2.3.1 Markov models for credit default
The essential element of Markov models for credit default is a classification of a counterparty in terms of a credit rating, with a probability of transition from one rating to another. If there are J credit ratings, denote the probability that a counterparty X (or ’agent’ in complexity terminology) has a credit rating j (\(0 \le j \le J\)) at time t by \(P(X_t=j)\). Let the conditional probability of a transition from a rating j at time t to a rating k at time t be \(p_{jk}=P(X_t=k| X_t=j)\). Then, the probabilities \(p_{jk} (0 \le j,k \le J)\) define a transition matrix P. When all possible states at time t for the Markov process are organised in a vector \(S_t\), the probabilistic evolution of the Markov process can be represented by the equation \(S_{t+1}={{\varvec{P}}} S_t\).
At any given time, the total loss to the system can be calculated by noting which counterparties are in the default state at time t, their probabilities of default (such as \(P(X_t=defaulted)\)) and multiplying by the amount of capital lost.
Markov models are particularly appealing because of their simplicity, but they are rather restrictive. They are limited to the credit risk context and require the imposition of a credit rating. The idea of systemic risk is essentially a secondary consideration, and the concepts of complexity only enter in more advanced models. In practice, it can be difficult to determine the transition probabilities. The literature abounds with extensions of the basic model described.
2.3.2 Systemic risk and contagion models
This category explicitly models systemic risk from the outset, rather than noting that a systemic effect is a consequence of the model. The method of Cont et al. (
2013)—hereinafter the
Cont model—is a prime example, as it also references a specific network topology, with a transmission mechanism for contagion over that network. However, the topology is still static and deterministic. Cont et al. (
2013) also define a metric for the systemic importance of institutions: the
Contagion Index. This is defined as the expected loss to the network triggered by the default of one financial institution in a macroeconomic stress scenario. The
Contagion Index takes into account both market shocks to portfolios and contagion through counterparty exposures. The premise is that most financial institutions present only a negligible risk of contagion, but a few are capable of generating a significant risk of contagion through their failure. We note this model in some detail because it has a bearing on the complexity model which is the subject of this paper. Many of the expressions used in our complexity model have equivalents in the
Cont model. The notation has been amended to correspond more closely to our complexity model. Equations (
1), (
2) and (
3) below all have parallels in Sect.
3 with corresponding results in Sect.
4.
With reference to a network in which the nodes represent financial institutions, let
\(E_{ij}\) represent the exposure of node
i to node
j, let the initial capital for node
i be
\(C_i(0)\), and let the recovery rate for node
i be
\(R_i\). The capital for node
i after time
\(t+1\) is given as a function of the capital after time
t and the sum of exposures to nodes with zero capital (the
defaulted ones) by:
$$\begin{aligned} C_{i}(t+1)=\mathrm{max}(C_{i}(t)-\sum _{j,C_{j}(t)=0}{(1-R_{j})}E_{ij}, 0). \end{aligned}$$
(1)
The extent of contagion due to node
i at time
t is measured by the default impact
\(\varDelta _{i}(t)\): the capital lost by node
i plus the contagion capital lost by other nodes.
$$\begin{aligned} \varDelta _{i}(t)=\sum _{\tau \le {t}}{}C_{i}(\tau ) +\sum _{j\ne {i}}{\sum _{\tau \le {}t}{}}C_{j}(\tau ). \end{aligned}$$
(2)
Then, to measure the total systemic impact of an institution, the
Contagion Index,
\(\hat{\varDelta }_{i}(t,z)\), is defined as its expected default impact when the network is subject to a macroeconomic stress
z, applied to the institution. The value
z is determined from a random variable
Z which has a predefined expected value that represents a large shock.
$$\begin{aligned} \hat{\varDelta }_{i}(t,z)=\mathbb {E}(\varDelta _{i}(t) |C_{i}(t)\le C_{i}(0)-z). \end{aligned}$$
(3)
2.3.3 Systemic effect models: variations
The Cont model contains the elements that many similar models rely on: a network, an interaction mechanism and a contagion propagation mechanism. In this section, we summarise some of the variations on the Cont theme.
Gai and Kapadia (
2010) describe a base model in which the focus is on the ‘terms and conditions’ of the interaction between a pair of banks. Their model starts with a predefined network in which each bank is represented by a node on a directed and weighted graph, where the weights represent exposure size. For bank
i, those exposures comprise (simplifying the notation): interbank assets,
\(A_i\), interbank liabilities,
\(L_i\), illiquid external assets such as mortgages,
\(M_i\) and external liabilities such as deposits,
\(D_i\). A solvency condition is expressed in terms of two parameters
\(\phi \) and
q, where
\(\phi \) is the fraction of banks with obligations to bank
i that have defaulted, and
q is the resale price of the illiquid asset. The solvency condition is
$$\begin{aligned} (1-\phi ) A_{i}+qM_{i}-L_{i}-D_{i}>0. \end{aligned}$$
(4)
Equation
4 can be rewritten in terms of the capital buffer (assets–liabilities),
\(K_i\), for bank
i,
$$\begin{aligned} K_{i}= & {} A_{i}+M_{i}-L_{i}-D_{i} \nonumber \\ \phi< & {} \frac{K_{i}-(1-q)M_{i}}{A_{i}}. \end{aligned}$$
(5)
Contagion then spreads in the following way. If bank
i is linked to
j others, it will lose a fraction 1 /
j of its interbank assets when a single counterparty defaults. Therefore, Eq. (
5) implies that the only way default can spread is if there is a neighbouring bank for which
$$\begin{aligned} \frac{K_{i}-(1-q)M_{i}}{A_{i}}<\frac{1}{j}. \end{aligned}$$
(6)
The probability that Eq. (
6) applies for all
\(j > 1\) gives a measure of the vulnerability of bank
i to default. The task is then to estimate that probability using any data available. Equation
6 is a more specific form of Eq. (
1), which does not give any details of how a default might occur. Equation
6 is also the equivalent of our default condition in Sect.
3.
May and Arinaminpathy (
2010) concentrate on how failure-causing shocks can arise in a network, and how they can be propagated by interbank lending–borrowing or by liquidity effects. In a similar way to Gai and Kapadia (
2010), they use a simpler solvency condition (with the same notation as above, with
\(\gamma \) meaning ‘net worth’):
$$\begin{aligned} \gamma =A_{i}+M_{i}-L_{i}-D_{i}>0. \end{aligned}$$
(7)
If the probability that any one of
j banks is linked to any other is
p, the mean number of connections is
\(z = p (j - 1)\). Now let
\(\theta \) be the ratio of outgoing loans to assets.
$$\begin{aligned} \theta =\frac{M_{i}}{M_{i}+A_{i}}. \end{aligned}$$
(8)
With this definition, the failure point is the condition
\((1-\theta ) f > \gamma \), in which
f is the fraction of assets wiped out by the shock. The loss is then distributed equally among the defaulting bank’s creditors. Therefore, each of the
z creditors experiences on average a shock of magnitude
\(\frac{(1-\theta ) f - \gamma }{z}\). This provides a mechanism for contagion propagation.
The failure condition implies an assumption that once a bank fails, all its external assets are lost. May and Arinaminpathy (
2010) justified this as an extreme liquidity effect. If some recovery is possible, not all of the difference is lost. They argue that a necessary and sufficient condition for not all the difference to be lost is
\(\frac{\theta }{z} > \gamma \).
Nier et al. (
2007) present a hierarchical picture of the immediate consequence of default. There is an assumed priority of (insured) customer deposits over bank deposits which, in turn, take priority over net assets. Let
\(s_i\) be the size of an initial shock applied to bank
i. That loss is first absorbed by bank
i’s net assets (capital buffers)
\(K_i\), then its interbank liabilities
\(L_i\) and lastly its deposits
\(D_i\). The bank defaults if
\(s_i > K_i\) leaving a residual loss
\(s_i - K_i\). If
\(s_i - K_i > L_i\), a further residual loss
\(s_i - K_i - L_i\) is transmitted to depositors. Then with the assumption that all of the
j creditor banks receive an equal amount of the initial residual loss, each creditor bank
J absorbs a loss
\(s_{J}=\frac{s_{i} -K_{i}}{j}\). If the amount
\(s_J \le K_J\), bank
J can absorb the loss. Otherwise bank
J’s residual loss
\(s_J - K_J\) is transmitted to bank
J’s depositors and the contagion spreads further. Simulations show that contagion does not decrease linearly with bank capitalisation. For high levels of capitalisation only the first bank defaults. When the capitalisation decreases to between 1 and 4% of a benchmark level, second-round defaults occur. Third-round defaults only occur if capital decreases below 1% of the benchmark level.
2.3.4 Which banks are systemically important?
Little has been said so far about which banks pose systemic risk and which do not. Battiston (see Battiston et al. (
2012) and Battiston et al. (
2013)) attempts to do this using a metric
DebtRank (
DR).
DR is intended to measure the systemic importance of a bank even when default cascade models predict no impact at all on amounts lost. It is defined as the fraction of the total economic value of a network due to a shock that hits a bank. Specifically, take a chain of unsecured loans granted by bank
k to bank
j and then by bank
j to bank
i. The amounts of these loans are
\(x_{kj}\) and
\(x_{ji}\), respectively. Let
\(E_i\),
\(E_j\) and
\(E_k\) be the equity of the three banks. Then, their total equity is
\(E=E_{i}+ E_{j}+ E_{k}\). Then, the
DebtRank for bank
i is the following weighted sum, with weights
\(W_{i,j}=\frac{x_{j,i}}{E_{j}}\) and
\(W_{j,k}=W_{i,j}\frac{x_{k,j}}{E_{k}}\):
$$\begin{aligned} \mathrm{DR}_{i}=W_{i,j} \frac{E_{j}}{E-E_{i}}+W_{j,k} \frac{E_{k}}{E-E_{i}}. \end{aligned}$$
(9)
The result of applying the DR metric to Italian banks is that the DR predicts sizable contagion effects. Measures which only account for a single transfer of default between two banks typically show no impact at all.
In cases discussed so far, the criterion for determining susceptibility to systemic risk has been taken as ‘zero capital’ or similar. That type of criterion only works by looking back at what has happened. It is not forward-looking. As an alternative, Huang et al. (
2009) suggest an indicator of systemic risk that attempts to be more predictive, although it is much more complicated to calculate. This is their
distress insurance premium—the theoretical price of insurance against financial distress. This indicator is calculated by constructing a hypothetical portfolio that consists of debt instruments (mainly bonds) issued by banks. The indicator of systemic risk is defined as the theoretical insurance premium that protects against distressed (credit) losses of this hypothetical portfolio in the coming three months. The components of the
distress insurance premium are risk-neutral probabilities of default (PDs) and equity returns (as a proxy for asset returns). The steps in the calculation are:
1.
For each bond i that defaulted at time t, calculate the risk neutral probability of default (\(\mathrm{PD}_{i,t}\)) using published credit default swap spreads (\(S_{i,t}\)) and assumed recovery rates \(R_{i,t}\): \(\mathrm{PD}_{i,t}=\frac{-tS_{i,t}}{1-R_{i,t}}\). PD implied from the CDS market is a forward-looking measure.
2.
Form a vector \({{\varvec{X}}}_t\) of bond prices and economic measures \(E_{i,t}\) such as the S&P500 and VIX indices
3.
Formulate 12-week future projections of \({{\varvec{X}}}_{t+12}\) using correlations based on the \({{\varvec{X}}}_t\) and the \(E_{i,t}\)
4.
Formulate 12-week future projections of \(\mathrm{PD}_{I,t+12}\) using correlations based on the \({{\varvec{X}}}_{t+12}\) and the \(\mathrm{PD}_{I,t}\)
5.
Define Monte Carlo bond pricing scenarios, each linked to a particular bond. Define distress as a situation in which at least 15% of total liabilities of those bonds are defaulted
6.
Run the scenarios defined by \(\mathrm{PD}_{I,t+12}\) and \({{\varvec{X}}}_{t+12}\), and determine which have defaulted at time \(t+12\). For each defaulted bond \(b_i\), calculate the loss given default (\(\mathrm{LGD}_i\)) as bond value at time \(t+12\) less its value at time t
The required insurance premium for each bond is then the indicator of systemic risk. The trend in insurance premium was found to follow the average PD series and correlations in the banking system very closely. This result is consistent with the conventional view that higher default rates and higher exposures to common factors are both symptoms of higher systemic risk.
2.3.5 Models of population dynamics
Models of population dynamics are more appropriate for very large populations. Examples are cases of runs on banks, or bond transactions, or even the spread of rumours that a bank is in trouble.
The discussion by Hatchett and Kuhn (
2006) is an example of a discrete time model. Their model is rooted in the idea of a population of size
n, although they do not explicitly mention a numerical size. However, their model deviates immediately from a traditional population-based treatment because they use a principal random variable to denote whether or not a bank has defaulted. Thus, let
\(n_{i,t}\) be an indicator variable that bank
i has defaulted at time
t (it takes two values: 1 means ‘defaulted’ and 0 means ‘not defaulted’). Then, equation
10 governs default. In that equation,
\(W_{i,t}\) is a ‘wealth’ variable, such that default is assumed if the wealth drops below zero, and
H is the Heaviside function.
$$\begin{aligned} n_{i,t+1}= n_{i,t}+(1-n_{i,t})H(-W_{i,t}). \end{aligned}$$
(10)
The wealth variable is a composite that comprises an initial wealth, stochastic components that represent changes in wealth (including defaults) and a Gaussian random component. Development of this model indicates that the effect of interactions is relatively weak in typical economic scenarios, but is pronounced in times of large economic stress. Thus, contagion is restricted to economic shock scenarios.
Contagion models based on continuous time have focussed on the spread of disease. Such models have been known for some time. Here, we summarise the base model, known as the SIR (susceptible-infected-recovered) model, details of which may be found in, for example, Hethcote (
2000). Following a brief statement of the SIR model, we discuss an augmented version of it, which incorporates a time delay. Equation (
11) shows the base SIR differential equations. All three components with their total
N denote numbers of individuals, and all are functions of time,
t. Initial conditions and other constant parameters are shown in greek typeface.
$$\begin{aligned} \frac{\mathrm{d}S}{\mathrm{d}t} =-\frac{\beta IS}{N};~~~~~~~~~~~~S(0)= & {} S_0\ge 0 \nonumber \\ \frac{\mathrm{d}I}{\mathrm{d}t} = \frac{\beta IS}{N}-\gamma ~I;~~~~~~~~~~~~I(0)= & {} I_0\ge 0 \nonumber . \\ \frac{\mathrm{d}R}{\mathrm{d}t} =\gamma ~I;~~~~~~~~~~~~R(0)= & {} R_0\ge 0 \nonumber . \\ S(t)+I(t)+R(t)=N ~~~~~~~~~~~~&\end{aligned}$$
(11)
In order to apply them to the context of financial contagion, we associate:
-
Susceptible \(\sim \) Susceptible to credit default
-
Infected \(\sim \) In default
-
Recovered \(\sim \) Recovered from default.
Wang et al. (
2018) propose an updated version of the base continuous time model (Eq.
11) in which the same three states
S,
I and
R are present. The original purpose of the model was to study the spread of a rumour in a social media context. With some adaptions, it can be used to model the spread of interbank financial contagion. In the context of social media, a person may be affected by his/her neighbours such that having been ‘stimulated’ several times by those neighbours, he/she will acquire the opinion of those neighbours. This is effectively a
voter model with a time delay. In the context of financial contagion, the ‘stimulus’ is the default of a counterparty. A bank may be forced into default only if fewer than a critical threshold of its counterparties have already defaulted. In general, thresholds are likely to be lower in the context of banking than they are in the social context. Otherwise, the situations are very similar. The same three distinct states are recognised at each time step, although in Wang et al. (
2018) the
infected state is called
affected and is designated the
A-state. A bank in the
S-state has not yet been affected by a systemic shock because the stimulus received from its neighbours is below its adoption threshold. A bank in the
A-state has been affected by such a shock, as the stimulus received exceeded its adoption threshold. A bank in the
R-state has recovered from any stimulus applied to it. For simplicity, set all banks at the same adoption threshold
T. If
\(p(X_i)\) is the probability that a bank
i is in state
X (where
X is one of
S,
A or
R), then, since there are only three states,
$$\begin{aligned} p(S_i) + p(A_i) + p(R_i) = 1 . \end{aligned}$$
(12)
Suppose that a bank has to receive up to
m stimulae from its counterparties before it is affected (i.e. it has been shocked). The probability of having received
k out of
m stimulae by time
t is
$$\begin{aligned} \phi (t,m,k)=(1-\rho )~_mC_k~(1-\theta (t))^m ~\theta (t)^{m-k} \end{aligned}$$
(13)
where
\(\theta (t)\) is the probability that a randomly chosen counterparty bank has not transmitted a systemic effect to bank
i by time
t, and
\(1- \rho \) is the probability that bank
i is initially susceptible.
If there is a time-delay
\(\tau \) in transmitting a contagion, the equivalent ‘time-delay’ probability is obtained by replacing
t by
\(t-\tau \) to get the following equation.
$$\begin{aligned} \psi (t,\tau ,m,k)=(1-\rho )~(1-\theta (t-\tau ))^m ~\theta (t-\tau )^{m-k} \end{aligned}$$
(14)
Then, if
u is the probability that bank
i can survive for a time-delay
\(\tau \), the probability that bank
i receives the shock by time
t is
$$\begin{aligned} \chi (t,\tau ,m,k)=u \phi (t,m,k) +(1-u) \psi (t,\tau ,m,k) \end{aligned}$$
(15)
In that case the proportion of banks in the
S-state at time
t is, where
\(p_k\) is the probability that
k stimulae have been received:
$$\begin{aligned} \hat{s}(t)=\sum _{k=0}^{m}\sum _{m=0}^{T-1}p_k ~\chi (t,\tau ,m,k) . \end{aligned}$$
(16)
Given this expression for the proportion of banks in the
S-state at time
t, the proportion of banks in the
A-state at time
t is governed by the differential equation
$$\begin{aligned} \frac{\mathrm{d}\hat{a}(t)}{\mathrm{d}t}=-\frac{\mathrm{d}\hat{s}(t)}{\mathrm{d}t}-\gamma \hat{a}(t) . \end{aligned}$$
(17)
Lastly, the proportion of banks in the
R-state at time
t is governed by the differential equation
$$\begin{aligned} \frac{\mathrm{d}\hat{r}(t)}{\mathrm{d}t}=\gamma \hat{a}(t) . \end{aligned}$$
(18)
Solving Eqs. (
17) and (
18) for
\(\hat{a}\) and
\(\hat{r}\) then gives, together with the expression in Eq. (
16) for
\(\hat{s}\), the complete time-delayed dynamic of the system. The results indicate that with no time delays,
R(
t) is monotonic increasing with
t. With time delays, both
S(
t) and
A(
t) show jumps in the direction of the
R-state at the values of the time delay
\(\tau \). Typically, there are three major jumps of this kind, followed by more smaller jumps, before all banks have recovered over the time horizon considered. It is not immediately clear how long that time horizon is, but given the 2008 banking crisis, the time horizon should be in the order of at least ten years.