Optimal reinsurance under VaR and CTE risk measuressas.uwaterloo.ca/~c2weng/papers/2008_Cai_Tan_Weng_Zhang.pdf · convex risk measures, Gajek and Zagrodny (2004) demonstrated that

Insurance: Mathematics and Economics 43 (2008) 185–196

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Insurance: Mathematics and Economics

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Optimal reinsurance under VaR and CTE risk measuresJun Cai a, Ken Seng Tan a,b, Chengguo Weng a, Yi Zhang c,∗

a Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, N2L 3G1, Canadab China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, 10081, Chinac Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

a r t i c l e i n f o

Article history:Received March 2007Received in revised formJuly 2007Accepted 21 May 2008

JEL classification:C02C61

SIBC classification:IM52IE10IB90

Keywords:Value-at-risk (VaR)Conditional tail expectation (CTE)Ceded lossRetained lossIncreasing convex functionExpectation premium principleStop-loss reinsuranceQuota-share reinsuranceChange-loss reinsurance

a b s t r a c t

Let X denote the loss initially assumed by an insurer. In a reinsurance design, the insurer cedes part of itsloss, say f (X), to a reinsurer, and thus the insurer retains a loss If (X) = X − f (X). In return, the insureris obligated to compensate the reinsurer for undertaking the risk by paying the reinsurance premium.Hence, the sum of the retained loss and the reinsurance premium can be interpreted as the total costof managing the risk in the presence of reinsurance. Based on a technique used in [Müller, A., Stoyan, D.,2002. ComparisonMethods for StochasticModels and Risks. In:Willey Series in Probability and Statistics]and motivated by [Cai J., Tan K.S., 2007. Optimal retention for a stop-loss reinsurance under the VaRand CTE risk measure. Astin Bull. 37 (1), 93–112] on using the value-at-risk (VaR) and the conditionaltail expectation (CTE) of an insurer’s total cost as the criteria for determining the optimal reinsurance,this paper derives the optimal ceded loss functions in a class of increasing convex ceded loss functions.The results indicate that depending on the risk measure’s level of confidence and the safety loadingfor the reinsurance premium, the optimal reinsurance can be in the forms of stop-loss, quota-share, orchange-loss.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Reinsurance is a commonly employed risk managementstrategy to ensure that the insurer’s earnings remain relativelystable or to protect the insurer against potentially large losses.Let X be the (aggregate) loss initially assumed by an insurer.Suppose that X is a non-negative random variable with cumulativedistribution function FX (x) = Pr{X ≤ x}, survival functionSX (x) = 1 − FX (x) = Pr{X > x}, and mean 0 < E[X] < ∞.Under a reinsurance arrangement, the insurer cedes part of itsloss, say f (X) with 0 ≤ f (X) ≤ X , to a reinsurer, and thus theinsurer retains a loss If (X) = X − f (X), where the function f (x),satisfying 0 ≤ f (x) ≤ x, is known as a ceded loss function andthe function If (x) = x − f (x) is called a retained loss function.

∗ Corresponding author. Tel.: +86 0571 87953667.E-mail address: [email protected] (Y. Zhang).

0167-6687/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2008.05.011

By transferring part of the risk exposure to a reinsurer, the insureralso incurs an additional cost in the form of reinsurance premiumthat is payable to the reinsurer. Let δf (X) denote the reinsurancepremium which corresponds to a ceded loss function f (x) and letTf (X) represent the resulting total cost or the total risk exposureof the insurer in the presence of reinsurance. Then we obtain thefollowing relationship:

Tf (X) = If (X) + δf (X). (1.1)

This demonstrates that in the presence of reinsurance, the insureris now concerned with the risk exposure Tf (X), instead of theground up loss X . This also suggests that an appropriate choice ofthe ceded loss function can provide an effective way of reducingthe risk exposure of an insurer.

In practice, there are a variety of reinsurance designs fromwhich an insurer can choose. These include quota-share reinsur-ance with f (x) = ax and If (x) = (1 − a)x, 0 < a ≤ 1; stop-lossreinsurance with f (x) = (x − d)+ = max{0, x − d} and If (x) =

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186 J. Cai et al. / Insurance: Mathematics and Economics 43 (2008) 185–196

min{x, d}, d ≥ 0; change-loss reinsurance with f (x) = a(x − d)+and If (x) = (1 − a)x + amin{x, d}. Various optimization criteriahave been proposed for the determination of the optimal reinsur-ance. For example, it is well known that the stop-loss reinsuranceis the optimal one that minimizes the variances of retained lossesamong the class of ceded loss functions that have the same expec-tations; see, for example, Bowers et al. (1997), Kaas et al. (2001) andGerber (1979). By using the criterion of minimizing some specificconvex risk measures, Gajek and Zagrodny (2004) demonstratedthat change-loss design is the optimal reinsurance. In a series ofpublished papers, Kaluszka (2001, 2004a,b, 2005) also makes im-portant contributions to the optimal reinsurance models.

More recently, using risk measures such as the value-at-risk(VaR) and the conditional tail expectation (CTE), Cai and Tan(2007) derived explicitly the optimal retention level of a stop-lossreinsurance under the expectation premium principle. Their work,in part, was sparked by an unprecedented surge in the usage ofthese measures as risk management tools among banks, financialinstitutions, and insurance companies in recent years. See, forexample, Artzner et al. (1999), Basak and Shapiro (2001), McNeilet al. (2005), Cai and Li (2005), Inui and Kijima (2005) and Yamaiand Yoshiba (2005).

The VaR of a non-negative random variable X at a confidencelevel 1 − α, 0 < α < 1, is defined asVaRX (α) = inf{x : Pr{X > x} ≤ α}. (1.2)Note that ifα ≥ SX (0), thenVaRX (α) = 0, and ifX has a continuousstrictly increasing distribution function on (0, ∞), then for 0 <α < SX (0), VaRX (α) = S−1

X (α), where S−1X is the inverse functions

of SX . The conditional tail expectation (CTE) of a random variable Xat its VaR is defined asCTEX (α) = E[X |X ≥ VaRX (α)]. (1.3)See Proposition A.1 in the Appendix for important propertiesassociated with VaR and CTE that are relevant to our subsequentdiscussion.

Since Tf (X) defined in (1.1) captures the overall cost of insuringa loss for a ceded loss function f , a prudent risk management foran insurer is therefore to ensure that the risk measures associatedwith Tf (X) are as small as possible.Motivated byCai and Tan (2007)and by assuming the expectation premium principle1 for settingthe reinsurance premium δf (X), i.e.

δf (X) = (1 + ρ)E[f (X)], (1.4)where ρ > 0 is the safety loading, this paper strives to determinethe optimal ceded loss functions that, respectively, minimize VaRand CTE of the total cost Tf (X) in some classes of ceded lossfunctions. More specifically, our objective is to seek the optimalreinsurance in the following class of ceded loss functions:

Definition 1.1. Let F denote the class of ceded loss functions,which consists of all increasing convex functions f (x) defined on[0, ∞) and satisfying 0 ≤ f (x) ≤ x for x ≥ 0 but excludingf (x) ≡ 0.

Mathematically, our optimal reinsurance model can be formu-lated as follows, depending on the chosen risk measure:

VaR-optimization : VaRTf ∗ (X)(α) = minf∈F

{VaRTf (X)(α)

}, (1.5)

andCTE-optimization : CTETf ∗ (X)(α) = min

f∈F

{CTETf (X)(α)

}, (1.6)

where VaRTf (X)(α) and CTETf (X)(α) are defined analogously as in(1.2) and (1.3) except for the total risk random variable Tf (X).

1 There are many other principles that have been proposed in connection todetermining the premium. The expectation premium principle has remained themost fundamental andwidely used and hence we continue to use this in this paper.

To facilitate the discussion of the optimal ceded loss function inF of the above optimization problems, we introduce the followingclass of ceded loss functions:

Definition 1.2. Let H denote the class of ceded loss functions,which consists of all non-negative functions h(x) defined on [0, ∞)with the following form

h(x) =

n∑j=1

cn,j(x − dn,j)+, x ≥ 0, n = 1, 2, . . . , (1.7)

where cn,j > 0 and dn,j ≥ 0 are constants such that

0 <

n∑j=1

cn,j ≤ 1 and 0 ≤ dn,1 ≤ dn,2 ≤ · · · ≤ dn,n

for all n = 1, 2, . . . .

The importance of analyzing the optimal reinsurance in H canbe argued as follows. First, note that H is a subclass of F and allfunctions in F are continuous on [0, ∞) since increasing convexfunctions are continuous. Second, by adopting a technique similarto the proof of Theorem 1.5.7 in (Müller and Stoyan (2002), p18) onincreasing convex order, we formally show that any function in Fis the limit of a sequence of functions in H , namely the class H is adense subclass ofF (see Lemma 3.1). Consequently, by using someconvergence results on VaR and CTE, we prove that the optimalfunctions in H , which minimize the VaR and CTE of the total costTh(X) for h ∈ H , also optimally minimize the VaR and CTE of thetotal cost Tf (X) for f ∈ F . The above arguments imply that we candeduce the solutions to the optimal reinsurance models (1.5) and(1.6) by first confining the optimal ceded loss functions within theclassH . Following this strategy, we derive the optimal reinsuranceanalytically as presented in Theorems 3.1 and 4.1. These resultsindicate that optimal reinsurance can be in the forms of stop-loss,quota-share, or change-loss, depending on the risk measure’s levelof confidence and the safety loading for the reinsurance premium.

Throughout this paper, the terms ‘‘increasing’’ and ‘‘decreas-ing’’ mean ‘‘non-decreasing’’ and ‘‘non-increasing’’, respectively.To simplify our discussions, we assume that X has a continuousstrictly increasing distribution function on (0, ∞) with a possiblejump at 0,which allowsX to be a randomsum

∑Ni=1 Xi and is an im-

portant special case in actuarial loss model. See Cai and Tan (2007)for details. Furthermore, to avoid discussing trivial cases, we fur-ther assume that the parameter α associated with the definitionsof VaR and CTE satisfies

0 < α < SX (0). (1.8)

The rest of the paper is organized as follows. Section 2 providesadditional notations and preliminary results on the VaR of Th(X)for h ∈ H . Sections 3 and 4 analyze the solutions to theoptimal reinsurance models (1.5) and (1.6), respectively. Section 5concludes our paper and the Appendix presents some of thetechnical propositions, lemmas and proofs.

2. Preliminaries

By defining VaRIf (X)(α) and CTEIf (X)(α) as, respectively, the VaRand CTE of the retained loss random variable If (X), then it followsfrom the translation invariance property of VaR and CTE that:

VaRTf (X)(α) = VaRIf (X)(α) + δf (X), (2.1)

and

CTETf (X)(α) = CTEIf (X)(α) + δf (X). (2.2)

J. Cai et al. / Insurance: Mathematics and Economics 43 (2008) 185–196 187

The above relations can also be justified as follows: (2.1) followsfrom (A.1) and (1.1), which in turn leads to the second relationtogether with the definitions of CTE.

Under our assumption that the reinsurance premium isdetermined using the expectation premium principle, this impliesthat for any function h(x) =

∑nj=1 cn,j(x − dn,j)+ ∈ H , the

reinsurance premium on the ceded loss h(X) can be written as

δh(X) = (1 + ρ)E[h(X)] = (1 + ρ)

{n∑

j=1

cn,j

∫∞

dn,jSX (x)dx

}. (2.3)

Furthermore, by defining An,i = 1 −∑i

j=1 cn,j, and Bn,i =∑ij=1 cn,j dn,j, i = 1, . . . , n, it is easy to show that the retained loss

is

Ih(X) = X − h(X) = X −

n∑j=1

cn,j(X − dn,j)+

=

{X, X ≤ dn,1,An,i X + Bn,i, dn,i ≤ X ≤ dn,i+1, i = 1, . . . , n − 1,An,n X + Bn,n, X ≥ dn,n.

(2.4)

We point out that Ih(x) in the above expression is an increasingcontinuous concave function of x and the coefficients of Ih(x) satisfy

An,i dn,i+1 + Bn,i = An,i+1 dn,i+1 + Bn,i+1 for all i = 1, . . . , n − 1.

It also follows from (2.4) that the survival function of Ih(X) can berepresented as

SIh(X)(x) =

SX (x), x ≤ dn,1,

SX

(x − Bn,i

An,i

),

An,i dn,i + Bn,i ≤ x ≤ An,i dn,i+1 + Bn,i,

i = 1, . . . , n − 1,

SX

(x − Bn,n

An,n

), x ≥ An,n dn,n + Bn,n,

(2.5)

where SX(

x−Bn,iAn,i

)= SX (∞) = 0 when An,i = 0 for some

i = 1, . . . , n. Moreover, by (2.5), we derive the VaR of Ih(X) at aconfidence level 1 − α as

VaRIh(X)(α) =

S−1X (α), S−1

X (α) ≤ dn,1,An,i S−1

X (α) + Bn,i, dn,i ≤ S−1X (α) ≤ dn,i+1,

i = 1, . . . , n − 1,An,n S−1

X (α) + Bn,n, dn,n ≤ S−1X (α),

(2.6)

which together with (2.1), leads to the following expressions forthe VaR of Th(X):

VaRTh(X)(α) =

S−1X (α) + δh(X), S−1

X (α) ≤ dn,1,An,i S−1

X (α) + Bn,i + δh(X),

dn,i ≤ S−1X (α) ≤ dn,i+1,

i = 1, . . . , n − 1,An,n S−1

X (α) + Bn,n + δh(X), dn,n ≤ S−1X (α).

(2.7)

We conclude this section by introducing the following nota-tions:

ρ∗=

11 + ρ

, d∗= S−1

X (ρ∗), (2.8)

g(x) = x +1ρ∗

∫∞

xSX (t) dt, x ≥ 0, (2.9)

u(x) = S−1X (x) +

1ρ∗

∫∞

S−1X (x)

SX (t) dt, x ≥ 0. (2.10)

See Proposition A.2 in the Appendix for some useful propertiesassociated with functions g(·) and u(·).

3. Optimal reinsurance under VaR risk measure

In this section, we will derive the optimal ceded loss functionf ∗ in the class F under the VaR criterion; i.e., the solutions to theVaR-optimization problem (1.5). As mentioned earlier we achievethis by first establishing that the optimal ceded loss functions inthe class H are also optimal in the class F (see Lemma 3.2). Thisresult suggests that it would be sufficient to just seek the solutionsto the following optimal reinsurance model:

VaRTh∗ (X)(α) = minh∈H

{VaRTh(X)(α)

}. (3.1)

We now present the following two lemmas. The former lemma,the proof of which we delay to the Appendix, is important forproving the latter lemma.

Lemma 3.1. For any f ∈ F , there exists a sequence of functions{hn, n = 1, 2, . . .} inH such that limn→∞ hn(x) = f (x) for all x ≥ 0and

hn(x) ≤ f (x) ≤ x for all x ≥ 0 and n = 1, 2, . . . .

Lemma 3.2. Optimal ceded loss functions which minimize the VaR ofinsurer’s total risk in the class H are also optimal in the class F .Proof. Let h∗ be any optimal ceded loss function in the classH un-der the VaR criterion, i.e., the solution to the optimization problem(3.1). We need to show that

VaRTf (X)(α) ≥ VaRTh∗ (X)(α), for any f ∈ F . (3.2)

By Lemma 3.1, we take a sequence of functions {hn, n = 1, 2, . . .}in H satisfying

limn→∞

hn(x) = f (x) for all x ≥ 0 (3.3)

and

hn(x) ≤ f (x) ≤ x for all x ≥ 0 and n = 1, 2, . . . . (3.4)

Then it follows from the dominated convergence theoremand (3.3)that

limn→∞

Ehn(X) = Ef (X), (3.5)

and the optimality of h∗ in the class F implies

VaRThn (X)(α) ≥ VaRTh∗ (X)(α), (3.6)

for any n = 1, 2, . . .. Note that the retained loss functions Ihn(x) =

x − hn(x) and If (x) = x − f (x) are increasing and continuous.Thus, by the continuous mapping theorem we see that Thn(x) =

Ihn(x)+δhn(X) = Ihn(x)+(1+ρ)E[hn(X)] → If (x)+(1+ρ)E[f (X)]as n → ∞. Thus, it follows from part (b) of Proposition A.1 thatlimn→∞ VaRThn (X)(α) = VaRTf (X)(α), which, together with (3.6),implies (3.2) and hence we complete the proof. �

We proceed with deriving the solutions to the optimalreinsurance model (3.1). For each n = 1, 2, . . ., and a given 0 <α < 1, let us first define the following sets of (dn,1, . . . , dn,n):

Dn = {(dn,1, . . . , dn,n) : 0 ≤ dn,1 ≤ · · · ≤ dn,n},D0n = {(dn,1, . . . , dn,n) : S−1

X (α) ≤ dn,1 ≤ · · · ≤ dn,n},Din = {(dn,1, . . . , dn,n) : 0 ≤ dn,1 ≤ · · · ≤ dn,i ≤ S−1

X (α)≤ dn,i+1 ≤ · · · ≤ dn,n}, i = 1, . . . , n − 1,

Dnn = {(dn,1, . . . , dn,n) : 0 ≤ dn,1 ≤ · · · ≤ dn,n ≤ S−1

X (α)}.

(3.7)

Note that D0n,D

1n, . . . ,D

nn form a partition of Dn and Dn =

⋃ni=0 D

in.

For any h(x) =∑n

j=1 cn,j (x − dn,j)+ ∈ H , we use the notationVaRTh(X)(dn,1, . . . , dn,n, α) to denote the VaR of the insurer’s totalcost associated with the ceded loss h(X) as a function of dn,j, j =

1, . . . , n. Combining (2.3) and (2.7),we obtain the following lemmafor the explicit expressions of VaRTh(X)(dn,1, . . . , dn,n, α) in termsof dn,1, . . . , dn,n.


Lemma 3.3. For any h(x) =∑n

j=1 cn,j (x − dn,j)+ ∈ H and a givenconfidence level 1 − α with 0 < α < SX (0):

(a) When S−1X (α) ≤ dn,1, i.e., (dn,1, . . . , dn,n) ∈ D0

n,

VaRTh(X)(dn,1, . . . , dn,n, α) = S−1X (α)

+1ρ∗

{n∑

j=1

cn,j

∫∞

dn,jSX (x)dx

}. (3.8)

(b) When dn,i ≤ S−1X (α) ≤ dn,i+1, i.e., (dn,1, . . . , dn,n) ∈ Di

n, and fori = 1, . . . , n − 1,

VaRTh(X)(dn,1, . . . , dn,n, α) =

(1 −

i∑j=1

cn,j

)S−1X (α)

+

i∑j=1

cn,j g(dn,j) +1ρ∗

n∑j=i+1

cn,j

∫∞

dn,jSX (x)dx. (3.9)

(c) When dn,n ≤ S−1X (α), i.e., (dn,1, . . . , dn,n) ∈ Dn

n,

VaRTh(X)(dn,1, . . . , dn,n, α) =

(1 −

n∑j=1

cn,j

)S−1X (α)

+

n∑j=1

cn,j g(dn,j). � (3.10)

Based on the above expressions of VaRTh(X)(dn,1, . . . , dn,n, α),we analyze its minimum on set Dn for n = 1, 2, . . ., by discussingits infimum on Di

n for each i = 0, 1, 2, . . . , n. The results aresummarized in the following lemma (see the Appendix for theproof):

Lemma 3.4. Given a confidence level 1− α with 0 < α < SX (0), forany function h(x) =

∑nj cn,j(x − dn,j)+ ∈ H with given coefficients

cn,j, j = 1, . . . , n:(a) If

ρ∗ < SX (0) and S−1X (α) ≥ u(ρ∗), (3.11)

then

minDn

VaRTh(X)(dn,1, . . . , dn,n, α)

= S−1X (α) +

n∑j=1

cn,j[u(ρ∗) − S−1

X (α)]

(3.12)

and the minimum VaR is attained at dn,1 = · · · = dn,n = d∗ or at

h∗(x) =

n∑j=1

cn,j (x − d∗)+ =

n∑j=1

cn,j(x − S−1

X (ρ∗))+

. (3.13)

(b) If

ρ∗≥ SX (0) and S−1

X (α) ≥ g(0), (3.14)

then

minDn

VaRTh(X)(dn,1, . . . , dn,n, α)

= S−1X (α) +

n∑j=1

cn,j[g(0) − S−1

X (α)]

(3.15)

and the minimum VaR is attained at dn,1 = · · · = dn,n = 0 or at

h∗(x) =

n∑j=1

cn,j x. (3.16)

(c) For all other cases, mindn∈Dn VaRTh(X)(dn,1, . . . , dn,n, α) doesnot exist.

We are now ready to present the key results of this sectionwhich are stated in Theorem 3.1. The above lemma is usedto obtain the solution to (3.1) by comparing the minimum ofVaRTh(X)(dn,1, . . . , dn,n, α) on Dn for each n = 1, 2, . . .. Lemma 3.2,in turn, asserts that these solutions are also the solutions to theproposed optimal reinsurance model (1.5).

Theorem 3.1. For a given confidence level 1 − α with 0 < α <SX (0):

(a) If ρ∗ < SX (0) and S−1X (α) > u(ρ∗), thenminf∈F VaRTf (X)(α)

= u(ρ∗) and the minimum VaR is attained at

f ∗(x) =(x − d∗

)+

. (3.17)

(b) If ρ∗ < SX (0) and S−1X (α) = u(ρ∗), thenminf∈F VaRTf (X)(α)

= S−1X (α) and the minimum VaR is attained at

f ∗(x) = c(x − d∗

)+

(3.18)

for any constant c such that 0 < c ≤ 1.(c) If ρ∗

≥ SX (0) and S−1X (α) > g(0), then minf∈F VaRTf (X)(α)

= g(0) and the minimum VaR is attained at

f ∗(x) = x. (3.19)

(d) If ρ∗≥ SX (0) and S−1

X (α) = g(0), then minf∈F VaRTf (X)(α)

= S−1X (α) and the minimum VaR is attained at

f ∗(x) = cx (3.20)

for any constant c such that 0 < c ≤ 1.

Proof. Note that all ceded loss functions given in the theorem areincluded in H . Hence, by Lemma 3.2, we only need to show theoptimality of these ceded loss functions in class H ; i.e., for anyh(x) =

∑nj cn,j(x − dn,j)+ ∈ H ,

VaRTh(X)(α) ≥ VaRTf ∗ (X)(α). (3.21)

(a) By (3.12) of Lemma 3.4 and the assumption S−1X (α) > u(ρ∗),

we have

VaRTh(X)(α) ≥ minDn

VaRTh(X)(dn,1, . . . , dn,n, α)

= S−1X (α) +

n∑j=1

cn,j[u(ρ∗) − S−1

X (α)]

≥ S−1X (α) + u(ρ∗) − S−1

X (α) = u(ρ∗). (3.22)

Furthermore, by (3.12) and (3.13), VaRTf ∗ (X)(α) = u(ρ∗) when∑nj=1 cn,j = 1 in (3.13) or when f ∗(x) = (x − d∗)+. Hence, (3.22)

implies (3.21).(b) By (3.12) of Lemma 3.4 and the assumption S−1

X (α) = u(ρ∗),we have

VaRTh(X)(α) ≥ minDn

VaRTh(X)(dn,1, . . . , dn,n, α)

= S−1X (α) +

n∑j=1

cn,j[u(ρ∗) − S−1

X (α)]

≥ S−1X (α). (3.23)

Furthermore, by (3.12) and (3.13), VaRTf ∗ (X)(α) = S−1X (α) when∑n

j=1 cn,j = c in (3.13) or when f ∗(x) = c(x − d∗)+ for any0 < c ≤ 1. Hence, (3.23) implies (3.21).

Parts (c) and (d) are proved similarly as in Parts (a) and (b) byusing (3.15) of Lemma 3.4 and the assumptions S−1

X (α) > g(0) andS−1X (α) = g(0), respectively. �


Remark 3.1. Theorem 3.1 establishes that for our proposedoptimal reinsurance model, the optimal reinsurance is a stop-lossreinsurance in case (a), a change-loss reinsurance in case (b), and aquota-share reinsurance in cases (c) and (d).

We conclude this section by examining several special functionsin F , as illustrated in the following examples:

Example 3.1. Let f̄ (X) = X so that an insurer cedes all its lossesto a reinsurer. In this case, the retained loss is If̄ (X) = 0. The totalcost for the insurer is Tf̄ (X) = (1 + ρ)E(X). However, the VaR of aconstant is itself. Hence, VaRTf̄ (X)(α) = (1 + ρ)E(X) = g(0). It iseasy to verify that

g(0) = (1 + ρ)E(X) > u(ρ∗). (3.24)

Indeed, noticing that SX (x) is decreasing in x and SX (d∗) =

SX(S−1X (ρ∗)

)= ρ∗, we have

(1 + ρ)E(X) = (1 + ρ)

(∫ d∗

0SX (x)dx +

∫∞

d∗

SX (x)dx

)

> S−1X (ρ∗) +

1ρ∗

∫∞

S−1X (ρ∗)

SX (x)dx = u(ρ∗).

Hence, VaRTf̄ (X)(α) is bigger than the minimum VaRs in cases (a)

and (b) of Theorem 3.1. However, f̄ is optimal in cases (c) and (d).�

Example 3.2. Let fa(X) = aX for 0 < a ≤ 1 so that an insureruses a quota-share reinsurance. In this case, the retained loss isIfa(X) = (1 − a)X . The total cost for the insurer is Tfa(X) =

(1 − a)X + a(1 + ρ)E(X). By (A.1), we have

VaRTfa (X) = (1 − a)VaRX (α) + a(1 + ρ)E(X)

= (1 − a)S−1X (α) + a(1 + ρ)E(X).

It follows from (3.24) that in cases (a) and (b) of Theorem 3.1,VaRTfa (X) > (1−a)u(ρ∗)+au(ρ∗) = u(ρ∗). In case (c), VaRTfa (X) >(1−a)g(0)+ag(0) = g(0).However, fa is optimal in case (d). �

Example 3.3. Let fd(X) = (X − d)+ for d > 0, namely, aninsurer uses a stop-loss reinsurance. In this case, the retained lossis Ifd(X) = min{X, d}. The total cost for the insurer is Tfd(X) =

min{X, d} + (1 + ρ)E[(X − d)+]. It has been proved in Cai andTan (2007) under the conditions of case (a) of Theorem 3.1 thatmind>0{VaRTfd (X)(α)} = u(ρ∗) and the minimum VaR is attainedat d∗

= S−1X (ρ∗). �

4. Optimal reinsurance under CTE risk measure

The last section derived explicitly the optimal reinsurance thatminimizes the VaR of the total cost of an insurer for reinsuringits losses. In this section, we extend the analysis by assuming CTEas the relevant risk measure. More specifically, we are interestedin the optimal ceded loss functions for the CTE-optimization asformulated in (1.6). Our procedure to identify the optimal cededfunctions under the CTE criterion will be parallel to the situationof VaR criterion in the previous section. We will also first establishthe fact that optimal ceded loss functions in the class H , whichminimize the CTE of the insurer’s total cost, are also optimal in thelarger class F (see Lemma 4.2), and then consider the followingoptimization problem over the class of ceded loss functions in H :

CTETh∗ (X)(α) = minh∈H

{CTETh(X)(α)

}. (4.1)

It should be emphasized that it is considerably more compli-cated to discuss the optimal ceded loss functions under the CTEcriterion than the VaR criterion. The difficulty lies mainly in the

proof of the following lemma (Lemma 4.1), which will be crucialto the proof of Lemma 4.2. We defer the proof of Lemma 4.1 to theAppendix.

Lemma 4.1. For any f (x) ∈ F , there exists a sequence of functionshn(x) ∈ H such that

limn→∞

hn(x) → f (x) for all x ≥ 0 (4.2)

and for α < SX (0),

Pr{Ihn(X) ≥ VaRIhn (X)(α)} → Pr{If (X)

≥ VaRIf (X)(α)} as n → ∞. (4.3)

Lemma 4.2. Optimal ceded loss functions which minimize the CTE ofinsurer’s total risk in class H are also optimal in class F .

Proof. Let h∗ be the optimal ceded loss function in class H and fbe any ceded loss function in the class F . We need to show

CTETf (X)(α) ≥ CTETh∗ (X)(α). (4.4)

Recall that by combining (A.3) of Proposition A.1 and (2.1), we have

CTEThn(X)(α) = VaRIhn (X)(α)

+

∫∞

VaRIhn (X)(α)SIhn (X)(x)dx

P{Ihn(X) ≥ VaRIhn (X)(α)}+ (1 + ρ)E[hn(X)], (4.5)

and

CTETf (X)(α) = VaRIf (X)(α)

+

∫∞

VaRIf (X)(α)SIf (X)(x)dx

P{If (X) ≥ VaRIf (X)(α)}+ (1 + ρ)E[f (X)]. (4.6)

By Lemma 4.1, there exists a sequence of functions hn(x) ∈ H suchthat limn→∞ hn(x) = f (x)with 0 ≤ hn(x) ≤ f (x) ≤ x for any x ≥ 0and

Pr{Ihn(X) ≥ VaRIhn (X)(α)} → Pr{If (X)

≥ VaRIf (X)(α)}, as n → ∞. (4.7)

Thus, using the dominated convergence theorem yields

limn→∞

Ehn(X) = Ef (X), (4.8)

since 0 < EX < ∞. Recall that If (x) = x − f (x) is increasingand continuous in x ≥ 0 and hence it follows from part (b) ofProposition A.1 that

limn→∞

VaRIhn (X)(α) = VaRIf (X)(α). (4.9)

Now, by denoting Yhn = Ihn(X)−VaRIhn (X)(α) and the continuity of

function κ(y) = yI{y≥0}, we have κ(Yhn)a.s.

−→ κ(Yf ). Hence, basedon the fact that

0 ≤ v(Yhn) = [X − VaRIhn (X)(α)]I{X−VaRIhn (X)(α)≤0} ≤ X, a.s.,

we conclude

limn→∞

∫∞

VaRIhn (X)(α)

SIhn (X)(x)dx = limn→∞

E[κ(Yhn)] = E[κ(Yf )]

=

∫∞

VaRIf (X)(α)

SIf (X)(x)dx. (4.10)

using the dominated convergence theorem. Consequently, by(4.5)–(4.10), we obtain

limn→∞

CTEThn (X)(α) = CTETf (X)(α). (4.11)


On the other hand, the optimality of h∗ in H implies

CTEThn (X)(α) ≥ CTETh∗ (X)(α), (4.12)

for n = 1, 2, . . ., which together with (4.11) implies (4.4). �

To proceed with the discussion of the solution to the optimalreinsurance model (4.1), it is convenient to first determine theexplicit expressions for the corresponding CTE of the total cost.Recall that from (A.3) of Proposition A.1 and together with (2.1),we obtain

CTETf (X)(α) = VaRIf (X)(α) +

∫∞

VaRIf (X)(α)SIf (X)(x)dx

Pr{If (X) ≥ VaRIf (X)(α)}+ δf (X).

(4.13)Similarly to the VaR-optimization, for any function h(x) =

∑nj=1

cn,j (x − dn,j)+ ∈ H , we express the CTE of the cededloss h(X) as a function of dn,1, . . . , dn,n, which we denoteas CTETh(X)(dn,1, . . . , dn,n, α). Then, by defining the followingfunction

v(x) = S−1X (x) +

1α

∫∞

S−1X (x)

SX (t) dt, x ≥ 0, (4.14)

we obtain explicit expressions of CTETh(X)(dn,1, . . . , dn,n, α), asshown in Lemma 4.3. Capitalizing on this lemma, we determinethe minimum of CTETh(X)(dn,1, . . . , dn,n, α) on the set Dn for eachi = 0, 1, 2, . . . , n and a given integer n. The results are given inLemma 4.4. The proofs of these two lemmas are presented in theAppendix.

Lemma 4.3. Given a confidence level 1− α with 0 < α < SX (0), forany h(x) =

∑nj=1 cn,j (x − dn,j)+ ∈ H :

(a) When S−1X (α) ≤ dn,1, i.e., (dn,1, . . . , dn,n) ∈ D0

n,

CTETh(X)(dn,1, . . . , dn,n, α)

= v(α) +

[1ρ∗

−1α

] n∑j=1

cn,j

∫∞

dn,jSX (x)dx. (4.15)

(b) When dn,i ≤ S−1X (α) ≤ dn,i+1, i.e., (dn,1, . . . , dn,n) ∈ Di

n, fori = 1, . . . , n − 1,

CTETh(X)(dn,1, . . . , dn,n, α) =

(1 −

i∑j=1

cn,j

)v(α)

+

i∑j=1

cn,j g(dn,j) +

[1ρ∗

−1α

] n∑j=i+1

cn,j

∫∞

dn,jSX (x)dx. (4.16)

(c) When dn,n ≤ S−1X (α), i.e., (dn,1, . . . , dn,n) ∈ Dn

n,

CTETh(X)(dn,1, . . . , dn,n, α) =

(1 −

n∑j=1

cn,j

)v(α)

+

n∑j=1

cn,j g(dn,j). (4.17)

Lemma 4.4. Consider any function h(x) =∑n

j=1 cn,j (x−dn,j)+ ∈ Hwith given coefficients cn,j, j = 1, . . . , n, and a confidence level 1−αsuch that 0 < α < SX (0).

(a) If α < ρ∗ < SX (0), then

minDn

{CTETh(X)(dn,1, . . . , dn,n, α)}

= v(α) +

n∑j=1

cn,j[u(ρ∗) − v(α)] (4.18)

and the minimum CTE is attained at dn,1 = · · · = dn,n = d∗ or at

h∗(x) =

n∑j=1

cn,j(x − d∗)+. (4.19)

(b) If α = ρ∗ < SX (0), then

minDn

{CTETh(X)(dn,1, . . . , dn,n, α)} = v(α) (4.20)

and the minimum CTE is attained at any (dn,1, . . . , dn,n) or at any

h∗(x) =

n∑j=1

cn,j(x − dn,j)+ (4.21)

provided that (dn,1, . . . , dn,n) satisfy d∗≤ dn,1 ≤ · · · ≤ dn,n. In

particular, theminimumCTE is attained at (d∗, . . . , d∗) or at h∗(x) =∑nj=1 cn,j(x − d∗)+.

(c) If α < SX (0) ≤ ρ∗, then

minDn

{CTETh(X)(dn,1, . . . , dn,n, α)}

= v(α) +

n∑j=1

cn,j[g(0) − v(α)] (4.22)

and the minimum CTE is attained at dn,1 = · · · = dn,n = 0 or at

h∗(x) =

n∑j=1

cn,jx. (4.23)

(d) For all other cases, minDn CTETh(X)(dn,1, . . . , dn,n, α) does notexist.

Finally, by comparing the minimum of CTETh(X)(dn,1, . . . , dn,n,α) for each n, we obtain the solutions to the optimization problem(4.1). Lemma 4.2, in turn, asserts that these solutions are also theoptimal ceded loss functions in the classF and hence the solutionsto our proposed optimal reinsurance model (1.6). The results aresummarized in the following theorem.

Theorem 4.1. For a given confidence level 1 − α with 0 < α <SX (0):

(a) If α < ρ∗ < SX (0), then minf∈F CTETf (X)(α) = u(ρ∗) andthe minimum CTE is attained at

f ∗(x) = (x − d∗)+. (4.24)

(b) If α = ρ∗ < SX (0), then minf∈F CTETf (X)(α) = u(ρ∗) andthe minimum CTE is attained at any

f ∗(x) =

n∑j=1

cn,j(x − dn,j)+ ∈ H (4.25)

such that d∗≤ dn,1 ≤ · · · ≤ · · · ≤ dn,n and n = 1, 2, . . ..

(c) If α < SX (0) ≤ ρ∗, then minf∈F {CTETf (X)(α)} = u(ρ∗) andthe minimum CTE is attained at

f ∗(x) = x. (4.26)

Proof. The proof is based on Lemmas 4.2 and 4.4 and parallels theproof of Theorem 3.1. Note that all ceded loss functions given inthe theorem are included in the class H . Hence, by Lemma 4.2, weonly need to show the optimality of these ceded loss functions inthe class H ; i.e., for any h(x) =

∑nj cn,j(x − dn,j)+ ∈ H ,

VaRTh(X)(α) ≥ VaRTf ∗ (X)(α). (4.27)

(a) First, note that when α < ρ∗ < SX (0), we have

u(ρ∗) < v(α). (4.28)

Indeed, in this case, S−1X > S−1

X (ρ∗) ≡ d∗, which, together with(A.4), implies g(S−1

X (α)) ≥ g(d∗). Moreover, it follows from thedefinitions of the functions u(·), v(·), and g(·) that v(α) > u(α) =

g(S−1X (α)) and g(d∗) = u(ρ∗). Consequently, we obtain (4.28),


which, together with (4.18) of Lemma 4.4, implies for any functionh ∈ H ,

CTETh(X)(α) ≥ minDn

CTETh(X)(dn,1, . . . , dn,n, α)

= v(α) +

n∑j=1

cn,j[u(ρ∗) − v(α)] ≥ u(ρ∗). (4.29)

Furthermore, (4.18) and (4.19) lead to CTETf ∗ (X)(α) = u(ρ∗) when∑nj=1 cn,j = 1 in (4.19) or when f ∗(x) = (x − d∗)+. Hence, (4.29)

implies (4.27).(b) By (4.20) of Lemma 4.4, we have for any function h ∈ H ,


CTETh(X)(dn,1, . . . , dn,n, α) = v(α). (4.30)

Furthermore, by (4.20) and (4.21), CTETf ∗ (X)(α) = v(α) when f ∗

is of the form such that d∗≤ dn,1 ≤ · · · ≤ · · · ≤ dn,n, and

n = 1, 2, . . .. Hence, (4.30) implies (4.27).(c) By thedefinitions ofv(·) and g(·), wehave g(S−1

X (α)) < v(α)

and by (A.6) of Proposition A.2, we have g(0) ≤ g(S−1X (α)). Thus

g(0) < v(α), which together with (4.22) of Lemma 4.4, leads to


CTETh(X)(dn,1, . . . , dn,n, α)

= v(α) +

n∑j=1

cn,j[g(0) − v(α)] ≥ v(α), (4.31)

for any function h ∈ H . Furthermore, by (4.22) and (4.23),CTETf ∗ (X)(α) = v(α)when

∑nj=1 cn,j = 1 in (4.23) or when f ∗(x) =

x. Hence, (4.31) implies (4.27). �

5. Conclusion

It is well known that reinsurance can be an effective riskmanagement technique for insurers to transfer part of its risk tothe reinsurer. The key, however, hinges on an optimal choice ofreinsurance contracts. By formulating an optimization problemthat minimizes the VaR (or CTE) of the total cost of the reinsurer,this paper establishes the conditions for optimal reinsurancedesigns. In particular, depending on the confidence level 1 − αfor the risk measure and the safety loading ρ for the reinsurancepremium, we formally justified that in some cases, a stop-lossreinsurance is optimal while in some other cases, a quota-sharereinsurance or a change-loss reinsurance is optimal. It is also ofinterest to note that the conditions for optimal solutions for CTEare less restrictive than those for VaR.

It should be pointed out that a basic assumption in our proposedoptimal reinsurance model is adopting the expectation premiumprinciple for setting the reinsurance premium. In practice, thereexists a number of other premiumprinciples. Itwill be of interest toinvestigate the impact of these premium principles on the optimalreinsurance designs. We leave this for future research exploration.

Acknowledgments

Cai acknowledges research support from the Natural Sciencesand Engineering Research Council of Canada (NSERC). Tan thanksthe funding from the Cheung Kong Scholar Program (China),Canada Research Chairs Program and NSERC. Weng acknowledgesthe research support from the Institute of Quantitative Financeand Insurance and the CAS/SOA Ph.D. grant. The authors alsothank the anonymous referee for the constructive comments andsuggestions which resulted in a significantly improved paper.

Appendix

This appendix collects the proofs of some of the results statedin the previous sections. Other necessary propositions and lemmasare also presented and proved in this appendix.

Proposition A.1. (a) If function f is increasing and continuous, then

VaRf (X)(α) = f (VaRX (α)). (A.1)

(b) If functions fn and f are increasing and continuous and satisfylimn→∞ fn(x) = f (x) for all x ≥ 0, then

limn→∞

VaRfn(X)(α) = VaRf (X)(α). (A.2)

(c) CTE and VaR of X are related as

CTEX (α) = E[X |X ≥ VaRX (α)]

= VaRX (α) +

∫∞

VaRX (α)SX (x)dx

Pr{X ≥ VaRX (α)}. (A.3)

Proof. Parts (a) and (b) follow immediately from the definition ofVaR while for part (c), see Cai and Tan (2007). �

Proposition A.2. (a) If ρ∗ < SX (0), then the continuous functiong(x) defined in (2.9) is decreasing on (0, d∗) while increasing on(d∗, ∞) and satisfies

min0≤x≤a

{g(x)} = g(a) for 0 ≤ a ≤ d∗, (A.4)

min0≤x≤a

{g(x)} = g(d∗) = u(ρ∗) for d∗≤ a. (A.5)

(b) If ρ∗≥ SX (0), then the continuous function g(x) defined in

(2.9) is increasing on (0, ∞) and satisfies

min0≤x≤a

{g(x)} = g(0) for a ≥ 0. (A.6)

Proof. (a) The proof follows from the condition that if ρ∗ < SX (0),then g ′(x) < 0 for 0 < x < d∗ and g ′(x) > 0 for x > d∗.

(b) The proof follows from g ′(x) > 0 for x > 0 if ρ∗≥ SX (0).

�

Proof of Lemma 3.1. It is known that for any non-negativeincreasing convex function f defined on [0, ∞), there exists asequence of non-negative functions {hn, n = 1, 2, . . .} defined on[0, ∞) such that hn(x) =

∑nj=1 cn,j(x − dn,j)+ for some constants

cn,j ≥ 0 and dn,j ≥ 0 and limn→∞ hn(x) = f (x) from below for anyx ≥ 0, which implies that

hn(x) ≤ f (x) for all x ≥ 0 and n = 1, 2, . . . . (A.7)

See, for example, Case 1 of the proof of Theorem1.5.7 ofMüller andStoyan (2002), p 18.

For any f ∈ F , we have 0 ≤ f (x) ≤ x, which, together with(A.7), implies (3.4). Furthermore, it follows from (3.4) that for anyx > 0 and n = 1, 2, . . .,

0 ≤hn(x)x

=

n∑j=1

cn,j(x − dn,j)+

x≤ 1. (A.8)

Thus, letting x → ∞ in (A.8) yields 0 ≤∑n

j=1 cn,j ≤ 1 for all n =

1, 2, . . . .Finally, to show hn ∈ H , we just need to verify that cn,1 >

0, . . . , cn,n > 0 and 0 ≤ dn,1 ≤ · · · ≤ dn,n for all n = 1, 2, . . ..Since f ∈ F and f (x) ≡ 0 does not hold, there exists a positiveinteger n0 so that when n ≥ n0, there is at least one of cn,1, . . . , cn,nwhich is positive. Hence, for any n ≥ n0, we delete the termcn,j(x − dn,j)+ from hn(x) when cn,j = 0, and relabel the rest ofthe coefficients of {(cn,j, dn,j)} such that {dn,j} is in an increasingorder. Thus, the sequence of the functions {hn, n ≥ n0} satisfiesthe requirements for the lemma and this completes the proof ofthe lemma. �


Proof of Lemma 3.4. We will first establish the fact that VaRTh(X)

(dn,1, . . . , dn,n, α) has minimum on Dn if and only if

minDnn

{VaRTh(X)(dn,1, . . . , dn,n, α)}

≤ mini=0,1,...,n−1

{infDin

{VaRTh(X)(dn,1, . . . , dn,n, α)}

}. (A.9)

To this end, we consider the value of infDin{VaRTh(X)(dn,1, . . . , dn,n,

α)} if it is attainable on the indicated set Din for i = 0, 1, 2, . . . , n.

Note that∫

∞

d SX (x)dx is a decreasing function in d ≥ 0 and∫∞

d SX (x)dx → 0 as d → ∞. Thus, it follows from (3.8) that

infD0n

{VaRTh(X)(dn,1, . . . , dn,n, α)} = VaRTh(X)(∞, . . . ,∞, α)

= S−1X (α). (A.10)

ForDin, i = 1, 2, . . . , n, we consider the following three possible

situations:

(i) When ρ∗ < SX (0) and d∗= S−1

X (ρ∗) ≤ S−1X (α), then

from (A.5) of Proposition A.2, (3.9) and (3.10), we have fori = 1, . . . , n − 1,

infDin

{VaRTh(X)(dn,1, . . . , dn,n, α)}

= VaRTh(X)(d∗, . . . , d∗︸︷︷︸i variants

, ∞, . . . ,∞, α)

=

(1 −

i∑j=1

cn,j

)S−1X (α) +

i∑j=1

cn,j g(d∗)

= S−1X (α) +

[u(ρ∗) − S−1

X (α)] i∑

j=1

cn,j, (A.11)

and for i = n,

minDnn

{VaRTh(X)(dn,1, . . . , dn,n, α)} = VaRTh(X)(d∗, . . . , d∗, α)

=

(1 −

n∑j=1

cn,j

)S−1X (α) +

n∑j=1

cn,j g(d∗)

= S−1X (α) +

[u(ρ∗) − S−1

X (α)] n∑

j=1

cn,j. (A.12)

(ii) If ρ∗ < SX (0) and d∗= S−1

X (ρ∗) > S−1X (α), then from (A.4) of

Proposition A.2, (3.9) and (3.10), we have for i = 1, . . . , n−1,

infDin

{VaRTh(X)(dn,1, . . . , dn,n, α)}

= VaRTh(X)(S−1X (α), . . . , S−1

X (α)︸︷︷︸i variants

, ∞, . . . ,∞, α)

=

(1 −

i∑j=1

cn,j

)S−1X (α) +

i∑j=1

cn,j g(S−1X (α))

= S−1X (α) +

[u(α) − S−1

X (α)] i∑

j=1

cn,j, (A.13)

and for i = n,

minDnn

{VaRTh(X)(dn,1, . . . , dn,n, α)}

= VaRTh(X)(S−1X (α), . . . , S−1

X (α), α)

=

(1 −

n∑j=1

cn,j

)S−1X (α) +

n∑j=1

cn,j g(S−1X (α))

= S−1X (α) +

[u(α) − S−1

X (α)] n∑

j=1

cn,j. (A.14)

(iii) Whenρ∗≥ SX (0), then by Proposition A.2 (b), (3.9) and (3.10),

we have for i = 1, . . . , n − 1,

infDin

{VaRTh(X)(dn,1, . . . , dn,n, α)}

= VaRTh(X)(0, . . . , 0︸︷︷︸i variants

, ∞, . . . ,∞, α)

=

(1 −

i∑j=1

cn,j

)S−1X (α) +

i∑j=1

cn,j g(0)

= S−1X (α) +

[g(0) − S−1

X (α)] i∑

j=1

cn,j, (A.15)

and for i = n,

minDnn

{VaRTh(X)(dn,1, . . . , dn,n, α)} = VaRTh(X)(0, . . . , 0, α)

=

(1 −

n∑j=1

cn,j

)S−1X (α) +

n∑j=1

cn,j g(0)

= S−1X (α) +

[g(0) − S−1

X (α)] n∑

j=1

cn,j. (A.16)

Combining all the above, we immediately see that for any h(x) =∑nj=1 cn,j (x − dn,j)+ ∈ H with fixed coefficients cn,j, j = 1, . . . , n,

VaRTh(X)(dn,1, . . . , dn,n, α) has a minimum only on Dnn and has

infimum but no minimum on all other sets D0n,D

1n, . . . ,D

n−1n . This

implies that VaRTh(X)(dn,1, . . . , dn,n, α) has aminimumonDn if andonly if (A.9) holds.

Note that when the above inequality (A.9) holds then

minDn

{VaRTh(X)(dn,1, . . . , dn,n, α)}

= minDnn

{VaRTh(X)(dn,1, . . . , dn,n, α)} (A.17)

since Dn =⋃n

i=1 Din. In addition, we point out that

g(0) =1ρ∗

E[X] >1ρ∗

∫ S−1X (α)

0SX (x)dx >

α

ρ∗S−1X (α)

and u(ρ∗) > S−1X (ρ∗). Hence, if α ≥ ρ∗, then g(0) > S−1

X (α) andu(ρ∗) > S−1

X (α). These discussions are useful in proving our resultsin the lemma. Now we are ready to prove the results.

(a) Together with (2.10), the condition of S−1X (α) ≥ u(ρ∗)

implies the inequality d∗= S−1

X (ρ∗) < S−1X (α). Thus, if (3.11)

holds, noticing that 0 <∑i

j=1 cn,j <∑n

j=1 cn,j ≤ 1 for i =

1, . . . , n − 1, we see that the minimum in (A.12) is less than orequal to any of the infimums in (A.10) and (A.11) since u(ρ∗) −

S−1X (α) ≤ 0. This implies that inequality (A.9) holds and so does(A.17). Consequently, it follows from (A.12) that

minDn

VaRTh(X)(dn,1, . . . , dn,n, α) = minDnn

VaRTh(X)(dn,1, . . . , dn,n, α)

= VaRTh(X)(d∗, . . . , d∗, α)

= S−1X (α) +

n∑j=1

cn,j[u(ρ∗) − S−1

X (α)],

where VaRTh(X)(d∗, . . . , d∗, α) means that the minimum VaR isattained at dn,1 = · · · = dn,n = d∗, and this proves (3.13).


(b) Condition (3.14) implies g(0) − S−1X (α) ≤ 0. Then by

comparing (A.16) with (A.10) and (A.15) and using a similarargument as in the proof of Part (a), it is easy to conclude that

minDn

VaRTh(X)(dn,1, . . . , dn,n, α) = minDnn

VaRTh(X)(dn,1, . . . , dn,n, α)

= VaRTh(X)(0, . . . , 0, α)

= S−1X (α) +

n∑j=1

cn,j[g(0) − S−1

X (α)],

where VaRTh(X)(0, . . . , 0, α) means that the minimum VaR isattained at dn,1 = · · · = dn,n = 0. Hence, (3.16) holds.

(c) For all other cases, it is easy to check that (A.9) does not holdand hence VaRTh(X)(dn,1, . . . , dn,n, α) has no minimum on Dn. �

Proof of Lemma 4.3. (a)When S−1X (α) ≤ dn,1, it follows from (2.6)

that VaRIh(X)(α) = S−1X (α) ≤ dn,1. Thus, by (2.5) and after some

algebras, it is not difficult to show that∫∞

VaRIh(X)(α)

SIh(X)(x)dx =

∫∞

S−1X (α)

SIh(X)(x)dx

=

∫ dn,1

S−1X (α)

SX (x)dx +

n−1∑i=1

∫ An,i dn,i+1+Bn,i

An,i dn,i+Bn,iSX

(x − Bn,i

An,i

)dx

+

∫∞

An,n dn,n+Bn,nSX

(x − Bn,n

An,n

)dx

=

∫∞

S−1X (α)

SX (x)dx −

n∑j=1

cn,j

∫∞

dn,jSX (x)dx.

Moreover, by (2.4) and noticing that Ih(X) is increasing in X , wehave

Pr{Ih(X) ≥ VaRIh(X)(α)} = Pr{Ih(X) ≥ S−1X (α)}

= Pr{X ≥ S−1X (α)} = α.

Together with (4.13), (2.3) and (2.6), we obtain (4.15).(b) When dn,i ≤ S−1

X (α) ≤ dn,i+1, it follows from (2.6) that

VaRIh(X)(α) = An,i S−1X (α) + Bn,i (A.18)

and An,i dn,i + Bn,i ≤ An,i S−1X (α) + Bn,i ≤ An,i dn,i+1 + Bn,i. Hence

by (2.5), we have∫∞

VaRIh(X)(α)

SXIh(X)(x)dx =

∫∞

An,i S−1X (α)+Bn,i

SIh(X)(x)dx

=

∫ An,i dn,i+1+Bn,i

An,i S−1X (α)+Bn,i

SX

(x − Bn,i

An,i

)dx

+

n−1∑j=i+1

∫ An,j dn,j+1+Bn,j

An,j dn,j+Bn,jSX

(x − Bn,j

An,j

)dx

+

∫∞

An,n dn,n+Bn,nSX

(x − Bn,n

An,n

)dx

= An,i

∫∞

S−1X (α)

SX (x)dx −

n∑j=i+1

cn,j

∫∞

dn,jSX (x)dx. (A.19)

Note that 0 < An,i < 1 for i = 1, . . . , n − 1. Furthermore,substituting (2.3), (A.18) and (A.19) and the following result

Pr{Ih(X) ≥ VaRIh(X)(α)} = Pr{Ih(X) ≥ An,i S−1X (α) + Bn,i}

= Pr{An,i X + Bn,i ≥ An,i S−1X (α) + Bn,i}

= Pr{X ≥ S−1X (α)} = α

into (4.13), we obtain the required expression (4.16).

(c) When dn,n ≤ S−1X (α), it follows from (2.6) that

VaRIh(X)(α) = An,n S−1X (α) + Bn,n (A.20)

and An,n S−1X (α) + Bn,n ≥ An,n dn,n + Bn,n. Hence, by (2.5)∫

∞

VaRIh(X)(α)

SIh(X)(x)dx =

∫∞

An,n S−1X (α)+Bn,n

SX

(x − Bn,n

An,n

)dx

= An,n

∫∞

S−1X (α)

SX (x)dx.

Note that 0 ≤ An,n < 1. If An,n = 0, then SX(

x−Bn,nAn,n

)= 0 so that∫

∞

VaRIh(X)(α)

SIh(X)(x)dx = 0. (A.21)

Moreover, in this case it follows from (2.4) and (2.6)

Pr{Ih(X) ≥ VaRIh(X)(α)}

=

{Pr{An,n X + Bn,n ≥ An,n S−1

X (α) + Bn,n}, if 0 < An,n < 1Pr{Ih(X) ≥ Bn,n}, if An,n = 0

=

{Pr{X ≥ S−1

X (α)} = α, if 0 < An,n < 1Pr{X ≥ dnn}, if An,n = 0 (A.22)

Substituting (2.3) and (A.20)–(A.22) into (4.13) for cases of 0 <An,n < 1 and An,n = 0, respectively, we obtain (4.17). �

Proof of Lemma 4.4. The proof is similar to Lemma3.4 by noticingthat

∫∞

d SX (x)dx is a decreasing function in d ≥ 0,∫

∞

d SX (x)dx → 0as d → ∞, and applying Proposition A.2 to the CTEs in Lemma 4.3.The details are as follows.

(a) When α < ρ∗, we have 1ρ∗ −

1α

< 0 and d∗= S−1

X (ρ∗) <

S−1X (α). Thus, it follows from (4.15) that

minD0n

{CTETh(X)(dn,1, . . . , dn,n, α)}

= CTETh(X)(S−1X (α), . . . , S−1

X (α), α)

= v(α) +

[1ρ∗

−1α

] n∑j=1

cn,j

∫∞

S−1X (α)

SX (x)dx. (A.23)

Furthermore, by (A.5), (4.16) and (4.17), we have for i =

1, . . . , n − 1,

minDin

{CTETh(X)(dn,1, . . . , dn,n, α)}

= CTETh(X)(d∗, . . . , d∗︸︷︷︸i variants

, S−1X (α), . . . , S−1

X (α), α)

=

(1 −

i∑j=1

cn,j

)v(α) +

i∑j=1

cn,j g(d∗)

+

[1ρ∗

−1α

] n∑j=i+1

cn,j

∫∞

S−1X (α)

SX (x)dx

= v(α) +[u(ρ∗) − v(α)

] i∑j=1

cn,j

+

[1ρ∗

−1α

] n∑j=i+1

cn,j

∫∞

S−1X (α)

SX (x)dx, (A.24)

and for i = n,

minDnn

{CTETh(X)(dn,1, . . . , dn,n, α)} = CTETh(X)(d∗, . . . , d∗, α)


=

(1 −

n∑j=1

cn,j

)v(α) +

n∑j=1

cn,j g(d∗)

= v(α) +[u(ρ∗) − v(α)

] n∑j=1

cn,j. (A.25)

Note that the condition α < ρ∗ implies

u(ρ∗) − v(α) = S−1X (ρ∗) − S−1

X (α)

+

[1ρ∗

−1α

] ∫∞

S−1X (α)

SX (x)dx < 0. (A.26)

It is therefore easy to see that the minimum in (A.25) is less thanany of the minimums in (A.23) and (A.24), which leads to

minDn

{CTETh(X)(dn,1, . . . , dn,n, α)}

= mindn∈Dn

n{CTETh(X)(dn,1, . . . , dn,n, α)}.

Hence this justifies both (4.18) and (4.19).(b) When α = ρ∗, we have d∗

= S−1X (ρ∗) = S−1

X (α) andu(ρ∗) = g(d∗) = v(α) so that in this case, (4.15) reduces to

CTETh(X)(dn,1, . . . , dn,n, α) = v(α) (A.27)

for any (dn,1, . . . , dn,n) ∈ D0n. Furthermore, by (A.5), (4.16) and

(4.17), we have for i = 1, . . . , n − 1,

minDin

{CTETh(X)(dn,1, . . . , dn,n, α)}

= CTETh(X)(d∗, . . . , d∗︸︷︷︸i variants

, dn,i+1, . . . , dn,n, α)

=

(1 −

i∑j=1

cn,j

)v(α) +

i∑j=1

cn,j g(d∗) = v(α), (A.28)

and for i = n,

minDnn

{CTETh(X)(dn,1, . . . , dn,n, α)} = CTETh(X)(d∗, . . . , d∗, α)

=

(1 −

n∑j=1

cn,j

)v(α) +

n∑j=1

cn,j g(d∗)

= v(α), (A.29)

where (A.28) holds for any dn,i+1, . . . , dn,n such that d∗=

S−1X (α) ≤ dn,i+1 ≤ · · · ≤ dn,n.Thus, CTETh(X)(dn,1, . . . , dn,n, α) has the sameminimumof v(α)

on all sets Din, i = 0, 1, . . . , n. Hence, minDn CTETh(X)(dn,1, . . . ,

dn,n, α) = v(α) and the minimum CTE is attained at any

(dn,1, . . . , dn,n) ∈ D0n

⋃(n−1⋃i=1

{(d∗, . . . , d∗, dn,i+1, . . . , dn,n) :

d∗≤ dn,i+1 ≤ · · · ≤ dn,n}

)⋃{(d∗, . . . , d∗)}.

Since d∗= S−1

X (α) in this case, we obtain

D0n

⋃(n−1⋃i=1

{(d∗, . . . , d∗, dn,i+1, . . . , dn,n) :

d∗≤ dn,i+1 ≤ · · · ≤ dn,n}

)⋃{(d∗, . . . , d∗)} = D0

n,

which consists of all (dn,1, . . . , dn,n) such that d∗≤ dn,1 ≤ · · · ≤

dn,n. This completes the proof of case (b).

(c) In this case,α < ρ∗ implies 1ρ∗ −

1α

< 0 and d∗= S−1

X (ρ∗) <

S−1X (α). Thus, it follows from (4.15) that

minD0n

{CTETh(X)(dn,1, . . . , dn,n, α)}

= CTETh(X)(S−1X (α), . . . , S−1

X (α), α)

= v(α) +

[1ρ∗

−1α

] n∑j=1

cn,j

∫∞

S−1X (α)

SX (x)dx. (A.30)

Furthermore, by (A.6), (4.16) and (4.17), we have for i = 1, . . . ,n − 1,

minDin

{CTETh(X)(dn,1, . . . , dn,nα)}

= CTETh(X)(0, . . . , 0︸︷︷︸i variants

, S−1X (α), . . . , S−1

X (α), α)

=

(1 −

i∑j=1

cn,j

)v(α) +

i∑j=1

cn,j g(0)

+

[1ρ∗

−1α

] n∑j=i+1

cn,j

∫∞

S−1X (α)

SX (x)dx

= v(α) + [g(0) − v(α)]i∑

j=1

cn,j

+

[1ρ∗

−1α

] n∑j=i+1

cn,j

∫∞

S−1X (α)

SX (x)dx, (A.31)

and for i = n,

minDnn

{CTETh(X)(dn,1, . . . , dn,n, α)} = CTETh(X)(0, . . . , 0, α)

=

(1 −

n∑j=1

cn,j

)v(α) +

n∑j=1

cn,j g(0) = v(α)

+

n∑j=1

cn,j [g(0) − v(α)] . (A.32)

Now, note that from the condition α < SX (0) ≤ ρ∗, we have1ρ∗

∫ S−1X (α)

0 SX (x)dx − S−1X (α) <

(1ρ∗

)S−1X (α)SX (0) − S−1

X (α) ≤ 0

and 1ρ∗ −

1α

< 0. This leads to

g(0) − v(α) = (1 + ρ)E[X] − S−1X (α) −

1α

∫∞

S−1X (α)

SX (x)dx

=1ρ∗

∫ S−1X (α)

0SX (x)dx − S−1

X (α)

+

[1ρ∗

−1α

] ∫∞

S−1X (α)

SX (x)dx < 0.

So it is easy to conclude that theminimum in (A.32) is less than anyof the minimums in (A.30) and (A.31). Consequently, we have

minDn

{CTETh(X)(dn,1, . . . , dn,n, α)}

= minDnn

{CTETh(X)(dn,1, . . . , dn,n, α)}

which confirms both (4.22) and (4.23).(d) It is easy to verify that mindn∈Dn{CTETh(X)(dn,1, . . . , dn,n, α)}

does not exist for all other cases. �

Lemma A.1. For any f (x) ∈ F , If (x) = x − f (x) is increasing andconcave in x.


Proof. The concavity of If (x) comes immediately from the fact thatf (x) is convex. Now suppose there exist two points x1 and x2 suchthat 0 ≤ x1 < x2 satisfying If (x1) − If (x2) > 0, i.e.,

f (x2) − f (x1)x2 − x1

> 1. (A.33)

Moreover, it follows from the convexity of f (x) that f (x2) ≤x−x2x−x1

f (x1) +x2−x1x−x1

f (x) for x ≥ x2, or equivalently f (x) ≥

f (x2)−f (x1)x2−x1

x+x2f (x1)−x1f (x2)

x2−x1. Hence, it follows from (A.33) that there

exists a constant x0 such that f (x0) > x0, which contradicts theassumption that f (x) ≤ x for all x ≥ 0. Therefore we conclude thatIf (x) is increasing. �

Lemma A.2. If If (x) is an increasing concave function, then thedistribution function of If (x), FIf (X)(x), has at most one discontinuityon (0, ∞). If such a discontinuity exists, then

If (x) =

{g(x), 0 < x ≤ e0,g(e0), x > e0,

(A.34)

for some strictly increasing function g(x) and some constant e0 ∈

(0, ∞), and g(e0) is the only discontinuity of FIf (X)(x).

Proof. Since If (x) is increasing concave, If (x) must be eitherstrictly increasing on [0, ∞) or has the following representationof (A.34) for some strictly increasing function g and some constante0 ∈ (0, ∞). Otherwise, if there exists an interval [x1, x2] such thatIf (x) is constant while strictly increasing on [x2, x2 + ε) (for anyreal number ε > 0), then there exists a real number 0 < δ < εsuch that

δ

δ + x2 − x1If (x1) +

x2 − x1δ + x2 − x1

If (δ + x2)

=δ

δ + x2 − x1If (x2) +

x2 − x1δ + x2 − x1

If (δ + x2)

>δ

δ + x2 − x1If (x2) +

x2 − x1δ + x2 − x1

If (x2)

= If (x2),

which violates the concavity of the function If (x).If If (x) is strictly increasing globally, by the assumption that

FX (x) is a one-to-one distribution function on (0, ∞), then FIf (X)(x)has no discontinuous point. On the other hand, if If (x) takes theform of (A.34), then

FIf (X)(x) =

FX (0), x = 0,FX (g−1(x)), 0 < x ≤ g(e0),1, x > g(e0).

Therefore g(e0) is the only discontinuous point on (0, ∞). �

Proof of Lemma 4.1. We verify the result by considering twocases respectively. First of all, we suppose the distribution functionof If (x), as denoted by FIf (X)(x), is continuous at VaRIf (X)

(α). ByLemma 3.1, there exists a sequence of functions hn(x) ∈ H suchthat limn→∞ hn(x) → f (x) for all x ≥ 0 and hence it follows fromProposition A.1(b) and the fact that hn(x) and f (x) are increasingand continuous that

limn→∞

VaRIhn (X)(α) = VaRIf (X)(α).

Consequently, by the dominated convergence theorem we obtain

Ihn(X) − VaRIhn (X)(α)a.s.

−→ If (X) − VaRIf (X)(α),

which immediately leads to (4.3).Next we suppose VaRIf (X)(α) is one discontinuity of FIf (X)

(x).By Lemma A.1, If (x) is increasing concave in x, and hence it

follows from Lemma A.2 that VaRIf (X)(α) is the only discontinuityof FIf (X)(x), and

If (x) =

{g(x), 0 < x ≤ e0,g(e0), x > e0,

(A.35)

for some strictly increasing and continuous function g(x) and aconstant e0 such that g(e0) = VaRIf (X)(α). Consequently

Pr{If (X) ≥ VaRIf (X)(α)} = Pr{If (X) ≥ g(e0)} = Pr{X ≥ e0}.

To verify the result in this case, we adopt the technique ofconstructing a sequence of functions hn(x) in class H satisfying(4.2) and (4.3) as follows.

Denote x′

2n,i = i · e0/2n, a′

2n,i = f ′+(x′

2n,i), and li(x) =

a′

2n,i(x − b′

2n,i) with b′

2n,i ∈ R, to be a tangent curve of f (x), i =

0, 1, 2, . . . , 2n. Then,

b′

2n,i = x′

2n,i −f (x′

2n,i)

f ′+(x′

2n,i)=

i2n

e0 −f ( i

2n e0)

f ′+( i

2n e0)

and l2n(x) = x − g(e0) = f (x), i.e. a′

2n,2n = f ′+(e0) = 1, b′

2n,2n =

g(e0) = e0−f (e0). Note that a′

2n,i−1 < a′

2n,i−1 ≤ 1, i = 1, 2, . . . , 2n

due to the assumption that f (x) is strictly increasing convex oninterval (0, e0] and the fact that 0 ≤ f ′

+(x) ≤ 1. Take

hn(x) = max{0, l1(x), l2(x), . . . , l2n−1(x), l2n(x)}.

Then for any given ε > 0, it follows from the continuity of f (x) andthe construction of hn(x) that there exists a large enough integern such that |f (x) − li(x)| ≤

ε2 and |hn(x) − li(x)| ≤

ε2 for x ∈

[x′

2n,i, x′

2n,i+1), i = 0, 1, 2, . . . , 2n− 1, and hn(x) = l2n(x) = f (x)

for x ∈ [e0, ∞). Hence it follows

limn→∞

hn(x) → f (x) for all x ≥ 0.

Furthermore, hn(x) can be rewritten as follows.

hn(x) =

2n∑i=1

(li(x) − li−1(x))+

=

2n∑i=1

(a′

2n,i − a′

2n,i−1)

(x −

b′

2n,ia′

2n,i − b′

2n,i−1a′

2n,i−1

a′

2n,i − a′

2n,i−1

)+

=

2n∑i=1

c2n,i(x − d2n,i)+,

where

c2n,i = a′

2n,i − a′

2n,i−1, and d2n,i =b′

2n,ia′

2n,i − b′

2n,i−1a′

2n,i−1

a′

2n,i − a′

2n,i−1,

for i = 1, 2, . . . , 2n. Obviously,∑2n

i=1 c2n,i =∑2n

i=1(a′

2n,i −

a′

2n,i−1) = a′

2n,2n = 1 with c2n,i > 0. Due to the convexity of f (x),we have

b′

2n,ia′

2n,i − b′

2n,i−1a′

2n,i−1 =

[i2n

e0 −f( i2n e0

)f ′+( i

2n e0)

]f ′

+

(i2n

e0

)

−

[i − 12n

e0 −f ( i−1

2n e0)

f ′+( i−1

2n e0)

]f ′

+

(i − 12n

e0

)=

i − 12n

e0

[f ′

+

(i2n

e0

)− f ′

+

(i − 12n

e0

)]+

e02n

[f ′

+

(i2n

e0

)−

f ( i2n e0) − f ( i−1

2n e0)e0/2n

]> 0

which means d2n,i > 0 for i = 1, 2, . . . , 2n. Therefore hn(x) ∈ H .


Now concentrate on d2n,i, which can be expressed as follows.

d2n,i =i2n

e0 +

e02n f

′+( i−1

2n e0) −[f ( i

2n e0) − f ( i−12n e0)

]f ′+( i

2n e0) − f ′+( i−1

2n e0). (A.36)

By the convexity of f (x) we have

d2n,i − d2n,i−1 =e02n

+

e02n

[f ′+( i−1

2n e0) −f ( i

2n e0)−f ( i−12n e0)

e0/2n

]f ′+( i

2n e0) − f ′+( i−1

2n e0)

+

e02n

[f ( i−1

2n e0)−f ( i−22n e0)

e0/2n− f ′

+( i−2

2n e0)]

f ′+( i−1

2n e0) − f ′+( i−2

2n e0)

=

e02n

[f ′+( i2n e0) −

f ( i2n e0)−f ( i−1

2n e0)e0/2n

]f ′+( i

2n e0) − f ′+( i−1

2n e0)

+

e02n

[f ( i−1

2n e0)−f ( i−22n e0)

e0/2n− f ′

+( i−2

2n e0)]

f ′+( i−1

2n e0) − f ′+( i−2

2n e0)> 0,

which means d2n,i is increasing in i and hence d2n,2n is themaximum among {d2n,i, i = 1, 2, . . . , n}. As a result, we know thatIhn can be represented in the form of (A.35) by replacing e0 withd2n,2n , i.e.,

Ihn(x) =

{k(x), 0 < x ≤ d2n,2n ,k(d2n,2n), x > d2n,2n ,

(A.37)

for some strictly increasing and continuous function k(x). More-over, it follows from (A.36) and the convexity of f (x) that

d2n,2n = e0 +

e02n f

′+( 2n−1

2n e0) −

[f (e0) − f ( 2n−1

2n e0)]

1 − f ′+( 2n−1

2n e0)≤ e0, (A.38)

and

d2n,2n = e0 +

e02n f

′+( 2n−1

2n e0) −

[f (e0) − f ( 2n−1

2n e0)]

1 − f ′+( 2n−1

2n e0)

= e0 −

e02n

[1 − f ′

+( 2n−1

2n e0)]

1 − f ′+( 2n−1

2n e0)

+

e02n f

′+(e0) −

[f (e0) − f ( 2n−1

2n e0)]

1 − f ′+( 2n−1

2n e0)

≥ e0 −e02n

. (A.39)

Combining (A.38) and (A.39) immediately yields

limn→∞

d2n,2n = e0.

Now turn to investigate Pr{Ihn(X) ≥ VaRIhn (X)(α)}. From (A.37)and (A.38) we have

Pr{Ihn(X) ≥ Ihn(d2n,2n)} = Pr{X ≥ d2n,2n} ≥ Pr{X ≥ e0} ≥ α,

and

Pr{Ihn(X) > Ihn(d2n,2n)} = 0.

These results in turn imply VaRIhn (X)(α) = Ihn(d2n,2n), andconsequently,

limn→∞

Pr{Ihn(X) ≥ VaRIhn (X)(α)} = limn→∞

Pr{Ihn(X) ≥ Ihn(d2n,2n)}

= limn→∞

Pr{X ≥ d2n,2n}

= Pr{X ≥ e0} = Pr{If (X) ≥ VaRIf (X)(α)}.

This completes the proof. �

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