Figure 1: Timeline of the model.
Note: Nature determines
the shock arrival time \(t_l\). If
\(\tau<t_l\), settlement occurs
before the shock: payments are netted, liquidity costs are incurred, all
payments are made in full, and surviving agents consume. If \(\tau\geq t_l\), the shock arrives before
settlement: agents may default, deadweight losses may arise, equilibrium
payments \(x\) are made, and surviving
agents consume.
This figure summarizes the timeline of the model. The nature decides the arrival time of the shock. If the settlement time is less than the arrival time of the shock, then there will be no defaults, payments are netted, reducing the actual payment network, and liquidity costs are paid by agents. Agents will settle all the payments, and surviving agents will consume their resulting wealth at the last period. If the settlement time is greater than or equal to the arrival time of shock, then some agents may default (possibly from spillovers), depending on the equilibrium payments. Surviving agents consume but defaults also incur deadweight loss multiplied by the amount of payment shortfalls.
Figure 2: The ring network and the complete network
Note:
The left graph depicts the ring network, in which each agent owes the
full liability amount of \(\bar{d}\) to
the next agent. The right graph depicts the complete network, in which
each agent owes \(\dfrac{\bar{d}}{n-1}\) to every other
agent.
The left graph depicts the ring network, in which each agent owes the full liability amount to the next agent. The right graph depicts the complete network, in which each agent owes an equal fraction to every other agent.
Figure 3: Number of defaults and aggregate payment shortfall: Complete vs Ring
Note: This figure illustrates the results in Proposition 1. The top panels have the number of defaults on the
y-axis and the settlement time τ on the x-axis. The top-left panel
illustrates the phase transition property of contagion in the complete
network, as the number of defaults jumps at the threshold τ value. In contrast, the top-right
panel illustrates the gradual increase in the number of defaults for the
ring network as τ decreases.
The bottom panels have the aggregate payment shortfall on the y-axis and
the settlement time τ on the
x-axis. The bottom-left panel illustrates the phase transition property
in the complete network. The bottom-right panel illustrates smooth
changes in the payment shortfall in the ring network as τ decreases.
(a) Number of defaults: Complete network
(b) Number of defaults: Ring network
(c) Aggregate payment shortfall: Complete network
(d) Aggregate payment shortfall: Ring network
This figure illustrates the results in Proposition 1. The top panels have the number of defaults on the y-axis and the settlement time on the x-axis. The top-left panel illustrates the phase transition property of contagion in the complete network, as the number of defaults jumps at the threshold value. In contrast, the top-right panel illustrates the gradual increase in the number of defaults for the ring network as the settlement time decreases. The bottom panels have the aggregate payment shortfall on the y-axis and the settlement time on the x-axis. The bottom-left panel illustrates the phase transition property in the complete network. The bottom-right panel illustrates smooth changes in the payment shortfall in the ring network as the settlement time decreases.
Figure 4: Ex ante Social Welfare: Complete vs Ring
Note: This figure
illustrates the results in Proposition 2. Both panels have the ex ante social welfare on the y-axis and
the settlement time τ on the
x-axis. The left panel shows the changes in ex ante social welfare for
the complete network. The ex ante social welfare shows a kink, a
discrete jump in the slope, at the threshold value. The right panel
shows the changes in the ex ante social welfare for the ring network
that changes smoothly. Thus, if τ is just below the threshold value,
the marginal change in the ex ante social welfare is positive for the
complete network, while it is negative for the ring network.
(a) Ex ante Social Welfare: Complete network
(b) Ex ante Social Welfare: Ring network
This figure illustrates the results in Proposition 2. Both panels have the ex ante social welfare on the y-axis and the settlement time on the x-axis. The left panel shows the changes in ex ante social welfare for the complete network. The ex ante social welfare shows a kink, a discrete jump in the slope, at the threshold value. The right panel shows the changes in the ex ante social welfare for the ring network that changes smoothly. Thus, if the settlement time is just below the threshold value, the marginal change in the ex ante social welfare is positive for the complete network, while it is negative for the ring network.
Figure 5: Impact of a faster settlement speed on financial stability
Note: This figure visualizes the effect of a faster
settlement speed (lower \(\tau\)) on
financial stability through three different channels—the cost channels,
(A) and (B), and the benefit channel (C). Faster settlement increases
liquidity cost and required payment due to less netting, while
decreasing the likelihood of shock arrival, i.e., the benefit channel
(C). However, the first two effects lead to an increase in payment
shortfalls and deadweight loss, i.e., the cost channel (A). Moreover, if
\(\tau\) is a default threshold point,
the first two effects can lead to an increase in the number of defaults,
leading to a further increase in deadweight loss, i.e., the cost channel
(B). Therefore, whether \(\tau\) is a
default threshold point or not is important, as it determines the
importance of the cost channel (B), highlighted by Theorem 1.
This figure visualizes the effect of a faster settlement speed (lower settlement time) on financial stability through three different channels---the cost channels, (A) and (B), and the benefit channel (C). Faster settlement increases liquidity cost and required payment due to less netting, while decreasing the likelihood of shock arrival, i.e., the benefit channel (C). However, the first two effects lead to an increase in payment shortfalls and deadweight loss, i.e., the cost channel (A). Moreover, if the settlement time is a default threshold point, the first two effects can lead to an increase in the number of defaults, leading to a further increase in deadweight loss, i.e., the cost channel (B). Therefore, whether the settlement time is a default threshold point or not is important, as it determines the importance of the cost channel (B), highlighted by Theorem 1.
Figure 6: An example of non-zero full netting matrix
Note:
Before netting (left), agents form a cycle: \(A\) owes \(B\) 1, \(B\) owes \(C\) 2, \(C\) owes \(A\) 1. After full netting (right), only
\(B\) owes \(C\) 1.
The left graph depicts the network, in which agent A owes 1 to agent B, who owes 1 to agent C, before netting. The right graph depicts the same network after full netting, in which agent B still owes 1 to agent C.