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The distances to default, derived from structural models, in particular from systemic models in the strict sense, can be used to measure systemic risk. In principle, the joint distribution of distances to default for all involved entities contains (together with the definition of distress barriers) all the relevant information. We assume that the joint distribution is continuous and let denote the joint density of the distances to default for all entities.

Note that the risk measures discussed in the following are often defined in terms of asset value, which is fully appropriate for systemic models in the broader sense. In view of the previous discussion of systemic models in the strict sense, we instead prefer to use the distances to default or loss variables derived from the distance to default.

The first group of risk measures is based directly on unconditional and conditional default probabilities. See Guerra et al. (2013) for an overview of such measures. The simplest approach considers the individual distress probabilities

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The term in squared brackets is the marginal density of , which means that it is not necessary to estimate the joint density for this measure. In similar manner, one can consider joint distributions for any subset of entities by using the related (joint) marginal density , which can be obtained by integrating the joint density over all other entities, that is, .

Joint probabilities of distress for a subset I can be achieved by

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where the set I contains the elements . Of special interest are the default probabilities of pairs of entities (see, e.g., Guerra, et al., 2013). Joint probabilities of distress describe tail risk within the chosen set I. If I represents the whole system (i.e., it contains all the entities), then the joint probability of distress can be considered as a tail risk measure for systemic risk (see, e.g., Segoviano & Goodhart, 2009).

Closely related are conditional probabilities of distress, that is, the probability that entity j is in distress, given that entity j is in distress, which can be written as

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These conditional probabilities can be presented by a matrix with as its ijth matrix element, the distress dependency matrix.

While conditional distress probabilities contain important information, it should be noted that they only reflect the two-dimensional marginal distributions. Conditional probabilities are often used for analyzing the interlinkage of the system and the likelihood of contagion. However, such arguments should not be carried to extremes. Finally, conditional probabilities do not contain any information about causality.

Another systemic measure related to probabilities is the probability of at least one distressed entity; see Segoviano and Goodhart (2009) for an application to a small system of four entities. It can be calculated as

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Guerra et al. (2013) propose an asset-value-weighted average of individual probabilities of distress as an upper bound for the probability of at least one distressed entity. Probabilities of exactly one, two, or another number of distressed entities are hard to calculate for large systems because of the large number of combinatorial possibilities.

An important measure based on probabilities is the banking stability index, measuring the expected number of entities in distress, given that at least one entity is in distress. This measure can be written as

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Other systemic risk measures based directly on the distribution of distances to default. Adrian and Brunnermeier (2009) propose a measure called conditional value at risk,1 . It is closely related to value at risk, which is the main risk measure for banks under the Basel accord.

is based on conditional versions of the quantile at level for an entity given that entity i reaches the -quantile. In terms of distances to default, this reads

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where

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The contribution of entity i to the risk of entity j then is calculated as

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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

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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

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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

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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:

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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,

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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,

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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).