CIID Frailty Models and Implied Copulas

Abstract

A unified stochastic framework for all portfolio default models with conditionally independent and identically distributed (CIID) default times is presented. Desirable statistical properties of dependent default times are introduced in an axiomatic manner and related to the unified framework. It is shown how commonly used models, stemming from quite different mathematical and economic motivations, can be translated into a multivariate frailty model. After a discussion of popular specifications, two new models are introduced. The vector of default times in the first approach has an Archimax survival copula. The second innovation is capable of producing default pattern with interesting statistical properties. The motivation for the latter approach is to add an additional source of jump frailty to a classical intensity-based approach. An approximation of the portfolio-loss distribution is available in all cases.

Appendix

10.1.1 Lévy Subordinators

A Lévy subordinator Λ = { Λ _t } _t ≥ 0 is a non-decreasing stochastic process. It starts at zero, is stochastically continuous, and has stationary and independent increments. Standard textbooks on this theory comprise [6, 11, 12, 52, 55]. A Lévy subordinator is uniquely characterized by its Laplace transforms, which admit the form

$$\displaystyle{ \mathbb{E}\big[{e}^{-x\,\varLambda _{t} }\big] = {e}^{-t\,\varPsi (x)},\quad \forall \,x \geq 0,\,t \geq 0, }$$

for a function Ψ : [0, ∞) → [0, ∞) which has a completely monotone derivative and satisfies Ψ(0) = 0, see [25, p. 450]. The function Ψ is called Laplace exponent of Λ and is strictly increasing unless Λ _t  ≡ 0.

10.1.1.1 Proof of Lemma 10.1

The first statement follows immediately from the Theorem of Glivenko–Cantelli, see [41, p. 20]: conditioned on {F _t } _t ≥ 0, {L _t } _t ≥ 0 is precisely the empirical distribution function of the law {F _t } _t ≥ 0 based on d samples. Hence,

$$\displaystyle\begin{array}{rcl} \mathbb{P}\Big(\lim _{d\rightarrow \infty }\,\sup _{t\geq 0}\big\vert F_{t} - L_{t}\big\vert = 0\Big)& =& \mathbb{E}\Big[\mathbb{P}\Big(\lim _{d\rightarrow \infty }\,\sup _{t\geq 0}\big\vert F_{t} - L_{t}\big\vert = 0\,\Big\vert \,\{F_{t}\}_{t\geq 0}\Big)\Big] {}\\ & =& \mathbb{E}[1] = 1. {}\\ \end{array}$$

For the second statement, immediate computations show that

$$\displaystyle{ \mathbb{E}\big[L_{t}^{2}\big] = \frac{1} {d}\,\mathbb{E}[F_{t}] + \frac{d - 1} {d} \,\mathbb{E}[F_{t}^{2}],\quad \mathbb{E}[L_{ t}\,F_{t}] = \mathbb{E}[F_{t}^{2}], }$$

which implies that

$$\displaystyle{ \int _{[0,T]}\mathbb{E}\big[{(L_{t} - F_{t})}^{2}\big]\,dt = \frac{1} {d}\,\int _{[0,T]}\big(\mathbb{E}[F_{t}] - \mathbb{E}[F_{t}^{2}]\big)\,dt\quad { d \rightarrow \infty \atop \rightarrow } \quad 0. }$$

The claim is established. □ 

10.1.1.2 Proof of Zero Tail Dependence in the Model of Sect. 10.2.5

The first step is to compute for β and α, given in (10.8) and (10.9), that

$$\displaystyle{{ \alpha }^{^{\prime}}(x,0) :=\lim _{t\downarrow 0} \frac{d} {dt}\alpha (x,t){ ({\ast}) \atop =} 0,{\quad \beta }^{^{\prime}}(x,0) :=\lim _{t\downarrow 0} \frac{d} {dt}\beta (x,t){ ({\ast}{\ast}) \atop =} - x,\quad x > 0. }$$

Both ( ∗ ) and ( ∗ ∗ ) are tedious computations that become simpler with the identities

$$\displaystyle\begin{array}{rcl} c(x) + d(x)& =& -\frac{\sqrt{{\kappa }^{2 } + {2\sigma }^{2 } x}} {x} ,\quad c(x) - d(x) = - \frac{\kappa } {x}, {}\\ c(x) - d(x)& =& \frac{{\sigma }^{2}} {2x},\quad \frac{b(x)} {c(x) + d(x)} = x. {}\\ \end{array}$$

Then, using formula (10.7), one computes with L’Hospital’s rule that¹²

$$\displaystyle\begin{array}{rcl} \varLambda _{l}& =& \lim _{t\downarrow 0}\frac{\mathbb{E}\big[F_{t}^{2}\big]} {\mathbb{E}[F_{t}]} =\lim _{t\downarrow 0}\Bigg\{1 + \frac{{e}^{\alpha (2,t)+\beta (2,t)\,\varLambda _{0}} - {e}^{\alpha (1,t)+\beta (1,t)\,\varLambda _{0}}} {1 - {e}^{\alpha (1,t)+\beta (1,t)\,\varLambda _{0}}} \Bigg\} {}\\ & =& 2 {-\frac{{\alpha }^{^{\prime}}(2,0) {+\beta }^{^{\prime}}(2,0)\,\varLambda _{0}} {\alpha }^{^{\prime}}(1,0) {+\beta }^{^{\prime}}(1,0)\,\varLambda _{0}} = 0. {}\\ \end{array}$$

10.1.1.3 Proof of Theorem 10.1

For $t_{1},\ldots ,t_{d} \in [0,\infty )$ with ordered list t ₍₁₎ ≤ … ≤ t _(d) and t ₍₀₎ : = 0 one has

$$\displaystyle{ \sum _{i=1}^{d}(d\,+\,1\,-\,i)\,\big(\varLambda _{ t_{(i)}}\,-\,\varLambda _{t_{(i-1)}}\big)\,=\,\sum _{i\,=\,1}^{d}(d\,+\,1\,-\,i)\,\varLambda _{ t_{(i)}}\,-\,\sum _{i=0}^{d-1}(d\,-\,i)\,\varLambda _{ t_{(i)}}\,=\sum _{i=1}^{d}\varLambda _{ t_{i}}. }$$

Since Λ is a Lévy process, the vector of increments $\big(\varLambda _{t_{(d)}} - \varLambda _{t_{(d-1)}},\ldots ,\varLambda _{t_{(1)}} - \varLambda _{t_{(0)}}\big)$ has independent components and $\varLambda _{t_{(i)}} - \varLambda _{t_{(i-1)}}$ is equal in distribution to $\varLambda _{t_{(i)}-t_{(i-1)}}$. Consequently

$$\displaystyle{ \mathbb{E}\left [{e}^{-\sum _{i=1}^{d}\varLambda _{ t_{i}}}\right ] =\prod _{ i=1}^{d}\mathbb{E}\left [{e}^{-(d+1-i)\,\varLambda _{(t_{(i)}-t_{(i-1)})}}\right ] =\prod _{ i=1}^{d}{e}^{-(t_{(i)}-t_{(i-1)})\,\varPsi (d+1-i)}. }$$

Secondly, compute the joint survival function (using the above identity)

$$\displaystyle\begin{array}{rcl} & & G(t_{1},\ldots ,t_{d}) := \mathbb{P}\big(\tau _{1} > t_{1},\ldots ,\tau _{d} > t_{d}\big) {}\\ & & \quad = \mathbb{P}\big(\epsilon _{1} > \varLambda _{\bar{{M}\,\varphi }^{-1}(1-p(t_{1}))/\varPsi (1)},\ldots ,\epsilon _{d} > \varLambda _{\bar{{M}\,\varphi }^{-1}(1-p(t_{d}))/\varPsi (1)}\big) {}\\ & & \quad = \mathbb{E}\left [\exp \Big(-\sum _{i=1}^{d}\varLambda _{\bar{{ M}\,\varphi }^{-1}(1-p(t_{i}))/\varPsi (1)}\Big)\right ] {}\\ & & \quad = \mathbb{E}\left [\mathbb{E}\left [\exp \Big(-\sum _{i=1}^{d}\varLambda _{\bar{{ M}\,\varphi }^{-1}(1-p(t_{i}))/\varPsi (1)}\Big)\Big\vert \bar{M}\right ]\right ] {}\\ & & \quad = \mathbb{E}\left [\exp \Big(\,-\, \frac{\bar{M}} {\varPsi (1)}\sum _{i=1}^{d}\varPsi (d + 1\,-\,i)\big{(\varphi }^{-1}(1\,-\,p(t_{ (i)})) {-\varphi }^{-1}(1\,-\,p(t_{ (i\,-\,1)})\big)\Big)\right ] {}\\ & & \quad =\varphi \Big ( \frac{1} {\varPsi (1)}\sum _{i=1}^{d}\varPsi (d + 1 - i)\big{(\varphi }^{-1}(1 - p(t_{ (i)})) {-\varphi }^{-1}(1 - p(t_{ (i-1)})\big)\Big) {}\\ & & \quad =\varphi \Big ( \frac{1} {\varPsi (1)}\sum _{i=1}^{d}{\varphi }^{-1}(1 - p(t_{ (d+1-i)}))\big(\varPsi (i) - \varPsi (i - 1)\big)\Big). {}\\ \end{array}$$

The last step involves expanding the sum of differences to two sums and shifting the summation index in the second sum by one. The resulting sums can then be recombined using Ψ(0) = 0.

Thirdly, for the margins one obtains using similar arguments

$$\displaystyle{ \mathbb{P}\big(\tau _{i} > t\big) = \mathbb{P}\big(\epsilon _{i} > \varLambda _{\bar{{M}\,\varphi }^{-1}(1-p(t))/\varPsi (1)}\big) = 1 - p(t),\quad i = 1,\ldots ,d,\quad t \geq 0. }$$

Thus, τ _i is distributed according to p(t). Finally, using the survival analogue of Sklar’s Theorem, see [49, p. 195], there exists a unique copula $\hat{C}$, called the survival copula of $(\tau _{1},\ldots ,\tau _{d})$, which satisfies

$$\displaystyle{ G(t_{1},\ldots ,t_{d}) =\hat{ C}\big(1 - p(t_{1}),\ldots ,1 - p(t_{d})\big). }$$

Testing the copula claimed in (10.10), one finds

$$\displaystyle{ \hat{C}\big(1 - p(t_{1}),\ldots ,1 - p(t_{d})\big) =\varphi \Big ( \frac{1} {\varPsi (1)}\sum _{i=1}^{d}{\varphi }^{-1}(1 - p(t_{ (d+1-i)}))\big(\varPsi (i) - \varPsi (i - 1)\big)\Big). }$$

Thus, the claim is established by the uniqueness of the survival copula. □ 

10.1.1.4 Proof of Lemma 10.2

The claimed composition of different distribution functions to a new one is based on an elementary decomposition of a distribution function. Considering only two intervals [0, T ₁] and (T ₁, T ₂], it is easy to verify for t ∈ [0, T ₂] that

$$\displaystyle\begin{array}{rcl} \mathbb{P}(\tau \leq t)& =& \nVdash _{\{t\in [0,T_{1}]\}}\,\mathbb{P}(\tau \leq t) + {}\\ & & \qquad \nVdash _{\{t\in (T_{1},T_{2}]\}}\,\mathbb{P}(\tau \leq T_{1})\,\Big(1 + \frac{\mathbb{P}(\tau > T_{1})} {\mathbb{P}(\tau \leq T_{1})}\,\mathbb{P}(\tau \leq t\,\vert \,\tau > T_{1})\Big). {}\\ \end{array}$$

The crucial observation from this elementary computation is that having determined the distribution $p_{1}(t) := \mathbb{P}(\tau \leq t)$ on [0, T ₁] already, to determine the distribution on [0, T ₂] it suffices to determine $p_{2}(u) := \mathbb{P}(\tau \leq u + T_{1}\,\vert \,\tau > T_{1})$ for u ∈ (0, T ₂ − T ₁]. However, the function p ₂ is a proper distribution function on [0, ∞). Hence, starting with two given distribution functions p ₁, p ₂, the claimed composition of those yields a proper distribution function with the interpretation that p ₂ is the conditional distribution in case of survival until time T ₁. The general case K > 2 is now easily obtained by iterating the above argument.