2.1 The economy

The economy consists of N banks, N + 1 households, and the central bank. The model can be thought of as a two-period model, starting at t = 0, and with two possible states for the macroeconomy (henceforth the “macro-state” ) at time t + 1 = 1, the good state Up (U) and the bad state Down (D). The banks are heterogeneous in the sense that they have different initial equity, risk-aversion, and attitude toward the cost of defaulting on their liabilities. Each household is assigned to borrow from a single bank. This limited participation assumption may come from history or informational constraint. The households H¹, H², H³, ..., H^N borrow from bank B¹, B², B³, ..., B^N, respectively, and the remaining household H^N+1 supplies deposits to all the banks. The risk-free loan rates and deposit rates are exogenous, but the repayment rates are endogenous and different across states of nature. In addition, there is an interbank market where the banks are allowed to borrow from and deposit in other banks. The Central Bank conducts open market operations (OMOs) to set the reference rate for the interbank market. Figure 4 provides a stylized presentation of the economy with its different agents and markets and the time structure of the model is shown in Figure 5.

The Economy

Citation: IMF Working Papers 2020, 054; 10.5089/9781513536170.001.A001

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The Economy

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The Economy

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The Time Structure

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The Time Structure

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The Time Structure

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At the beginning of period 0, the interbank, loan and deposit markets open. Banks determine how much to lend to households, and how much to deposit or borrow in the interbank market. The central bank conducts OMOs in the interbank market to fix the risk-free rate. Finally, nature decides whether a good or a bad macro-state occurs.

At the beginning of period 1, financial contracts are settled. In addition, the households and the banks choose the probability of default on their liabilities, a decision formalized by the repayment rate, which can be interpreted as an expected repayment rate (the bad macro-state of nature implicitly represents a set of sub-states of nature where the occurrence of default is exogenous and a probability in inverse proportion to the repayment rate). In the paper, we will use the wording “probability of default” of the bank to capture its intuition as a risk-taking decision, but the formal representation is a fraction of default on the banks’ liabilities. At the end of the period, financial markets re-open. ⁹ All banks in the model are assumed to operate in a competitive environment, and thus they take all the interest rates as given. The structure of each bank’s balance sheet is shown in Table 1.

^{Table 1:}

Bank’s balance sheet

^{Table 1:}

Bank’s balance sheet

Assets		Liabilities
Loans	L	Deposits	D
Interbank loans	LI	Interbank deposits	D
Market book	A	Equity	e
		Others	ε

View Table

^{Table 1:}

Bank’s balance sheet

Assets		Liabilities
Loans	L	Deposits	D
Interbank loans	LI	Interbank deposits	D
Market book	A	Equity	e
		Others	ε

2.2 The banks’ optimization problem

The banks choose how much to leverage, by taking deposits from households and borrowing from the interbank market. They also choose how much credit to allocate to consumer loans, and they choose the probability/fraction of default on their liabilities. We also assume that deposits and interbank loans have equal seniority, i.e. the default rates are the same. The timing of the model is such that the asset allocation by each bank is done before the macro-state is realized. Banks expect a higher return from investing in riskier assets (consumer loans) than in safer ones (interbank loans and market book), but consequently face greater risks. The decision on the value of the probability of default is done after the macro-state is known. A higher probability of default yields higher profits, but carries a utility cost, formalized as a default penalty.

The banks’ optimization problem is given below. For the sake of simplicity, we follow Goodhart, Sunirand, and Tsomocos (2006) in assuming that optimization is done over a two-period horizon, i.e. at time t the banker maximizes expected utility for time t + 1. Goodhart, Sunirand, and Tsomocos (2006) also argue it is a reasonable assumption since bank managers, who can switch jobs relatively easily, tend to have short horizons. The bank b maximizes its expected utility, minus the non-pecuniary penalties that it incurs if its bank defaults on its deposit and interbank obligations. Table 2 presents the notation for all the variables in the model.

^{Table 2:}

Variables in the model

^{Table 2:}

Variables in the model

Subscripts and Superscripts
t ∈ T = {0,1, 2, ...}	time periods
s ∈ S = {U, D}	the set of possible states after each node
b ∈ B	the set of all the banks
Exogenous Variables
ps	probability of state s ∈ S
γb	risk-aversion coefficient in the utility function of bank b ∈ B
λb	bank b ∈ B’s non-pecuniary cost of default coefficient for bank b ∈ B
$A_t^b$	the value of risk-free asset held by bank b ∈ B at time t
$D_t^{I,b}$	interbank deposit taken by bank b ∈ B at time t
$L_t^{I,b}$	interbank loan issued by bank b ∈ B at time t
$e_t^b$	amount of capital that bank b ∈ B holds at time t
${ϵ}_t^b$	the un-modelled other items in the balance sheet of bank b ∈ B at time t
$r_t^b$	interest rate of the consumer loans that bank b ∈ B extends at time t
$r_t^{D,b}$	deposit rate offered by bank b ∈ B at time t
$r_t^A$	the rate of return on the safe asset at time t
ρt	interbank rate at time t
${\delta }_{t+1,s}^b$	dividend distribution of bank b ∈ B at time t + 1 in state s ∈ S
gb,1,gb,2,gb,3	coefficients for consumer loans repayment rate
us,1,us,2	coefficients for GDP level
Endogenous Variables
$U_{t+1}^b$	utility of bank b ∈ B at t + 1
${\mathrm{\Pi }}_{t+1,s}^b$	profit/loss of bank b ∈ B at time t + 1 in state s ∈ S
$\mathrm{\Delta }{\mathrm{\Pi }}_{t+1,D}^b$	expected shortfall in profit of bank b ∈ B at time t + 1
${\mathrm{\Lambda }}_{t+1,s}^b$	default of bank b ∈ B at time t + 1 in state s ∈ S
$L_t^b$	consumer loans of bank b ∈ B at time t
$D_t^b$	deposit taken by bank b ∈ B at time t
$u_{t+1,s}^b$	repayment rate of bank b ∈ B at time t + 1
$u_{t+1,s}^{b,h}$	repayment rate of bank b’s consumer loans in state s ∈ S
Yt+1,s	GDP level at time t + 1 in state s ∈ S
Mt	central bank’s supply of base money at time t
${\tilde{R}}_{t+1}$	pooled repayment rate in the interbank market

View Table

^{Table 2:}

Variables in the model

Subscripts and Superscripts
t ∈ T = {0,1, 2, ...}	time periods
s ∈ S = {U, D}	the set of possible states after each node
b ∈ B	the set of all the banks
Exogenous Variables
ps	probability of state s ∈ S
γb	risk-aversion coefficient in the utility function of bank b ∈ B
λb	bank b ∈ B’s non-pecuniary cost of default coefficient for bank b ∈ B
$A_t^b$	the value of risk-free asset held by bank b ∈ B at time t
$D_t^{I,b}$	interbank deposit taken by bank b ∈ B at time t
$L_t^{I,b}$	interbank loan issued by bank b ∈ B at time t
$e_t^b$	amount of capital that bank b ∈ B holds at time t
${ϵ}_t^b$	the un-modelled other items in the balance sheet of bank b ∈ B at time t
$r_t^b$	interest rate of the consumer loans that bank b ∈ B extends at time t
$r_t^{D,b}$	deposit rate offered by bank b ∈ B at time t
$r_t^A$	the rate of return on the safe asset at time t
ρt	interbank rate at time t
${\delta }_{t+1,s}^b$	dividend distribution of bank b ∈ B at time t + 1 in state s ∈ S
gb,1,gb,2,gb,3	coefficients for consumer loans repayment rate
us,1,us,2	coefficients for GDP level
Endogenous Variables
$U_{t+1}^b$	utility of bank b ∈ B at t + 1
${\mathrm{\Pi }}_{t+1,s}^b$	profit/loss of bank b ∈ B at time t + 1 in state s ∈ S
$\mathrm{\Delta }{\mathrm{\Pi }}_{t+1,D}^b$	expected shortfall in profit of bank b ∈ B at time t + 1
${\mathrm{\Lambda }}_{t+1,s}^b$	default of bank b ∈ B at time t + 1 in state s ∈ S
$L_t^b$	consumer loans of bank b ∈ B at time t
$D_t^b$	deposit taken by bank b ∈ B at time t
$u_{t+1,s}^b$	repayment rate of bank b ∈ B at time t + 1
$u_{t+1,s}^{b,h}$	repayment rate of bank b’s consumer loans in state s ∈ S
Yt+1,s	GDP level at time t + 1 in state s ∈ S
Mt	central bank’s supply of base money at time t
${\tilde{R}}_{t+1}$	pooled repayment rate in the interbank market

A bank b ∈ B maximizes the following expected utility, which consists of a quadratic utility from profits, and quadratic disutility from default,

where Π is the bank’s profit and Λ are the unpaid obligation. In addition, the maximization problem is subject to the budget constraint at time t, $B_t^b(r_t^b,r_t^{D,b})$ defined by equation (2.4). At time t, the sum of risky consumer loans, risky interbank loans, and safe assets equals the sum of deposits, interbank deposits, equity, and the other items that are not explicitly modelled,

The profit/loss at state s of time t + 1 equals the proportional repayments from the consumer loans and interbank loans, and the risk-free assets, minus the proportional repayments for the deposits and interbank deposits, as well as the initial equity and some other exogenous items,

and the expected shortfall in profit is defined as,

The bank chooses the probability (or fraction) of default $u_{t+1}^b$ after the macro-state is realized, under the constraint,

Unpaid obligations for deposits and interbank deposits amount to

Bank fe’s equity in period t + 1 is the sum of equity in the previous period, of profits, minus the dividends distributed at t + 1,

2.3 Central bank

The decisions of the central bank are taken as exogenous. The central bank sets the interbank rate (ρ_t), and base money is then endogenous (M_t). A regulator could also affect each bank’s risk-taking parameters, but the utility function of each bank, in particular the default penalty coefficients λ^b, are taken as exogenous.

2.4 Households

Each borrowing household asks for funding from the bank it is allocated to, and after the macro-state is revealed, chooses how much to default on its debt. More specifically, the i^th household borrows from the i^th bank (i = 1, 2, 3,..., N), and the (N +1)^th household supplies deposit to all the banks. We do not explicitly model the optimization problem of the households, but assume them to be strategic dummies on the loan and deposit market. This assumption simplifies the model, which allows us to focus on the banking sector. The repayment rates of the consumer loans is assumed to be simple function of GDP and of aggregated credit supply.

with exogenous coefficients gb,1, 9b,2,9b,s that will be calibrated below.

2.5 GDP

GDP in each state is assumed to be a simple function of aggregate credit supply available in the previous period:

where u_s,1 and u_s,2 have to be calibrated.

2.6 Market clearing conditions

In equilibrium, the central bank clears the interbank market by setting the money supply M_t to the difference between total interbank deposits and total interbank loans:

For the sake of simplicity, the repayment rate in the interbank market is assumed to be identical across banks, and is the determined by the following equation:

2.7 Equilibrium

The equilibrium is defined as a set of asset allocations and repayment rates such that the banks optimize their expected utility under the given constraints, and the interbank market clears. The first-order conditions are given in Appendix A.

Definition 1. The Equilibrium of the economy

is

iff:

1. $${\sigma }_t^b\in \underset{{\sigma }_t^b\in B_t^b}{\arg \max }{E}_t\left(U_{t+1}^b\right)b\in B$$

2. Household repayments satisfy equation (2.10)

3. GDP evolves by equation (2.11)

4. Interbank market clears by equation (2.12)

5. Banks are correct in their expectation about the repayment rates on the interbank market, given by equation (2.13)

The model can be solved numerically as a system of non-linear equations. Which variables are exogenous and which variables and endogenous depends on the closure of the model, which will be discussed below. The model’s results of most interest is each banks’ expected shortfall, which will be matched to its empirical counterpart, estimated using the method described in the next section.

2.8 Financial distress, systemic risk and the Minsky cycle

We conclude the description of the model by explaining how systemic risk is captured. At the bank’s individual level, the fraction (or probability) of default is endogenous in the model since banks choose the (expected) repayment level on their obligations. The lending banks’ first order condition w.r.t. the repayment rate is

and the equation for a borrowing bank is similar. The left-hand-side is the marginal benefit of default and the right-hand-side is the non-pecuniary marginal cost of default (which can be understood as the banker’s cost of bankrupting the bank). After the macro-state is realized, and repayment and profits from consumer loans are known, each bank chooses the fraction of default $$\left(u_{t+1}^b\right)$$. The bank’s optimal solution is such that the fraction of default is lower when profits are higher, ceteris paribus.

At the banking-sector level, we interpret the good macro-state in the model as a scenario where all the banks are healthy, while in the bad state at least one bank is in distress. The good state in the equilibrium model represents the scenario where no bank defaults, while the bad state represents the combination/average of the remaining 2^N — 1 scenarios, where at least one bank is in distress. Due to the existence of direct and indirect contagion, financial distress in one bank can affect the other banks. Thus all the banks have positive probabilities (fraction) of default in the bad state. The measure of exposure to systemic risk used in the model is the conditional expected shortfall defined in equation (2.6). This measure captures, for each bank, the difference in P&L between the good macro-state and the bad macro-state.

The equilibrium model explicitly incorporates two sources of systemic risk: contagion in the interbank market and a common asset exposure channel. As illustrated by Figure ??, when a borrowing bank suffers a loss on its loan portfolio in the bad macro-state $$(u_{t+1,D}^{b,h}<1)$$, the banks’ fraction/probability of default increases $$(0<u_{t+1,D}^{b,h}<1)$$, affecting deposits and the interbank market in particular. Thus, the banks that were lending in the interbank market suffer losses. The lending banks profits $$\left({\mathrm{\Pi }}_{t+1,D}^{b\prime }\right)$$ fall, as do their capital available for the next period $$\left(e_{t+1}^{b\prime }\right)$$.

The model also includes a common asset exposure channel, magnified by a credit channel similar to the typical fire-sale channel (Figure 2). First, the repayment rates of loans across households are correlated (because no default occurs in the good state, whereas repayment rates are lower than 100 percent in the bad state for all banks). Second, when a bank suffers losses, it extends less credit, which reduces GDP (since GDP depends on the sum of all bank credit) and repayment rates across the entire banking system fall (see equation (2.10)).

Finally, a Minsky cycle of optimism is possible since we allow banks’ subjective probabilities that a good macro-state is realized to differ from the true probability, and the bank’s beliefs affect its portfolio choice. Because there are only two macro-states of nature in the model, the bank portfolio choice is restricted to choosing how to invest between two asset classes (loans to households vs. safe assets), or equivalently, banks choose their leverage and the proportion of investment in the risky asset. We do not specify a rule for the evolution for the optimism¹⁰ but we use the model and the empirical moments to shed light on how optimism, formalized as a higher value of the probability of the good state p, might explain changes in the leverage and investment strategies of banks (higher value of $$m_t^b$$) through time.

The following proposition shows how the banks adjust their assets when subjective beliefs change.

Proposition 1. The banks described in Section 2.2 invest more in the risky consumer loans when becoming more optimistic, i.e., $$\frac{\partial L_t^b}{\partial p}>0$$. his leads to a higher volatility in the banks profit/loss, i.e. $$\frac{\partial {\mathrm{\Pi }}_{t+1,U}^b}{\partial p}>0and\frac{\partial {\mathrm{\Pi }}_{t+1,D}^b}{\partial p}<0$$, thus larger expected shortfalls in profit.

The proof is given in Appendix B.

3.1 Method

CIMDO infers the unknown multivariate distribution of the equity value for a system M banks, p(l¹, ...,l^M) (the posterior distribution), from the observed Probability of Distress (PoD) of each bank making up the system and from a prior multivariate distribution q(l¹, ...,l^M). The cross-entropy approach recovers the distribution that is closest to the prior distribution but also consistent with the empirically observed probabilities of default of each bank making up the banking system.

We illustrate an example of two assets below, while all results are directly applicable when M > 2. The two banks are characterized by their equity’s logarithmic returns x and y and the optimization problem is simply defined as

where q(x, y) ∈ ℝ² is the prior distribution and p(x, y) ∈ ℝ² the posterior distribution. The Kullback cross-entropy criteria C [p(x,y), q(x,y)] can be thought of as the weighted average of the relative distance between p and q and is a measure of distance between the prior distribution q and the posterior distribution p.

The moment-consistency constraints incorporate information about the probabilities of distress of each bank. In line with Merton (1974)’s structural approach, the constraint is such that the probability of distress is equal to the probability that the value of the bank falls below a certain threshold. Imposing this constraint on the optimization problem guarantees that the posterior multivariate distribution is consistent with the probability of default observed for each bank:

where —x, —y are the equity annualized returns, and 1 is the indicator function, and the thresholds for banks x and y are $$X_d^{x}and{X}_d^y$$, respectively. In addition, the posterior density p should sum to 1. Segoviano and Espinoza (2017) solve the optimization problem, and give the posterior multivariate density as

where the lagrange multipliers λ_x and λ_y correspond to the constraints (4.2). The lagrange multiplier µ corresponds to the constraint that the posterior distribution sums to 1. Numerically, information to infer values for the unknown probability density.

λ_x, λ_y, and µ are the solutions of the following system (which requires numerical integration):

Intuitively, CIMDO adjusts the prior distribution with available information on moments using the factor $\exp \{-[1+\mu +({\lambda }_x{1}_x\ge X_d^x]+\left({\lambda }_y\right]1_{y\ge X_d^y}\left)\right]\}$, which depends on the domain of the density (see Figure 6). Segoviano and Espinoza (2017) show that CIMDO tends to strengthen dependence when marginal PoDs are underestimated by the prior.

CIMDO-density, adjustment factor

Citation: IMF Working Papers 2020, 054; 10.5089/9781513536170.001.A001

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CIMDO-density, adjustment factor

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CIMDO-density, adjustment factor

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The prior density is calibrated using information on equity returns. The stock market returns’ moments are used to calibrate a prior’s t-distribution. The observed PoDs are crucial inputs to CIMDO. For the application to systemic risk as in this paper, 1-year ahead CDS spreads are used as proxies. The default thresholds $X_d^{x}and{X}_d^y$ are fixed to an average through time that is consistent with a historical average of the probability of default for each bank, and with the prior distribution.¹²

3.2 Expected shortfalls in profits

The expected shortfall in profit, which is the difference between the profits/loss in the good macro-state and the P&L in the bad macro-state when banks are in distress, is our bank-specific measure of exposure to systemic risk. This measure is an outcome of the theoretical model but it can also be matched to its empirical counterpart, estimated using market data. In the good macro-state, households do not default on their loans, and thus all the banks are healthy. In the bad macro-state, however, households partially default on their loans, damaging the banks’ profitability and financial stability. The model solves for the conditional P&L at each state and thus for the expected shortfall in profits, as a function of the banks’ initial endowments, utility parameters, and the monetary conditions.

The P&L can also be estimated using the empirical model. We start with an example with two banks A and B. Let us call p(x,y) the joint density function of equity values’ annualized returns obtained by CIMDO. The expected value of the equity of bank A given the distress of bank B is,

where EQ_A,t is the equity value of bank A at time t, and event B represents the case where bank B is in distress. Given the initial value of equity, we can calculate the annualized return on equity. The change of equity value through time represents the profit/loss in each scenario.¹³

We generalize the method to a banking system with N banks. Since each bank can be either health or in distress, there are 2^N scenarios in total. S is the set of all 2^N scenarios. The scenario where no bank is in distress is denote H. Equation (3.6) gives the expected value of equity for each bank in cases where at least one bank is in distress, whereas equation (3.7) gives the expected value of equity for each bank in the case where no bank is in distress. The difference between the two values is the expected shortfall of profits (∆Π) due to systemic risk.

4.1 Data

We apply the model to the UK banking sector, which we simplify by focusing on its five largest banks (as per their asset value for 2005): Barclays plc, HSBC Bank plc, Lloyds Bank plc, the Royal Bank of Scotland plc, HBOS plc, and Standard Chartered plc.¹⁵ The first period of the model is June 2005-June 2006, while the second period is June 2006-June 2007. This period, before the financial crisis, was characterized by growing leverage and overall optimism following the years of the “Great Moderation”.

We collect data from FitchConnect and Moody’s on semi-annual balance sheets, daily stock prices, and EDFs. We set L^b to the value of total gross loans from the balance sheet data. A^b is set to the value of total securities, D^b is set to the value of total deposits, e^b is total equity, L^I is the value of loans from other banks, and D^I is the value of deposits from other banks. The data is shown in Table 5 in Appendix C (unavailable data points are linearly interpolated). We take the stock prices for the holding companies of each bank, and the expected default frequency (EDF) since 2004 as the input variables for the empirical estimation. The EDFs are uses as a measure of probability of default for each bank.

4.2 Results

We first calibrate the model with data from June 2005 in order to set each bank’s utility parameters (risk aversion and marginal cost of defaulting). The model is inverted as described in Table 3. The variables are classified by column according to whether they are exogenous (RHS column) or endogenous (LHS column) according to theory. The rows show how the model is inverted: for each variable, the row shows whether the variable is solved for endogenously, is observed empirically, is estimated using market data and CIMDO, or is selected arbitrarily.

When solving the system of simultaneous equations, we calibrate the loans extended (L), deposits taken (D), safe asset holdings (A), capital level (e), and interbank loans/deposits (L^I/D^I) against the real balance sheet data in June 2005, the GDP in the good state against the nominal

^{Table 3:}

Variables for model calibration

Notes: This table presents the variables for the equilibrium model. The values of these variables are presented in Table 6 in Appendix C. Remember that the variables which are exogenous when solving the system of simultaneous equations, do not necessarily have to be those which are exogenous in the model. The column on the left shows the endogenous variables in the equilibrium model, while the column on the right shows the exogenous variables. This is consistent with the classification of the variables in Table 2. When solving the simultaneous equations, the value for some exogenous variables in the model such as the default penalty coefficient (λ^b,b ∈ B) and the risk aversion coefficients (γ^b,b ∈ B) cannot be empirically observed or estimated, while some endogenous variables in the model such as the position in the consumer loans (L^b,b ∈ B) and deposit taken (D^b,b ∈ B) can be observed empirically. Variables in the first row are endogenously computed when solving the simultaneous equations; the balance sheet items and the macroeconomic data in the second row are empirically observed; the conditional expected profits/losses in the third row are estimated by CIMDO, which captures the information about distress dependence structure; the variables in the forth row are “manually” selected.

^{Table 3:}

Variables for model calibration

Equilibrium Model
	Model Endogenous Variables	Model Exogenous Variables
Endogenously Solved	$${\mathrm{\Pi }}_U^b$$,∀b ∈ B: banks’ conditional P/L in good state $${\mathrm{\Pi }}_D^b$$, ∀b ∈ B: banks’ conditional profits/losses in the bad state vb,h,∀b ∈ B: repayment level of the consumer loans M: Money supply by the central bank $$\tilde{R}$$: Pooled interbank deposit repayment rate in the bad state	λb, ∀b ∈ B: the non-pecuniary default penalty coefficient γb, ∀b ∈ B: each bank’s risk-aversion coefficient εb, ∀b ∈ B: un-modelled balance sheet items $$g_1^b$$, ∀b ∈ B: the elasticity of consumer loan repayment rate with respect to GDP us,1,s ∈ {U,D}: coefficients for GDP
Empirically Observed	Lb,∀b ∈ B: banks’ holding of consumer loans Db, ∀b ∈ B: banks’ liability of deposits YU: the GDP level at the good state	Ab, ∀b ∈ B: banks’ holding of safe assets eb, ∀b ∈ B: banks’ initial equity level DI,b’ ‘, ∀b ∈ B: banks’ liability of interbank deposits LI,b, ∀b ∈ B: banks’ holding of interbank loans ρ: the interbank interest rate
Estimated by CIMDO	∆Πb, ∀b ∈ B: banks’ expected shortfalls in profits
Manually Selected	ub,∀b ∈ B: banks’ repayment rate in the bad state YD: the GDP level at the bad state	rD,b , ∀b ∈ B: deposit interest rate rb, ∀b ∈ B: consumer loan interest rate rA : banks’ market book return p: probability of good state $$g_2^b,g_3^b$$: coefficients for consumer loan repayment us,2,s ∈ {U,D}: coefficients for GDP

Notes: This table presents the variables for the equilibrium model. The values of these variables are presented in Table 6 in Appendix C. Remember that the variables which are exogenous when solving the system of simultaneous equations, do not necessarily have to be those which are exogenous in the model. The column on the left shows the endogenous variables in the equilibrium model, while the column on the right shows the exogenous variables. This is consistent with the classification of the variables in Table 2. When solving the simultaneous equations, the value for some exogenous variables in the model such as the default penalty coefficient (λ^b,b ∈ B) and the risk aversion coefficients (γ^b,b ∈ B) cannot be empirically observed or estimated, while some endogenous variables in the model such as the position in the consumer loans (L^b,b ∈ B) and deposit taken (D^b,b ∈ B) can be observed empirically. Variables in the first row are endogenously computed when solving the simultaneous equations; the balance sheet items and the macroeconomic data in the second row are empirically observed; the conditional expected profits/losses in the third row are estimated by CIMDO, which captures the information about distress dependence structure; the variables in the forth row are “manually” selected.

View Table

^{Table 3:}

Variables for model calibration

Equilibrium Model
	Model Endogenous Variables	Model Exogenous Variables
Endogenously Solved	$${\mathrm{\Pi }}_U^b$$,∀b ∈ B: banks’ conditional P/L in good state $${\mathrm{\Pi }}_D^b$$, ∀b ∈ B: banks’ conditional profits/losses in the bad state vb,h,∀b ∈ B: repayment level of the consumer loans M: Money supply by the central bank $$\tilde{R}$$: Pooled interbank deposit repayment rate in the bad state	λb, ∀b ∈ B: the non-pecuniary default penalty coefficient γb, ∀b ∈ B: each bank’s risk-aversion coefficient εb, ∀b ∈ B: un-modelled balance sheet items $$g_1^b$$, ∀b ∈ B: the elasticity of consumer loan repayment rate with respect to GDP us,1,s ∈ {U,D}: coefficients for GDP
Empirically Observed	Lb,∀b ∈ B: banks’ holding of consumer loans Db, ∀b ∈ B: banks’ liability of deposits YU: the GDP level at the good state	Ab, ∀b ∈ B: banks’ holding of safe assets eb, ∀b ∈ B: banks’ initial equity level DI,b’ ‘, ∀b ∈ B: banks’ liability of interbank deposits LI,b, ∀b ∈ B: banks’ holding of interbank loans ρ: the interbank interest rate
Estimated by CIMDO	∆Πb, ∀b ∈ B: banks’ expected shortfalls in profits
Manually Selected	ub,∀b ∈ B: banks’ repayment rate in the bad state YD: the GDP level at the bad state	rD,b , ∀b ∈ B: deposit interest rate rb, ∀b ∈ B: consumer loan interest rate rA : banks’ market book return p: probability of good state $$g_2^b,g_3^b$$: coefficients for consumer loan repayment us,2,s ∈ {U,D}: coefficients for GDP

Notes: This table presents the variables for the equilibrium model. The values of these variables are presented in Table 6 in Appendix C. Remember that the variables which are exogenous when solving the system of simultaneous equations, do not necessarily have to be those which are exogenous in the model. The column on the left shows the endogenous variables in the equilibrium model, while the column on the right shows the exogenous variables. This is consistent with the classification of the variables in Table 2. When solving the simultaneous equations, the value for some exogenous variables in the model such as the default penalty coefficient (λ^b,b ∈ B) and the risk aversion coefficients (γ^b,b ∈ B) cannot be empirically observed or estimated, while some endogenous variables in the model such as the position in the consumer loans (L^b,b ∈ B) and deposit taken (D^b,b ∈ B) can be observed empirically. Variables in the first row are endogenously computed when solving the simultaneous equations; the balance sheet items and the macroeconomic data in the second row are empirically observed; the conditional expected profits/losses in the third row are estimated by CIMDO, which captures the information about distress dependence structure; the variables in the forth row are “manually” selected.

GDP in 2006, and the interbank rate against the 12-month LIBOR at June 2005. The P&L expected shortfalls for each bank (∆Π) is estimated using the empirical CIMDO-distribution.¹⁶ In the model, the interest rate on interbank loans and on loans to households are equal; thus we assume the rate r^D is the same as the interbank rate ρ. The average spread between loans and deposits rates for commercial banks was 3.66% in 2005¹⁷, so we set r to be r^D plus the spread. The probability of a good state to occur (p) is selected manually to 0.94, but we run a robustness check afterwards. The banks’ repayment rate (v) is set to be consistent with the empirically observed Expected Default Frequency (EDFs)¹⁸. The banks’ market book return (r^A), the level of GDP in the bad macro-state of nature (Y_D), and the parameters driving the household’s repayment rates (g₂,g₃) and the sensitivity of growth to credit (u₂) are selected manually, following the parametrization in Goodhart, Sunirand, and Tsomocos (2005). The other variables are solved endogenously by inverting the model. The values of the input variables and the simulation results are presented in Table 6 of Appendix C.

The calibration yields two parameters for each bank: a coefficient setting the bank’s risk aversion (γ^b,b ∈ B), and a coefficient setting the marginal cost of default (λ^b,b ∈ B) see Table 3. These parameters are essential as they drive the banks’ portfolio decision process, but they are not directly observable. Overall, the model is thus calibrated so that the bankers’ utility parameters are consistent with the banks’ actual portfolio choices in 2005.

4.3 Analysis of channels of contagion

We then assume that these coefficients are constant between 2005 and 2007 and use the model as well as June-2006 data to predict the banks’ portfolio choices and expected P&L for June-2007. In the first exercise, we set the subjective probabilities that the good-macro state is realized to its true value, i.e. 0.94 (the Minsky cycle is added in the second exercise presented below).

The balance sheet items (banks’ equity e^b, position in the interbank market D^I,b, L^I,b, and other items e^b) and the interbank rate (ρ) are updated to their values on June 30th, 2006. The data used is presented in Table 8 in the appendix. The model is solved in its natural way. The exogenous parameters and variables according to theory are set as exogenous. The endogenous variables according to theory are the outcome of the model. These are: the banks conditional expected P&L in the good and the bad macro-states $$({\mathrm{\Pi }}_U^b,{\mathrm{\Pi }}_D^b,b\in B)$$, the expected shortfalls in profits (∆Π^b,b ∈ B); the banks’ choice to invest in the risk-free asset (A^b,b ∈ B), to issue loans to households (L^b,b ∈ B) and to take deposits (D^b,b ∈ B) the banks’ fraction of default (v^b,b ∈ B) the default rate on loans to households (v^b,h,b ∈ B) money supply (M); the repayment rate in the interbank market in the bad state $$\left(\tilde{R}\right)$$ and the level of GDP in each macro-state (Y_s,s ∈ {U,D}).

The numerical solution is shown in Table 7, with the same presentation as before. The model predicts that all the banks are making profits in the good macro-state, while some banks suffer from losses in the bad state. The model estimates of expected shortfall are compared to the empirical estimates (see Table 4 and Figure 8). Overall, the empirical model estimates larger expected shortfalls in profits than what theoretical model predicts. At this stage, the theoretical model only accounts for the interbank lending channel and the common asset exposure channel. We interpret the model’s underestimation of expected shortfall as showing the possibility that markets capture additional channels of distress dependence across banks. There are many candidate channels, for example liquidity contagion, maturity mismatch, etc., but in the next section we focus on whether the Minsky cycle could help improve the performance of the model.

Expected Shortfalls in Profits, June 2007

Citation: IMF Working Papers 2020, 054; 10.5089/9781513536170.001.A001

Notes: This figure compares the prediction of the theoretical model without Minsky cycle and CIMDO on the expected shortfalls in profits for each bank at June 2007. Agg. Bank represents the combination of HBOS and Standard Chartered. For each bank, the bar on the left is the theoretical model’s result, while the bar on the right is CIMDO’s result. CIMDO predicts larger expected shortfalls in profits in general. We know that the theoretical model incorporates the interbank lending channel and the common asset exposure channel for systemic risk transmission and amplification, while the CIMDO model should be able to capture the effects of all the channels, given the market data. Thus the difference between the two methods represents the missing channels of the theoretical model, which we attributes to the Minsky cycle.

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Expected Shortfalls in Profits, June 2007

Citation: IMF Working Papers 2020, 054; 10.5089/9781513536170.001.A001

Notes: This figure compares the prediction of the theoretical model without Minsky cycle and CIMDO on the expected shortfalls in profits for each bank at June 2007. Agg. Bank represents the combination of HBOS and Standard Chartered. For each bank, the bar on the left is the theoretical model’s result, while the bar on the right is CIMDO’s result. CIMDO predicts larger expected shortfalls in profits in general. We know that the theoretical model incorporates the interbank lending channel and the common asset exposure channel for systemic risk transmission and amplification, while the CIMDO model should be able to capture the effects of all the channels, given the market data. Thus the difference between the two methods represents the missing channels of the theoretical model, which we attributes to the Minsky cycle.

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Expected Shortfalls in Profits, June 2007

Citation: IMF Working Papers 2020, 054; 10.5089/9781513536170.001.A001

Notes: This figure compares the prediction of the theoretical model without Minsky cycle and CIMDO on the expected shortfalls in profits for each bank at June 2007. Agg. Bank represents the combination of HBOS and Standard Chartered. For each bank, the bar on the left is the theoretical model’s result, while the bar on the right is CIMDO’s result. CIMDO predicts larger expected shortfalls in profits in general. We know that the theoretical model incorporates the interbank lending channel and the common asset exposure channel for systemic risk transmission and amplification, while the CIMDO model should be able to capture the effects of all the channels, given the market data. Thus the difference between the two methods represents the missing channels of the theoretical model, which we attributes to the Minsky cycle.

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4.4 The Minsky cycle

We now invert the model to take the empirical estimates of each bank’s conditional expected shortfall as input, and thus to solve for each bank’s subjective probability that a good state occurs $$(p_1^b,b\in B)$$. Figure 1 and Table 9 summarizes the procedure. Table 10 in Appendix C presents the model results.

The model finds that $$p_1^b>p_0^b$$,∀b ∈ B, i.e. according to the model, all the five banks in the had become more optimistic in June 2006 than they were in June 2005. This is how the model interprets the worsening of systemic risk —i.e., the increase in expected shortfalls estimated empirically— observed through that period. The model’s prediction for banks’ (risky) loans to households (L^b, b ∈ B) is thus higher than when the Minsky cycle is turned off. Individually, the banks rationally increase their leverage and their investment in the risky asset if they believe that the probability of a good state occurring is high.

^{Table 4:}

Banks’ expected shortfalls in profits in June 2007, measured by CIMDO

Notes: CIMDO generates the percentage change in equity value for each period. In this simulation, CIMDO predicts the change from June 2006 to June 2007. Given the balance sheet data of equity level for each bank in June 2006, the profit at the end of the period can be computed. The difference across states represents the expected shortfall in profit.

^{Table 4:}

Banks’ expected shortfalls in profits in June 2007, measured by CIMDO

∆Π1 = 3.8507

∆Π2 = 1.3481

∆Π3 = 4.8913

∆Π4 = 3.0423

∆Π5 = 2.8821

Notes: CIMDO generates the percentage change in equity value for each period. In this simulation, CIMDO predicts the change from June 2006 to June 2007. Given the balance sheet data of equity level for each bank in June 2006, the profit at the end of the period can be computed. The difference across states represents the expected shortfall in profit.

View Table

^{Table 4:}

Banks’ expected shortfalls in profits in June 2007, measured by CIMDO

∆Π1 = 3.8507

∆Π2 = 1.3481

∆Π3 = 4.8913

∆Π4 = 3.0423

∆Π5 = 2.8821

Notes: CIMDO generates the percentage change in equity value for each period. In this simulation, CIMDO predicts the change from June 2006 to June 2007. Given the balance sheet data of equity level for each bank in June 2006, the profit at the end of the period can be computed. The difference across states represents the expected shortfall in profit.

In addition, such risk-taking increases the frequency/fraction of default, which is an externality for each bank: the banks do not factor in the impact of their decision on the repayment rate in the interbank market and on future credit growth and thus the repayment rate of loans. When all the banks change their beliefs in the same direction simultaneously, their common exposure increases and the feedback loop between credit growth and loans’ repayment rates is stronger.

The model’s prediction for the leverage ratios¹⁹ is closer to actual data when the Minsky cycle is turned on. Figure 9 shows the actual leverage ratios for the main UK banks along those predicted by the equilibrium model when the Minsky cycle is turned on and off. This shows that the observed behavior in risk-taking can be represented relatively well by varying the subjective probability of the good macro-state occurring.

Leverage

Citation: IMF Working Papers 2020, 054; 10.5089/9781513536170.001.A001

This figure presents the comparison among the banks’ leverage ratios, following the definition in Equation ??. For each bank, the bar on the left, in the middle, and on the right represents the leverage computed with data from GST model without Minsky effect, with Minsky effect, and with real data, respectively.

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Leverage

Citation: IMF Working Papers 2020, 054; 10.5089/9781513536170.001.A001

This figure presents the comparison among the banks’ leverage ratios, following the definition in Equation ??. For each bank, the bar on the left, in the middle, and on the right represents the leverage computed with data from GST model without Minsky effect, with Minsky effect, and with real data, respectively.

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Leverage

Citation: IMF Working Papers 2020, 054; 10.5089/9781513536170.001.A001

This figure presents the comparison among the banks’ leverage ratios, following the definition in Equation ??. For each bank, the bar on the left, in the middle, and on the right represents the leverage computed with data from GST model without Minsky effect, with Minsky effect, and with real data, respectively.

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