Quantitative Financial Risk Management. Galariotis Emilios

      That is, the conditional value at risk at level

is compared to the conditional value at risk at the median level. From all the

values, it is possible to construct another kind of dependency matrix.

This idea can also be applied to the system as a whole: If

is replaced by

the distance to default of the whole system, (1.16) to (1.18), leads to a quantity

that measures the impact of entity i on the system. In this way one is able to analyze notions like “too big to fail” or “too interconnected to fail.”

      In contrast to probability-based measures,

emphasizes the role of potential monetary losses. This approach can be carried forward, leading to the idea that systemic risk should be related to the losses arising from adverse events. Given a model for the distances to default

, the overall loss of the system can be written as

1.19

      

covers all credit losses in the whole system, both from interbank credits and from credits to the public.

From the viewpoint of a state, this notion of total loss may be seen as too extensive. One may argue that only losses guaranteed by the state are really relevant. Definition (1.19) therefore depicts a situation in which a state guarantees all debt in the system, which can be considered as unrealistic. However, in most developed countries, the state guarantees saving deposits to a high extend, and anyhow society as a whole will have to bear the consequences of lost debt from outside the banking system. Therefore, a further notion of loss is given by

      1.20

which describes the amount of lost nonbanking debt For the structural model, which has been described in the previous section, loss given default can be calculated using (1.8) and (1.9).

      In general, the notion of loss depends on the exact viewpoint (loss to whom). We will therefore use the symbol L to represent any kind of loss variable in the following discussion of systemic risk measures.

An obvious measure is expected loss – that is, the (discounted) expectation of the risk variable L. For simple structural models like (1.2), this measure can be calculated from the marginal distribution of asset values, respectively, of distances to default. Modeling the joint distributions is not necessary. Note that this is different for the strict systemic model (1.9).

      The expectation can be calculated with respect to an observed (estimated) model, or with respect to a risk-neutral (martingale) model. Using observed probabilities may account insufficiently for risk, which contradicts the aim of systemic risk measurement. Using risk-neutral valuation seems reasonable from a finance point of view and has been used, for example, in Gray and Jobst (2010) or Gray et al. (2010). However, it should be kept in mind that the usual assumptions underlying contingent claims analysis – in particular, that the acting investor is a price taker – are not valid if the investor has to hedge the whole financial system, which clearly would be the case when hedging the losses related to systemic risk.

      Using expectation and the concept of loss cascades, Cont et al. (2010) define a contagion index as follows: They define first the total loss of a loss cascade triggered by a default of entity i and the contagion index of entity i as the expected total loss conditioned on all scenarios that trigger the default of entity i.

      Clearly, the expectation does not fully account for risk. An obvious idea is 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