an agent means that its performance falls below an α-quantile, then 
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
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, VaR0.99 = 25; Right: All correlations are
Figure 1.4 Left: All correlations are
Figure 1.5 The system consists of two independent subsystems with internal correlations
Figure 1.6 The system consists of two independent subsystems with internal correlations
Conclusions
Systemic financial risk is an important issue in view of the distress the banking systems all over the world have experienced in the recent years of crises. Even if breakdowns are prevented by the government, the related societal costs are extremely high.
We described the measurement of systemic risk, based on the structural approach originating from structural credit risk models. In particular, the cascading effects that are caused by mutual debt between the individual banks in the system were analyzed in detail. Furthermore, we related the notion of systemic risk to the copula structure, modeling dependency between the performances of the individual banks. The effects of different levels of dependency on the total systemic risk in terms of the value at risk of total losses were demonstrated by examples.
References
Acharya, V., L. Pedersen, T. Phillipon, and M. Richardson. 2009. Regulating systemic risk. In Restoring Financial Stability: How to Repair a Failed System. Hoboken, NJ: John Wiley and Sons.
Adrian, T., and M. K. Brunnermeier. 2009. CoVar. In: Staff Report 348: Federal Reserve Bank of New York.
Chan-Lau, J., J. M. Espinosa-Vega, and J. Sole. 2009a. On the use of network analysis to assess systemic financial linkages. Washington, D.C.: International Monetary Fund, IMF.
Chan-Lau, J., M. A. Espinosa-Vega, K. Giesecke, and J. Sole. 2009b. A. Assessing the systemic implications of financial linkages. In: Global Financial Stability Report. Washington, D.C.: International Monetary Fund, IMF.
Cont, R., A. Moussa, and E.e.S. Bastos. 2010. Network structure and systemic risk in banking systems, s.l.: Preprint, electronic copy available at http://ssrn.com/abstract=1733528.
Crosbie, P. and J. Bohn. 2003. Modeling default risk: Moody's KMV.
European Central Bank. 2004. Annual Report, Frankfurt, available at
