Modelling cascading effects for systemic risk: Properties of the Freund copula

Abstract We consider a dependent lifetime model for systemic risk, whose basic idea was for the first time presented by Freund. This model allows to model cascading effects of defaults for arbitrarily many economic agents. We study in particular the pertaining bivariate copula function. This copula does not have a closed form and does not belong to the class of Archimedean copulas, either.We derive some monotonicity properties of it and show how to use this copula for modelling the cascade effect implicitly contained in observed CDS spreads.


Introduction
We consider a system of n entities and their dependent lifetimes. The term "entity" can be understood as broadly as possible, i.e., the system can consist of banks, nancial institutions, electrical devices, state sovereigns, living beings, etc., but is always assumed to be homogeneous in the sense that all entities are of the same type. We call the end of the life of an entity a "default", even if the entity is not a rm.
The fundamental idea of the model is that the individual lifetime distributions are a ected by the defaults of other entities. To be more precise, we assume initially individual exponential lifetimes with default intensity λ k for every entity k. The choice of an exponential lifetime is motivated by the fact that the corresponding hazard function is constant and therefore conditional residual lifetimes do not depend on the conditioning. Notice that the same assumption is made for the celebrated Marshall-Olkin-type models.
The main idea for such type of models is that the default of one entity puts more stress on the other entities. In a competitive market one may sometimes observe the opposite: the default of an entity eliminates a competitor and reduces the stress for others. However, we motivate our model by an application to nancial institutions, where the default of an entity typically implies losses for the other entities and therefore increases the stress.
If entity k defaults, then the residual default intensity of all other entities increases by a value of a k,l . If (X , . . . , Xn) is the resulting vector of lifetimes, then their n-dimensional copula C is determined by the vector of intensities (λ , . . . , λn) and the cascading e ects a k,l for k ≠ l. The n parameters can be arranged in a (typically asymmetric) matrix . . a ,n a , λ . . . a ,n . . . . . . . . . . . . a n, a n, . . . λn The described model is related to the well-known Marshall-Olkin model [8], where certain subsets of entities in the system can receive simultaneous shocks, i.e., default at the same time. We argue that simultaneous defaults do not happen in many applications, especially not in nancial systems and we consider our model as more appropriate. A cascading default model appears already in an earlier work of Yu [12]. Freund [4] has suggested the same model for n = that we consider in this current paper. In his honour, we call the pertaining copula after him.
The paper is organised as follows. In Section 2, we provide the formal de nition and the fundamentals of the model for n = , and we elaborate on some details, including also the copula of the lifetime variables. In the last part of the section, we show how the setting can be generalized for more entities (n ≥ ). In Section 3, we examine how the dependency structure changes as the model parameters vary. Our ultimate question is: does any monotonic behaviour hold for the lifetime variables and their copula with respect to some stochastic dependence order relation? The reader will nd a positive answer for the upper orthant order. In Section 4, we give a numerical illustration using CDS-data of three European banks. Section 5 concludes the paper.

The fundamentals of the model
Since our main application is the systemic risk of nancial institutions, we use from now on the term institution for the entities.
In Subsections 2.1 to 2.4 we present a detailed analysis for the bivariate model, parts of which were already published by Freund [4]. In Subsection 2.5, we sketch the idea of a multivariate (n ≥ ) setting. . The bivariate model (n = ) Consider a system of two entities, and let Y k ∼ Exp(λ k ) (k = , ) be independent random variables. They are attributed as auxiliary lifetime variables (if one wishes as pre-lifetime variables) to the two entities of the system. When in a certain realization the rst entity defaults earlier, i.e., Y < Y , then the second entity will continue its operation according to another exponentially distributed random variable Z ∼ Exp(λ + a ) , which is independent of Y and Y . The parameter a ≥ is called the shock parameter, and it expresses the e ect of the default of the rst institution on the second institution. Z is de ned analogously: when Y < Y , then Z ∼ Exp(λ + a ) , where a ≥ is a shock parameter.
The actual lifetime variables of the two entities are denoted by X , X , and -in the light of the above mechanism -they can be written as follows.
The new lifetime variables X , X can be expressed explicitly in terms of Y , Y , Z , Z : The case Y = Y does not need to be taken into account, since it has probability zero.

. Cumulative distribution functions and probability density functions
In this subsection, we explore the joint distribution and the marginal distributions of the new -already dependent -bivariate lifetime variable (X , X ) given in (2), as well as some remarkable properties of the joint and marginal cumulative distribution functions and probability density functions. We note that the joint density and the marginal densities (with another parameter setting) directly appear in Freund's work (look at formulas (1.9), (2.5) and (2.6) in [4]). We prove the formula for the joint density in a di erent way than he did. We emphasize again, that the model we consider can be described by the quadruple [λ , λ , a , a ]. Focusing now on the above mentioned densities and cumulative distribution functions, via some elementary computation one gets the following.
Since formula (5) given in statement (ii) is valid for all parameter constellations, we will prove this statement directly. The formulas given in (i) (a), (i) (b), (i) (c) and (i) (d), as well as the formulas in (iv) (a), (iv) (b) and (iv) (c) can be derived from (5) by integration. Finally, the formulas in (iii) can be derived (for instance) by taking the suitable limits in the formulas given in (i).
Turning to the proof of (ii), let us assume that x < y, and let ∆x > , ∆y > such that x + ∆x < y. (The proof in the case when x ≥ y is analogue.) . Dividing by ∆x · ∆y, and then letting ∆x → , ∆y → , we get that = λ · e −λ ·x · e −λ ·x · (λ + a ) · e −(λ +a )·(y−x) = λ · (λ + a ) · e −(λ −a )·x · e −(λ +a )·y , as it was stated in (ii). The second last equality holds, because Figure 1 depicts the joint density function of (X , X ) for two parameter settings. All the cdfs and pdfs in formulas (3)-(13) continuously depend on the parameter a and a . For instance, this continuity is trivial for the marginal density functions (10) and (11) when a ≠ λ and a ≠ λ . At the places a = λ resp. a = λ one can see the continuity by taking the limits a → λ and a → λ in formula (10) and (11), which then yield formula (12) and (13).
The marginal density (10) reduces to f (x) = λ · e −λ x if a = , so in this case X is exponentially distributed with parameter λ . It might be surprising at rst glance, when someone only considers the construction (2) of X . The background of this feature is the constant hazard rate property of the exponential distribution. Nevertheless, a disturbance parameter a of value has indeed no e ect on the marginal distribution of the rst entity, since in this case X ∼ Y . However, X and X are not independent, unless a = also holds.
We examine the other extreme case as well, i.e., when a → ∞ . Then Z a.s.
= ∼ "Exp(∞)" is added to the (truncated) Y , which means nally that either the rst entity expires earlier ( Y < Y ), or X takes the value of Y , so all in all X = min{Y , Y } , which is exponentially distributed with parameter λ + λ . This fact is also re ected by the marginal density f (look at (10)), which reduces to f (x) = (λ + λ ) · e −(λ +λ )x in this case.
It is also worth to see that the joint cdf reduces to the following symmetric function when x = y: (a) A symmetric setting: a = a = .
(b) An asymmetric setting: a = . , a = . We remark that the parameter constellation a = ∞, a = ∞ corresponds to the special case of the Marshall-Olkin model, when λ A = , λ B = , λ {A,B} > , i.e., the system of two entities can face only a common shock. (No separate individual shocks are present.)

Inverse marginal cumulative distribution functions.
Notice that the univariate quantile functions (i.e., the inverse functions of the marginal distribution functions (8), (9)) are smooth, but they cannot be written in a closed, analytical form (except in some very special cases). Since in this current work the quantile functions are mainly used in connection with the copula function, we will further elaborate this question in Subsection 2.3.

. Copula function and copula density
We have discussed the features of the lifetimes variables X , X , but in the end we are mainly interested in their copula. While the lifetimes are given by the pair (X , X ) , their copula is de ned on the pair of uniform marginals (U, V), where U = F(X ), V = G(X ) and F, G are the marginal cdfs (8) and (9).
The copula function (15) and copula density (16) in our model -in accordance with the standard literature -are de ned as follows.
where F − (u) and G − (v) are the generalized inverse functions of the cumulative distribution functions (8) and (9), namely they are the true inverse functions for ≤ u, v < , and F − ( ) = ∞, G − ( ) = ∞ . The copula density is given by Notice that the formula (16), strictly speaking, cannot be extended to the entire [ , ] , since lim , the copula density is unbounded around ( , ). To see the unboundedness, we provide a sketch of the argument. The details are left to the reader.
The formulas (8) and (9) show that F and G are, roughly speaking, of type Similarly, one can argue that Altogether, the above considerations and formulas mean that c(u, v) is unbounded when both u → and v → , and in all other cases it is bounded.
If one wishes to compute explicitly the copula function (15), then an explicit formula for the inverse marginal cumulative distribution functions F − (u) , G − (v) would be needed as well, but in our model this is impossible in most cases (look at (8) and (9)). Therefore we will use numerical methods, as the reader will see in the following. In Figure 2, the copula density is shown for a symmetric and for an asymmetric case.
(a) A symmetric setting: a = a = .

Figure 2:
Copula density of (X , X ) for two di erent parameter settings.

. Scatter plots of copulas for di erent parameter settings
A great advantage of our model is the exibility that it can easily handle asymmetric situations, too, i.e., when the e ect of the default of an institution on another institution is larger than vice versa. Figure 3 gives an insight into the dependence structure by scatter plots, which are common tools for visualizing bivariate (or even three-variate) copulas.  In Figure 3b, we recognize a narrow region where many observations accumulate. We can call this a "line mass". Such a region is also present on Figure 3a and 3c, but the most visible on 3b. Notice that this line mass corresponds to the ridge which can be seen at the copula density plots (Figure (2a) and (2b)). Two questions arise: what is the interpretation of the line mass and which curve describes this line? Looking at our particular parameter values in Figure 3b , a = is much larger than a = , which causes that in the case when Y expires earlier than Y , the (new) remaining lifetime of the second entity will be typically very short, since it follows an exponential distribution with parameter λ + a = . Loosely speaking, it results in X ≈ X (or if one wishes X X ), so this is the interpretation of the line mass. The corresponding probability, i.e., the weight of the line mass, is P( The theoretical equation of the curve of the line mass (see also Figure 4 ) is obtained by setting a = ∞ , and then we can use the exact equality X = X (which event has probability due to the above mentioned fact), and then we obtain v(u) = G(F − (u)) , where u and v are the variables of the copula function (see also (15) ) . It may be shown that this copula family does not exhibit the Archimedean property (for the de nition of Archimedeanity look at for instance Nelsen [9]), since the associativity does not hold. The details of this analysis are omitted.

. The idea of a multivariate setting
Let Y , . . . , Yn be independent exponential random variables with Y k ∼ Exp( λ k ), k = , . . . , n . We construct the actual lifetime variables X (q) k (k = , . . . , n) for the q-th phase of an m-step cascading e ect (q = , . . . , m; m < n ) via the following mechanism. Note that de ning an m-step cascade in our model means to de ne an ordered m-tuple of indices (k , . . . km) which indicates the defaulting institution in each step.
For the rst step ( rst phase) of the cascade let i.e., the institution that defaults rst in a certain realization is denoted by k . Let us introduce the variables Z k,k ∼ Exp(λ k + a k,k ) with parameters a k,k ∈ [ , ∞) for k ≠ k, and with a k,k = ∞ . (In this latter case the corresponding random variable is degenerated, namely Z k,k = with probability .) The variables Z k,k are independent of each other and of all Y k s. We de ne the modi ed lifetime variables X ( ) k (k = , . . . , n) via The random variable Z k,k , more precisely the shock parameter a k,k , expresses the e ect of the default of institution k on institution k . We introduce the notation I , the index set of defaulted institutions after one step of the cascade. With this notation I = {k } .
As already mentioned, the parameters can be organized as a n × n matrix. We always assume in this paper that Z k,k = with probability for all k = , . . . , n , which corresponds to a k,k = ∞, i.e., there is no possibility for governmental or other kind of bailout, when an institution has already defaulted. Note that the random variables X ( ) k are not exponentially distributed anymore (except when a k,l = , l = , . . . , n, l ≠ k for some k ), and also no longer independent (unless all a k,l = for k ≠ l ) .
After the rst step of the cascading e ect described in ( ) and ( ), the institutions continue operating until the next default happens. Let i.e., the institution that defaults in the second step of the cascading e ect is denoted by k . Then where I = {k , k } is the set of defaulted institutions after two steps of the cascade, Z k,I = Z k,k ,k ∼ Exp(λ k + a k,k + a k,k ), and the variables Z k,k ,k are independent of each other and also independent of any other variables. The random variable Z k,k ,k , more precisely the parameter a k,k + a k,k , expresses the e ect of the defaults of institutions k and k on institution k . We also assume here (like in the rst step) that Z k,I = with probability , when k ∈ I . Note that the setting in (20) does not distinguish the order of defaults regarding institutions k and k . Furthermore, by the de nition of Z k,I , we impose a simple and well-tractable additivity for modelling the e ect of consecutive defaults.
Finally, to put it more generally, in the q -th step of the cascading e ect ( q = , . . . , m ), let where I q− = {k , . . . , k q− } is the index set of the already defaulted institutions. So we call (label) the institution which defaults in the q -th phase by kq . ( X ( ) k = Y k for all k = , . . . , n and I = ∅. ) Then where Z k,Iq ∼ EXP(λ k + p∈Iq a k,p ) , and the variables Z k,Iq are independent of each other and also independent of any other variables. The random variable Z k,Iq , more precisely the parameter a k,k + . . . + a k,kq , expresses the e ect of the defaults of institutions k , . . . , kq on institution k . We also assume here that if k ∈ Iq , then Z k,Iq = with probability . (Similarly as we have stressed it after step (20), the order within the index set Iq in (22) does not play any role.) We also emphasize that the index sets Iq (q = , . . . , m) are random in the sense that they depend on the particular realizations of the random variables X (q− ) k (k = , . . . , n).

Examining the change in the dependency structure in a symmetric case
In this section, we consider the monotonicity properties of the copula for the symmetric model [λ, λ, a, a]. Notice that the copula is invariant with respect to scaling of the time axis, implying that the copulas of the models [λ, λ, a, a] and [ , , a/λ, a/λ] are identical.
Without loss of generality, we concentrate therefore on the model [ , , a, a].
The dependency structure between the lifetime variables X , X is given by the joint cumulative distribution function (3), or alternatively (but not equivalently) by the copula function (15). Both of them depend on four parameters ( λ , λ , a , a ). Our ultimate goal is to study how the joint cdf resp. the copula changes, when we let these parameters vary.
The one-parametric setting is also re ected in the notation. First we list the functions for a ≠ in formulas (23)-(29), then for a = in formulas (30)-(35) .
We will examine the change in the dependency structure given by (23)-(29) in two di erent ways. First we will consider some indicators (like expectation, variance, correlation coe cients of several kinds, etc.) extracted from the bivariate distribution. We will present these in Subsection 3.1. Secondly, we attempt to catch the dependency structure as a whole, and in Subsection 3.2 we will prove monotonicity result in the upper orthant order as parameter a varies. In this subsection we will examine the most common correlation coe cients studied in the literature, namely the usual product moment correlation (also known as Pearson's correlation coe cient), Spearman's ρ and Kendall's τ . As Figure 5 shows, each of them is increasing function of the model parameter a . We provide analytic formulas for the (Pearson's) correlation and for Spearman's ρ (see also Figure 5), from which the increasing property can be clearly veri ed. It seems impossible for us to derive an analytic formula for Kendall's τ (we will explain the reason for that in the corresponding paragraph). However, by sampling from our model and by numerically evaluating Kendall's τ for the samples, we obtained a curve for it. Furthermore, the increasing property of Kendall's τ will be proved in Proposition 2.
The previous formulas show that for a = the covariance and the correlation of X and X is . It is also obvious from the more general fact that they are independent, which can be seen by substituting a = a = in the general formula (5) of the joint density function.
The formula Ea(X ) = Ea(X ) = · a+ a+ has a nice interpretation as a → ∞ . In this case, the realizations of the two lifetime variables di er less and less from each other, and their marginal distributions can be approximated better and better with min{Y , Y }, which is distributed according to Exp(λ + λ ) , i.e., in our case Exp( ) .
We can also see that cova(X , X ) → as a → ∞ . It is more informative to examine the limit of the correlation: corra(X , X ) → as a → ∞ .

Spearman's ρ .
For (bivariate) samples Spearman's ρ is de ned through the order statistics, namely the correlation of the ranked data. Accordingly, for distributions we need to compute the following (in notation we immediately use our variables): Using the fact that U = F(X), V = G(Y), and formulas (24), (25) and (27) consist of (sums of) exponential functions, through a cumbersome, but elementary computation we get that Finally we get ρa = · a + a + a + a + a + a + − = a + a + a a + a + a + , which is pictured in Figure 5 .
Recall that for a bivariate general copula C, Kendall's τ is de ned as If the copula is the empirical one based on a sample (X (i), X (i)) N i= , then Kendall's τ can be also de ned as (A pair (X (i), X (i)) is called concordant with another pair (X (j), X (j)), if sgn(X (i) − X (j)) = sgn(X (i) − X (j)), otherwise they are discordant.) Now we are ready to present the increasing property of Kendall's τ in Proposition 2. Then the reader nds the main result of this paper in Proposition 3, namely the upper orthant ordering concerning the copulas Ca. We notice that by Theorem 5.1.9 in Nelsen [9], Proposition 2 is a direct consequence of Proposition 3. We still present them in this order, because the proof of Proposition 2 only focuses on (concordant) pairs in a sample, while the proof of Proposition 3 deals with the entire order statistics, and in this way it can be considered as an extension of the proof of Proposition 2.
For a given sample S a N , let U = {i : X a (i) < X a (i)} and L = {i : X a (i) > X a (i)} . Notice that the event X a (i) = X a (i) has probability zero. Notice now that if Z ∼ Exp( + a), then +a +a · Z ∼ Exp( + a ). Therefore we can easily modify the sample S a N to get a valid sample S a N = (X a (i), X a (i)) N i= for model [ , , a , a ].
To this end, let • for i ∈ U X a (i) = X a (i) + (X a (i) − X a (i)) · + a + a ; X a = X a .
We claim that the number of concordant pairs in sample S a N is not less than the number of concordant pairs in sample S a N . To prove this assertion, let (i, j) be a concordant pair in S a N , i.e., X a (i) − X a (j) · X a (i) − X a (j) > . We now have to distinguish four cases: (a), (b), (c) and (d) .
We have illustrated the situation in Figure 6, where the original pairs (X a (i), X a (i)) and (X a (j), X a (j)) are shown as little circles • and the modi ed pairs (X a (i), X a (i)) and (X a (j), X a (j)) are shown as dots • . W.l.o.g. we may assume that X a (j) − X a (i) > and X a (j) − X a (i) > . Let us look e.g., at case (a). Here X a (j) − X a (i) = a − a + a · X a (j) − X a (i) + + a + a · X a (j) − X a (i) > , and X a (j) = X a (j) , X a (i) = X a (i) , so X a (j) − X a (i) = X a (j) − X a (i) > . In case (b) only the roles of the coordinates are exchanged.
In case (c) we have X a (i) < X a (i) < X a (j) = X a (j) and X a (i) = X a (i) < X a (i) < X a (j) = X a (j) < X a (j) .
The last inequality holds, because the point (X a (j), X a (j)) lies above the diagonal line y = x . Again, interchanging the roles of the coordinates shows also the validity of the statement in case (d).
Finally, we argue that the empirical copula converges a.s. to the true one (see e.g., Gaensslar and Stute [5]) and that the empirical τ converges to the true τ. Thus we obtain the statement.

. Monotonicity of the copula in upper orthant order w.r.t. parameter a
Our general purpose is to examine how the copula (26) (and slightly more general the copula (15) for the model [λ, λ, a, a] ) changes as we change the value of parameter a, i.e., to determine and describe (some properties of) the function a → Ca.

Upper orthant order for copulas.
De nition 1. Let C and C be two bivariate copulas and let (U , V ) be distributed according to C and (U , V ) be distributed according to C . We say that C is dominated by C in upper orthant order (in symbol . In other words one may say that C is more (co-monotone) dependent than C .
, one sees that that C UO C is equivalent to for all u, v ∈ [ , ] . Some authors say that the random vector (U , V ) is smaller than the random vector (U , V ) in lower orthant order, if (38) holds (see e.g., Denuit et al. [3], De nition 3.3.80). Nevertheless, the notion of lower orthant order is not needed in our work.
We now formulate the main result of this section. Ca(u, v) be the copula of the model [ , , a, a] and let C a (u, v) the copula of the model  [ , , a , a ], where a ≤ a . Then

Proposition 3. Let
Proof. We have to show that Ca(u, v) ≤ C a (u, v) for all u, v. We use the same construction for the samples S a N resp. S a N as in the proof of Proposition 2. (Look at (36) and (37).) Notice that for all s, t #{i : X a (i) ≤ s, X a (i) ≤ t} ≤ #{i : X a (i) ≤ s, X a (i) ≤ t} .
We will show rst that for the empirical copula C (N) a of the sample (X a (i), X a (i)), i = , . . . , N we have the upper orthant order and m a = #(I a ). Notice that C (N) a (u, v) = m a /N. We have to show that m a is monotonically increasing with a. We may assume w.l.o.g. that there are no ties in the sample. If the index sets I a do not change for some a > a, the empirical copula also does not change. Let now a be such that exactly one index changed from I a to I a , because one of the the indices a , a changed. Suppose for instance that X a ( a ) is no longer the i-th largest among the X a (.), but there is a ∉ I a such that X a ( ) < X a ( a ). Then two situations may occur: • If X a ( ) > X a ( a ), then m a = m a .
• If X a ( ) ≤ X a ( a ), then m a = m a + .
In both cases is m a non-decreasing. One may repeat the argument for two indices changing, three indices changing and so on to see that (40) is proved. Now, again invoking the argument that the empirical copulas converge to the true copulas as in Proposition 2, one sees that

An application for measuring systemic risk
It is a usual approach in nancial theory and practice that the strength and vulnerability of the institutions is quanti ed by indirect manners. This is because an actual default or bankruptcy of a nancial institution occurs very rarely, however the stability of the institutions can vary signi cantly. In this way, the lifetime model presented in Sections 1, 2 and 3 can also be considered as an indirect tool to measure the stability and potential strength of entities in a nancial system.
In this section we have two aims. First, we relate our lifetime variables to loss variables, since mostly these latter ones are in the focus of interest for nancial institutions. Secondly, we also establish a relation between lifetime intensities and CDS spreads, which are widely used indicators for the nancial strength of entities in banking systems. This enables us to provide numerical illustration for our model using real nancial data.
We will formulate the de nitions for arbitrary n, and in the numerical case study we will restrict our analysis to n = .

. Relation between lifetime and loss variables; a model for systemic risk
In the previous sections, we dealt with de ning and exploring a joint lifetime model, where we considered the random vector of lifetimes (X , . . . , Xn) . In this subsection, we translate the distribution of lifetimes into the distribution of losses, so we introduce the random vector of losses (L , . . . , Ln) . The basic idea is simple: the longer the lifetime is, the less the losses are. There are several ways how to formulate this. In the following we list a few of them.

A few possible de nitions of loss variables de ned via lifetime variables.
(i) L i := min X i , c , (i = , . . . , n), where c > is a constant.
(ii) L i := c i · e −r·X i , (i = , . . . , n), where c i is the initial capital of institution i , and r is the risk-free interest rate.
(iii) . . . , n), where t i > is a threshold, and ξ i is a random variable with given distribution.

Quantifying systemic risk.
While the individual risk refers to the fact that random losses may occur to a nancial institution, the notion of systemic risk measures the extra risk which can be attributed to the interdependence of several institutions. Let R be a risk functional, which assigns a real value to the risk of a potential loss variable L. If the system consists of n institutions, then L + . . . + Ln is the total loss of the whole system. The distribution of this sum depends on the marginal distributions and the copula. For xed marginal distribution, let us write a C L i for the total loss variable, when the individual losses are coupled by copula C.

De nition 2.
(See also P ug and Pichler [10]). For a given risk functional R, de ne the systemic risk by where L = (L , . . . , Ln) is the vector of (univariate) marginal loss variables and Π is the independent copula. (41) compares the total loss under the copula C with the (hypothetical) total loss of independent institutions. In particular, one may consider C as the copula of the lifetimes (X , . . . , Xn) (e.g., a Freund copula) and losses depending on the lifetimes via the above formulas (i)-(iv).
In De nition 3 we recall the notion of average-value-at-risk, which can be found in several sources, e.g., in P ug and Römisch [11]. We would like to stress that technically there are two variants of the AV@R, the lower-AV@R, which focuses on the left tail of the distribution, and the upper-AV@R, which focuses on the right tail. We will need this latter one.
De nition 3. Let X be a random variable with cdf F, and let ≤ α < . The (upper) average-value-at-risk at level α is de ned as where F − (u) is the generalized inverse function of F.

Remark 2.
When it is clear from the context which signi cance level α is meant, or a certain statement holds for all α, then the lower index can be omitted from the notation, and we simply write AV@R(X).

Remark 3.
It was shown in P ug and Pichler [10] that C UO C implies that In particular, by Proposition 3, for our bivariate copula Ca (26) we have that for a ≤ a . Notice also that if a = a = in our model, then R = , so the system does not possess any systemic risk. It does not mean that the overall risk in the system would be zero, but it means that the part of the risk, which is attributed only to the dependencies of the institutions, is zero.

Examples.
We where L is a loss variable, and t is a threshold, whose excess is considered as a "bad" event. This risk functional is closely related to the stop-loss transform, which is a popular risk functional in insurance mathematics (look at for instance Denuit et al. [3], De nition 1.7.1.1) . In accordance with our systemic risk de nition L := L + L , and let us set the threshold t = . It means that we consider a situation risky, when the market loses half of its capital or more. Example 2. Let (X , X ) ∼ H , where H is according to ( ), speci ed by (43). Let us de ne now the loss variables according to (ii) from the above list, i.e., via L = c · e −r·X and L = c · e −r·X , where c = c . .= , and the risk-free interest rate r is set up to r = . . The risk functional is de ned as in Example 1, with threshold t = .
We can observe again the increase in systemic risk as we increase the shock parameters a , a , but in a much more moderate way.   Here LGD is the so-called (relative) Loss-Given-Default and r is the interest rate. (In nancial applications LGD is a commonly used notion, which expresses the maximal proportion of the rm's capital, which can be lost in case of default.) For simplicity, we assume that the credits are long-term and set T = ∞. Equating the costs and the bene ts in this case one gets the relationship Now, in order to set up a data set for lifetime data, to each CDS-spread observation {s(i)} N i= of a certain institution we create an observation {X(i)} N i= using (44). (Note that the argument i refers to the ordinal number of the observation, as it was also the case in the proof of Proposition 2 and Proposition 3.) where LGD is the Loss-Given-Default and r is the risk-free interest rate. In our analysis LGD = . , r = . . In Subsection 4.1 we already discussed some alternatives how to link lifetime and loss variables. Consider now (ii) from the list in Subsection 4.1, i.e., L := e −r·X .
Combining (44) and (46) we obtain with which we gained one possible way for having a direct connection between CDS-spreads and losses, i.e., between the available inputs and objects of interest. Notice that the function L given by (47) is a monotonically increasing function. This implies that the spreads and the losses have exactly the same copula. Notice, however, that the copula of the lifetimes is the survival copula of the spreads:C(u, v) = − u − v + C(u, v), but monotonicity w.r.t. parameter a is preserved between C andC.

Parameter estimation.
Our next task is to estimate the model parameters λ , λ , a , a , i.e., to nd the copula in our model which ts the best to the data according to some criterion. We will perform this analysis for the pair Erste Bank (k = ) and Alpha Bank (k = ). For illustration purpose we have selected a subset of our data set from 21.3.2008. to 19.1.2012, which consists of 1000 observations. The realized lifetimes (X (i), X (i)) i= are in uenced by all the four parameters λ , λ , a , a . We consider the optimization problem, which is in fact a Cramér-von Mises type minimum distance estimation: min λ ,λ ,a ,a i= C λ ,λ ,a ,a (U(i), V(i)) − Cemp(U(i), V(i)) , where • C λ ,λ ,a ,a is given by (15) via (6) and (7), (although in these formulas the parameters λ , λ , a , a were not indicated directly in the notation); • U(i) = F λ ,λ ,a (X (i)), V(i) = G λ ,λ ,a (X (i)), where F λ ,λ ,a and G λ ,λ ,a are given by (6) and (7) (although in these formulas the parameters λ , λ , a , a were not indicated directly in the notation); • Cemp is the empirical copula which corresponds to the sample (X (i), X (i)) i= .
As a result we get λ = .
, which can be interpreted as follows: indicate -roughly speaking -the expected lifetimes of the institutions. A measure of unit does not need to be attached, because the copula is invariant under simultaneous positive scaling of the model parameters, i.e., the models [λ , λ , a , a ] and [α · λ , α · λ , α · a , α · a ] have the same copula. In the light of this remark on scale invariance, we might also say that the expected lifetime of Erste Bank is 23.8 expected lifetime of Alpha Bank. Furthermore, the estimated value a = .
is the e ect of the default of Alpha Bank on Erste Bank according to our analysis, while a = .
is the e ect of the default of Erste Bank on Alpha Bank. Using again the comment on scale invariance, one might say that the expected remaining lifetime of Alpha Bank after Erste Bank defaults is λ +â = . , which is . of the expected lifetime of Alpha Bank without Erste defaulting. Similarly, the expected remaining lifetime of Erste Bank after Alpha Bank defaults is λ +â = . , which is . of the expected lifetime of Erste Bank without Alpha defaulting. For properties of Cramér von-Mises type minimum distance estimates like consistency and asymptotic normality we refer to the work of Boos [2], Klugman and Parsa [6] or Koul [7].

Conclusion
As we have seen, our lifetime-based cascading model is able to catch the dynamics of dependence structures in nancial systems, although does not contain time-dependency explicitly. However, in a latent way the model successfully replaces the explicit appearance of a time variable. Further work will focus on the analysis and numerical illustration of the multivariate Freund copula, which may accomodate multi-step cascades.