Precautionary Demand and Liquidity in Payment Systems
Gara Afonso Federal Reserve Bank of New York
Hyun Song Shin Princeton University
Banco de México – March 4, 2010
The views expressed in this paper are those of the authors and not necessarily those of the Federal Reserve Bank of New York or the Federal Reserve System.
Afonso & Shin
Precautionary Demand and Liquidity in Payment Systems
Flows in high-value systems
⋄ Characterized by high velocity. Flows I receive from A depend on flows A receives from B, C, etc.
But
flows leaving B or C may depend on flows coming from me.
⋄ High degree of coordination and synchronization is needed.
⋄ Payment systems: Rely heavily on inflows to fund outflows. McAndrews and Potter (2002)
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Precautionary Demand and Liquidity in Payment Systems
This paper
⋄ Presents a model of a payment system to examine the impact of shocks and their endogenous propagation within the system. Key element: fully endogenous transitions in the reaction function of banks
⋄ Studies: 1. Delays in payments to a potentially vulnerable bank. 2. Increase in precautionary demand.
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Precautionary Demand and Liquidity in Payment Systems
Precautionary demand
Figure 1: Ashcraft, McAndrews and Skeie (2009).
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Precautionary Demand and Liquidity in Payment Systems
Framework ⋄ n banks in the payment system. ⋄ Every member maintains an account which contains: bit
i’s time t balances.
cit
i’s time t (remaining) credit capacity.
where θt = (bt, ct )
⋄ Let yti be the time t payments from i to other members and xit the time t payments received by i: yti = f i(xit, θt) Banco de México
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Precautionary Demand and Liquidity in Payment Systems
Incoming and outgoing flows
⋄ Assume: 1. i only pays out a proportion of its incoming funds. 2. Transfers do not decrease as incoming funds rise.
⋄ Incoming funds depend on all payments sent over the payment system, so yt can be written as: yt = F yt−1, θt
iT h iT 1 2 n 1 2 n where yt = yt , yt , · · · , yt and F = f , f , · · · , f . h
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Precautionary Demand and Liquidity in Payment Systems
Unique solution
Two-step argument:
⋄ Step 1: There exists at least one fixed point of mapping F .
⋄ Step 2: There exists a unique fixed point of the mapping F .
Comparative Statics - Milgrom and Roberts (1994)
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Precautionary Demand and Liquidity in Payment Systems
A stylized payment system We examine a simplified RTGS system reminiscent of Fedwire. ⋄ 21.5 operating hours. ⋄ 50 institutions: 87.27% of total value over Fedwire in 2008 (78.27%) ⋄ Can send and receive payments to and from any member. ⋄ Subject to idiosyncratic shocks which determine final payments. ⋄ Balances as in Figure 2(a). ⋄ Access to daylight overdraft up to net debit caps (Figure 2(b)). Banco de México
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Precautionary Demand and Liquidity in Payment Systems
Balances and net debit caps
(a)
(b)
35
20
15
25
Number of banks
Number of banks
30
20 15 10
10
5
5 0 0
2
4 6 Average daily balances
8
0 0
50
100 150 200 250 Average daily net debit caps
300
Figure 2: Distribution of the average daily balances and average daily net debit caps ($ billion) of the 50 institutions. Banco de México
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Precautionary Demand and Liquidity in Payment Systems
The payment system
A
B C
G H
I
J
K
Figure 3: An example of payment system.
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Precautionary Demand and Liquidity in Payment Systems
Bank i’s payment decisions at time t. '
$
“Normal Conditions” at t − 1? Yes
Daylight Overdraft <
1 2
No
Balance > 0? Yes No
Net Debit Cap?
Yes
No Back to NORMAL
N ORMAL C ONDITIONS :
CAUTIOUS C ONDITIONS :
80% Incoming Outgoing < + ? 50% ND Cap
Daylight Overdraft < ND Cap?
Yes Pay at most: 80% Incoming + up to 50% ND Cap
Yes
Remain CAUTIOUS
No
No Queue payments
20% Incoming Outgoing < + ? 0.5% ND Cap Yes
No
Stop using intraday credit Outgoing < 20% Incoming? Yes
&
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Pay at most: 20% Incoming + up to 0.5% ND Cap
Queue payments
Pay at most: 20% Incoming
No Queue payments
%
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Afonso & Shin
Precautionary Demand and Liquidity in Payment Systems
Payments
Banks pay at most: N ORMAL C ONDITIONS CAUTIOUS C ONDITIONS 80% of its cumulative receipts
20% of its cumulative receipts
and
and
reserves and intraday credit
reserves and intraday credit
up to
up to
50% bank’s net debit cap
0.5% of bank’s net debit cap
Figure 4: Outgoing payments.
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Precautionary Demand and Liquidity in Payment Systems
Standard Functioning of the Payment System 90
Value of payments
80 70 60 50 40 30 20 10 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Figure 5: Value of all payments (McAndrews and Rajan (2000) & Coleman (2002))
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Precautionary Demand and Liquidity in Payment Systems 8
90 7 6
Balance at CB accounts
Value of payments
80 70 60 50 40 30
Bank A Bank B Bank C
5 4 3 2
20
1
10
0 −1 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
0.7
Queued payments
0.8 Bank A Bank B Bank C
0.6 0.5
4
Bank A Bank B Bank C
2
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
0.4 0.3 0.2 0.1 0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Slope of reaction function
Daylight overdrafts
Eastern Time
Eastern Time 1
0.5 Bank A Bank B Bank C 0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Figure 6: Standard Functioning of the Payment System. Banco de México
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Precautionary Demand and Liquidity in Payment Systems
Example of Sudden Inflows Dry-up
⋄ Examine the consequences of a reduction in payments sent to a bank (bank D) identified as vulnerable to failure.
– Bank D initially behaves as in baseline case.
– All other banks postpone payments to bank D.
⋄ Case of bank D being a large player.
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Precautionary Demand and Liquidity in Payment Systems
Value of payments
Sudden Inflows Dry-up
60 40 20 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Balance at CB accounts
Eastern Time 0 −100 −200
Bank A Bank B Bank C Bank D
−300 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Figure 7: Value of payments and balances at CB accounts.
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Precautionary Demand and Liquidity in Payment Systems Inflows Dry-up
Baseline 90
80
80
70
70
Value of payments
Value of payments
90
60 50 40 30
60 50 40 30
20
20
10
10
21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
50
50
0
0
Balance at CB accounts
Balance at CB accounts
Eastern Time
−50 −100 −150 −200 −250
Bank A Bank B Bank C Bank D
Bank A Bank B Bank C
−50 −100 −150 −200 −250
−300 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
−300 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Figure 8: Value of payments and balances at CB accounts. Banco de México
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Precautionary Demand and Liquidity in Payment Systems Inflows Dry-up
Baseline
300
250
Daylight overdrafts
Daylight overdrafts
250
300 Bank A Bank B Bank C Bank D
200
150
100
50
Bank A Bank B Bank C
200
150
100
50
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
900 800
Eastern Time
Bank A Bank B Bank C Bank D
900 800 700
Queued payments
Queued payments
700
Bank A Bank B Bank C
600 500 400 300
600 500 400 300
200
200
100
100
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Figure 9: Total overdrafts and queued payments. Banco de México
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Precautionary Demand and Liquidity in Payment Systems
Banks’ decision to cancel payments...
1. Reduction in the overall value of payments.
2. Increase in queued and unsettled payments.
3. Bank not receiving payments: – Demands enormous intraday credit. – Negative end-of-day balance.
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Precautionary Demand and Liquidity in Payment Systems
Example of Increased Precautionary Demand
⋄ Examine the consequences of a shift by a small group of banks towards a more conservative decision rule.
– Banks M - R concerned about a liquidity shortage: pay only 20% of the funds they receive and up to 0.5% of their net debit caps. – Group comprise of at least a large domestic bank, a small domestic bank, a foreign financial holding company and a government sponsored enterprise. Banco de México
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Precautionary Demand and Liquidity in Payment Systems
Value of payments
Increased Precautionary Demand 40 30 20 10 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Balance at CB accounts
Eastern Time
400 200 0 −200 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Figure 10: Value of payments and balances at CB accounts.
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Afonso & Shin
Precautionary Demand and Liquidity in Payment Systems Precautionary Demand
Baseline 90
80
80
70
70
Value of payments
Value of payments
90
60 50 40 30
60 50 40 30
20
20
10
10
21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
500
500
400
400
Balance at CB accounts
Balance at CB accounts
Eastern Time
300 200 100 0 −100
300 200 100 0 −100
−200 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
−200 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Figure 11: Value of payments and balances at CB accounts. Banco de México
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Precautionary Demand and Liquidity in Payment Systems Baseline
700
700
600
600
500
500
Daylight overdrafts
Daylight overdrafts
Precautionary Demand
400 300 200 100
400 300 200 100
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
3000
3000
2500
2500
Total queued payments
Total queued payments
Eastern Time
2000
1500
1000
500
2000
1500
1000
500
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
0 21:00 23:00 1:00 3:00 5:00 7:00 9:00 11:00 13:00 15:00 17:00
Eastern Time
Figure 12: Total overdrafts and queued payments. Banco de México
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Precautionary Demand and Liquidity in Payment Systems
Banks’ concern about a liquidity shortage...
1. Disruption of payments: Payment System freezes.
2. Increases size of banks’ balances.
3. Enormous use of intraday credit.
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Precautionary Demand and Liquidity in Payment Systems
Conclusions Study liquidity and precautionary demand in payment systems, where the ability to send payments relies heavily on other participant’s capability to send out funds.
We present a flexible setting that shows the importance of the interdependence of flows in high-value payment systems and allows us to analyze: ⋄ Delay in payments to a vulnerable bank, ⋄ Banks’ attempt to conserve liquidity.
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Precautionary Demand and Liquidity in Payment Systems
Appendix
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Precautionary Demand and Liquidity in Payment Systems
Step 1 (Existence) Theorem 1 (Tarski (1955) Fixed Point Theorem) Let (Y, ≤) be a complete lattice and F be a non-decreasing function on Y . Then there are y ∗ and y∗ such that F (y ∗) = y ∗, F (y∗) = y∗, and for any fixed point y, we have y∗ ≤ y ≤ y ∗.
A complete lattice is an ordered set (Y, ≤) which satisfies that each subset S ⊆ Y has both a greatest lower bound inf (S ) and a least upper bound sup (S ) in Y .
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Precautionary Demand and Liquidity in Payment Systems
In our payments setting, ⋄ (Y, ≤) is a complete lattice: Y = [0, y 1] × [0, y 2] × . . . × [0, y n] ⋄ F is non decreasing on Y (payments do not decrease as incoming funds rise). There is at least one fixed point of F .
Step 2 (Uniqueness) Theorem 2 There exists a unique profile of payments flows yt that solves yt = F (yt−1, θt). Proof by contradiction. Banco de México
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Precautionary Demand and Liquidity in Payment Systems
Sketch of the proof ⋄ Suppose there are two distinct solutions y ∗ and y∗. ⋄ By Tarski, yi∗ ≥ yi∗ ∀i and yk∗ > yk∗ for some component k. ⋄ Slope of outgoing payments is bounded above by 1: yi∗ − yi∗ ≤ x∗i − xi∗ ∀i
and
yk∗ − yk∗ < x∗k − xk∗ for some k.
⋄ Then, summing across institutions, n X
i=1
xi∗ −
n X
i=1
yi∗ <
n X
i=1
x∗i −
n X
yi∗
i=1
Total value of balances of the system is strictly larger under y ∗ Contradiction. Banco de México
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Comparative Statics (Milgrom and Roberts (1994)) Theorem 3 Let yt∗(θt) be the unique fixed point of the mapping F . If for all yt ∈ Y , F is increasing in θt, then yt∗(θt) is increasing in θt.
Comparative statics provides the framework for policy issues: ⋄ target balances, ⋄ delays in payments, ⋄ early and/or multiple settlement periods. Banco de México
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