Leveraging the network: A stress-test framework based on DebtRank

A.1 Balance-sheet dynamics

In the framework, we consider a financial system composed of n institutions (banks). Each institution i in the system can invest in either m external assets or in the funding of the other n-1 financial institutions. The focus of our analysis is on the dynamics of the balance sheets of each institution (at each time t=0,1,2,…) and, in particular, of their equity levels. The balance sheet is modelled as follows: Ei⁢(t) is the equity value of institution i at time t, Ai⁢(t) is the value of its total assets and Di its total liabilities. Consistently with much of the literature, we assume that assets are marked-to-market whereas liabilities are written at their face value. We can classify assets and liabilities into external and interbank. In particular, we consider the n×n interbank lending matrix, whose element Ai⁢jb is the amount bank i lends to bank j in the interbank market and the n×m external assets matrix, whose element Ai⁢ke is the amount invested by bank i in the external asset k. The sum Aib=∑j=1nAi⁢jb is the total amount of interbank assets of bank i and the sum Aie=∑k=1mAi⁢ke is the total amount of external assets of bank i. In this framework, we consider external liabilities as exogenous and do not specifically model them: to simplify the notation, these liabilities do not carry a time index. The balance sheet identity at each time t=0 reads: Ai⁢(t)=Di⁢(t)+Ei⁢(t) or, equivalently, Aie⁢(t)+Aib⁢(t)=Die+Dib⁢(t)+Ei⁢(t). We define the total leverage of bank i at time t as the ratio between its total assets and its equity: li⁢(t)=Ai⁢(t)/Ei⁢(t), which can disaggregated into its additive subcomponents:

where the element li⁢jb⁢(t)=Ai⁢jb/Ei⁢(t) is the leverage of bank i towards bank j at time t and the element li⁢ke⁢(t)=Ai⁢ke/Ei⁢(t) is the external leverage of bank i with respect to the external asset k. By considering these two matrices as weighted adjacency matrices, we can then envision two leverage networks: (i) a mono-partite interbank leverage network and (ii) a bipartite external leverage network. By summing along the columns of these matrices, we can obtain the total interbank leveragelib⁢(t)=∑jli⁢jb⁢(t) (the interbank leverage out-strength) and the total external leveragelie=∑kli⁢ke⁢(t) (the external leverage out-strength). These quantities are the key variables in our framework. In particular, we will show that interbank and external leverage produce compounded effects when the dynamic of losses for the second round is considered.

A.2 The distress process

As banks deplete capital in order to face losses in both interbank and external assets, in the stress-test framework we are mainly concerned with the dynamics of the relative loss in equity for each institution, with respect to a baseline level at t=0. This dynamics is captured by the following process:

which represents the individual cumulative relative equity loss in time. We assume that either no replenishment of capital or positive cash flow are possible, therefore Ei⁢(t)≤Ei⁢(t-1) for all t. In this way, the relative equity loss is a non-decreasing function of time. Further, hi⁢(t)∈[0,1] for all t. A bank defaults (i.e. the bank reaches the maximum distress possible) if hi⁢(t)=1. When hi⁢(t)=0 the bank is undistressed. All values of hi⁢(t) between 0 and 1 imply that the bank is under distress. Similarly, we can compute the global cumulative relative equity loss at each time t as the weighted average of each individual level of distress:

where the weights are given by wi=Ei⁢(0)/∑jEj⁢(0), i.e. the fraction of equity of each bank at the baseline level (t=0). Notice that hi⁢(t) is a pure number and so is H⁢(t). The monetary value (e.g. in Euros or Dollars) of the loss can be obtained by hi⁢(t)×Ei⁢(0) (individual loss) and Hi⁢(t)×∑iEi⁢(0) (global loss).

Using the terminology introduced in the main text, equations (2) and (3) allow to measure the individual and global vulnerability respectively. The entire distress process featured in the framework can be outlined in the following steps.

A.3 Third round and fire sales

After the second round, banks have experienced a certain level of equity loss that has completely reshaped the initial configuration of the balance sheets at time t=0. Banks are now attempting to restore, at least partially, this initial configuration. In particular, we assume [64] that each bank i will try to move to the original leverage level li⁢(0). This implies that banks will try to sell external assets in order to obtain enough cash to repay their obligations and therefore reduce the size of their balance sheet. Because of the vast quantity of external assets sold by the banking system in aggregate, the impact on the prices of external assets is also relevant, which will reduce accordingly. Banks therefore will experience further losses due to fire sales and we label such losses as third-round effects. Here, we provide a minimal model for the scenario described above.

Consider the leverage dynamics at t=1,2,…,T,T+1,T+2. The leverage at t is

We assume that, at t=0, each bank had a quantity of external assets Qi and, without loss of generality, that the initial price of the asset is unitary (p⁢(0)=1). Hence, the asset values at t=0 can be written as Ai⁢(0)=Qi⁢(0)=li⁢(0)⁢Ei⁢(0). The asset price after the first round is therefore simply p⁢(1)=p⁢(T)=(1-r). Recalling that the first round affects only the external asset and that the second round affects only interbank assets, the leverage of each bank i immediately after the second round can be written as

where, for ease of notation, lie=lie⁢(0) and lib=lib⁢(0). First, we need to prove that the new leverage levels are higher with respect to the initial conditions. It is easy to prove that li⁢(T)>li⁢(0) (as long as i has not defaulted):

where li=li⁢(0). The above inequality leads to the condition hi⁢(2)⁢(li-1)>0, which is always verified in our setting.

At t=T+1, banks attempt to restore the target leverage li*=li⁢(0)=lie+lib, by selling a fraction si∈[0,1] of their external assets at the price (1-r) replenishing their equity of an amount Qi⁢(1-r)⁢s. Therefore, we modify (7) as follows:

After some passages, we obtain the value for si:

which satisfies (8). The relative amount of assets sold is given by

We further assume that the simultaneous selling of external assets in the market produces a further linear impact on the price. Given the impact of fire sales, the new price is further reduced as follows:

and the relative change in price is therefore proportional to the relative change in quantity of sold assets through a constant η∈[0,1]. Finally, by computing the additional loss given by the decline in price following (9), we obtain the final individual relative equity loss at t=T+2:

and the global equity loss at the third round (assuming no defaults):

A.4 Loss distribution

The distress process allows to capture, at each time t, the relative equity loss for both the individual institution and the system as a whole. This implies the possibility to compute, at each time t, a (continuous) relative equity loss distribution conditional to a certain shock. The equity loss distribution can be characterized, for example, by two typical risk measures: Value at Risk (VaR) and Conditional Value at Risk (CVaR) (also known as Expected Shortfall, ES). Since hi⁢(t) and H⁢(t) are nonnegative variables in [0,1] for all i,t, the individual Value at Risk for bank i at time t at level α is defined as the 1-α quantile [47, 31]:

and the Conditional Value at Risk for bank i at time t at level α is defined as the expected value of the losses exceeding the VaR:

Considering the system as a whole, we can likewise analyze the global relative equity losses H⁢(t) at each time t, therefore obtaining a global VaR:

and the global CVaR:

1. Total exposure re-balancing.

Since we are considering a subset of the entire interbank market, we observe an inconsistency: the total interbank assets A=∑iAi are systematically smaller than the total interbank liabilities L=∑iLi for each year (EU banks are net borrowers from the rest of the world). To adopt a conservative scenario, we assume that the total lending volume in the network is the minimum between the two (A in the exercise). Let Ai/A and Li/∑jLj be respectively the lending and borrowing propensity of i.

2. Exposure link assignment.

The fitness model, when applied to interbank networks [25] attributes to each bank a so-called fitness level xi (typically a proxy of its size in the interbank network). We can estimate the probability that an exposure between i and j exists via the following formula, where z is a free parameter:

Notice that pi⁢j=pj⁢i. Consistently with a recent stream of literature [52, 51], for each bank we take as fitness xi the average between its total lending and borrowing propensity, implying that, the greater this value, the higher will be the number of counterparties (the degree of a node). Considering empirical evidence on the density of different interbank networks [39], we assume on average a density of 5% (i.e. about 1670 over the n⁢(n-1) possible links).^[8] Since it can be proved that the total number of links is equal to the expected value of

we can determine the parameter z and compute the matrix of link probabilities pi⁢j. We now generate 100 network realizations. For each of these realizations, we assign a link to the pair of banks (i,j) with probability pi⁢j. The link direction (which determines whether i or j is the lender or the borrower) is chosen at random with probability 0.5.

3. Exposure volume allocation.

Last, we need to assign weights to the edges (the volumes of each exposure). We impose the fundamental constraint that the sum of the exposures of each bank (out-strength) equals its total interbank asset Ai. To achieve this, we implement an iterative proportional fitting algorithm on the interbank exposure matrix ai⁢j. We wish to estimate the matrix πi⁢j=Ai⁢j/A, which is the relative value of each exposure with respect to the total interbank volume. We begin the estimation π^i⁢j of πi⁢j, at each iteration:

i.e. π^i⁢j is divided by its relative lending propensity and multiplied by the total relative assets of i; and

We repeat the two steps (12) and (13) until ∑jπ^i⁢j-Ai/A and ∑jπ^j⁢i-Li/L are below 1%. Last, the exposure network can be estimated by πi⁢j×A.