 4 4 4 3 3 3 2 2 2 1 1 1 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 5 Intraday Distribution of Loan Sizes, 28 1, 5 Intraday Distribution of Loan Sizes, 29 8 Intraday Distribution of Loan Sizes, 21 4 6 1, 3 4 2 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

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, 15 5 1 5 25 26 27 28 29 21 25 26 27 28 29 21 9th Percentile, 15:45 16:15 9th Percentile, 17:45 18:15 1th Percentile, 15:45 16:15 1th Percentile, 17:45 18:15

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.8.8.6.6.4.4.2.2 25 26 27 28 29 21 25 26 27 28 29 21 Loans by 16:15 Loans by 17:15 Loans by 18:15 Loans by 16:15 Loans by 17:15 Loans by 18:15

1 Normalized Balances, 29 Intraday evolution of the distribution of reserve balances 1 Normalized Balances, 25 1 Normalized Balances, 26 1 Normalized Balances, 27.5.5.5.5 16: 16: 3 17: 17: 3 18: 18: 3.5 16: 16: 3 17: 17: 3 18: 18: 3.5 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 1.5 Normalized Balances, 28 1 Normalized Balances, 29 15 Normalized Balances, 21 1 8 1 6.5 4 5 2.5 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

Intraday evolution of the distribution of reserve balances 1 S tandard Deviation of Normalized B alances 8 6 4 2 25 26 27 28 29 21 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) 1.2 5.4.6 25 26 27 28 29 21 25 26 27 28 29 21 1th P ercentile, 16:15 1th P ercentile, 18:15 9th P ercentile, 16:15 9th P ercentile, 18:15

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 4.5 2.5 25 26 27 28 29 21 FOMC Target Fed Funds Rate 1 25 26 27 28 29 21

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 1 4.5 2.5 1 25 26 27 28 29 21 FOMC Target Effective Fed Funds Rate 1.5 25 26 27 28 29 21

Percent Daily fed funds rate dispersion 6 Daily Dispersion of Fed Funds Rate 4 2 25 26 27 28 29 21 FOMC Target 1th Percentile of Fed Funds Rate 9th Percentile of Fed Funds Rate

Percent Fed funds rate vs. effective fed funds rate.6 Daily Spread between Fed Funds and Effective Rate.4.2.2 25 26 27 28 29 21

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, 27. 5.1.1. 5.1.1.1.2. 15 16: 16: 3 17: 17: 3 18: 18: 3.2 16: 16: 3 17: 17: 3 18: 18: 3.3 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.6 Intraday Distribution of Fed Funds Spreads, 28. 15 Intraday Distribution of Fed Funds Spreads, 29.1 Intraday Distribution of Fed Funds Spreads, 21.4.1. 5.2. 5.2. 5.4 16: 16: 3 17: 17: 3 18: 18: 3.1 16: 16: 3 17: 17: 3 18: 18: 3. 5 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

Percent Intraday distribution of fed funds spreads (over time) Intraday Distribution of Fed Funds Spreads (over time) 6 4 2 2 4 25 26 27 28 29 21 1th Percentile, 16:15 9th Percentile, 16:15 1th Percentile, 18:15 9th Percentile, 18:15

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, 27.2.2.4.5.6.2 16: 16: 3 17: 17: 3 18: 18: 3 1 16: 16: 3 17: 17: 3 18: 18: 3.8 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 Spreads to FOMC Target, 28.2 Intraday Distribution of Spreads to FOMC Target, 29.1 Intraday Distribution of Spreads to FOMC Target, 21. 15. 8.5. 6.1 1. 4 1.5. 5. 2 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

Billions of Dollars Millions of Dollars Daily intermediation 15 E xcess Funds Reallocation.6 P roportion of Intermediated Funds.5 1.4.3 5.2.1 25 26 27 28 29 21 25 26 27 28 29 21 8 Daily Distribution of E xcess Funds Reallocation.8 Daily Distribution of P roportion of Intermediated Funds 6.6 4.4 2.2 25 26 27 28 29 21 25 26 27 28 29 21 1th P ercentile Median 1th P ercentile Median 9th P ercentile Mean 9th P ercentile Mean

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, 21 5 4 4 4, 3 3 2, 2 2 1 1 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

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, 27 8 4 4 6 3 3 4 2 2 2 1 1 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 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, 6 4 4 1, 2 3 3 8 2 2 4 1 1 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

Percent Daily fed funds rate vs. IOR.5 Daily Dispersion of Fed Funds Rate and IOR.4.3.2.1 29 21 1th Percentile of Fed Funds Rate 9th Percentile of Fed Funds Rate IOR

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.2.2.15.15.1.1 29 21.5 29 21 IOR Fed Funds Rate IOR Effective Fed Funds Rate

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

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 )

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

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 )

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)] )

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 >

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 )

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)]

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

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)]}

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)]}

Fed funds rate R (τ) = e ρ(τ+ ) 1

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 +

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 ()

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

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)

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.

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

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

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

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

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).1.5.9.5 36.75 36.1 36 25 5 1

Surplus Surplus ρ (%) ρ (%) Bargaining power 1.6 x 1 5 θ=.1 θ=.5 1.4 θ=.9 1.2 1.18 θ=.1 θ=.5 θ=.9.8 θ=.1 θ=.5 θ=.9 1.2 1.16.7 1.8 V 2 V 1 1.14 1.12.6.6.5.4.2 16: 16:3 17: 17:3 18: 18:3 Eastern Time 1.1 1.8 1.6 16: 16:3 17: 17:3 18: 18:3 Eastern Time.4.3 16: 16:3 17: 17:3 18: 18:3 Eastern Time 1.6 x 1 5 θ=.1 θ=.5 1.4 θ=.9 1.9 θ=.1 θ=.5 θ=.9.8 θ=.1 θ=.5 θ=.9 1.2 1 1.85.7.8 V 2 V 1 1.8.6.6 1.75.5.4.4.2 1.7.3 16: 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

Surplus Surplus ρ (%) ρ (%) Contact rate 1.6 x 1 5 α=25 α=5 1.4 α=1 1.2 1.18 α=25 α=5 α=1.7.68 α=25 α=5 α=1 1.2 1.16.66 1.8.6 V 2 V 1 1.14 1.12.64.62.6.4 1.1.58.2 1.8.56.54 16: 16:3 17: 17:3 18: 18:3 Eastern Time 1.6 16: 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=1 1.2 1 1.85.5.8 V 2 V 1 1.8.45.6 1.75.4.4.2 1.7.35 16: 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

Surplus Surplus ρ (%) ρ (%) Discount-Window lending rate 2.5 x 1 5 2 w i =.5% f w i =.75% f w i =1% f 1.2 1.18 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% 1.5 1 V 2 V 1 1.16 1.14 1.12 1.1.75.7.65.6.55.5.5 1.8.45 1.6.4 16: 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 1.1 1.95 i f w =.75% i f w =1%.7.65 i f w =.75% i f w =1% 1.5 V 2 V 1 1.9 1.85.6.55.5 1 1.8.45.5 16: 16:3 17: 17:3 18: 18:3 Eastern Time 1.75 1.7 16: 16:3 17: 17:3 18: 18:3 Eastern Time.4.35.3 16: 16:3 17: 17:3 18: 18:3 Eastern Time