Stochastic orders and non-Gaussian risk factor models

Abstract

The main results of this paper are monotonicity statements about the risk measures value-at-risk (VaR) and tail value-at-risk (TVaR) with respect to the parameters of single and multi risk factor models, which are standard models for the quantification of credit and insurance risk. In the context of single risk factor models, non-Gaussian distributed latent risk factors are allowed. It is shown that the TVaR increases with increasing claim amounts, probabilities of claims and correlations, whereas the VaR is in general not monotone in the correlation parameters. To compare the aggregated risks arising from single and multi risk factor models, the usual stochastic order and the increasing convex order are used in this paper, since these stochastic orders can be interpreted as being induced by the VaR-concept and the TVaR-concept, respectively. To derive monotonicity statements about these risk measures, properties of several further stochastic orders are used and their relation to the usual stochastic order and to the increasing convex order are applied.

Appendix

Proof of Theorem 1

• 1. It holds that D _i  = ${1}\kern-0.28em{\rm l}${c _i  − B _i  ≥ 0} and D′ _i  = ${1}\kern-0.28em{\rm l}${c′ _i  − B′ _i  ≥ 0} for $i=1,\ldots,n$. $(\mathbf{c} \leq \mathbf{c}^{\prime},\mathbf{B} =_{{\rm st}} \mathbf{B}^{\prime}) \Rightarrow \mathbf{c} - \mathbf{B} \leq_{{\rm st}} \mathbf{c}^{\prime} - \mathbf{B}^{\prime}$, which follows from the fact that $\mathbf{X} \leq_{{\rm st}} \mathbf{Y}$ is equivalent to ${\mathbb{P}}(\mathbf{X}\in U) \leq {\mathbb{P}}({\mathbf{Y}} \in U)$ for all upper sets $U \subseteq \hbox{I\!R}^n$, i.e. sets with the property that $\mathbf{x} \in U, \mathbf{x} \leq \mathbf{y} \Rightarrow \mathbf{y} \in U$, cf. Shaked and Shanthikumar (2007, p. 266). Function $f:\hbox{I\!R}^n \to \hbox{I\!R}^n, f(x_1,\ldots,x_n) = ({1}\kern-0.28em{\rm l}\{ x_1 \geq 0 \}, \ldots,{1}\kern-0.28em{\rm l}\{ x_n \geq 0 \} )$ is non-decreasing so that

  $$ {\mathbf{D}}= f({\mathbf{c}} - {\mathbf{B}} ) \leq_{{\rm st}}\, f({\mathbf{c}}^{\prime} - {\mathbf{B}}^{\prime} ) = {\mathbf{D}}^{\prime} $$

  follows from Lemma 4.

• 2. $\mathbf{B} =_{{\rm st}} \mathbf{B}^{\prime}$ implies $F_{B_i}=F_{B_i^{\prime}}$ for $i=1,\ldots,n$. Then $\mathbf{c} \leq \mathbf{c}^{\prime} \iff \varvec{\pi} \leq \varvec{\pi}^{\prime}$ so that the second statement follows from the first statement. $\square$

Proof of Theorem 2

Since ${\mathbb{P}}(\mathbf{D}\geq {\bf 0}) = 1$ and $\mathbf{a}\leq \mathbf{a}^{\prime}$, from

$$ {\mathbb{P}}\left(\sum_{i=1}^n a_{i} D_{i} \leq \sum_{i=1}^n a_{i}^{\prime} D_{i} \right) = {\mathbb{P}}\left(\sum_{i=1}^n (a_{i}^{\prime} - a_{i})D_{i} \geq 0 \right) =1, \\ $$

it follows that $\sum_{i=1}^n a_{i} D_{i} \leq \sum_{i=1}^n a_{i}^{\prime} D_{i}$ almost surely, and therefore

$$ S_n = \sum_{i=1}^n a_{i} D_{i} \leq_{{\rm st}} \sum_{i=1}^n a_{i}^{\prime} D_i, $$

cf. Remark 1.2.5 in Müller and Stoyan (2002). From $\mathbf{a}^{\prime}\geq {\bf 0}, \mathbf{D}\leq_{\rm st} \mathbf{D}^{\prime}$ and the fact that function $g:\hbox{I\!R}^n \to \hbox{I\!R}, g(x_1,\ldots,x_n) = \sum_{i=1}^n a_i^{\prime} x_i$ is non-decreasing, it follows that

$$ \sum_{i=1}^n a_{i}^{\prime}D_{i} \leq_{{\rm st}} \sum_{i=1}^n a_{i}^{\prime}D_{i}^{\prime} = S_n^{\prime} $$

by using Lemma 4. The transitivity of the relation $\leq_{{\rm st}}$ implies $S_n \leq_{{\rm st}} S_{n^{\prime} }$.$\square$

Proof of Theorem 3

For $i=1,\ldots,n$, it holds that

$$ \begin{aligned} D_i &= {1}\kern-0.28em{\rm l}\left\{B_i \leq c_i \right\} \\ & = {1}\kern-0.28em{\rm l}\left\{\sqrt{\rho_i} Z + \sqrt{1 -\rho_i} U_{i} \leq c_i \right\} \\ & = {1}\kern-0.28em{\rm l}\left\{ \delta_i \sqrt{\rho_i}\delta Z + \delta \sqrt{1 -\rho_i} \delta_i U_{i} \leq c_i \delta\delta_i\right\} \\ & = {1}\kern-0.28em{\rm l}\left\{\delta_i \sqrt{\rho_i}(\gamma + \delta Z) + \delta \sqrt{1 -\rho_i}(\gamma_i + \delta_i U_{i}) \leq c_i\delta\delta_i + \gamma \delta_i \sqrt{\rho_i} + \gamma_i \delta \sqrt{1 -\rho_i} \right\} \\ & = {1}\kern-0.28em{\rm l}\left\{ \sqrt{\rho_i^*}(\gamma + \delta Z) + \sqrt{1 -\rho_i^*}(\gamma_i + \delta_i U_{i}) \leq c_i^*\right\} \\ & = {1}\kern-0.28em{\rm l}\left\{B_i^* \leq c_i^* \right\} \\ \end{aligned} $$

with

$$ \rho_i^* \,\overset{\mathrm{def}}{=}\, {\frac{ \rho_i\delta_i^2 }{ \rho_i\delta_i^2 + (1- \rho_i)\delta^2}} \quad\hbox {and}\quad c_i^* \,\overset{\mathrm{def}}{=}\, {\frac{ c_i\delta\delta_i + \gamma \delta_i \sqrt{\rho_i} + \gamma_i \delta \sqrt{1 -\rho_i} } { \sqrt{ \rho_i\delta_i^2 + (1- \rho_i)\delta^2} }}. $$

$\square$

The following Definitions 13–16, Lemmas 8–12 and Theorem 20 are used in the proof of Theorem 4.

Definition 13

Let X and Y be random variables with distribution functions F _X and F _Y . Then X is said to be less dangerous than Y (written $X \leq _{{\rm da}} Y$), if ${\mathbb{E}}[X] \leq {\mathbb{E}}[Y] < \infty$ and if there is some $t_0 \in \hbox{I\!R}$ such that $F_X(t) \leq F_Y(t)$ for all t < t ₀ and $F_X(t) \geq F_Y(t)$ for all $t \geq t_0$, see Müller and Stoyan (2002, Definition 1.5.16) and Denuit et al. (2005, p. 158).

Lemma 8

$X \leq _{{\rm da}} Y$ implies $X \leq_{{\rm icx}} Y$.

This is Theorem 1.5.17 in Müller and Stoyan (2002).

The next theorem is a useful tool in identifying the single crossing property of distribution functions, which is a necessary condition for the less dangerous order.

Theorem 20

Let Y and Y′ be random variables with the same continuous and increasing distribution function F _Y . Let G be a continuous and increasing distribution function. For the random variables

$$ X \,\overset{\mathrm{def}}{=}\, G( \gamma + \delta Y )\quad\, and\,\quad \hbox {X}^{\prime} \,\overset{\mathrm{def}}{=}\, G( \gamma^{\prime} + \delta^{\prime} Y^{\prime} ) $$

with $\gamma,\gamma^{\prime},\delta,\delta^{\prime} \in \hbox{I\!R}$, the relations $0 < \delta < \delta^{\prime}$ imply that there is some $t_0 \in \hbox{I\!R}$ such that $F_X(t) \leq F_{X^{\prime}}(t)$ for all t < t ₀ and $F_X(t) \geq F_{X^{\prime}}(t)$ for all $t \geq t_0$.

Proof of Theorem 20

For all t ≤ 0 and all t ≥ 1, it holds that $F_X(t) = F_{X^{\prime}}(t)$. For 0 < t < 1, it holds that

$$ \begin{aligned} F_X(t) \leq F_{X^{\prime}}(t) &\iff {\mathbb{P}}\left( G( \gamma + \delta Y) \leq t \right) \leq {\mathbb{P}}\left( G( \gamma^{\prime} + \delta^{\prime} Y^{\prime}) \leq t \right) \\ &\iff {\mathbb{P}}\left( \gamma + \delta Y \leq G^{-1}(t)\right) \leq {\mathbb{P}}\left( \gamma^{\prime} + \delta^{\prime} Y^{\prime} \leq G^{-1}(t)\right) \\ &\iff {\mathbb{P}}\left( Y \leq {\frac{ G^{-1}(t) - \gamma} { \delta}} \right) \leq {\mathbb{P}}\left( Y^{\prime} \leq {\frac{ G^{-1}(t) - \gamma^{\prime}}{ \delta^{\prime}}} \right) \\ &\iff F_Y\left( {\frac{ G^{-1}(t)- \gamma}{ \delta}}\right) \leq F_Y\left( {\frac{ G^{-1}(t) - \gamma^{\prime}} { \delta^{\prime}}}\right) \\ &\iff {\frac{ G^{-1}(t) - \gamma}{ \delta}} \leq {\frac{ G^{-1}(t) - \gamma^{\prime}} { \delta^{\prime}}} \\ &\iff (\delta^{\prime} - \delta)G^{-1}(t) + \delta\gamma^{\prime} - \delta^{\prime}\gamma \leq 0.\\ \end{aligned} $$

$\delta < \delta^{\prime} \Rightarrow \delta^{\prime} - \delta > 0$. Since G is continuous and increasing, the inverse G ⁻¹ is continuous and increasing with $\lim\limits_{t\to 0} G^{-1}(t) = -\infty$ and $\lim\limits_{t \to 1} G^{-1}(t) = \infty$. Therefore, some $t_0 \in\hbox{I\!R}$ exists so that $F_X(t) \leq F_{X^{\prime}}(t)$ for all t < t ₀ and $F_X(t) \geq F_{X^{\prime}}(t)$ for all $t \geq t_0.$ $\square$

Definition 14

Let X and Y be random variables with finite means. Then X is said to be less than Y in convex order (written $X \leq_{{\rm cx}} Y$), if ${\mathbb{E}}[f(X)] \leq {\mathbb{E}}[f(Y)]$ for all convex functions f such that the expectations exist, see Müller and Stoyan (2002, Definition 1.5.1(i)).

Lemma 9

Let X and Y be random variables with finite means. Then

$$ X \leq_{{\rm icx}} Y \; and\; {\mathbb{E}}[X]={\mathbb{E}}[Y] \iff X \leq_{{\rm cx}} Y. $$

This is Theorem 1.5.3 in Müller and Stoyan (2002).

Definition 15

Let $\mathbf{X}$ and $\mathbf{Y}$ be n-dimensional random vectors. Then $\mathbf{X}$ is said to precede $\mathbf{Y}$ in the directionally convex order (written $\mathbf{X} \leq_{{\rm dcx}} \mathbf{Y}$), if ${\mathbb{E}}[f(\mathbf{X})] \leq {\mathbb{E}}[f(\mathbf{Y})]$ for all directionally convex functions f such that the expectations exist, see Müller and Stoyan (2002, Definition 3.12.4a) and Shaked and Shanthikumar (2007, p. 336).

Since each increasing directionally convex function is a directionally convex function, the next lemma follows immediately from Definitions 8 and 15.

Lemma 10

Let $\mathbf{X}$ and $\mathbf{Y}$ be n-dimensional random vectors. Then

$$ {\mathbf{X}} \leq_{{\rm dcx}} {\mathbf{Y}} \Rightarrow {\mathbf{X}} \leq_{{\rm idcx}} {\mathbf{Y}}. $$

Definition 16

Let F be the distribution function of an n-dimensional random vector $\mathbf{X}$ with marginal distribution functions $F_1,\ldots,F_n$. The vector $\mathbf{X}$ is called comonotone, if

$$ F({\mathbf{x}}) = \min\{F_1(x_1), \ldots,F_n(x_n)\}\quad \hbox{for all}\; {\mathbf{x}} \in \hbox{I\!R}^n, $$

see Müller and Stoyan (2002, p. 87) and Denuit et al. (2005, Proposition 1.9.4).

Lemma 11

If $\mathbf{X}$ and $\mathbf{Y}$ are comonotone n-dimensional random vectors, then

$$ X_i \leq_{{\rm cx}} Y_i \quad \hbox {for}\; i=1,\ldots,n \Rightarrow {\mathbf{X}} \leq_{{\rm dcx}} {\mathbf{Y}}. $$

This is Lemma 3.12.13 in Müller and Stoyan (2002).

Lemma 12

For $i=1,\ldots,n$ let $G^{(i)}_{\theta_i}$ be n families of univariate distribution functions parameterized by $\theta_i \in \Uptheta_i$ and let Y _i denote a random variable with distribution function $G^{(i)}_{\theta_i}$. Let $\tilde{\varvec{\theta}}$ and $\tilde{\varvec{\theta}}^{\prime}$ be two n-dimensional random vectors with supports in $\Uptheta = \Uptheta_1 \times \ldots \times \Uptheta_n$ and distribution functions $F_{\tilde{\varvec{\theta}}}$ and $F_{\tilde{\varvec{\theta}}^{\prime}}$, respectively. Let $\mathbf{X}$ and $\mathbf{X}^{\prime}$ be two n-dimensional random vectors with distribution functions H and H′ given by

$$ H(x_1,\ldots,x_n) = \int\limits_{\Uptheta} \prod_{i=1}^n G^{(i)}_{\theta_i}(x_i){\rm d}F_{\tilde{\varvec{\theta}}}(\theta_1,\ldots,\theta_n), \quad x_1,\ldots,x_n \in \hbox{I\!R} $$

and

$$ H^{\prime}(x_1,\ldots,x_n) = \int\limits_{\Uptheta} \prod_{i=1}^n G^{(i)}_{\theta_i}(x_i){\rm d}F_{\tilde{\varvec{\theta}}^{\prime}}(\theta_1,\ldots,\theta_n), \quad x_1,\ldots,x_n \in \hbox{I\!R}. $$

If for every non-decreasing convex function ${f, {\mathbb{E}}_{\theta_i}\left[f(Y_i)\right]}$ is non-decreasing and convex in $\theta_i$ for $i=1,\ldots,n$, then

$${\tilde{\varvec{\theta}}} \leq_{{\rm idcx}}{\tilde{\varvec{\theta}}}^{\prime} \Rightarrow {\mathbf{X}} \leq_{{\rm idcx}} {\mathbf{X}}^{\prime}. $$

This is Theorem 7.A.37 in Shaked and Shanthikumar (2007).

Proof of Theorem 4

Let the random variables $Z, U_{1},\ldots,U_{n}$ be stochastically independent with continuous and increasing distribution functions $F_Z, F_{U_{1}}$,$\ldots$,$F_{U_{n}} \in \hbox{I\!F}_{0,1}$. Let the random vector $(Z^{\prime}, U_{1}^{\prime},\ldots, U_{n}^{\prime})$ be distributed as the random vector $(Z, U_{1},\ldots, U_{n})$. Let $\mathbf{B}$ and $\mathbf{B}^{\prime}$ be random vectors with components $B_i \,\overset{\mathrm{def}}{=}\, \sqrt{\rho_i}Z + \sqrt{1-\rho_i} U_i$ and $B_i^{\prime} \,\overset{\mathrm{def}}{=}\, \sqrt{\rho_i^{\prime}}Z^{\prime} + \sqrt{1 -\rho_i^{\prime}} U_i^{\prime}$, respectively. Let $\mathbf{D}$ and $\mathbf{D}^{\prime}$ be random vectors with components $D_i \,\overset{\mathrm{def}}{=}\, {1}\kern-0.28em{\rm l}\{ B_i \leq F_{B_i}^{-1}(\pi_i) \}$ and $D_i^{\prime} \,\overset{\mathrm{def}}{=}\, {1}\kern-0.28em{\rm l}\{ B_i^{\prime} \leq F_{B_i^{\prime}}^{-1}(\pi_i) \}$, respectively. Under Assumption 1, the random variables B _i are conditionally independent given Z. Therefore, the random variables D _i are also conditionally independent given Z. As a result, the distribution of random vector $\mathbf{D}$ is given by

$$ {\mathbb{P}}\left(D_1 = d_1,\ldots,D_n =d_n \right) = \int\limits_{-\infty}^\infty \prod\limits_{i=1}^n {\mathbb{P}}(D_i = d_i|Z =z) {\rm d}F_Z(z) $$

with

$$ {\mathbb{P}}(D_i =d_i|Z =z) = p_i(z)^{d_i}(1-p_i(z))^{1 -d_i}, \quad d_i \in\{0,1\} $$

for $i=1,\ldots,n$, where functions p _i (z) are defined as

$$ p_i(z) \,\overset{\mathrm{def}}{=}\, F_{U_i}\left( {\frac{F_{B_i}^{-1}(\pi_i) - \sqrt{\rho_i} z } { \sqrt{1-\rho_i} }} \right). $$

With change of variables, the mixture representation

$$ {\mathbb{P}}\left(D_1 = d_1,\ldots,D_n =d_n \right) = \int\limits_{]0,1[ ^n} \prod\limits_{i=1}^n \theta_i^{d_i}(1-\theta_i)^{1 - d_i} {\rm d}F_{\tilde{\varvec{\theta}}}(\theta_1,\ldots,\theta_n; \varvec{\pi},\varvec{\rho}) $$

(28)

results, where $F_{\tilde{\varvec{\theta}}}(\theta_1,\ldots,\theta_n; \varvec{\pi},\varvec{\rho})$ is the distribution function of the random vector $\tilde{\varvec{\theta}} = ({\tilde \theta}_1,\ldots,{\tilde \theta}_n)$ with components

$$ {\tilde \theta}_i = p_i(Z) = F_{U_i}\left( {\frac{F_{B_i}^{-1}(\pi_i) - \sqrt{\rho_i} Z } { \sqrt{1-\rho_i} }} \right) $$

(29)

and

$$ {\mathbb{E}}[{\tilde \theta}_i] = {\mathbb{E}}\left[{\mathbb{E}}[D_i|Z]\right] = {\mathbb{E}}[D_i] = \pi_i. $$

(30)

The distribution function $F_{B_i}$ is the convolution of the distribution functions of the two stochastically independent random variables $\sqrt{\rho_i}Z$ and $\sqrt{1-\rho_i} U_i$ and therefore depends on ρ _i . Equation 29 implies that the random vector ${\tilde{\varvec{\theta}}}$ is comonotone.

The proof of implication (8) is based on the proof of the two implications

$$ \varvec{\rho} \leq \varvec{\rho}^{\prime} \Rightarrow {\tilde{\varvec{\theta}}} \leq_{{\rm idcx}} {{\tilde{\varvec{\theta}}}^{\prime}}, $$

(31)

and

$$ {\tilde{\varvec{\theta}}} \leq_{{\rm idcx}} {{\tilde{\varvec{\theta}}}^{\prime}} \Rightarrow {\mathbf{D}}\leq_{{\rm idcx}} {\mathbf{D}}^{\prime}, $$

(32)

where ${\tilde{\varvec{\theta}}}$ and ${{\tilde{\varvec{\theta}}}^{\prime}}$ have distribution functions $F_{\tilde{\varvec{\theta}}}\left(\cdot; \varvec{\pi},\varvec{\rho}\right)$ and $F_{{\tilde{\varvec{\theta}}}^{\prime}}(\cdot; \varvec{\pi},\varvec{\rho}^{\prime})$, respectively.

• $\varvec{\rho} \leq \varvec{\rho}^{\prime}$ implies $\rho_i \leq \rho_i^{\prime}$ for $i=1,\ldots,n$. If $\rho_i = {\rho^{\prime}}_i$ then $B_i =_{{\rm st}} B_i^{\prime}$, and ${\tilde \theta}_i =_{{\rm st}} {\tilde \theta}_i^{\prime}$ follows from Eq. 29. If $\rho_i < {\rho^{\prime}}_i$ then Theorem 20 can be applied with $X = {\tilde \theta}_i,\; X^{\prime} ={\tilde \theta}_{i}^{\prime},\; G = F_{U_i} = F_{U_i}^{\prime}, \;\gamma = {\frac{F_{B_i}^{-1}(\pi_i)}{\sqrt{1 - \rho_i}}}, \; \gamma^{\prime} = {\frac{F_{B_i^{\prime}}^{-1}(\pi_i)}{\sqrt{1 - \rho_{i^{\prime}}}}}, \; \delta = \sqrt{{\frac{\rho_i}{1 - \rho_i}}}, \; \delta^{\prime} =\sqrt{{\frac{\rho_i^{\prime}}{1 - \rho_i^{\prime}}}}, \; Y =-Z, \; Y^{\prime} =-Z^{\prime}$. Then $0 < \delta < \delta^{\prime}$ so that there is some $t_0 \in \hbox{I\!R}$ such that $F_{{\tilde \theta}_i}(t) \leq F_{{\tilde \theta}_i^{\prime}}(t)$ for all t < t ₀ and $F_{{\tilde \theta}_i}(t) \geq F_{{\tilde \theta}_i^{\prime}}(t)$ for all $t \geq t_0$. From Eq. 30, it follows that ${\mathbb{E}}[{\tilde \theta}_i] = \pi_i = {\mathbb{E}}\left[{\tilde \theta}_i^{\prime}\right]$; from Definition 13, it follows that ${\tilde \theta}_i \leq_{{\rm da}} {\tilde \theta}_i^{\prime}$; from Lemma 8, it follows that ${\tilde \theta}_i \leq_{{\rm icx}} {\tilde \theta}_i^{\prime}$ for $i=1,\ldots,n$. Further ${\mathbb{E}}[{\tilde \theta}_i] = {\mathbb{E}}\left[{\tilde \theta}_i^{\prime}\right]$ and ${\tilde \theta}_i \leq_{{\rm icx}} {\tilde \theta}_i^{\prime}$ implies ${\tilde \theta}_i \leq_{{\rm cx}} {\tilde \theta}_i^{\prime}$ for $i=1,\ldots,n$, cf. Lemma 9. Using the fact that ${\tilde{\varvec{\theta}}}$ and ${{\tilde{\varvec{\theta}}}^{\prime}}$ are comonotone random vectors, ${\tilde \theta}_i \leq_{{\rm cx}} {\tilde \theta}_i^{\prime}$ for $i=1,\ldots,n$ implies ${\tilde{\varvec{\theta}}} \leq_{{\rm dcx}} {{\tilde{\varvec{\theta}}}^{\prime}}$, cf. Lemma 11. Further, ${\tilde{\varvec{\theta}}} \leq_{{\rm dcx}} {{\tilde{\varvec{\theta}}}^{\prime}} \Rightarrow {\tilde{\varvec{\theta}}} \leq_{{\rm idcx}}{ {\tilde{\varvec{\theta}}}^{\prime}}$, cf. Lemma 10. This proves the implication (31).

• Let $\varvec{\theta} = (\theta_1,\ldots,\theta_n) \in\ ]0,1[^n$. Since $D_i |{\tilde{\varvec{\theta}}} = \varvec{\theta}$ is a Bernoulli distribution with parameter $\theta_i,$

  $$ {\mathbb{E}}[f(D_i)|{\tilde{\varvec{\theta}}} = \varvec{\theta}] = (1- \theta_i) f(0)+\theta_i f(1) = f(0) + \theta_i (f(1)- f(0)) $$

  holds for every function f. For any non-decreasing convex function f, it holds that $f(1)- f(0) \geq 0$ and then function ${\mathbb{E}}_{\theta_i}[f(D_i)] \,\overset{\mathrm{def}}{=}\, {\mathbb{E}}\left[f(D_i)|{\tilde{\varvec{\theta}}} = \varvec{\theta}\right]$ is non-decreasing and convex in $\theta_i$. Let $G_{\theta_i}^{(i)}$ denote the conditional distribution function of D _i given ${\tilde{\varvec{\theta}}} = \varvec{\theta}$. From the mixture representation for the n-dimensional probability function of $\mathbf{D}$ given in Eq. 28, the mixture representation

  $$ {\mathbb{P}}\left(D_1 \leq x_1,\ldots,D_n \leq x_n \right) = \int \limits_{]0,1[ ^n} \prod\limits_{i=1}^n G_{\theta_i}^{(i)}(x_i) {\rm d}F_{\tilde{\varvec{\theta}}}\left(\theta_1,\ldots,\theta_n; \varvec{\pi},\varvec{\rho}\right) $$

  (33)

  for the n-dimensional distribution function of $\mathbf{D}$ follows. Using the fact that the expectation ${\mathbb{E}}_{\theta_i}[f(D_i)]$ is nondecreasing and convex in $\theta_i$ for every non-decreasing convex function f, and using the mixture representation given in Eq. 33, Lemma 12 leads to the implication ${\tilde{\varvec{\theta}}} \leq_{{\rm idcx}} {{\tilde{\varvec{\theta}}}^{\prime}} \Rightarrow \mathbf{D}\leq_{\rm idcx}{\mathbf{D}^{\prime}}$. This proves Eq. 32.

Implication (8) follows from Eqs. 31 and 32. $\square$

Proof of Theorem 5

According to Lemma 5, $\mathbf{D}\leq_{\rm idcx} {\mathbf{D}^{\prime}}$ implies $\mathbf{D}\leq_{\rm iplcx}{\mathbf{D}^{\prime}}$, which means that $S_n = \sum_{i=1}^n a_{i} D_{i} \leq_{\rm icx} \sum_{i=1}^n a_{i} D_{i}^{\prime}$ for all $a_i \geq 0$ ($i = 1,\ldots,n$), cf. Definition 9. Theorem 2 implies $\sum_{i=1}^n a_i D_i^{\prime} \leq_{{\rm st}} \sum_{i=1}^n a_i^{\prime} D_i^{\prime}$, if $\mathbf{a}\leq \mathbf{a}^{\prime}$ and therefore $\sum_{i=1}^n a_i D_i^{\prime} \leq_{{\rm icx}} \sum_{i=1}^n a_i^{\prime} D_i^{\prime} = S_n^{\prime}$, cf. Lemma 3. Noting that $\leq_{\rm icx}$ is transitive, it follows that $S_n \leq_{{\rm icx}} S_n^{\prime}.$ $\square$

Proof of Theorem 6

Applying Lemma 3 on Eq. 7, it follows that

$$ ({\mathbf{a}}\leq {\mathbf{a}}^{\prime}, \varvec{\pi} \leq \varvec{\pi}^{\prime}, \varvec{\rho} = \varvec{\rho}^{\prime}) \Rightarrow S_n \leq_{\rm icx} S_n^{\prime}. $$

By using Theorems 4 and 5, it follows that

$$ \left({\mathbf{a}}\leq {\mathbf{a}}^{\prime}, \varvec{\pi} = \varvec{\pi}^{\prime}, \varvec{\rho} \leq \varvec{\rho}^{\prime}\right) \Rightarrow S_n \leq_{{\rm icx}} S_n^{\prime}. $$

Noting that the order $\leq_{{\rm icx}}$ is transitive, the two implications above can be combined to derive implication (9). $\square$

Proof of Theorem 7

• 1. The monotonicity of the VaR in a _i and π _i follows from Eq. 7 and Lemma 6.

• 2. The monotonicity of the TVaR in a _i , π _i and ρ _i follows from Eq. 9 and Lemma 7.

• 3. The monotonicity of $\mu_{G}[S_n]$ in a _i , π _i and ρ _i follows from Eqs. 3 and 9.$\square$

Proof of Theorem 8

Implications (11) and (12) can be derived from implications (7) and (9), respectively, by noting that $\bar{D}_n$ ($\bar{D}_n^{\prime}$) has the same distribution as S _n (S′ _n ) under Assumption 1 with a _i  = 1/n (a′ _i  = 1/n) for $i=1,\ldots,n$. $\square$

Proof of Theorem 9

Theorem 9 is a special case of Theorem 7 with a _i  = 1/n for $i=1,\ldots,n.$ $\square$

The proof of Theorem 10 uses the following lemma about closure properties of the usual stochastic order and of the increasing convex order with respect to convergence in distribution (denoted by $\to_{{\rm st}}$). For any random variable X, let (X)₊ denote the random variable $\max\{X,0\}$.

Lemma 13

Let $X_1,X_2,\ldots$ and $Y_1,Y_2,\ldots$ be two sequences of random variables such that $X_n \to_{{\rm st}}X$ and $Y_n \to_{{\rm st}} Y$.

• 1. Then

  $$ X_i \leq_{{\rm st}} Y_i \quad \hbox {for}\; i=1,2,\ldots \Rightarrow X \leq_{{\rm st}} Y. $$

• 2. If ${\lim\limits_{n\to\infty} {\mathbb{E}}[(X_n)_+] = {\mathbb{E}}[X_+]}$ and ${\lim\limits_{n\to\infty} {\mathbb{E}}[(Y_n)_+] = {\mathbb{E}}[Y_+]}$ and if ${\mathbb{E}}[X_+]$ and ${\mathbb{E}}[Y_+]$ are finite, then

  $$ X_i \leq_{{\rm icx}} Y_i \quad \hbox {for}\; i=1,2,\ldots \Rightarrow X \leq_{{\rm icx}} Y. $$

The first statement is Theorem 1.A.3(c) and the second statement is Theorem 4.A.8(c) in Shaked and Shanthikumar (2007).

Proof of Theorem 10

Consider two sequences of random variables $\bar{D}_n$ and $\bar{D}_n^{\prime}$ for $n= 1,2,\ldots$, both fulfilling Assumption 1 with parameters $\pi_i=\pi, \pi_i^{\prime}=\pi^{\prime}, \rho_i =\rho, \rho_i^{\prime}=\rho^{\prime}$ and distribution functions $F_{U_i}=F_{U_i^{\prime}}=F_U, F_Z=F_{Z^{\prime}}$ for $i=1,\ldots,n$. Then $\bar{D}_n \to_{{\rm st}} X \sim \mathrm{Vas}(\pi,\rho; F_Z,F_U)$ and $\bar{D}_n^{\prime} \to_{{\rm st}} X^{\prime} \sim \mathrm{Vas}(\pi^{\prime},\rho^{\prime}; F_Z,F_U)$, cf. Theorem 3.13(v) in Höse and Huschens (2008). By using the fact that the random variable which is described in Eq. 29 follows a generalized Vasicek distribution and has the mean given in Eq. 30, it follows that ${\mathbb{E}}[X]=\pi$ and ${\mathbb{E}}[X^{\prime}]=\pi^{\prime}$.

• Assume that $\pi \leq \pi^{\prime}$ and $\rho = \rho^{\prime}$, then from Theorem 8, it follows that $\bar{D}_n \leq_{{\rm st}} \bar{D}_n^{\prime}$ for all $n = 1,2, \ldots$. Hence $X \leq_{{\rm st}} X^{\prime}$, cf. Lemma 13.1. This proves Eq. 13.

• Assume that $\pi \leq \pi^{\prime}$ and $\rho \leq \rho^{\prime}$, then from Theorem 8, it follows that $\bar{D}_n \leq_{{\rm icx}} \bar{D}_n^{\prime}$ for all $n = 1,2, \ldots$. Since $X, X^{\prime}, \bar{D}_n, \bar{D}_n^{\prime}\geq 0$, it follows that ${(X)_+ = X,(X^{\prime})_+ = X^{\prime}, (\bar{D}_n)_+ = \bar{D}_n, (\bar{D}_n^{\prime})_+ = \bar{D}_n^{\prime}, \lim_{n\to\infty} {\mathbb{E}}[\bar{D}_n]=\pi={\mathbb{E}}[X]}$ and ${\lim_{n\to\infty} {\mathbb{E}}[\bar{D}_n^{\prime}]=\pi^{\prime}={\mathbb{E}}[X^{\prime}]}$. Hence $X \leq_{{\rm icx}} X^{\prime}$ follows from Lemma 13.2. This proves Eq. 14. $\square$

Proof of Theorem 11

• 1. The monotonicity of the VaR in π follows from Eq. 13 and Lemma 6.

• 2. The monotonicity of the TVaR in π and ρ follows from Eq. 14 and Lemma 7.

• 3. The monotonicity of $\mu_G[S_n]$ in π and ρ follows from Eqs. 3 and 14.$\square$

Proof of Theorem 12

The monotonicity of the VaR in a _i and π _i follows from Eq. 18 and Lemma 6. The monotonicity of the TVaR in a _i and π _i follows from Eq. 18, Lemmas 3 and 7. The monotonicity of $\mu_G[S_n]$ in a _i and π _i follows from Eq. 18, Lemma 3 and Eq. 3. $\square$

The proof of Theorem 13 is based on the supermodular order, cf. Definition 6, and on the following three propositions.

Lemma 14

Let $\mathbf{X}$ and $\mathbf{X}^{\prime}$ be n-dimensional random vectors.

• 1. If $\mathbf{X} \leq_{{\rm sm}} \mathbf{X}^{\prime}$, then $\mathbf{X}$ and $\mathbf{X}^{\prime}$ have the same marginals.

• 2. If $\mathbf{X}$ and $\mathbf{X}^{\prime}$ are normally distributed random vectors with the same marginals, then

  $$ {\rm Cov}(X_i,X_j) \leq {\rm Cov}(X_i^{\prime},X_j^{\prime})\quad \hbox {for}\; 1 \leq i < j \leq n \quad \Rightarrow \quad {\mathbf{X}} \leq_{{\rm sm}} {\mathbf{X}}^{\prime}. $$

These statements result from Theorems 3.9.5.a) and 3.13.5 in Müller and Stoyan (2002).

Theorem 21

Let Assumption 2 hold for the random vectors $\mathbf{B} = (B_1,\ldots,B_n)$ and $\mathbf{B}^{\prime} = (B_1^{\prime},\ldots,B_n^{\prime})$ with parameter vectors $(\varvec{\rho},\varvec{\alpha})$ and $(\varvec{\rho}^{\prime},\varvec{\alpha}^{\prime})$, respectively. Suppose that $F_{Z_k} = F_{Z_k^{\prime}} =\Upphi$ for $k=0,1,\ldots,K$ and that $F_{U_i} = F_{U_i^{\prime}} = \Upphi$ for $i=1,\ldots,n$. Then

$$ \left(\varvec{\rho} \leq \varvec{\rho}^{\prime},\varvec{\alpha} \leq \varvec{\alpha}^{\prime} \right) \Rightarrow {\mathbf{B}} \leq_{{\rm sm}} {\mathbf{B}}^{\prime}. $$

Proof of Theorem 21

For the model defined in Assumption 2, it holds that

$$ \left(\varvec{\rho} \leq \varvec{\rho}^{\prime},\varvec{\alpha} \leq \varvec{\alpha}^{\prime} \right) \Rightarrow {\rm Cov}(B_i,B_j) \leq {\rm Cov}(B_i^{\prime},B_j^{\prime}), \quad i \neq j, $$

since ${\rm Cov}(B_i,B_j) = {\rm Corr}(B_i,B_j)$, where the correlations are given by Eqs. 16 and 17. Since all risk factors are Gaussian distributed and stochastically independent, the vectors $\mathbf{B}$ and $\mathbf{B}^{\prime}$ are multivariate Gaussian distributed and the ordering of covariances implies the supermodular ordering of the random vectors, cf. Lemma 14.2. $\square$

Theorem 22

Let $(\mathbf{B},\mathbf{c})$ and $(\mathbf{B}^{\prime},\mathbf{c}^{\prime})$ be threshold models generating the random vectors $\mathbf{D}$ and $\mathbf{D}^{\prime}$, respectively. Let $\pi_i = F_{B_i}(c_i), c_i = F_{B_i}^{-1}(\pi_i), \pi_i^{\prime} = F_{B_i^{\prime}}(c_i^{\prime})$ and $c_i^{\prime} = F_{B_i^{\prime}}^{-1}(\pi_i^{\prime})$ for $i=1,\ldots,n$. Then

$$ \left(\varvec{\pi} = \varvec{\pi}^{\prime}, {\mathbf{B}} \leq_{{\rm sm}} {\mathbf{B}}^{\prime}\right) \Rightarrow {\mathbf{D}}\leq_{{\rm sm}} {\mathbf{D}}^{\prime}. $$

Proof of Theorem 22

Since $\mathbf{B} \leq_{{\rm sm}} \mathbf{B}^{\prime}$, the marginals are identical, i.e. $F_{B_i} = F_{B_i^{\prime}}$, cf. Lemma 14.1. Therefore, $\varvec{\pi} = \varvec{\pi}^{\prime}$ implies $\mathbf{c}=\mathbf{c}^{\prime}$. Since

$$ D_i = g_i(B_i) = {1}\kern-0.28em{\rm l}\{ B_i \leq c_i\}, \quad D_i^{\prime} = g_i(B_i^{\prime}) = {1}\kern-0.28em{\rm l}\{ B_i^{\prime} \leq c_i \}, $$

where the n functions $g_i: \hbox{I\!R} \to \hbox{I\!R}$ are all non-increasing, from Theorem 9.A.9(a) in Shaked and Shanthikumar (2007, p. 395), it follows that $\mathbf{D}\leq_{{\rm sm}} \mathbf{D}^{\prime}.$ $\square$

Proof of Theorem 13

• 1. By combining Theorems 21 and 22 the implication (19) follows.

• 2. The proof of implication (20) is based on

  $$ ({\mathbf{a}}\leq {\mathbf{a}}^{\prime}, \varvec{\pi} \leq \varvec{\pi}^{\prime}, \varvec{\rho} = \varvec{\rho}^{\prime},\varvec{\alpha} = \varvec{\alpha}^{\prime}) \Rightarrow S_n \leq_{{\rm icx}} S_n^{\prime} $$

  (34)

  and

  $$ ({\mathbf{a}}= {\mathbf{a}}^{\prime}, \varvec{\pi} = \varvec{\pi}^{\prime}, \varvec{\rho} \leq \varvec{\rho}^{\prime},\varvec{\alpha} \leq \varvec{\alpha}^{\prime}) \Rightarrow S_n \leq_{{\rm icx}} S_n^{\prime}. $$

  (35)

  Noting that $\leq_{{\rm st}}$ implies $\leq_{{\rm icx}}$ (cf. Lemma 3), the implication (34) follows from Eq. 18. Combining implication (19) with Lemma 5, it follows that $(\varvec{\pi} = \varvec{\pi}^{\prime},\varvec{\rho} \leq \varvec{\rho}^{\prime},\varvec{\alpha} \leq \varvec{\alpha}^{\prime} ) \Rightarrow \mathbf{D}\leq_{{\rm iplcx}} \mathbf{D}^{\prime}$. Implication (35) follows from Definition 9. Noting that $\leq_{{\rm icx}}$ is transitive, Eq. 20 follows from the implications given in Eqs. 34 and 35. $\square$

Proof of Theorem 14

The monotonicity of the TVaR in $a_i, \pi_i, \rho_i$ and α _k follows from Eq. 20 and Lemma 7. The monotonicity of $\mu_G[S_n]$ in a _i , π _i , ρ _i and α _k follows from Eqs. 3 and 20.$\square$

Proof of Theorem 15

The monotonicity of the VaR in a _i and π _i follows from Eq. 22 and Lemma 6. The monotonicity of the TVaR in a _i and π _i follows from Eq. 22, Lemmas 3 and 7. The monotonicity of $\mu_G[S_n]$ in a _i and π _i follows from Eq. 22, Lemma 3 and Eq. 3. $\square$

Proof of Theorem 16

• 1. Since it is assumed that ${\rm Corr}(V_i,V_j)\geq 0$, each correlation ${\rm Corr}(B_i,B_j)$ is non-decreasing in each component of the vector $\varvec{\rho}$, cf. Eq. 21. Thus, $\varvec{\rho} \leq \varvec{\rho}^{\prime} \Rightarrow {\rm Corr}(B_i,B_j) \leq {\rm Corr}(B_i^{\prime},B_j^{\prime})$ for $1\leq i<j\leq n$. Since $\mathbf{B}$ and $\mathbf{B}^{\prime}$ are multivariate standard normal distributed with the same marginals, it holds that $\mathbf{B} \leq_{{\rm sm}} \mathbf{B}^{\prime}$, cf. Lemma 14.2. Using $\varvec{\pi}=\varvec{\pi}^{\prime}$ and $c_i = c_i^{\prime}={\rm \Upphi}^{-1}(\pi_i)$, implication (23) follows from Theorem 22.

• 2. Recall that $\mathbf{P}=\mathbf{P}^{\prime}$ with ${\rm Corr}(V_i,V_j) = {\rm Corr}(V_i^{\prime},V_j^{\prime})\geq 0$ for $1 \leq i < j \leq n$. Then the proof of implication (24) is based on

  $$ ({\mathbf{a}}\leq {\mathbf{a}}^{\prime}, \varvec{\pi} \leq \varvec{\pi}^{\prime}, \varvec{\rho} = \varvec{\rho}^{\prime}) \Rightarrow S_n \leq_{{\rm icx}} S_n^{\prime} $$

  (36)

  and

  $$ ({\mathbf{a}}= {\mathbf{a}}^{\prime}, \varvec{\pi} = \varvec{\pi}^{\prime}, \varvec{\rho} \leq \varvec{\rho}^{\prime}) \Rightarrow S_n \leq_{{\rm icx}} S_n^{\prime}. $$

  (37)

  Noting that $\leq_{{\rm st}}$ implies $\leq_{{\rm icx}}$ (cf. Lemma 3), the implication (36) follows from Eq. 22. Combining implication (23) with Lemma 5, it follows that $(\varvec{\pi} = \varvec{\pi}^{\prime},\varvec{\rho} \leq \varvec{\rho}^{\prime}) \Rightarrow \mathbf{D}\leq_{{\rm iplcx}} \mathbf{D}^{\prime}$. Then implication (37) follows from Definition 9. Noting that $\leq_{{\rm icx}}$ is transitive, Eq. 24 follows from the implications given in Eqs. 36 and 37. $\square$

Proof of Theorem 17

Using Eq. 24, the monotonicity of the TVaR and of $\mu_G[S_n]$ in a _i , π _i and ρ _i follows from Lemma 7 and Eq. 3, respectively. $\square$

Proof of Theorem 18

• 1. The variance of the aggregated risk is given by

  $$ {\mathbb{V}}[S_n] = {\mathbb{E}}[S_n^2] - {\mathbb{E}}[S_n]^2, $$

  where ${\mathbb{E}}[S_n] = \sum_{i=1}^n a_i {\mathbb{P}}(D_i=1)$ and

  $$ {\mathbb{E}}[S_n^2] = \sum_{i=1}^n a_i^2 {\mathbb{P}}(D_i=1) + 2\sum_{1 \leq i<j \leq n} a_i a_j {\mathbb{P}}(D_i=1, D_j=1). $$

  It holds that ${\mathbb{P}}(D_i=1) = \pi_i$ and

  $$ {\mathbb{P}}(D_i=1, D_j=1) = {\mathbb{P}}(B_i \leq c_i, B_j \leq c_j) = \Upphi_2(c_i,c_j;\rho_{ij}), \quad i \neq j. $$

  It is well known that the simultaneous probability ${\mathbb{P}}(D_i=1, D_j=1)$ is increasing in parameter ρ _ij for all thresholds c _i and c _j (Tong 1980, Lemma 2.1.2). From $B_i \sim B_i^{\prime} \sim N(0,1)$ for all $i=1,\ldots,n$ and $\mathbf{c}=\mathbf{c}^{\prime}$, it follows that ${\mathbb{P}}(D_i=1) = {\mathbb{P}}(D_i^{\prime}=1)$, which leads together with $\mathbf{a}=\mathbf{a}^{\prime}$ to ${\mathbb{E}}[S_n]={\mathbb{E}}[S_n^{\prime}]$. Then $\rho_{ij} \leq \rho_{ij}^{\prime}$ implies ${\mathbb{E}}[S_n^2] \leq {\mathbb{E}}[S_n^{\prime 2}]$, which proves Eq. 25. Since $\mathbf{B}$ and $\mathbf{B}^{\prime}$ are multivariate standard normal distributed with the same marginals, it holds that $\rho_{ij} \leq \rho_{ij}^{\prime} \Rightarrow \mathbf{B} \leq_{{\rm sm}} \mathbf{B}^{\prime}$, cf. Lemma 14.2. Using $\mathbf{c}=\mathbf{c}^{\prime}$ and $\pi_i = \pi_i^{\prime}= \Upphi(c_i), \mathbf{D}\leq_{{\rm sm}} \mathbf{D}^{\prime}$ follows from Theorem 22. Recall $\mathbf{a}= \mathbf{a}^{\prime}$. Then applying Lemma 5 and Definition 9, it follows that $\mathbf{D}\leq_{{\rm sm}} \mathbf{D}^{\prime} \Rightarrow \mathbf{D}\leq_{{\rm iplcx}} \mathbf{D}^{\prime} \Rightarrow S_n \leq_{{\rm icx}} S_n^{\prime}$, respectively.

  Using $S_n \leq_{{\rm icx}} S_n^{\prime}$, the monotonicity of the TVaR and of $\mu_G[S_n]$ in ρ _ij follows from Lemma 7 and Eq. 3, respectively. This proves Eqs. 26 and 27.

• 2. If in addition $\rho_{ij} < \rho_{ij}^{\prime}$ for at least one pair (i, j), then ${\mathbb{V}}[S_n] < {\mathbb{V}}[S_n^{\prime}]$. From ${\mathbb{E}}[S_n]={\mathbb{E}}[S_n^{\prime}]$ and ${\mathbb{V}}[S_n] < {\mathbb{V}}[S_n^{\prime}]$, it follows that neither $S_n \leq_{{\rm st}} S_n^{\prime}$ nor $S_n^{\prime} \leq_{{\rm st}} S_n$. This follows from the well-known fact that $X \leq_{{\rm st}} Y$ and ${\mathbb{E}}[X] ={\mathbb{E}}[Y]$ implies that X and Y have the same distribution, cf. Theorem 1.2.9 b) in Müller and Stoyan (2002), and therefore can not have different variances. By applying Lemma 6, neither VaR _p [S _n ] ≤ VaR _p [S′ _n ] for all 0 < p < 1 nor VaR _p [S′ _n ] ≤ VaR _p [S _n ] for all 0 < p < 1.

$\square$

Proof of Theorem 19

For two different claims $i \neq j$ the simultaneous probability P(D _i  = 1, D _j  = 1) is given by

$$ \begin{aligned} {\mathbb{P}}(D_i=1,D_j=1) &= {\mathbb{P}}({\tilde B}_i \leq c_i, {\tilde B}_j \leq c_j) \\ & = {\mathbb{P}}(W_iB_i \leq c_i, W_jB_j \leq c_j) \\ & = \int\limits_{]0,\infty[ ^2} {\mathbb{P}}(w_iB_i \leq c_i,w_jB_j \leq c_j) {\rm d}F_{W_i,W_j}(w_i,w_j) \\ & = \int\limits_{]0,\infty[ ^2} {\mathbb{P}}\left(B_i \leq {\frac{c_i} {w_i}},B_j \leq {\frac{c_j}{w_j}}\right) {\rm d}F_{W_i,W_j}(w_i,w_j), \\ \end{aligned} $$

where $F_{W_i,W_j}$ denotes the distribution function of the random vector $(W_i,W_j)$. Since each probability ${\mathbb{P}}\left(B_i \leq {\frac{c_i} {w_i}},B_j \leq {\frac{c_j}{w_j}}\right)$ is increasing in ρ _ij for all thresholds ${\frac{c_i}{w_i}}$ and ${\frac{c_j}{w_j}}$, cf. the proof of Theorem 18, the probabilities P(D _i  = 1, D _j  = 1) are increasing in ρ _ij . The same reasoning as in the proof of Theorem 18 leads to the propositions of Theorem 19. $\square$