3. Estrategia Empírica
3.2 Datos y Variables
p1 = 1 N N X i=1 P ✓ (1 ✓)µA+ ✓µA µL+ q 2 A+ L2✏i < 0 ◆ , (3.17)
where P (.), as before, is the CDF of the location-scale distribution. The fraction of operating banks after r iterations then can be written as:
pr = 1 N N X i=1 P ✓ (1 ✓)µA+ ✓µApr 1+ q✓µA(1 pr 1) µL+ q 2 A+ 2L✏i < 0 ◆ , (3.18) For N ! 1, the above equation can be simplified further to:
pr = 1 P
✓
a0 b0pr 1
◆
, (3.19)
where we reduced the multi-parameter system to the following two parameters given
0 =p 2 A+ 2L: a0 = (1 ✓)µA+ µ0 L q✓µA, (3.20) and b0 = (1 q)✓µ0 A. (3.21)
We have two reasons for introducing ✓. First, we use ✓ to calibrate the simulation- based model of counterparty failure described in the next section. When comparing the simulation solution with the solution of the mean-field model, a0 and b0 are used to
calculate the mean-field solution. Second, estimates for ✓ can be obtained from other studies (M¨uller, 2006; Upper, 2011). Whereas it is more difficult to retrieve zp from publicly available data.
3.3 Simulation-Based Counterparty Risk Model
The mean-field assumption of the interbank market implies that each bank lends the same amount to all other banks and that the size of banks’ balance sheets are roughly the same, with similar capital reserves to counteract a shock to banks’ non-interbank assets and liabilities. In Chapter 5, we test the robustness of the results of the mean-
3.3. Simulation-Based Counterparty Risk Model 43 field model by comparing it with a simulation-based model. In Chapter 6, we use two regulatory datasets from the BoE to calibrate a simulation-based model.
The banking system discussed in Chapter 5 is still highly stylised using standard network structures and random distributions to initialize the banking system. Whereas when constructing the banking system studied in Chapter 6, we use real-world expo- sure networks and balance sheet data. For this reason, the initialization processes of the simulation-based models are stated in the beginning of Chapters 5 and 6 in more detail. However, the assumptions about balance sheet variables during the insolvency propa- gation are mostly the same in both simulation models and outlined in the following.
For both simulation-based models, the number of banks N is fixed and, as in the general model layout, each bank i is assigned a state Si(0) 2 {0, 1}, stating whether
bank i is solvent (Si(0) = 1) or not (Si(0) = 0). As in the mean-field model, we also
consider the instantaneous impact of counterparty failure. Hence, we assume liabilities, Li(t) = Li, non-interbank assets, ˆAi(t) = ˆAi, interbank exposure, gij(t) = gij, and
recovery rates, qij(t) = qij do not change in time. The total liabilities, Li, and the initial
value of total assets, Ai(0), of bank i are assumed to be random variables.
For the same reasons as in the mean-field model, we assume that interbank liabil- ities to insolvent banks have to be returned also to an insolvent bank. Hence, we use total liabilities, Li, in the simulation models only. Interbank assets are the sum of the
exposure from bank i to bank j multiplied by the state of banks j plus the recovery value of the exposure:PN
i=1gijSj(0) + qijgij(1 Sj(0)). The total assets of bank i in
round r are then
Ai(r) = ˆAi+ N
X
i=1
gijSj(r) + qijgij(1 Sj(r)) (3.22)
The insolvency cascade algorithm describes the propagation of insolvency of banks in the banking system. Specifically, we use a similar algorithm as proposed in Furfine (2003), where at each iteration round r:
1. The total assets, Ai(r), are calculated using Eq. 3.22 for each bank i.
2. For all banks i, the difference between total liabilities, Li, and total assets, Ai(r),
3.3. Simulation-Based Counterparty Risk Model 44 3. For any bank i with negative capital, Ai(r) Li, the state of bank i is set to zero.
4. The iteration is repeated until no bank changes its state.
When the process stops, all surviving banks are counted and the fraction of surviv- ing banks, p, is computed as the total number of surviving banks divided by the total number of banks.
3.3.1 Differences in the Assumptions of the Simulation Models
In Chapter 5, each bank i is initially calibrated with liabilities Liand assets Ai(0)drawn
from random distributions with location µLand scale Lfor liabilities, and location µA
and scale Afor total assets.
We construct the interbank network using standard network structures (Erd˝os- R´eny networks, Small-World networks (Watts and Strogatz, 1998) and a Barab´asi- Albert networks (Barab´asi and Albert, 1999)). The adjacency matrix of the standard network, X = { 1ijN}, indicates whether a bank i is exposed to a bank j (i.e. ij = 1) or not (i.e. ij = 0). The total interbank assets of a bank i are computed using
a fixed fraction, ✓ 2 (0, 1), of interbank assets to total assets, such that the total inter- bank exposure is: ✓Ai(0). To extract the value gij of interbank assets from bank i to
bank j, ✓Ai(0)is divided by the degree of bank i, zi =PNj=1 ij. Hence, the individual
loan from bank i to its neighbouring banks j is ✓Ai ij/zi. The difference between total
initial assets, Ai(0), and total interbank assets, ✓Ai(0), are the non-interbank assets, ˆAi,
of bank i.
The recovery rate is constant for all exposures from bank i to bank j, such that an element of the recovery matrix Q can be written as qij = q. Hence, the value of
recovered loans are given as q✓Ai(0)gij.
In Chapter 6, each bank i is initially calibrated with liabilities Liand assets Ai(0)
drawn from random distributions with location µLi and scale Li for liabilities, and
location µAi and scale Ai for total assets, i.e. µAi, µLi, Ai and Li are chosen individ-
ually for each bank. This creates a highly heterogeneous banking system where banks vary greatly in the size of their balance sheets.
The value of interbank exposure, gij, from bank i to bank j is taken from the
regulatory data. The non-interbank assets, ˆAi, are the difference between the total
3.3. Simulation-Based Counterparty Risk Model 45 We disregard the recovery rate, qij, in the simulation model of the heterogeneous
banking system. The reason for this is quite practical: we do not have data on the recovery rate of exposure between specific banks. Thus, the recovery rate is set to zero.
3.3.2 Changes in the Value of Balance Sheet Quantities Over Time
Fire-sales, QE, bank runs and bail-outs are examples that potentially change the value of banks’ balance sheet quantities during the insolvency propagation. To investigate the effects of value changes in time of balance sheet quantities, we incorporate functions that alter the value of assets or liabilities following the initial shock to banks’ balance sheets.
For the simulation model in Chapter 5, we apply a function that changes the bal- ance sheet values of a bank i at time t. If a bank becomes insolvent, its assets are liquidated to satisfy debtors demands. This might cause a change in the price value of specific assets due to an over supply of these assets. When mark-to-market accounting is used to evaluate the value of the asset side of balance sheets, other banks experience a shock to their balance sheets because of asset devaluation. In particular, we use an inverse demand curve for the illiquid asset to simulate a reduction in the asset value caused by the insolvency of banks:
ˆ
Ai(r + 1) = exp( 1(1 pr)) ˆAi(r), (3.23)
where 1 2 [0, 1] is a constant. Thus, exp( 1(1 pr)) reduces the value of non-
interbank assets of a bank i in round r proportional to the fraction of insolvent banks, 1 pr.
Eq. 3.23 captures the effect of asset value reduction by multiplying the initial value of non-interbank assets with the exponential function. It should be noted that there is no evidence that a price reduction indeed corresponds to an exponential function that is proportional to the fraction of insolvent banks. We nonetheless use this form as it is in accordance with the literature. Other studies (Cifuentes et al., 2005; M¨uller, 2006; May and Arinaminpathy, 2010; Tsatskis, 2012) use the same function to investigate the impact of price reduction of assets and liquidity shortages.
3.4. The Initial Shock to Banks’ Balance Sheets 46