Quantitative Financial Risk Management. Galariotis Emilios

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to augment expectation by some risk measure , which, with weight a, leads to

      1.21

      Typical choices of

are dispersion measures like the variance or the standard deviation. Such measures are examples of classical premium calculation principles in insurance. Further, more general premium calculation principles are for example, the distortion principle or the Esscher premium principle. For an overview on insurance pricing, see Furmann and Zitikis (2008). In the context of systemic risk, the idea to use insurance premiums was proposed in Huang et al. (2009). In this chapter, empirical methods were used for extracting an insurance premium from high-frequency credit default swap data. Even more generally, it should be noted that any monetary risk measure – in particular, coherent measures of risk – can be applied to the overall loss in a system. See Kovacevic and Pflug (2014) for an overview and references.

      In this broad framework, an important class of risk measures is given by the quantiles of the loss variable L:

1.22

      With probability

, the loss will not be higher than the related quantile.

Quantiles are closely related to the value at risk (VaR), which measures quantiles for the deviation of the loss from the expected loss. Note the slight difference between (1.22) and (1.17), because (1.17) is stated in terms of distance to default and (1.12) in terms of loss.

      

can also be interpreted in an economic way, as follows. Assume that a fund is built up in order to cover systemic losses in the banking system. If we ask how large the fund should be, such that it is not exhausted, with probability over the planning period, then the answer will be . This idea can also be reversed. Assume now that a fund of size q has been accumulated to deal with systemic losses. Then the probability that the fund is not exhausted,

      1.23

      is a reasonable systemic risk measure. Clearly,

is the distribution function of the loss, and q is the quantile at level .

      Unfortunately, quantiles do not contain any information about those

percent cases, in which the loss lies above the quantile. Two different distributions, which are equal in their negative tails, but very different in the positive tails, are treated equally.

      The average value at risk (AVaR) avoids some drawbacks of quantiles. It is defined for a parameter α, which again is called level. The AVaR averages the bad scenario,

      1.24

      The latter formula justifies the alternative name conditional value at risk (CVaR), which is frequently used in finance. In insurance, the AVaR is known as conditional tail expectation or expected tail loss.

      The effect of individual banks 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

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