Quantitative Financial Risk Management. Galariotis Emilios
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That is, the conditional value at risk at level
This idea can also be applied to the system as a whole: If
In contrast to probability-based measures,
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
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Typical choices of
In this broad framework, an important class of risk measures is given by the quantiles of the loss variable L:
With probability
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.
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is a reasonable systemic risk measure. Clearly,
Unfortunately, quantiles do not contain any information about those
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