Bivariate lower and upper orthant value-at-risk - Springer Link

Eur. Actuar. J. (2013) 3:321–357 DOI 10.1007/s13385-013-0079-3 ORIGINAL RESEARCH PAPER

Bivariate lower and upper orthant value-at-risk He´le`ne Cossette • Me´lina Mailhot • E´tienne Marceau • Mhamed Mesfioui

Received: 10 June 2013 / Revised: 7 September 2013 / Accepted: 18 September 2013 / Published online: 31 October 2013 DAV / DGVFM 2013

Abstract Value-at-risk (VaR) is an important risk measure widely used in actuarial science and quantitative risk management. Embrechts and Puccetti (J Multivar Anal 97(2):526–547, 2006a) have introduced the multivariate lower and upper orthant VaR. The practical applications of these risk measures is very promising, especially in actuarial science and quantitative risk management. Our objective is to study in details the multivariate lower and upper orthant VaR in the bivariate setting, their properties and their applications. In particular, new characterizations of the bivariate lower and upper orthant VaR and desirable properties are given, such as translation invariance, positive homogeneity and comonotonic additivity. Lower and upper confidence regions for random vectors are developed and used to provide new results on the convexity conditions and to suggest capital allocation techniques. We provide bounds on functions of random pairs and derive interesting relations with existing results. We motivate the use of the bivariate lower and upper ortant VaR for risk allocation, to represent bivariate ruin probabilities and for risk comparison. Practical illustrations and examples of the results are presented throughout the article. Keywords Convexity

Multivariate value-at-risk Risk measures Copulas Bounds

H. Cossette E´. Marceau E´cole d’Actuariat, Universite´ Laval, Quebec, Canada M. Mailhot (&) Department of Mathematics and Statistics, Concordia University, Montreal, QC, Canada e-mail: [email protected] M. Mesfioui De´partement de mathe´matiques et informatique, Universite´ du Que´bec a` Trois-Rivie`res, Trois-Rivie`res, Canada

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1 Introduction and motivation Under the univariate hypothesis, several tools have been studied to measure risk and allocate capital, especially for insurance companies and other financial institutions. Value-at-risk (VaR) is an important risk measure in actuarial science and in finance first because of regulatory reasons, but also because it is easy to understand. Let X be a random variable (rv) with cumulative distribution function (cdf) FX. The VaR, at level a, 0 \ a \ 1, of X is defined by VaRa ðXÞ ¼ inffx 2 R; FX ðxÞ ag: This risk measure is well understood and has been a point of interest for several years. See e.g. McNeil et al. [21] for a review. Some situations require that each component of a portfolio be fixed such that the joint cdf does not to exceed a given level a. Notably, one may want to describe relationships using inverse quantile functions, or allocate capital for each component of a portfolio. As noted in Jouini et al. [16], investors might not be able to aggregate their risks, due to liquidity problems and/or transaction costs between the different security markets. In those situations, the use of a multivariate VaR is more appropriate. Embrechts and Puccetti [10] have introduced the multivariate lower and upper orthant VaR associated to a loss X. Let X ¼ ðX1 ; . . .; Xk Þ be a random vector with joint cdf FX and joint survival function (sf) FX : For a 2 ð0; 1Þ; the multivariate lower orthant VaR at probability level a is the boundary of its a-level sets, defined by VaRa ðXÞ ¼ o x 2 Rk : FX ðxÞ a : ð1:1Þ Analogously, the multivariate upper orthant VaR at probability level a is given by VaRa ðXÞ ¼ o x 2 Rk : FX ðxÞ 1 a : ð1:2Þ The practical applications of these risk measures seem to be very promising, especially in finance, quantitative risk management and actuarial science. As stated in Cherubini and Luciano [5], one could be interested in a trade-off between the VaR of two stock indices or in comparing a-level sets. Gugan and Hassani [14] justify the use of multivariate risk measures by linking capital requirements with operational risk capital, using pair-copulas. Frees and Valdez [13] consider the bivariate allocation to losses and the related expense and adjustment costs. The example in this paper has also been used in Di Bernardino et al. [8] to illustrate the multivariate CTE presented in this article. The latter article studies the empirical bivariate distributions of risks, and provide an estimation method for a multivariate Conditional Tail Expectation. The applications of multivariate risk measures can be used in other contexts than multivariate capital allocation or risk comparison, as for studying reinsurance premiums and building asset portfolios. Our objective is to study the behavior of VaRa (X) and VaRa (X) in a bivariate setting to begin with. Results could be extended to higher dimensions (k [ 2) by, for example, successively fixing each rv. Such an extension to a higher dimension will be addressed in future work. Note that in certain contexts, such as vine copulas,

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higher dimensions result in a study of many two-dimensions. In this paper, we consider continuous rv’s and leave the non continuous case (suitable e.g. for layered risks or risks covered by reinsurance stop loss contracts) for further research work. The paper is organized as follows. In Sect. 2.1 definitions and characterizations of the bivariate lower and upper orthant VaR are given. Section 2.2 presents interesting and desirable properties of the bivariate lower and upper orthant VaR, such as behaviors under transformations of the bivariate set, translation invariance and positive homogeneity. In Sect. 2.3 lower and upper confidence regions are presented. In Sect. 2.4 the convexity conditions are studied, based on the joint and marginal distributions. Also, special cases are presented. In Sect. 2.5 the impact of the dependence and of marginal distributions on the bivariate lower and upper orthant VaR is provided. Section 3 studies the bivariate lower and upper VaR for sums of random pairs. This section also provides bounds for those sums. Finally, Sect. 4 illustrates some results of the previous sections about the lower and upper orthant VaR. More specifically, Sect. 4.1 presents two methods using VaRa (X) and VaRa (X) to obtain an allocation couple, Sect. 4.3 treats the special case of two lines of business covering the same number of risks, Sect. 4.2 links bivariate lower and upper orthant VaR with bivariate ruin probabilities and Sect. 4.4 illustrates the concept of confidence regions and allocation sets.

2 Characterization and resulting properties 2.1 Bivariate lower and upper orthant value-at-risk In this section, we propose an alternative way to define the lower and upper orthant VaR, i.e. VaRa (X) and VaRa (X), defined in Embrechts and Puccetti [10]. Let X = (X1, X2) be a random vector with joint cdf FX and sf FX : Denote Fx1 and Fx2 the marginal cdf’s of X. For fixed x1, define the functions x2 7! Fx1 ðx2 Þ ¼ FX ðx1 ; x2 Þ and x2 7! Fx1 ðx2 Þ ¼ FX ðx1 ; x2 Þ: Let Fx1 (a) and Fx1 ðaÞ be their corre1 1 sponding generalized inverse functions given by ðaÞ ¼ inf ft 2 R : Fx1 ðtÞ ag Fx1 1

Fx1 ðaÞ ¼ inf ft 2 R : Fx1 ðtÞ ag 1

and

respectively. Note that the inequality FX ðx1 ; x2 Þ a and FX ðx1 ; x2 Þ 1 a are ðaÞ and x2 Fx1 ð1 aÞ respectively. Moreover, if FX is equivalent to x2 Fx1 1 1 continuous, then we have X x1 ; Fx1 ð1 aÞ ¼ 1 a; x1 VaRa ðX1 Þ: FX x1 ; Fx1 ðaÞ ¼ a; F 1 1 Throughout the paper we adopt the notations ðaÞ ¼ VaRa;x1 ðXÞ Fx1 1

and

Fx1 ð1 aÞ ¼ VaRa;x1 ðXÞ: 1

Hereafter, we propose a convenient characterization of the boundary a-level sets VaRa (X) and VaRa (X) given in (1.1) and (1.2) in terms of the a-level curves x1 7! VaRa;x1 ðXÞ; x1 7! VaRa;x1 (X) and x2 7! VaRa;x2 ðXÞ; x2 7! VaRa;x2 (X), namely

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x1 ; VaRa;x1 ðXÞ ; x1 VaRa ðX1 Þ ; VaRa ðXÞ ¼ x1 ; VaRa;x1 ðXÞ ; x1 VaRa ðX1 Þ

VaRa ðXÞ ¼

ð2:1Þ ð2:2Þ

and VaRa;x2 ðXÞ; x2 ; x2 VaRa ðX2 Þ ; VaRa ðXÞ ¼ VaRa;x2 ðXÞ; x2 ; x2 VaRa ðX2 Þ : VaRa ðXÞ ¼

ð2:3Þ ð2:4Þ

The latter approach defines the values obtained to represent the bivariate lower and upper orthant VaR respectively, for X2 and X1. More precisely, (2.1) and (2.2) allow to establish the lower and upper curves for X2, by fixing x1 and isolating x2. The lower and upper curves for X1 are provided by (2.3) and (2.4). They are obtained by fixing x2 and isolating x1. To simplify the presentation, we will systematically use (2.1) and (2.2) to express our results. Definitions (2.1) and (2.2) of the bivariate VaR allow us to derive the results presented in this paper. We now investigate the behavior of the a-level curves x1 7! VaRa;x1 ðXÞ and x1 7! VaRa;x1 ðXÞ: For that purpose, denote the supports of X1 and X2 by supp(X1) and supp(X2). Let lX1 and uX1 be the essential supremum and essential infimum of X1 defined by lX1 ¼ inf fx : x 2 suppðX1 Þg and uX1 ¼ supfx : x 2 suppðX1 Þg; and define lX2 and uX2 for X2 similarly. Proposition 2.1 Let X ¼ ðX1 ; X2 Þ be a pair of rv’s with joint cdf’s FX and marginal cdf’s FX1 and FX2 . Then, the a-level curves x1 7! VaRa;x1 ðXÞ

and x1 7! VaRa;x1 ðXÞ

are decreasing functions. Moreover, if FX is strictly increasing, then ð1Þ ð2Þ Proof

lim VaRa;x1 ðXÞ ¼ VaRa ðX2 Þ

and

lim VaRa;x1 ðXÞ ¼ VaRa ðX2 Þ

and

x1 !uX1

x1 !lX1

lim

VaRa;x1 ðXÞ ¼ uX2 ; ð2:5Þ

lim

VaRa;x1 ðXÞ ¼ lX2 : ð2:6Þ

x1 !VaRa ðX1 Þ

x1 !VaRa ðX1 Þ

Since the joint cdf FX is assumed to be continuous, we have FX x1 ; VaRa;x1 ðXÞ ¼ a; FX x1 ; VaRa;x1 ðXÞ ¼ 1 a:

ð2:7Þ

Given that FX is an increasing function (respectively FX decreasing), then one has necessarily from (2.7) that x1 7! VaRa;x1 ðXÞ and x1 7! VaRa;x1 ðXÞ are decreasing functions of x1 (if not, we have a contradiction). In addition, since FX is continuous, then ðaÞ ¼ VaRa ðX2 Þ: lim VaRa;x1 ðXÞ ¼ FX1 2

x1 !uX1

Also, since FX is continuous and strictly increasing, then lim

x1 !VaRa ðX1 Þ

VaRa;x1 ðXÞ ¼ uX2 ;

hence (2.5). The result (2.6) is obtained similarly.

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The interest of Proposition 2.1 is to show that VaRa ðXÞ ½VaRa ðX1 Þ; uX1 ½ ½VaRa ðX2 Þ; uX2 ½;

ð2:8Þ

VaRa ðXÞ lX1 ; VaRa ðX1 ÞlX2 ; VaRa ðX2 Þ:

ð2:9Þ

Example 2.2 Consider X1*exponentialðk1 ¼ 0:2Þ and X2*exponentialðk2 ¼ 0:25Þ; linked by a Clayton copula with h = 2. Figure 1 represents the 95 %-level curves of VaRa ðXÞ (dashed line) and VaRa ðXÞ (solid line). Note that bivariate risks linked by an Archimedean copula have concave marginal cdf’s. The horizontal and vertical solid lines represent the univariate VaRa(X1) and VaRa(X2). This represents a typical situation. Nevertheless, we will see further that one prefers the situation where VaRa ðXÞ is convex and VaRa ðXÞ is concave. The desirable conditions are more easily obtained with VaRa ðXÞ than with VaRa ðXÞ. h Remark 2.3 It is also possible to verify that VaRa ðXÞ is always smaller than VaRa ðXÞ: Using q to represent the boundary of a set, we have that VaRa ðXÞ ¼ o x 2 R2 : FX ðxÞ a : Then, the boundary sets must verify T Analogously, we have that o x 2 R2 : PðX1 x1 X2 x2 Þ a : 2 VaRa ðXÞ ¼ o x 2 R : FX ðxÞ 1 a : Then, the boundary sets must verity S o x 2 R2 : PðX1 x1 X2 x2 Þ a : Example 2.4 Consider X1*Beta(a1 = 5, b1 = 1) and X2*Beta(a2 = 2, b2 = 1), linked by a Frank copula with h = -5. Figure 2 illustrates the bivariate lower and upper orthant VaR, using convex marginal cdf’s. One sees that a lower dependence parameter provides a lower curve for the bivariate upper orthant VaR and an upper curve for the bivariate lower orthant VaR. Also, we observe a concave bivariate upper orthant VaR, and slightly convex lower orthant VaR. In this case, it is clear that the convexity, based on the marginals and dependence structure, differs from that of Example 2.2. h 2.2 Properties of the bivariate lower and upper orthant VaR In this section, we present analogous properties of the bivariate lower and upper orthant VaR to those of the univariate VaR. The following proposition states that the bivariate lower and upper orthant VaR of a transformation of the bivariate set, through increasing functions, modifies the curves by this same transformation. It also shows analogous results for decreasing functions, where the bivariate lower (upper) orthant curve at level a, results in the bivariate upper (lower) orthant at level 1 - a. Proposition 2.5

Let X = (X1, X2) be a continuous random vector and /ðXÞ ¼ ð/1 ðX1 Þ; /2 ðX2 ÞÞ;

where /1 and /2 are real functions defined on the supports of X1 and X2 respectively.

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30

25

20

15

10

5

0

5

10

15

20

25

30

35

Fig. 1 Graphical representation of the bivariate lower and upper VaR. (a-curves of VaRa (X) (dashed) and VaRa (X) (solid))

1

1

θ=−5

↑ VaRα(X2)

0.9

θ=5 0.995

0.8 0.7

0.99 0.6

θ=−5 θ=5

0.5

VaRα(X1)→

0.985

←VaRα(X1)

0.4 0.3

0.98

0.2 0.975

0.1 0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

VaRα(X2) ↓ 0.99 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999

1

Fig. 2 Graphical representation of the upper and lower orthant VaR with h = -5 and h = 5

For increasing functions /i and /j, i, j = 1, 2, i = j, we have VaRa;/j ðxj Þ ð/ðXÞÞ ¼ /i VaRa;xj ðXÞ and VaRa;/j ðxj Þ ð/ðXÞÞ ¼ /i VaRa;xj ðXÞ :

1.

2.

For decreasing functions /i and /j, i, j = 1, 2, i = j, we have VaRa;/j ðxj Þ ð/ðXÞÞ ¼ /i VaR1a;xj ðXÞ and VaRa;/j ðxj Þ ð/ðXÞÞ ¼ /i VaR1a;xj ðXÞ :

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Proof Let us condition on X2 = x2 and consider increasing functions /i, i = 1, 2. Then, one has VaRa;x2 7! Fx2 ðVaRa;x2 Þ such that

1.

a ¼ FX ðVaRa;x2 ðXÞ; x2 Þ ¼ PðX1 VaRa;x2 ðXÞ; X2 x2 Þ ¼ Pð/1 ðX1 Þ /1 ðVaRa;x2 ðXÞÞ; /2 ðX2 Þ /2 ðx2 ÞÞ ¼ Pð/1 ðX1 Þ VaRa;/2 ðx2 Þ ð/ðXÞÞ; /2 ðX2 Þ /2 ðx2 ÞÞ: Analogous arguments are used for a similar result with the bivariate upper orthant VaR. 2. Let us condition on X2 = x2 and consider decreasing functions /i, i = 1, 2. Then, one has VaRa;x2 7! F x2 ðVaRa;x2 Þ such that 1 a ¼ F X ðVaRa;x2 ðXÞ; x2 Þ; and 1 a ¼ PðX1 [ VaR1a;x2 ðXÞ; X2 [ x2 Þ: Because /i, i = 1, 2 are decreasing functions, one has 1 a ¼ Pð/1 ðX1 Þ /1 ðVaR1a;x2 ðXÞÞ; /2 ðX2 Þ /2 ðx2 ÞÞ ¼ Pð/1 ðX1 Þ VaRa;/2 ðx2 Þ ð/ðXÞÞ; /2 ðX2 Þ /2 ðx2 ÞÞ: Analogous arguments are used for the result with the bivariate upper orthant VaR. h Corollary 2.6 ensures the translation invariance. For any additional risk to the set, the curves will also translate for this same values c ¼ ðc1 ; c2 Þ 2 R: Corollary 2.6 As a special case of Proposition 2.5, one has for all c ¼ ðc1 ; c2 Þ 2 R and i, j = 1, 2, i = j, then VaRa;xj þcj ðX þ cÞ ¼ VaRa;xj ðXÞ þ ci ;

VaRa;xj þcj ðX þ cÞ ¼ VaRa;xj ðXÞ þ ci :

Obtained with the application of Proposition 2.5 (item 1.), corollary 2.7 relates the conditions for the bivariate lower and upper orthant VaR to be homogeneously invariant. Through positive transformations of the bivariate set, the bivariate lower and upper orthant VaR will vary with the same transformations, with level a. Corollary 2.7

For all c ¼ ðc1 ; c2 Þ 2 Rþ Rþ and i, j = 1, 2, i = j, then

VaRa;cj xj ðcXÞ ¼ ci VaRa;xj ðXÞ;

VaRa;cj xj ðcXÞ ¼ ci VaRa;xj ðXÞ:

Corollary 2.8 results from an application of Proposition 2.5 (2.), for negative transformations of the bivariate set.

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Corollary 2.8

For all c ¼ ðc1 ; c2 Þ 2 R R and i, j = 1, 2, i = j, then

VaRa;cj xj ðcXÞ ¼ ci VaR1a;xj ðXÞ;

VaRa;cj xj ðcXÞ ¼ ci VaR1a;xj ðXÞ:

2.3 Lower and upper confidence regions In this section, we want to study other properties of the bivariate lower and upper orthant VaR and explain how to enlarge the use of the bivariate lower and upper orthant VaR, by using confidence regions. The bivariate lower and upper confidence regions of level a represents the sets of points covering a % of the possible values of a bivariate set of rv’s. Those regions can represent an acceptance region as defined in Jouini et al. [16] and Bentahar (2006), based on the bivariate VaR curves. The bivariate lower orthant confidence region is bounded by a bivariate lower orthant VaR, up to which a % of the sets are under that curve. The bivariate upper orthant confidence region is bounded by the bivariate upper orthant VaR, up to which (1 a)% of the sets are over that curve. Here and in the sequel, we denote CFX ;a ¼ ðx1 ; x2 Þ 2 R2 : FX ðx1 ; x2 Þ a and CFX ;a ¼ ðx1 ; x2 Þ 2 R2 : FX ðx1 ; x2 Þ 1 a : In this section, we discuss how one can derive a lower and upper confidence region based on the lower and upper orthant VaR. The objective is to find a level curve such that the probability that the random vector X is below (respectively above) this curve equals the level a. To this end, define the bivariate order, denoted ; by X VaRa ðXÞ if and only if X 2 CcFX ;a ; where CcFX ;a denotes the complement of the event CFX ;a : Clearly, X VaRa ðXÞ is equivalent to X being below the curve VaRa ðXÞ: One has PðX VaRa ðXÞÞ ¼ P X 2 CcFX ;a ¼ A1 þ A2 ¼aþ

Z1 FX1 ðaÞ 1

VaRa;s ðXÞ

Z

f ðs; tÞdtds ¼ kðaÞ;

1

ð2:10Þ where f(s, t) denotes the joint probability density function (pdf) of random vector X. Figure 3 illustrates the regions A1 and A2 : Consequently, one has P X VaRk1 ðaÞ ðXÞ ¼ a: In other words, the level curve VaRk1 ðaÞ ðXÞ can be considered as a lower confidence region of the random vector X at level a. Moreover, one observes that k1 : ½0; 1 7! ½0; 1 is an increasing function such that k-1(a) B a. This means that the lower confidence region curve VaRk1 ðaÞ (X) is smaller than the lower orthant a-level curve VaRa ðXÞ: Similarly, one can obtain the upper confidence region of random vector X at level 1 - a, in terms of the upper orthant VaR. In fact, let be a bivariate order defined by X VaRa ðXÞ if and only if X is above the a-curve VaRa ðXÞ: It follows that

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30 ψα(s) A1

25

A2

t

20

15

10

→ F−1 (α) 1

5

0

0

5

10

15

20

25

30

35

s Fig. 3 Lower orthant confidence region for rv’s with positive supports

P X VaRa ðXÞ ¼ 1 a þ

FX1 ðaÞ

Z1

1

Z1

f ðs; tÞdtds ¼ kðaÞ:

VaRa;s ðXÞ

The upper confidence region of random vector X is then given by VaRk1 ðaÞ ðXÞ; that is, P X VaRk1 ðaÞ ðXÞ ¼ 1 a: Hence, VaRk1 ðaÞ ðXÞ (respectively VaRk1 ðaÞ ðXÞ) can be viewed as a threshold curve such that the probability that the components of the loss X over the given time horizon are simultaneously below (respectively above) this curve with probability a (respectively 1 - a). Example 2.9 Let us consider the model chosen in Cherubini and Luciano [5] for the returns on two different indexes, S&P100 and FTSE100 (historical data downloaded from YahooFinance) represented by a Clayton copula (h = 0.5) and normal marginal distributions. The copula allows to separate the impact on the joint distribution of the marginal cdf’s. We use this example in order to illustrate the confidence region at level 1 %, that could be of interest to study the effect of moving capital from one desk to the other, using the trade-off of VaR. The confidence bivariate lower orthant confidence region illustrates the 1 % acceptable scenarios, for which trading from one desk to the other does not produce an undesirable outcome. The same exercise could be done using portfolios of assets.

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x 10

log−returns of FTSE100

0

−5

−10

−15

−12

−10

−8

−6

−4

log−returns of S&P100

−2

0 −3

x 10

Fig. 4 Lower orthant confidence region at level 1 %

Figure 4 represents the trade-off curves at level 1 % and area under 0.535 % level curve. It shows that 1 % of the sets under the bivariate lower orthant VaR at level a = 1 % can be represented by the bivariate lower orthant VaR at level 0.535 %. h 2.4 Convexity of the bivariate lower and upper orthant value-at-risk As shown by Examples 2.2 and 2.4, the convexity of the joint cdf of X ¼ ðX1 ; X2 Þ has an impact on the shape of the bivariate lower and upper orthant VaR. It will also affect the allocation sets, when one wants to select a bivariate vector from the lower or upper orthant VaR. This preoccupation is covered in Sect. 4.1 We need to study the convexity of the curves first. In what follows, we examine the shape of the a-level curves x1 7! VaRa;x1 ðXÞ and x1 7! VaRa;x1 ðXÞ: The following result establishes sufficient conditions to ensure the convexity and the concavity of these a-level curves. Proposition 2.10 sf FX : One has (1) (2)

Let X ¼ ðX1 ; X2 Þ be a random vector with joint cdf FX and joint

If FX is concave (respectively convex) then x1 7! VaRa;x1 ðXÞ is convex (respectively concave). If FX is convex (respectively concave) then x1 7! VaRa;x1 ðXÞ is convex (respectively concave).

Proof To show (1), suppose that FX is a concave function, with the confidence region

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CFX ;a ¼ ðx1 ; x2 Þ 2 R2 : FX ðx1 ; x2 Þ a and let x ¼ ðx1 ; x2 Þ 2 CFX ;a ; y ¼ ðy1 ; y2 Þ 2 CFX ;a and k 2 ½0; 1: Then, one has FX ðkx þ ð1 kÞyÞ kFX ðxÞ þ ð1 kÞFX ðyÞ ka þ ð1 kÞa ¼ a: Thus, the confidence region CFX ;a is a convex set and its boundary is a convex function, so x1 7! VaRa;x1 ðXÞ is convex. Now, if FX is convex, then the complement of the confidence region CFX ;a is a convex set so the boundary of CFX ;a is concave, thus (1) holds. Similar arguments may be used to show (2). h We propose a practical criterion that ensures the convexity of these risk measures and we set x1* = FX11 (a) and x2* = FX1 (a). 2 Proposition 2.11 Let X ¼ ðX1 ; X2 Þ be a random vector with joint cdf FX and joint sf FX : Denote FX1 and FX2 the marginal cdf’s. Assume that FX is twice differentiable. Then, (1)

If

o2 FX ox2i

ðx1 ; x2 Þ 0 for all x1 C x*1 and x2 C x*2, then the a-level curve

x1 7! VaRa;x1 ðXÞ is convex. (2)

If

o2 FX ox2i

ðx1 ; x2 Þ 0 for all x1 B x*1 and x2 B x*2, then the a-level curve

x1 7! VaRa;x1 ðXÞ is concave. Proof Using implicit differentiable calculus rules, one deduces that VaRa;x1 ðXÞ is also twice differentiable (because F is twice differentiable and FX ðx1 ; VaRa;x1 ðXÞÞ ¼ a). Now using the fact that d2 FX x1 ; VaRa;x1 ðXÞ =dx21 ¼ 0; then

d2 VaRa;x1 ðXÞ d FX x1 ; VaRa;x1 ðXÞ dx2 dx21 dVaRa;x1 ðXÞ d2 d2 ¼ 2 FX x1 ; VaRa;x1 ðXÞ þ 2 FX x1 ; VaRa;x1 ðXÞ ð2:11Þ dx1 dx1 dx2 dx1 dVaRa;x1 ðXÞ 2 d2 þ 2 FX x1 ; VaRa;x1 ðXÞ dx1 dx2

hence (1), because d 2 F X ðx 1 ; x2 Þ 0; dx1 dx2

d FX ðx1 ; x2 Þ 0; dx2

The statement (2) is obtained similarly.

dVaRa;x1 ðXÞ 0: dx1 h

Many bivariate distributions satisfy the criteria of Proposition 2.10, as for the bivariate Eynaud–Farlie–Gumbel–Morgenstein (EFGM) bivariate exponential distribution, as presented in Balakrishnan and Lai [1]. Example 2.12 Consider the bivariate EFGM exponential distribution, with parameters (b1 = 10, b2 = 15, h = 3). Figure 5 illustrates the curves of the bivariate lower and upper orthant VaR at level 95 %, and for h = –0.9 and

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θ=−0.9

80 70 60 50 ↑ VaRα(X2)

40 30

←VaRα(X1)

20 10 0

0

10

20

30

40

50

60

Fig. 5 Graphical representation of the lower and upper bounds with the bivariate FGM distribution, for h = -0.9 and h = 0.9

h = 0.9. One sees that the bivariate lower orthant VaR is convex for h [ 0 and for h \ 0. However, the bivariate upper orthant VaR is concave only when h [ 0, which is a desirable scenario to obtain optimized values, as we will see further in this chapter. The bivariate upper orthant VaR is convex when h \ 0. Moreover, Fig. 5 clearly shows that the bivariate upper orthant VaR is more affected by changes in the dependence parameter than the bivariate lower orthant VaR. h As shown in Example 2.12, when there is a positive dependence between the rv’s, the bivariate EFGM exponential distribution satisfies the two conditions of Proposition 2.11, in order to have convenient lower and upper orthant VaR curves. Note that the bivariate EFGM copula is generated from the bivariate EFGM exponential distribution. It is now well recognized that copulas provide a flexible approach to model the joint behavior of rv’s. In fact, they allow the representation of a multivariate distribution as a function of its univariate marginal cdf’s through a linking function called a copula. Let X ¼ ðX1 ; X2 Þ be a random vector with joint cdf FX and marginal cdf’s FX1 and FX2 . A well known theorem in Sklar [25] ensures that there exists a unique copula C : ½0; 12 ! ½0; 1 such that for all x1 ; x2 2 R FX ðx1 ; x2 Þ ¼ CðFX1 ðx1 Þ; FX2 ðx2 ÞÞ:

ð2:12Þ

As a consequence of Proposition 2.11, the shape of the a-level curves x1 7! VaRa;x1 ðXÞ and x1 7! VaRa;x1 ðXÞ may be studied in terms of copulas and marginal cdf’s as stated next.

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Corollary 2.13 Let X ¼ ðX1 ; X2 Þ be a random vector with joint cdf FX and marginal cdf’s FX1 and FX2 connected by a copula C. Suppose that the copula C and FXi , i = 1, 2 are twice differentiable. o2 C ðu1 ; u2 Þ 0 ou2i

for all u1 ; u2 2 ½a; 1; i ¼ 1; 2 and FX1 (x1) and FX2 (x2) are

(1)

If

(2)

concave for all x1 C x*1 and x2 C x*2, then the a-level curve x1 7! VaRa;x1 ðXÞ is convex. 2 If oouC2 ðu1 ; u2 Þ 0 for all u1 ; u2 2 ½0; a; i ¼ 1; 2 and FX1 (x1) and FX2 (x2) are i

convex for all x1 B x*1 and x2 B x*2, then the a-level curve x1 7! VaRa;x1 ðXÞ is concave. Proof has

The result is immediate from Sklar’s theorem stated in (2.12). In fact, one o2 F X o2 C ðx ; x Þ ¼ ðFX1 ðx1 Þ; FX2 ðx2 ÞÞðFX0 i ðxi ÞÞ2 1 2 oFX2 i ðxi Þ ox2i oC þ 2 ðFX1 ðx1 Þ; FX2 ðx2 ÞÞFX00i ðxi Þ; oFXi ðxi Þ

i ¼ 1; 2;

ð2:13Þ

so (1) holds. Similar arguments ensure (2). h Note that the assumption that F1(x1) and F2(x2) are concave for all x1 C x*1 and x2 C x*2 is fulfilled for many important univariate distributions in actuarial science and quantitative risk management, such as the exponential, Pareto and gamma distributions. Moreover, several multivariate distributions, as the EFGM bivariate exponential distribution, satisfy Corollary 2.13. An important class of copulas are the Archimedean copulas (e.g. Nelsen [23]). A copula C is archimedean with generator /, if for all u; v 2 ½0; 1 Cðu; vÞ ¼ /1 ð/ðuÞ þ /ðvÞÞ; where / : ½0; 1 ! Rþ is a continuous, possibly infinite, strictly decreasing convex function such that /(1) = 0. For this class of copulas, the convexity of the a-level curve x1 7! VaRa;x1 ðXÞ depends only of the behavior of the marginal cdf’s as stated below. Proposition 2.14 Let X ¼ ðX1 ; X2 Þ be a random vector with joint cdf FX and marginal cdf’s FX1 and FX2 connected by an archimedean copula C with generator /. If FX1 (x1) and FX2 (x2) are concave for all x1 C x*1 and x2 C x*2, then x1 7! VaRa;x1 ðXÞ is convex. Proof Let La(u) be the a-level curve of the copula C, that is, C(u, La(u)) = a. One can show that La(u) = /-1(/(a) - /(u)). Theorem 4.3.2 in Nelsen [23] states that for any archimedean copula, the a-level curve u 7! La ðuÞ is convex. Let (F La G)(x) = F(G(x)), be the composite function. Then, VaRa;x1 ðXÞ ¼ FX1 2 FX1 ðx1 Þ: If FX1 and FX2 are concave, then x1 7! VaRa;x1 ðXÞ is convex. h

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We refer the reader to Example 2.2 for an illustration of Proposition 2.14. We now investigate the concavity of the a-curve x1 7! VaRa;x1 ðXÞ in terms of the La FX1 ðx1 Þ where La ðuÞ is the generator /. Here also, we have VaRa;x1 ðXÞ ¼ FX1 2 a-level curve associated to the bivariate sf associated to the copula C, that is vÞ ¼ 1 u v þ Cðu; vÞ; so Cðu; La ðuÞÞ ¼ a; as mentioned in Remark 2.3. Cðu; Hereafter, we derive conditions that ensure the concavity of the upper orthant VaR. Proposition 2.15 Let X ¼ ðX1 ; X2 Þ be a random vector with joint cdf FX and marginal cdf’s FX1 and FX2 connected by an archimedean copula C with generator /. Suppose that /, FX1 and FX2 are twice differentiable. If FX1 and FX2 are convex for all x1 B x*1 and x2 B x*2, and t 7! /00 ðtÞ=ð/0 ðtÞÞ2 is increasing for t 2 ½0; a; then x1 7! VaRa;x1 ðXÞ is concave. Proof

We use (2) in Corollary 2.13 to prove this result. We have that ¼ 1; 2; if and only if /00 ðwÞ=ð/0 ðwÞÞ2 /00 ðui Þ=ð/0 ðui ÞÞ2 ; i ¼

o2 C ðu1 ; u2 Þ 0; i ou2i

1; 2; where w = /-1((/(u1) ? /(u2)) and w B min(u1, u2). It follows that if 2 t 7! /00 ðtÞ=ð/0 ðtÞÞ2 is increasing, then oouC2 ðu1 ; u2 Þ 0; i ¼ 1; 2; hence the desired i

result.

h

Refer to Example 2.4 for an illustration of Proposition 2.15. Note that it is easy to verify the monotonicity of the function t 7! /00 ðtÞ=ð/0 ðtÞÞ2 for copulas that are members of families with real parameters. For example, for Clayton’s family with generator /(t) = (th - 1)/h, one has /00 ðtÞ=ð/0 ðtÞÞ2 ¼ ðh þ 1Þth ; so that t 7! /00 ðtÞ=ð/0 ðtÞÞ2 is increasing if and only if h C 0. Also, for Frank’s family with generator /ðtÞ ¼ lnðeh 1Þ lnðeth 1Þ; one has /00 ðtÞ=ð/0 ðtÞÞ2 ¼ eht ; which is increasing if and only if h C 0. For these families of copulas, h C 0 generates a positive dependence.

2.5 Impact of dependence and marginals In this section, we study the effect of the dependence level and the marginal cdf’s on the bivariate upper and lower orthant VaR. Explicit bounds on these risk measures are also obtained. In the following, we introduce stochastic ordering in order to compare bivariate lower orthant VaR’s (respectively upper orthant VaR’s). The latter is based on the confidence regions, presented in Sect. 2.3. Definition 2.16 Let X1 = (X1,1, X2,1) and X2 = (X1,2, X2,2) be two pairs of risks with joint cdf’s FX1 and FX2 ; respectively. Then, VaRa ðX1 Þ is smaller than VaRa ðX2 Þ; denoted VaRa ðX1 Þ VaRa ðX2 Þ; if CFX2 ;a CFX1 ;a (equivalently VaRa;x1 ðX1 Þ VaRa;x1 ðX2 Þ for all x1). Similarly, VaRa ðX1 Þ is smaller than VaRa ðX2 Þ; denoted VaRa ðX1 Þ VaRa ðX2 Þ; if CFX2 ;a CFX1 ;a (or equivalently VaRa;x1 ðX1 Þ VaRa;x1 ðX2 Þ for all x1).

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Given two random vectors X1 = (X1,1, X2,1) and X2 = (X1,2, X2,2) with joint cdf’s FX1 and FX2 respectively, X1 is said to be more concordant than X2, denoted X1 co X2 ; if FX1 ðx1 ; x2 Þ FX2 ðx1 ; x2 Þ holds for all x1 ; x2 2 R: It is easy to see that if X1 co X2 ; then CFX1 ;a CFX2 ;a and CFX2 ;a CFX1 ;a : Moreover, if CX1 and CX2 denote the copulas of X1 and X2 respectively, then X1 co X2 if and only if CX1 ðu; vÞ CX2 ðu; vÞ for all u; v 2 ½0; 1: We define the Fre´chet class, denoted by CðFX1 ; FX2 Þ; as the set of joint cdf’s FX1 with fixed marginals FX 1 and FX 2 . We also denote by M(x1, x2) = min(FX1 (x1), FX2 (x2)) and W(x1, x2) = max(FX1 (x1) ? FX2 (x2) - 1, 0) the Frchet upper and lower bounds respectively. It is well known that Wðx1 ; x2 Þ FX ðx1 ; x2 Þ Mðx1 ; x2 Þ; for all F 2 CðFX1 ; FX2 Þ and x1 ; x2 2 R: The following result follows from the definition of the concordance ordering. It shows the impact when the dependence structures within the vectors X1 and X2 are different, but they have the same componentwise marginal cdf’s. Lemma 2.17 (Impact of dependence) Let X1 = (X1,1, X2,1) and X2 = (X1,2, X2,2) be two pairs of risks with joint cdf’s FX1 2 CðFX1 ; FX2 Þ and FX2 2 CðFX1 ; FX2 Þ; respectively. Then, we have X1 co X2

)

VaRa ðX2 Þ VaRa ðX1 Þ

for all a 2 ½0; 1;

ð2:14Þ

X1 co X2

)

VaRa ðX1 Þ VaRa ðX2 Þ

for all a 2 ½0; 1:

ð2:15Þ

Let us also discuss the effect of the marginal cdf’s on VaRa ðXÞ and VaRa ðXÞ when the dependence between the components of X is fixed. Lemma 2.18 (Impact of marginals) Let X1 = (X11, X21) and X2 = (X12, X22) be continuous random vectors with the same copula C, within X1 and X2. Also, consider the respective joint cdf’s FX1 2 CðFX1 ; FX2 Þ and FX2 2 CðGX1 ; GX2 Þ: Then, for fixed a 2 ð0; 1Þ; we have VaRa ðXi;1 Þ VaRa ðXi;2 Þ;

i ¼ 1; 2 , VaRa ðX1 Þ VaRa ðX2 Þ;

ð2:16Þ

VaRa ðXi;1 Þ VaRa ðXi;2 Þ;

i ¼ 1; 2 , VaRa ðX1 Þ VaRa ðX2 Þ:

ð2:17Þ

Proof To verify ()) in (2.16) and (2.17), we use the fact that if VaRa(X i,1) B VaRa(Xi,2), i = 1, 2, then C FX1 ;a C FX2 ;a and C FX1 ;a C FX2 ;a : To show (() in (2.16), let VaR1 x2 ;a ðXi Þ be the inverse function of x1 ! VaRx1 ;a ðXi Þ: One sees that lim VaRx1 ;a ðX1 Þ ¼ VaRa ðX1;1 Þ;

x1 !1

lim VaR1 x2 ;a ðX1 Þ ¼ VaRa ðX2;1 Þ;

x2 !1

VaRa ðX1 Þ VaRa ðX2 Þ 1 VaRx2 ;a ðX1 Þ VaR1 Thus, x2 ;a ðX2 Þ: Now,

lim VaRx1 ;a ðX2 Þ ¼ VaRa ðX1;2 Þ;

ð2:18Þ

lim VaR1 x2 ;a ðX2 Þ ¼ VaRa ðX2;2 Þ:

ð2:19Þ

x1 !1 x2 !1

implies from

VaRx1 ;a ðX1 Þ VaRx1 ;a ðX2 Þ (2.18)

and

(2.19),

we

and have

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VaRa(Xi1) B VaRa(Xi2), i = 1, 2, hence the result. The implication (() in (2.17) can be obtained similarly. h Lemma 2.18 considers a fixed dependence structure C between the components of the random sets, with fixed joint cdf FXi, i = 1, 2 within each random set. Based on the latter assumption, we established the order of the bivariate lower and upper orthant VaR of X1 and X2, based on the univariate VaR’s of the individual components of each random set, vice versa. Let X = (X1, X2) be a random vector with joint cdf FX and marginal cdf’s FX1 and FX2 . As a consequence of Lemma 2.17, one obtains VaRa ðXM Þ VaRa ðXÞ VaRa ðXW Þ;

VaRa ðXW Þ VaRa ðXÞ VaRa ðXM Þ: ð2:20Þ

An interesting property of positive dependence is the concept of positive quadrant dependence (PQD) introduced by Lehmann [18]. The random vector X = (X1, X2) is said to be positively quadrant dependent if and only if FX ðx1 ; x2 Þ Pðx1 ; x2 Þ ¼ FX1 ðx1 ÞFX2 ðx2 Þ for all x1 ; x2 2 R: In such a situation and using Lemma 2.18, inequalities in (2.20) become VaRa ðXM Þ VaRa ðXÞ VaRa ðXP Þ;

VaRa ðXP Þ VaRa ðXÞ VaRa ðXM Þ;

where XP is the random vector with the same marginal cdf’s than X, but with independent components.

3 Lower and upper orthant value-at-risk for sums of random pairs In this section, we motivate the use of bivariate VaR’s to obtain accurate values for risk allocation and comparison, in the case of sum of random pairs. Models should try to capture important characteristics such as the marginal cdf’s of homogeneous classes and the appropriate dependence structure between classes. Using bivariate VaR’s allows to consider each homogeneous structure of a dependent set during the modeling process. We initiate this section by setting the framework. Let X1 ¼ ðX1;1 ; X1;2 Þ; . . .; Xn ¼ ðXn;1 ; Xn;2 Þ be a sequence of n random pairs with distributions FX1 ; . . .; FXn and marginal cdf’s FX1;1 ; . . .; FXn;1 and FX1;2 ; . . .; FXn;2 : Denote S1 ¼ X1;1 þ þ Xn;1 and S2 ¼ X1;2 þ þ Xn;2 : In this section, we examine the lower and upper orthant VaR of the random vector. We have S ¼ X1 þ þ Xn ; where

S1 X11 ... Xn1 S¼ ¼ þ þ ; S2 X12 Xn2 ... and where X1 ; . . .; Xn are linked by a copula C. The computation of the joint cdf of S is not obvious even for specific dependence structures assumed for the random vectors X1 ; . . .; Xn : Therefore, it is not easy to evaluate the univariate VaR for sums of random variables. The problem is even more challenging when considering bivariate sums of random pairs.

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In Sect. 3.1 we study the sum of random vectors with comonotonic components. In Sect. 3.2, we develop bounds on the bivariate lower and upper orthant VaR for pairs representing random sums, in terms of the univariate VaR. We also establish the relation with existing stochastic bounds. In Sect. 3.3, we provide bounds for the sum of a bivariate set, where each component represents aggregated homogeneous risks. 3.1 Sum of random vectors with comonotonic components The objective of this subsection is to provide the bivariate version of Pn Pn VaRa i¼1 Xi ¼ i¼1 VaRa ðXi Þ; when X1 ; . . .; Xn are comonotonic, that is if Xi ¼ 1 FXi ðUÞ; i ¼ 1; . . .; n; where U is uniformly distributed on the interval [0, 1]. Hereafter, we have an analogous result in a bivariate setting. Proposition 3.1 Let X1 ¼ ðX1;1 ; X1;2 Þ; . . .; X n ¼ ðXn;1 ; Xn;2 Þ be a sequence of random pairs with distributions FX1 ; . . .; FXn and marginal cdf’s FX1;1 ; . . .; FXn;1 and FX1;2 ; . . .; FXn;2 : Suppose that X1;1 ; . . .; Xn;1 and X1;2 ; . . .; Xn;2 are comonotonic respectively, meaning that there exists uniform rv’s U1 and U2 such that ðU2 Þ; i ¼ 1; . . .; n: Suppose that U1 and U2 are Xi;1 ¼ FX1i;1 ðU1 Þand Xi;2 ¼ FX1 i;2 connected with a copula C. Then, n X VaRa;sj ðSÞ ¼ VaRa;xk;j ðXk Þ; sj VaRa ðSj Þ; ð3:1Þ k¼1

VaRa;sj ðSÞ ¼

n X

VaRa;xk;j ðXk Þ;

sj VaRa ðSj Þ;

ð3:2Þ

k¼1

P

n

Pn

k=1

xk,j = sj =

1

k=1Fxk;j

FSj (sj) and FS1 (u) = j

Pn

FX1 (u), j = 1, 2. k;j P n ðU1 Þ; FS1 ðU2 ÞÞ and Proof Using the fact that S ¼ ðS1 ; S2 Þ ¼ ðFS1 k=1 1 2 Pn 1 xk,j = sj = k=1FXk;j FSj (sj), then from Proposition 2.5, we have for i, j = 1, 2 and i = j, where

k=1

VaRa;sj ðSÞ ¼ VaRa;sj ðFS1 ðU1 Þ; FS1 ðU2 ÞÞ 1 2 1 ¼ FSi VaRa;FSj ðsj Þ ðU1 ; U2 Þ n X ¼ FX1k;i VaRa;FSj ðsj Þ ðU1 ; U2 Þ k¼1

¼ ¼ ¼

n X k¼1 n X k¼1 n X

FX1k;i

1 1 VaRa;FX1 ðFSj ðsj ÞÞ ðFXk;1 ðU1 Þ; FXk;2 ðU2 ÞÞ k;j

VaRa;xk;j ðFX1k;1 ðU1 Þ; FX1k;2 ðU2 ÞÞ VaRa;xk;j ðXk Þ:

k¼1

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Hence, one obtain (3.1). Similar arguments lead to (3.2). h This interesting proposition shows that the relation for the univariate VaR of a sum of comonotonic rv’s also holds for the bivariate lower and upper orthant VaR of a sum of comonotonic random couples. 3.2 Bounds on the bivariate lower and upper orthant value-at-risk for sums of random pairs Now, let us examine bounds on VaRa;si ðSÞ and VaRa;si ðSÞ; i ¼ 1; 2: We link univariate and multivariate results using the bivariate lower and upper orthant VaR and stochastic bounds for FSi , i = 1, 2. It allows to take into account homogeneous groups of risks, part of a global dependent set. One can easily derive the following bounds on the a-level curves si 7! VaRa;si ðSÞ and si 7! VaRa;si ðSÞ in terms of VaRa(Si) and FSi : VaRa ðSj Þ VaRa;si ðSÞ VaRaFSi ðsi Þþ1 ðSj Þ; VaRaFSi ðsi Þ ðSj Þ VaRa;si ðSÞ VaRa ðSj Þ;

si VaRa ðSi Þ; si VaRa ðSi Þ;

ð3:3Þ ð3:4Þ

for i, j = 1, 2, i = j. It is not easy to evaluate explicitly VaRa(Si) and FSi , i = 1, 2. However, several authors have examined explicit formulas and the estimation of these quantities by deriving stochastic bounds on the distribution of Si, i = 1, 2. Makarov [20] and independently Rschendorf (1982) obtained stochastic bounds on FSi , i = 1, 2. Williamson and Downs [26] also studied stochastic bounds and extended previous results, using the duality principle. Denuit et al. [7] applied the stochastic bounds in actuarial science, applying their results to insurance problems. Embrechts and Puccetti [11] uses stochastic bounds on FSi , i = 1, 2 to improve the results obtained in Embrechts et al. [9]. To recall these results, let Ci be the copula associated to the rv’s X1;i ; . . .; Xn;i ; i ¼ 1; 2: Denote by Cdi the dual of Ci, i = 1, 2 defined by ! n [ Cid ðu1;i ; . . .; un;i Þ ¼ P Uj;i uj;i ; i ¼ 1; 2; j¼1

where ðU1;i ; . . .; Un;i Þ denotes a random vector with distribution Ci, i = 1, 2. It will be supposed, however, that partial information is available about Ci, namely that there are copulas Ci,L and Ci,U such that Ci C Ci,L and Cdi B Cdi,U, i = 1, 2. Any multivariate distribution function can be represented in a way that emphasizes the separate roles of the marginal cdf’s and the dependence structure. For all s 2 R; Fmin;Si ðsÞ FSi ðsÞ Fmax;Si ðsÞ such that Fmin;Si ðsÞ ¼ sup Ci;L F1;i ðu1 Þ; . . .; Fn;i ðun Þ ; i ¼ 1; 2; ð3:5Þ u1 þþun ¼s

Fmax;Si ðsÞ ¼

inf

u1 þþun ¼s

d Ci;U F1;i ðu1 Þ; . . .; Fn;i ðun Þ ;

i ¼ 1; 2:

ð3:6Þ

Williamson and Downs [26] presented bounds for the VaR of the sum of two risks

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using the duality principle. The n-dimensional formulation of this result is stated formally by VaRmin;a ðSi Þ ¼

sup d ðu ;...;u Þ¼a Ci;U 1 n

VaRmax;a ðSi Þ ¼

inf

Ci;L ðu1 ;...;un Þ¼a

n X

1 Fj;i ðuj Þ;

i ¼ 1; 2;

ð3:7Þ

1 Fj;i ðuj Þ;

i ¼ 1; 2:

ð3:8Þ

j¼1 n X j¼1

This is in fact a special case of Theorem 3.1 of Embrechts et al. [9], where the VaR of a function wðx1 ; . . .; xn Þ of n-dependent risks was treated, applying the duality principle of Frank and Schweizer [12]. In practical situations, the dependence structure of ðX1i ; . . .; Xni Þ; i ¼ 1; 2 is often ~ d hold, unknown. However, for any copula, the inequalities Ci C W and Cid W where Wðu1 ; . . .; un Þ ¼ minðu1 þ þ un 1; 0Þ ~ d ðu1 ; . . .; un Þ ¼ minð1; u1 þ þ un Þ: W

and

~ d and W respecPractical bounds can be obtained by replacing Cdi,U and Ci,L by W tively in (3.7) and (3.8). Proposition 3.2 We obtain bounds for the lower and upper orthant VaR, using bounds on the univariate VaR, that is VaRmin;a ðS2 Þ VaRa;s1 ðSÞ VaRmax;aFmin;S1 ðs1 Þþ1 ðS2 Þ;

1 s1 Fmin;S ðaÞ; 1

ð3:9Þ

and VaRmin;aFmax;S1 ðs1 Þ ðS2 Þ VaRa;s1 ðSÞ VaRmax;a ðS2 Þ; Proof

1 s1 Fmax;S ðaÞ: 1

ð3:10Þ

Since a 7! VaRa ðS2 Þ is increasing, then VaRaFS1 ðs1 Þþ1 ðS2 Þ VaRaFmin;S1 ðs1 Þþ1 ðS2 Þ VaRmax;aFmin;S1 ðs1 Þþ1 ðS2 Þ

Moreover, one has VaRmin;a ðS2 Þ VaRa ðS2 Þ: These informations together with (3.3) imply (3.9). The inequalities (3.10) may be obtained analogously. h 3.3 Bounds on the sum of aggregated risks In this subsection, we consider a univariate framework. We establish bounds on the sum of two dependent classes of homogeneous aggregated risks. This setting differs from Sects. 3.1 and 3.2, since we obtain one-dimensional results that are related to the sum of all risks. Moreover, we can now deal with different lengths of aggregated risks. Our approach provides a simple way of bounding the sum of hypothetically homogeneous classes of risks, based on the bivariate lower and upper orthant VaR.

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Also, we provide results when the dependence structures between and within the classes of risks are known or not. Consider a portfolio divided into two classes comprising n1 and n2 contracts, and let Xi,j represent the risk associated to the ith contract in the jth class, j = 1, 2. In many situations, it is convenient to model separately the distribution of each random vector X1 ¼ ðX1;1 ; . . .; Xn1 ;1 Þ and X2 ¼ ðX1;2 ; . . .; Xn2 ;2 Þ instead of the distribution of the random vector X ¼ ðX1;1 ; . . .; Xn1 ;1 ; X1;2 ; . . .; Xn2 ;2 Þ: This is because the classes are often homogeneous, and it can be easier to identify the structure of each vectors X1 and X2 instead of the structure of the random vector X. In such a case, the lower and upper orthant VaR can be used to derive bounds on VaR(S1 ? S2), for example, instead of modeling the aggregate distribution of all the risks. Corollary 3.3 ( max

s1 \FS1 ðaÞ

When the structure of dependence of (S1, S2) is known, one obtains )

s1 þ VaRa;s1 ðSÞg VaRa ðS1 þ S2 Þ

1

min fs1 þ VaRa;s1 ðSÞ :

s1 [ FS1 ðaÞ 1

ð3:11Þ When the structure of the dependence of the random vector (S1, S2) is unknown, one can use Proposition 3.2 to derive bounds on VaRa(S1 ? S2) that are expressed in terms of the bounds on the cdf’s of S1 and S2 as shown next: max Amin a VaRa ðS1 þ S2 Þ Aa ;

where Amin ¼ a

max fs1 þ FS1 ða FS1 ðs1 ÞÞg 2

s1 \FS1 ðaÞ

and

1

Amax ¼ a

min fs1 þ FS1 ða FS1 ðs1 Þ þ 1Þg: 2

s1 [ FS1 ðaÞ 1

FS1 , i

Note that if FSi and i = 1, 2 are not available, then one can substitute them by their bounds Fmin;S1 ; Fmax;S1 , and Fmin;S1 ; Fmax;S1 , respectively. Hence, one obtains max Dmin a VaRa ðS1 þ S2 Þ Da ;

where Dmin ¼ a

n

max

1 s1 \Fmin;S ðaÞ

1 s1 þ Fmin;S ða Fmax;S1 ðs1 ÞÞ 2

o

1

and Dmax ¼ a

n

min

1 s1 [ Fmin;S ðaÞ

o 1 s1 þ Fmax;S ða F ðs Þ þ 1Þ : min;S 1 1 2

1

We want to highlight the fact that using our approach to provide bounds on the lower and upper orthant VaR gives the opportunity to consider the dependence within different sectors X1 and X2, and also to consider the model that represents the dependence between sectors. Traditional methods consider a different dependence

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structure, that is possibly hard to fit, because they consider heterogeneous variables, which is not the case in Eqs. (3.7) and (3.8). This method leaves aside the dependence within and between X1 and X2. For illustration, let us consider the case where X1;i ; . . .; Xn;i are identically distributed with common cdf FXi ; i ¼ 1; 2: Suppose that there exists x i 2 R such that the density function fXi ðxÞ ¼ dFXi ðxÞ=dx is non-increasing for all x x i ; i ¼ 1; 2: This assumption is fulfilled for many important models in actuarial science and quantitative risk management like exponential, Pareto and gamma models. Suppose also that X1;i ; . . .; Xn;i ; i ¼ 1; 2 are positively lower orthant dependent (PLOD). Then, from Remark 3.2 in Mesfioui and Quessy [22] we have ð3:12Þ VaRmax;a ðS2 Þ ¼ nFX12 a1=n ; Fmin;S1 ðs1 Þ ¼ ½FX1 ðs1 =nÞn : Combining (3.9) and (3.12), we get the next explicit upper bound of VaRa;s1 ðSÞ h i 1=n VaRa;s1 ðSÞ nFX12 ða ½FX1 ðs1 =nÞn þ1Þ ð3:13Þ ; s1 nFX11 ða1=n Þ: Example 3.4 In order to appreciate the influence of the dimension n on the upper bound given in (3.13), consider FX1 Expðk1 Þ; that is, FX1 ðxÞ ¼ 1 ek1 x ; x [ 0 and FX2 * Pareto(a), namely, FX2 ðxÞ ¼ 1 xa ; x [ 1 and a [ 0. Figure 6 provides the a-curves of this upper bound, with a ¼ 0:95; k ¼ 0:2 and a = 1.5, for n = 2, n = 3 and n = 4, respectively, and n1 = n2 = n. We remark that increasing the dimension n increases the proposed upper bound with respect to the order : h

400 n=2 n=3 n=4

Lower α−curve values

350 300 250 200 150 100 50 0

0

10

20

30

40

50

60

70

80

90

100

s1

Fig. 6 Graphical representation of the upper bounds of the lower orthant VaR Graphical representation of the upper bounds of VaRa;s1 (S) for n = 2, 3, 4

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4 Applications 4.1 Bivariate value-at-risk and allocation In this section, we suggest methods based on the bivariate lower and upper orthant VaR to obtain optimal capital allocation sets. Two optimization criteria are developed to select a bivariate set of values from these curves. The objective is to allocate a value to each homogeneous risk, that could be used for comparison or for capital requirements of a company with several business lines. These criteria are developed mainly to fulfil practical needs. The bivariate lower and upper orthant VaR curves are useful for risk comparison, but in several situations, companies or regulators need to allocate a single amount to each business line or risk of a portfolio. Therefore, a set has to be selected from the bivariate VaR. As shown in Sect. 2.4, we must have a convex lower orthant VaR and a concave upper orthant VaR, and when this condition is respected, one can obtain a bivariate set of values from the curves, based on two different approaches. Using the bivariate lower orthant VaR, the allocation couple has to be such that the probability that X1 is smaller than x1 and X2 is smaller than x2 equals a: To find an allocation couple from the curve ðx1 ; VaRa;x1 ðXÞÞ; we propose two different criteria of interest in finance, actuarial science and quantitative risk management. Analogous results can be obtained from the curve ðx1 ; VaRa;x1 ðXÞÞ: 4.1.1 Orthogonal projection We start by the Orthogonal projection allocation, which consists in finding the closest point from ðx 1 ; VaRa;x 1 ðXÞÞ to the couple (VaRa(X1), VaRa(X2)), by solving the following minimization problem, n 2 o ðVaRa ðX1 Þ x1 Þ2 þ VaRa ðX2 Þ VaRa;x1 ðXÞ min : x1 [ F11 ðaÞ

To find the solution, we have to solve dVaRa;x1 ðXÞ ¼ 0; dx1

ð4:1Þ

d2 VaRa;x1 ðXÞ dVaRa;x1 ðXÞ 2 þ2 2 þ 2 VaRa;x1 ðXÞ VaRa ðX2 Þ dx1 dx21

ð4:2Þ

2ðx1 VaRa ðX1 ÞÞ þ 2ðVaRa;x1 ðXÞ VaRa ðX2 ÞÞ and verify that the second derivative

is positive, in order to get a minimum. The convexity of VaRa;x1 ðXÞ plays a central role to obtain the optimal solution. In fact, we see that the last expression in (4.2) is positive if VaRa;x1 ðXÞ is convex. In this situation, Eq. (4.1) will provide the orthogonal projection from the couple ðVaRa ðX1 Þ; VaRa ðX2 ÞÞ to ðx 1 ; VaRa;x 1 ðXÞÞ: An intuitive interpretation of this minimization is to calculate the simultaneous set for which each rv reaches the closest values to the one that would be obtained on a

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stand-alone basis, with the strongest dependence level and the possibility of risk mitigation. This interpretation allows the user to quantify the impact of protecting each homogeneous risk, without the possibility of aggregation.

4.1.2 Proportional allocation The second approach is the Proportional allocation. The idea of this method is to preserve the same ratio of the univariate VaR, that is to consider ðx 1 ; VaRa;x 1 ðXÞÞ solution of

2 VaRa ðX1 Þ min x1 VaRa;x1 ðXÞ : VaRa ðX2 Þ x1 [ F11 ðaÞ Again, if VaRa;x1 ðXÞ is convex, the minimum is obtained by solving the equation d VaRa;x1 ðXÞ ¼ VaRa ðX2 Þ=VaRa ðX1 Þ: dx1 Explicit forms for inverse bivariate cdf’s do not exist for most cases. Copulas are more tractable for that purpose, as shown in an example in the last section. The intuitive interpretation of this minimization is to calculate the set that allocates the same proportion to each risk as if they were comonotonic and could be mitigated. The latter framework represents the strongest dependence level between risks that are aggregated, which is often considered for capital requirement purposes. If no closed-form expression exists, bounds can be found and optimization methods can be used, as shown in the next example. Example 4.1 Consider the random couple (X1, X2), following a bivariate mixture of Erlang distributions with the same scale parameter h. The joint pdf is fX1 ;X2 ðx1 ; x2 j sm ; hÞ ¼

1 X 1 X

sm1 ;m2

m1 ¼1 m2 ¼1

2 Y

hðxj ; mj ; hÞ;

j¼1

with

sm ¼

s1;1 s2;1

s1;2 s2;2

s1;3 s2;3

¼

0:2

0:1

0

0:4

0

0:3

;

and sm1 m2 = 0 for m1 = 3, 4, … and m2 = 4, 5, …. Also, hð; a; bÞ represents the pdf of a gamma distribution with shape and scale parameters a and b respectively. We obtain the following results for the allocation couples resulting from the two allocation criteria previously presented. Since VaRa(X1) [ VaRa(X2), the allocation to X1 is always higher in the Proportional allocation couples. The Orthogonal projection allocation provides the closest couple from (VaRa(X1), VaRa(X2)), resulting in a smaller total than with the Proportional allocation, but not preserving the proportion of each risk on the aggregate risks. In Table 1, we see that the sum of the components of the orthogonal

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Table 1 Couples resulting from the Orthogonal projection and Proportional allocation criteria a

Orthogonal projection

Total

Proportional

Total

0.95

(5.2762, 5.7244)

11.0006

(5.2252, 5.7788)

11.0040

9.2699

0.99

(7.1314, 7.8442)

14.9756

(7.0794, 7.8994)

14.9788

13.3138

0.995

(7.9071, 8.7167)

16.6238

(7.8581, 8.7687)

16.6268

14.9863

VaRa (X1 ? X2)

projection and proportional allocation couples are higher than the VaR of the sum of the components. This is because each component is always protected to the level a, without considering the value of the remaining rv. h Note that x 1 þ VaRa;x 1 ðXÞ VaRa ðX1 Þ þ VaRa ðX2 Þ and x 1 þ VaRa;x 1 ðXÞ VaRa ðX1 Þ þ VaRa ðX2 Þ: This is because the protection level a is dedicated to both risk, without embedding any possibility of risk mitigation. The minimal value of x 1 þ VaRa;x 1 ðXÞ and x 1 þ VaRa;x 1 ðXÞ are obtained in the situation where X1 and X2 are comonotonic. Then, as the dependence gets less positive between the rv’s, the bivariate lower orthant curve would be higher than in the comonotic scenario, and it means that if X1 takes a smaller value, X2 should take higher values with a higher probability, and a higher amount would be necessary to cover both risks at the same level a. 4.2 Bivariate value-at-risk and Ruin probabilites for a portfolio of bivariate risks We consider a portfolio with two lines of business. The aggregate claim amounts for the next period (e.g. a month, three months or a year) for the line i is defined by the rv Si and the corresponding premium income is denoted by pi, i = 1, 2. We assume that pi ¼ ð1 þ gi ÞE½Si ; i ¼ 1; 2: The initial reserve allocated to the line i is denoted by ui, i = 1, 2. Inspired from Chan et al. [4] and Cai and Li [2, 3], we define the two following ruin probabilities of the next periods ! 2 [ wor ðu1 ; u2 Þ ¼ Pr f S i pi [ u i g i¼1

and wand ðu1 ; u2 Þ ¼ Pr

2 \

! f S i pi [ u i g :

i¼1

We can relate wor ðu1 ; u2 Þ and wand ðu1 ; u2 Þ to VaRa ðSÞ and VaRa ðSÞ; where S ¼ðS1 ; S2 Þ: We fix a given value a 2 ð0; 1Þ and assume that the premium rates p1 and p2 are fixed such that pi \VaRa ðSi Þ; i ¼ 1; 2: Using the representations in (2.1) ðor Þ

ðor Þ

and (2.3) we can express for the given value a, the set of couples u1 ; u2

such

that wor ðu1 ; u2 Þ ¼ 1 a coincides with the curve of VaRa ðS1 p1 ; S2 p2 Þ: It means that either S1 or S2 stay smaller than their respective (1 - a)% largest value, without considering the other rv’s value, but consideringthat its valuemight affect the rv of interest. Similarly, the set of couples

123

ðandÞ

u1

ðand Þ

; u2

such that


345

wand ðu1 ; u2 Þ ¼ a coincides with the curve of VaRa ðS1 p1 ; S2 p2 Þ: This means that both S1 and S2 do not exceed the level a. For example, at a level of 99 %, if S1 reaches its 99 % greatest value but S2 is smaller than its 99 % greatest value, wor is respected, but not wand. Since there are more scenarios where either S1 or S2 can ðor Þ

ðor Þ

will always provide reach their level a, using wand, the set of couples u1 ; u2 ðandÞ ðand Þ higher values than u1 ; u2 ; as mentioned in Remark 2.3.

In Sect. 2.5, we have demonstrated the impact of the dependence and the marginal cdf’s on the bivariate also true for the values VaR’s. This is therefore ðor Þ ðor Þ ðand Þ ðandÞ and u1 ; u2 : within the sets of u1 ; u2 We illustrate this situation in the following example, using a bivariate compound Poisson model. We assume ðS1 ; S2 Þ to be a vector of rv’s following a bivariate compound distribution where (P Mi Mi [ 0 ji ¼0 Bi;ji ; ; ð4:3Þ Si ¼ 0; Mi ¼ 0 where the joint probability mass function (pmf) of ðM1 ; M2 Þ is given by fM1 ;M2 ðm1 ; m2 Þ ¼ PrðM1 ¼ m1 ; M2 ¼ m2 Þ ¼ qm1 ;m2 ; for m1 ; m2 2 N: Also, for each i; Bi;1 ; Bi;2 ; . . .; form a sequence of independent and rv’s, and Bi,1, Bi,2, …* Bi, i = 1, 2. We have that identically distributed B1;1 ; B1;2 ; . . .; ; B2;1 ; B2;2 ; . . .; are independent between themselves and independent of ðM1 ; M2 Þ: The rv’s B1, B2 are assumed to be continuous and strictly positive. We assume that ðM1 ; M2 Þ follows a multivariate Poisson distribution which is defined with a common shock as explained in e.g. Johnson et al. [15], Lindskog and McNeil [19]. We briefly recall the definition of this distribution. Let J0, J1, J2 be independent rv’s with J0 Poisðc0 Þ and Ji Poisðci ¼ ki c0 Þ with 0 c0 minðk1 ; k2 Þ: The components of the random vector ðM1 ; M2 Þ are defined by M1 ¼ J1 þ J0 ; M2 ¼ J2 þ J0 : It implies that Mi Poisðki Þ; i ¼ 1; 2: Also, the joint pmf and the joint probability generating function of ðM1 ; M2 Þ are respectively given by qm1 ;m2 ¼

minX ðm1 ;m2 Þ j¼0

ec0

2 c0j Y ðki c0 Þðmi jÞ ; eðki c0 Þ j! i¼1 ðmi jÞ!

2 Y 1 M2 c0 ðs1 s2 1Þ ¼ e PM1 ;M2 ðs1 ; s2 Þ ¼ E sM s eðki c0 Þðsi 1Þ ; 1 2

ð4:4Þ

i¼1

where c0 corresponds to the dependence parameter. When c0 = 0, it means that the components of ðM1 ; M2 Þ are independent.

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Since ðM1 ; M2 Þ follows a bivariate Poisson distribution, it implies that ðS1 ; S2 Þ has a bivariate compound Poisson distribution. We also assume that Bi gammaðsi ; bi Þ with E½Bi ¼ bsi ; i ¼ 1; 2: Then the joint cdf of ðS1 ; S2 Þ is given i by 1 X 1 X qm1 ;m2 H ðx1 ; s1 ; b1 ÞH ðx2 ; s2 ; b2 Þ: FS1 ;S2 ðx1 ; x2 Þ ¼ m1 ¼0 m2 ¼0

Using the previous model, we illustrate the bivariate VaR’s in a context of ruin probabilities for a portfolio of bivariate risks. Example 4.2 Let us fix k1 = 2, k2 = 3, s1 = s2 = 1, b1 = b2 = 0.6, g1 = g2 = 25 %. It implies that p1 ¼ p2 ¼ 1:25 E½Si : We obtain the following values Tables 2 and 3 both illustrate Lemma 2.17. The allocation values obtained from the bivariate lower and upper orthant VaR are always smaller for S1 than for S2, because the marginal cdf of S2 that is always smaller than the marginal cdf of S1. Note that this is the same result as for the univariate VaR. Also, one sees that the sum of the components of the allocation couples is always smaller from the orthogonal projection than for the proportional allocation. Moreover, as the dependence parameter increases, the bivariate lower orthant VaR decreases and the bivariate upper orthant increases (Tables 2, 3).

Table 2 Couples

ðorÞ ðorÞ u1 ; u2 resulting from the Orthogonal projection and Proportional allocation

criteria for wor ðu1 ; u2 Þ ¼ 1 %: c


Total

Proportional

Total

0

(18.3767, 23.9027)

42.2794

(18.7214, 23.5904)

42.3118

1

(18.3547, 23.8858)

42.2405

(18.7045, 23.5693)

42.2738

2

(18.3221, 23.8442)

42.1663

(18.6723, 23.5293)

42.2016

ðandÞ ðandÞ resulting from the Orthogonal projection and Proportional allocation Table 3 Couples u1 ; u2 criteria for wand ðu1 ; u2 Þ ¼ 1 %: c


0

(9.2946, 15.7539)

25.0785

(10.2339, 14.4112)

24.6451

1

(10.4030, 16.3737)

26.7767

(11.1460, 15.5437)

26.6897

2

(11.3596, 17.1804)

28.5400

(11.9447, 16.5355)

28.4802

123

Total

Proportional

Total


347

Table 2 having higher values than Table 3 show that to make sure that both risks do not exceed their 99 % respective worst possible values, without restriction on the values of the remaining risk, more has to be kept aside than to make sure that the worst 1 % is not reached by any of the two risks. h 4.3 Two lines of business with dependent risks Now, we consider an insurance company with two lines of business (i = 1, 2). The two vectors of rv’s (risks) associated to the lines 1 and 2 are 0 1 0 1 X1;1 X2;1 BX C BX C B 1;2 C B 2;2 C X1 ¼ B C and X2 ¼ B C; @ ... A @ ... A X1;n

X2;n

Line1

Line2

where Xi,j corresponds to the cost associated to the policy j in line i, j = 1, 2, …, n and i = 1, 2. Such a framework is suitable for insureds in non-life insurance companies exposed to the same pair of risks (e.g. net costs and loss adjustment expenses) or same pair of coverages (e.g. car and property insurance). Another context is in life insurance where insureds have invested in the same two financial products. We assume that the components within Xi are dependent and that X1 and X2 are also dependent. P Define the random pair S ¼ ðS1 ; S2 Þ where Si = nj=1Xi,j corresponds to the aggregate claim amount for the whole line i, i = 1, 2. Also, Y = S1 ? S2 is the aggregate claim amount for the whole portfolio. In order to examine this problem, we use a risk model based on multivariate mixed Erlang. For more information on the multivariate mixed Erlang, see Lee and Lin [17]. Let H be a positive mixing rv with cdf FH and mgf MH : Given H ¼ h; we assume FX1;1 ;...;X1;n ;X2;1 ;...;X2;n jH¼h x1;1 ; . . .; x1;n ; x2;1 ; . . .; x2;n m X m n Y X ðhÞ p1;2 ðl1 ; l2 Þ H x1;j ; l1 ; b1 H x2;j ; l2 ; b2 ; ¼ l1 ¼1 l2 ¼1

j¼1

MX1;1 ;...;X1;n ;X2;1 ;...;X2;n jH¼h t1;1 ; . . .; t1;n ; t2;1 ; . . .; t2;n

l1

l2 ! m X m n Y X b1 b2 ðhÞ p1;2 ðl1 ; l2 Þ ¼ : b1 t1;j b2 t2;j j¼1 l ¼1 l ¼1 1

2

It implies that

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H. Cossette et al. m X m X ðhÞ FX1;j ;X2;j jH¼h x1;j ; x2;j ¼ p1;2 ðl1 ; l2 ÞH x1;j ; l1 ; b1 H x2;j ; l2 ; b2 ; l1 ¼1 l2 ¼1

MX1;j ;X2;j jH¼h t1;j ; x2;j ¼

m X m X

ðhÞ p1;2 ðl1 ; l2 Þ

l1 ¼1 l2 ¼1

b1 b1 t1;j

l1

b2 b2 t2;j

l2 :

Let ðhÞ

p1 ð l 1 Þ ¼

m X

ðhÞ

p1;2 ðl1 ; l2 Þ;

l2 ¼1

and ðhÞ

p2 ð l 2 Þ ¼

m X

ðhÞ

p1;2 ðl1 ; l2 Þ:

l1 ¼1

Also, we have

m X

FXi;1 ;...;Xi;n jH¼h xi;1 ; . . .; xi;n ¼

ðhÞ pi ðli Þ

li ¼1

MXi;1 ;...;Xi;n jH¼h ti;1 ; . . .; ti;n ¼

m X

! n Y H xi;j ; li ; bi ; j¼1

ðhÞ pi ð l i Þ

li ¼1

n

Y

bi bi ti;j

j¼1

li ! :

We obtain

Z

FX1;j ;X2;j x1;j ; x2;j ¼

h2AH

Z

MX1;j ;X2;j t1;j ; x2;j ¼

h2AH

m X m X

ðhÞ p1;2 ðl1 ; l2 ÞH x1;j ; l1 ; b1 H x2;j ; l2 ; b2 dFH ðhÞ;

l1 ¼1 l2 ¼1 m X m X l1 ¼1 l2 ¼1

Z

FXi;1 ;...;Xi;n xi;1 ; . . .; xi;n ¼

m X

h2AH

MXi;1 ;...;Xi;n ti;1 ; . . .; ti;n ¼

Z h2AH

and

123

ðhÞ

p1;2 ðl1 ; l2 Þ

b1 b1 t1;j

ðhÞ pi ð l i Þ

li ¼1 m X li ¼1

l1

b2 b2 t2;j

l2 dFH ðhÞ;

! n Y H xi;j ; li ; bi dFH ðhÞ; j¼1

ðhÞ pi ð l i Þ

n

Y j¼1

bi bi ti;j

li ! dFH ðhÞ


349

FX1;1 ;...;X1;n ;X2;1 ;...;X2;n x1;1 ; . . .; x1;n ; x2;1 ; . . .; x2;n ! Z X m X m n Y ðhÞ ¼ p1;2 ðl1 ; l2 Þ H x1;j ; l1 ; b1 H x2;j ; l2 ; b2 dFH ðhÞ; h2AH

l1 ¼1 l2 ¼1

j¼1

MX1;1 ;...;X1;n ;X2;1 ;...;X2;n t1;1 ; . . .; t1;n ; t2;1 ; . . .; t2;n l1

l2 ! Z X m X m n

Y b1 b2 ðhÞ ¼ p1;2 ðl1 ; l2 Þ dFH ðhÞ: b1 t1;j b2 t2;j j¼1 l ¼1 l ¼1 h2AH

1

2

The above leads to MS1 ;S2 jH¼h ðt1 ; t2 Þ ¼ MX1;1 ;...;X1;n ;X2;1 ;...;X2;n jH¼h ðt1 ; . . .; t1 ; t2 ; . . .; t2 Þ

l1

l2 ! m X m n X Y b1 b2 ðhÞ ¼ p1;2 ðl1 ; l2 Þ b1 t1 b2 t2 j¼1 l1 ¼1 l2 ¼1

nl2 nl1 m X m X b1 b2 ðhÞ p1;2 ðl1 ; l2 Þ ; ¼ b1 t1 b2 t2 l ¼1 l ¼1 1

2

which implies FS1 ;S2 jH¼h ðs1 ; s2 Þ ¼

m X m X

ðhÞ

p1;2 ðl1 ; l2 ÞH ðs1 ; n l1 ; b1 ÞH ðs2 ; n l2 ; b2 Þ:

l1 ¼1 l2 ¼1

Also, we have FS1 ;S2 ðs1 ; s2 Þ ¼

Z h2AH

¼

m X m X l1 ¼1 l2 ¼1

m X m X l1 ¼1 l2 ¼1

¼

ðhÞ

p1;2 ðl1 ; l2 ÞH ðs1 ; n l1 ; b1 ÞH ðs2 ; n l2 ; b2 ÞdFH ðhÞ

m X m X

0 B H ðs1 ; n l1 ; b1 ÞH ðs2 ; n l2 ; b2 Þ@

Z

1 C ðhÞ p1;2 ðl1 ; l2 ÞdFH ðhÞA

h2AH

p1;2 ðl1 ; l2 ÞH ðs1 ; n l1 ; b1 ÞH ðs2 ; n l2 ; b2 Þ;

l1 ¼1 l2 ¼1

where p1;2 ðl1 ; l2 Þ ¼

Z

ðhÞ

p1;2 ðl1 ; l2 ÞdFH ðhÞ:

h2AH

We consider the following example, to illustrate the allocation values for two dependent lines of business, based on the bivariate VaR. The dependence model is represented by a bivariate mixture of Erlang distributions, as presented above.

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Example 4.3 We suppose that b1 ¼ 1; b2 ¼ 0:5; H 2 fh1 ; h2 g; with PrðH ¼ h1 Þ ¼ s; PrðH ¼ h2 Þ ¼ 1 s: Let 0:75 0:05 0:65 0:15 ðh1 Þ ðh2 Þ ¼ ¼ p and p : 0:05 0:15 0:15 0:05

This bivariate model can be interpreted as being constituted of two possible bivariate mixture of Erlang distributions, based on the sets of weights pðh1 Þ and pðh2 Þ ; with probabilities s and 1 - s respectively. ðh Þ ðh Þ _ Also, It implies that pi k ð1Þ ¼ 0:8 and pi k ð2Þ ¼ 0:2; for i = 1, 2 and k ¼ 1; 2: we have pð1; 1Þ ¼ 0:75 s þ 0:65 ð1 sÞ; pð1; 2Þ ¼ pð2; 1Þ ¼ 0:05 s þ 0:15 ð1 sÞ; pð2; 2Þ ¼ 0:15 s þ 0:05 ð1 sÞ: Note that E½X1 ¼ 1:2; E½X2 ¼ 2:4; E½S1 ¼ 12 and E½S2 ¼ 24: Figure 7 illustrates the lower and upper VaR curves for the set (X1, X2), with a = 99 %. We see that the upper orthant VaR is more affected by s, that changes in the set of weights of the model. The lower orthant VaR is calculated in terms of the multivariate mixted Erlang, and the two different weight sets do not expose a strong difference, which is reflected by the three almost overlapping lower orthant VaR. The sum of the squared differences between the lower orthant VaR are the following :0.000363 for the curves with s = 0 and s = 0.5, 0.000365 for the curves with s = 0.5 and s = 1 and 0.001455 for the curves with s = 0 and s = 0.5. The following tables illustrates many features of the bivariate lower and upper VaR. Table 4 shows the values obtained from the orthogonal projection and proportional allocation methods for the bivariate lower orthant VaR, whereas Table 5 show the results for the bivariate upper orthant VaR. We see that the amounts from Table 4 are always higher than those from Table 5, as explained in Remark 2.3. As illustrated in Fig. 7, we can see that the values obtained for the 18

12 τ=0 τ=0.5 τ=1

17

τ=0 τ=0.5 τ=1

↑

10

VaRα(X2)

16

VaRα(X1)→

8

X 2,i

X 2,i

15 14 13 12

6

4 →VaRα(X1) 2 VaRα(X2) ↓

11

0 5.5

6

6.5

7

X1,i

7.5

8

8.5

9

0

1

2

3

4

5

6

X1,i

Fig. 7 Graphical representation of the bivariate lower and upper orthant VaR with a = 99 %, with different dependence levels, for single bivariate risks

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351

Table 4 Optimal couples based on the bivariate lower orthant VaR for individual risks (Xi,1, Xi,2) a 0.99

0.95

c

Orthogonal

Total

Proportional

Total 18.2770

0

(6.4489, 11.6468)

18.0957

(6.0924, 12.1846)

0.5

(6.4469, 11.6461)

18.0930

(6.0914, 12.1828)

18.2742

1

(6.4465, 11.6435)

18.0900

(6.0904, 12.1807)

18.2711 12.8867

0

(4.6533, 8.0513)

12.7046

(4.2956, 8.5911)

0.5

(4.6491, 8.0446)

12.6937

(4.0292, 8.5833)

12.6125

1

(4.6393, 8.0442)

12.6835

(4.2877, 8.5753)

12.8630

Table 5 Optimal couples based on the bivariate upper orthant VaR for individual risks (Xi,1, Xi,2) a

c

0.99

0.95

Orthogonal

Total

Proportional

Total

0

(1.2343, 8.4532)

9.6875

(2.7573, 5.5147)

8.2720

0.5

(1.2698, 8.4377)

9.7075

(2.8427, 5.6853)

8.5280

1

(1.5301, 8.4225)

9.9526

(2.9216, 5.8433)

8.7649

0

(0.7984, 5.6057)

6.4041

(1.8166, 3.6332)

5.4498

0.5

(0.8968, 5.5704)

6.4672

(1.8679, 3.7358)

5.6037

1

(0.9843, 5.5674)

6.5517

(1.9186, 3.8371)

5.7557

upper orthant VaR are more varying in terms of s, which means that the probability of exceeding a specified level for both rv’s is more affected by the dependence structure. Figure 8 illustrates the bivariate lower and upper orthant curves for the set (S1, S2). We have the same small difference between the lower orthant VaR curves. We can draw the same conclusions as for the individual bivariate risks. Tables 6 and 7 show that the same relation is obtained as for Tables 4 and 5. We also see that the amounts are higher for each aggregate line of business. Again, the total allocation for both lines is always smaller using the orthogonal projection method than the proportional allocation method. h 60 τ=0 τ=0.5 τ=1

74 72

55

τ=0 τ=0.5 τ=1

↑ VaRα(X2)

50

70 VaRα(X1)→

45 68 66

S2

S2

40 35

64 30 62 25 60

→VaRα(S1)

20

58 15

VaRα(S2) ↓

56

10 28

29

30

31

32

33

S1

34

35

36

37

38

10

12

14

16

18

20

22

24

26

28

30

S1

Fig. 8 Graphical representation of the bivariate lower and upper orthant VaR with a = 99 %, with different dependence levels for the business lines

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Table 6 Optimal couples based on the bivariate upper orthant VaR for lines of business (S1, S2), with n = 10 a

c

Orthogonal

Total

Proportional

Total

0.99

0

(30.5148, 58.0388)

88.5536

(29.6666, 59.3334)

89.0000

0.5

(30.5036, 58.0291)

88.5327

(29.6588, 59.3177)

88.9765

1

(30.4915, 58.0202)

88.5117

(29.6509, 59.3019)

88.9528

0

(26.3009, 48.6328)

74.9337

(25.1809, 50.3618)

75.5427

0.5

(26.2186, 48.5780)

74.7966

(25.1286, 50.2572)

75.3858

1

(26.1370, 48.5135)

74.6505

(25.0734, 50.1467)

75.2201

0.95

Table 7 Optimal couples based on the bivariate upper orthant VaR for lines of business (S1, S2), with n = 10 a

c

Orthogonal

Total

Proportional

Total

0.99

0

(16.9283, 45.9186)

62.8469

(20.3982, 40.7964)

61.1946

0.5

(19.0284, 48.1650)

67.1934

(21.8845, 43.7689)

65.6534

1

(20.0769, 49.5461)

69.6230

(22.6883, 45.3766)

68.0649

0

(11.4444, 39.0837)

50.5281

(15.6172, 32.2343)

47.8515

0.5

(15.2038, 38.6930)

53.8968

(17.5943, 35.1885)

52.7828

1

(16.9518, 40.6229)

57.5747

(18.8556, 37.7113)

56.5669

0.95

4.4 Confidence regions, optimal couples and bounds This subsection intends to illustrate the use of bivariate set-valued quantiles and motivate our results on confidence regions, optimal couples and bounds. Describing relationships among different dimensions of an outcome is a basic actuarial technique for explaining the behavior of financial security systems to concerned businesses and policy makers. A VaR trade-off between two stock indices have been studied by Cherubini and Luciano [5] and Cherubini et al. [6]. Studying the bivariate VaR is useful in the decision process, based on a predefine a-level. It shows the possible vectors that can be obtained if each risk reaches the a-level, considering the dependence between them. At level a, the VaR movement between two stock indices can be studied to understand and quantify the impact of transfering a stock from one class to another. Example 4.4 that

Consider a random vector X with joint cdf F/ 2 CðFX1 ; FX2 Þ such F/ ðx1 ; x2 Þ ¼ C/ FX1 ðx1 Þ; FX2 ðx2 Þ ;

where C/ is the archimedean copula with generator / defined as C/ u1 ; u2 ¼ /1 /ðu1 Þ þ /ðu2 Þ ;

ð4:5Þ

ð4:6Þ

for 0 B u1, u2 B 1. The generator / : ½0; 1 ! Rþ in (4.6) is a continuous, possibly

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353

infinite, strictly decreasing convex function such that /(1) = 0. Denote by VaRa;/;x1 ðXÞ; the a-curve of the bivariate lower orthant VaR of X. The latter may be expressed explicitly by using (4.5) and (4.6). Indeed, VaRa;/;x1 ðXÞ ¼ FX12 /1 /ðaÞ /ðFX1 ðx1 Þ ; x1 FX1 ðaÞ: ð4:7Þ 1 This situation could represent a simplified example of a financial institution having two different possible investments. Consider the first possibility to be investing in a fund (X1) composed of two assets and the second one to be one of the two assets from the fund (X2). Requirements in capital allocation should be calculated in order to cover each possible investment at level a. In that sense, the use of the bivariate VaR is appropriate. As illustration, consider exponential marginal cdf’s with parameters k1 = 0.2 and k2 = 0.25, that is Fi ðxi Þ ¼ 1 eki xi ; i ¼ 1; 2; linked by a Clayton copula with dependence parameter h = 2 defined by 1=h h Ch ðu1 ; u2 Þ ¼ uh ; h [ 0: 1 þ u2 1 This copula belongs to the archimedean class, generated by /h(x) = h-1(x-h - 1), h [ 0. Standard computations show that 1=h 1 1 h kx1 h VaRa;h;x1 ðXÞ ¼ ln 1 a 1 e þ1 ; x1 lnð1 aÞ: k2 k1 ð4:8Þ The aim of this example is to provide allocation couples and a lower orthant confidence region. As a consequence of the convexity of the bivariate risk measures, Fig. 1 represents the a-curves for a = 0.95. In this situation, values allocated from both criteria do not expose a large difference. In Table 8, the results from our capital allocation criteria are presented. Moreover, we use this example to examine the lower risk confidence curve at level a. Table 9 provides the level k-1(a) given by (2.10) in terms of a. This allows

Table 8 Couples resulting from the Orthogonal projection and Proportional allocation criteria a


Total

Proportional

Total

0.95

(17.11, 15.82)

32.81

(18.26, 14.61)

32.86

0.99

(26.02, 21.54)

47.56

(26.45, 21.16)

47.62

0.995

(29.51, 24.33)

53.84

(29.94, 23.95)

53.89

Table 9 Lower orthant confidence region levels

a

k-1(a)

0.95

.8114

0.99

.9172

0.995

.9417

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H. Cossette et al.

us to compute the confidence region at level a. The values in Table 9 represent the level curves covering a % of the area under VaRa ðXÞ: Figure 9 shows the confidence curves for a = 0.95. This curve coincides with VaR0:8114 ðXÞ: Note that VaR:8114 ðXÞ can be viewed as a threshold curve such that the probability that the components of the loss X over the given time horizon are simultaneously under this curve with probability 0.95. h Example 4.5 Consider a portfolio of 10 risks of a third party liability in motor insurance, where the risks Xi (i = 1, …, 10) have to be split into a physical P damage claim P Xi,1 and a material damage claim Xi,2. Let us define S1 = 10 i=1Xi,1 and S2 = 10 X , the rv’s representing the total amount for each type of claim. The i=1 i,2 random vector (S1, S2) is studied in order to establish the capital allocated to each party, knowing that Xi,j, i = 1, …, 10 and j = 1,2 are covered by different parties. Suppose that each physical damage rv Xi,1 follows an exponential distribution (k = 1.5) and each material damage rv Xi,2 follows a Pareto distribution (aP = 2). Also, Xi,1, i = 1, …, 10 and j = 1,2 are independent within each class and the dependence between S1 and S2 is unknown. The capital allocation for each party based on the optimal couple criterion provided in the previous section is a set (S1, S2) from the lower orthant VaR. The distance between each a-level curve is large, showing the effect of heavytails. Also, Fig. 10 illustrates that the bivariate risk measure increases with a, with respect to the order ; similarly to Fig. 6. It is interesting to note the impact of the

30 VaR.95(X1,X2)

28

VaR.8114(X1,X2)

26 24 22 20 18 16 14 12 10 8 10

15

20

25

30

35

Fig. 9 Graphical representation of the lower orthant confidence region VaR:8114 (X) coinciding for VaR:95 (X)

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355

800 α=.95 α=.99 α=.995

Lower α−curve values

700

600

500

400

300

200

100

40

50

60

70

80

90

100

110

120

130

s1

Fig. 10 Graphical representation of the upper bounds of the lower orthant VaR VaRa;s1 (S)

aggregation of fat-tailed distributions (Pareto) on VaRa;s1 ðSÞ and on the upper bounds compared to the optimal couples. Since the dependence between each party is unknown, the study of the worst-case scenario provides a considerable difference between the allocations that have to be made from each party (Table 10). Now, suppose that Xi,2, i = 1,…,10 follow a gamma distribution with shape and scale parameters a = 2 and b = 1 respectively, and that the dependence between the sums of risks S1 and S2 is represented by an archimedean copula. In such a situation, (4.7) provides an explicit form for VaRa;/;s1 ðSÞ; where S = (S1, S2). This can be used to establish from (3.11) an upper bound of VaRa(S1 ? S2) in terms of the cdf’s FS1 and FS2 , and the generator / of the copula of (S1, S2), namely n 1 o /ðaÞ /ðF s1 þ FS1 / ðs ÞÞ VaRa ðS1 þ S2 Þ min : ð4:9Þ S1 1 2 s1 FS1 ðaÞ 1

As an illustration, let us consider a Clayton copula with dependence parameter h. In Table 11, we provide the upper bound on VaR.95(S1 ? S2), for different dependence levels, such that Kendall’s tau is 0.25, 0.5 and 0.75 respectively. Table 11 allows us

Table 10 Optimal couples from the upper bound

a

Orthogonal

0.95

(35.3559, 859.8439)

(35.2579, 1242.4300)

0.99

(46.1428, 2360.9330)

(46.0628, 4036.78)

0.995

(50.7628, 3578.9410)

(50.6861, 6861.1220)

Proportional

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356 Table 11 Upper bound of VaR.95(S1 ? S2)

H. Cossette et al.

h

Upper bound

2/3

56.38

2

56.34

6

56.25

to establish the upper bound, using the bivariate lower orthant VaR, considering the dependence model of S1 and S2 (Table 10). h Acknowledgments The authors wish to thank the anonymous reviewers for their detailed and helpful comments. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada, the Fonds qubcois de la recherche sur la nature et les technologies, the Chaire en actuariat de l’Universite´ Laval, and the Faculty of Arts and Science of Concordia University.

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