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
1.13
These conditional probabilities can be presented by a matrix with
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
1.14
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
1.15
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
where
The contribution of entity i to the risk of entity j then is calculated as
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
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
Conditional value at risk should not be confused with general risk measure with the same name, which is also known as expected tail loss or average value at risk.
