Quantitative Financial Risk Management. Galariotis Emilios

can be analyzed in obvious manner by defining conditional versions of the quantile or

loss measures – that is, by conditioning the overall loss on the distance to default of an individual bank in the style of

; see (1.16).

      Systemic Risk and Copula Models

      The distinction between risk factors that are related to individual performances and risk factors that are a consequence of the interrelations of the economic agents has its parallel in a similar distinction for probability distributions or stochastic processes:

      Suppose that

describe the performance processes of k economic agents. The individual (marginal) processes are assumed to follow certain stochastic models as discrete Markov processes, diffusion models, or jump-diffusion models. The joint distribution, however, depends on the copula process, which links the marginal processes.

      To simplify, suppose only a single-period model is considered and that the performance after one period is X₁,… X_k. If this vector has marginal cumulative distribution functions F₁,… F_k (meaning that

), then the joint distribution of the whole vector can be represented by

      1.25

      where C is called the copula function. Typical families of copula functions are the normal copula, the Clayton copula, the Gumbel copula or – more generally – the group of Archimedean copulas.

      While the marginal distributions describe the individual performances, the copula function models the interrelations between them and can thus be seen as representing the systemic component. In particular, the relation between underperformance of agent i and agent j can be described on the basis of the copula. To this end, we use the notion of conditional value at risk (CoVaR), as already described. Following Mainink and Schaaning (2014) we use the notations

      for the notion of CoVAR introduced by Adrian and Brunnermeier (2009) and

      for the variant introduced by Girardi and Ergün (2012). Keep in mind that we work here with profit and loss variables and not with pure loss variables. The latter variant can be expressed in terms of the conditional copula

      Its inverse

      and the marginal distribution of X can be used to write the CoVaR in the following way:

      For the

, the conditional copula

      is needed. With

      one gets

      Notice that both notions of CoVaR depend only on the copula and the marginal distribution of X.

      If underperformance of an agent means that its performance falls below an α-quantile, then

indicates the probability that the other agent also underperforms.

gives the necessary risk reserve for agent i to survive a possible default of agent j. For a system of k agents, the notion of

can be generalized in a straightforward manner to k components.

       Example 1. Consider a financial institution A, which faces a gamma-distributed loss with mean 10 and variance 20. Then A's unconditional 99 % VaR is 23.8.

If A's performance related to B's performance with a normal copula with correlation ρ, then A's conditional VaR (the CoVaR) increases with increasing ρ, see Table 1.1.

Table 1.1 Conditional VaR and correlation for example 1

Example 2. Consider a system of seven banks, where the performances

are related by a normal copula stemming from a correlation matrix with all off-diagonal elements ρ (the diagonal elements are 1). Suppose that the first bank defaults if its performance drops below the 5 percent quantile. Given the copula, one may determine the number of other banks that also fall below the 5 percent quantile (i.e., default as a consequence of the first bank's default). Figures 1.1 and 1.2 show the distribution of these numbers for the choice of

and

. One may observe that in the independent case

the other banks are practically not affected by the default of one bank, while for higher correlated cases a contagion effect to other banks can be easily seen.

Figure 1.1 The distribution of the number of affected banks using the assumptions of Example 2. Left: the uncorrelated case (ρ = 0). Right: the weakly correlated case (ρ = 0.2).

Figure 1.2 The distribution of the number of affected banks using the assumptions of Example 2. Left: the medium correlated case (ρ = 5). Right: the highly correlated case (ρ = 0.8).

      A very interdependent banking system carries a high systemic risk. It has therefore been proposed to limit the dependencies by creating quite independent subsystems. Example 3 gives evidence for this argument.

       Example 3. Here, we consider seven banks, each of which has a performance given by a negative gamma distribution with mean 100 and variance 200, but shifted such that with probability 5 percent a negative performance happens, which means bankruptcy. The total losses of the system are calculated on the basis of a normal copula linking the individual losses. By assuming that the government (or the taxpayer) takes responsibility for covering total losses up to the 99 percent quantile, this quantile (the 99 percent VaR) can be seen as a quantization of the systemic risk.

In Figures 1.3 to 1.6, we show in the upper half a visualization of the correlations (which determine the normal copula) by the thickness of the arcs connecting the seven nodes representing the banks. The lower half shows the distribution of the total systemic losses, where also the 99 percent VaR is indicated. As one can see, the higher correlation increases the systemic risk. If the system is divided into independent subsystems, the systemic risk decreases.

Figure 1.3 Left: All banks are independent, VaR_0.99 = 25; Right: All correlations are

, VaR_0.99 = 29.