Marginal Indemnification Formulation for Optimal
Reinsurance
Sheng Chao Zhuang† , Chengguo Weng†∗, Ken Seng Tan† , Hirbod Assa‡
†
Department of Statistics and Actuarial Science
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
‡
Institute for Financial and Actuarial Mathematics
University of Liverpool, Peach Street, Liverpool, L69 7ZL, UK
July 3, 2015
Abstract
In this paper, we propose to combine the Marginal Indemnification Formulation (MIF) and
the Lagrangian dual method to solve optimal reinsurance model with distortion risk measure
and distortion reinsurance premium principle. The MIF method exploits the absolute continuity
of admissible indemnification functions and formulates optimal reinsurance model into a linear
programming of determining an optimal measurable [0, 1]-valued function. The MIF method
was recently introduced to analyze the reinsurance model but without premium budget constraint. In this paper, a Lagrangian dual method is developed to combine with MIF to solve for
optimal reinsurance solutions under premium budget constraint. Compared with the existing
literature, our proposed integrated MIF-based Lagrangian dual method provides a more technically convenient and transparent solution to the optimal reinsurance design. To demonstrate the
practicality of our proposed method, analytical solution is derived on a particular reinsurance
model that involves minimizing Conditional Value at Risk (a special case of distortion function) and with the reinsurance premium being determined by the inverse-S shaped distortion
principle.
JEL Classification: C61; G22; G32.
Key Words: optimal reinsurance; marginal indemnification function; Lagrangian dual method;
distortion risk measure; inverse-S shaped distortion premium principle.
∗
Corresponding author. Tel: +001(519)888-4567 ext. 31132.
Email: [email protected]
1
Introduction
Reinsurance, if exploited appropriately, can be an effective risk management tool for an insurer.
In a typical reinsurance contract, the insurer pays a certain amount of premium to a reinsurer in
return for some indemnification when losses occur from a designated risk. This indemnification is
always designed to be a function of the risk, and the premium is determined by this indemnification
function and a given premium principle. From the insurer’s perspective, the more the stipulated
indemnification, the less risk the insurer retains, and the more premium the insurer needs to pay.
The optimal reinsurance is to strike a balance between the risk retaining and risk transferring in an
optimal way to minimize its total risk.
Two pioneering works on optimal reinsurance are attributed to Borch (1960) and Arrow (1963).
Borch (1960) demonstrated that the stop-loss reinsurance is the best contract if the insurer measures
risks by variance and the reinsurer prices risks by the expected value premium principle. Arrow
(1963) also showed that the stop-loss reinsurance is an optimal one if the insurer is an expected
utility maximizer under the assumption of the expected value premium principle. These fundamental
results have been generalized in a number of interesting and important directions. Just to name a few,
Kaluszka (2001) extended the Borch’s result by considering the mean-variance premium principle,
while Young (1999) elaborated Arrow’s result by taking Wang’s premium principle into account.
More recently, there is a surge of interest in formulating the optimal reinsurance problem involving
more sophistical risk measures such as Value at Risk (VaR) and Conditional Value at Risk (CVaR).
Fo example, Cai and Tan (2007), Cai et al. (2008), Cheung (2010) and Tan et al. (2011) discussed
the minimization of VaR and CVaR of the insurer’s total risk exposure with expected value premium
principle. Cheung et al. (2014) further explored Tan et al. (2011)’s results under the general lawinvariant convex risk measure. Chi and Tan (2013), and Chi and Weng (2013) have considered VaR
and CVaR with a wide class of premium principles which preserve the convex ordering. Zheng et al.
(2014) designed the optimal reinsurance contract under distortion risk measure, but assuming that
the distortion function is piecewise concave or convex. Cui et al. (2013) studied a general model
involving the distortion risk measure and the distortion premium principle. As pointed out in Asimit
et al. (2013) that the approach of Cui et al. (2013) is not transparent and explicit. Assa (2015),
on the other hand, demonstrated that the optimal reinsurance model of Cui et al. (2013), without
the premium constraint, can be tackled more elegantly via a marginal indemnification formulation
(MIF).
The primary objective of the present paper is to extend the MIF-based method so as it can
be used to derive analytically the solution to the reinsurance model based on the distortion risk
measures but with the presence of the premium budget constraint. It is well-known that in many
optimization problems, the complexity of the optimization problem can be significantly increased by
merely imposing a constraint. In particular, one often discovers that while an optimization procedure
can be used to solved analytically an unconstrained optimization problem, the same procedure may
not be simply used to solve the same optimization problem but with an added constraint. This is
exactly the method of MIF proposed by Assa (2015). While in Assa (2015) MIF is used to derive
2
analytically the reinsurance model of Cui et al. (2013), without the premium constraint, the same
method can not be readily used if the premium budget constraint is imposed to the reinsurance
model.
The objective of this paper is to demonstrate that by integrating MIF with a Lagrangian method
one can design reinsurance policies for problems with budget constraints. Compared to the approach
of Cui et al. (2013) for solving the same reinsurance model with premium budget constraint, our
proposed integrated MIF and Lagranging method is much simpler and much more transparent.
More specifically, the approach of Cui et al. (2013) critically depends on a pre-conjectured candidate
solution. The method requires us to first guess an optimal solution and then apply certain comparison
analysis to prove its optimality. Their method, therefore, requires us to have a good understanding of
the shape of the optimal solution in order to justify its optimality. Such approach is clearly unrealistic,
particularly on a new reinsurance model where we have limited knowledge on the structure of the
optimal reinsurance treaty. In contrast, our integrated method does not require any preanalysis on
the shape of optimal solutions. Moreover, due to the nature of the procedure in searching for the
solution developed by Cui et al. (2013), it is even harder to discuss the uniqueness of solution to the
reinsurance model. On the other hand, the uniqueness of solutions of our proposed approach can be
easily established for certain cases, and the non-uniqueness of solutions can also be uncovered from
our optimization procedure.
To highlight the practicality of our proposed solution, we consider a particular reinsurance model
that minimizes CVaR (a special distortion risk measure) and with the premium being dictated by the
inverse-S shaped distortion (ISSD) premium principle. The ISSD premium principle is a distortion
premium principle with a distortion function such that it has derivative which changes from being
strictly decreasing to being strictly increasing derivative at certain point. Thus, it encompasses both
the concave and convex distortion premium principles as special cases. Indeed, as it will become
clear in Section 5, the optimal solutions for either a concave or a convex distortion premium principle
can be retrieved from those we obtained for the ISSD premium principle as special cases.
More interestingly, the ISSD premium principle enjoys an important economic interpretation
that the insurance provider may overweight not only large losses but also small ones in underwriting
the insurance risks. This is consistent with the empirically observed phenomena in psychological
experiments (Quiggin 1982, 1992; Tversky and Kahneman 1992; Tversky and Fox 1995; Gonzalez
and Wu, 1999). Furthermore, Kaluszka and Krzeszowiec (2012) introduce a premium principle from
the perspective of the Cumulative Prospect Theory (CPT). The ISSD premium principle can also
be viewed as a special CPT premium principle corresponding to a linear utility function and a zero
reference point. Unlike the concave distortion premium principle, the premium principle based on
ISSD has not received much attention in the insurance literature. This in part can be attributed
to the relatively new concept and its short history. Other reasons could be due to the possibility
that the ISSD introduces some technical hurdles, such as non-convex order property, for solving
optimization problems.
The rest of the paper proceeds as follows. The distortion risk measure minimizing reinsurance
model is specified in Section 2, and the MIF is introduced in Section 3. The Lagrangian dual
3
method is developed in Section 4 and the solutions to the general distortion risk measure minimizing
reinsurance model are derived. The in-depth analysis on the CVaR minimizing model with ISSD
premium principle is given in Section 5. Section 6 concludes the paper.
2
Model Setup
Throughout the paper, all the random variables involved in the paper are defined on a common
probability space (Ω, F, P). The indicator function is denoted by 1A (s), i.e., 1A (s) = 1 for s ∈ A and
1A (s) = 0 for s ∈
/ A.
Let X be the non-negative random variable representing the risk againt which the insurer seeks
reinsurance protection. The problem of optimal reinsurance is concerned with the optimal partitioning of X into f (X) and r(X) such that X = f (X) + r(X). Here f (X) represents the portion of
loss ceded to a reinsurer and r(X) is the residual loss retained by the insurer. The functions f and
r are respectively referred to as “indemnification function" (or “ceded loss function") and “retained
loss function". As in Chi and Tan (2013), Cui et al. (2013) and Assa (2015), among many others,
we consider the following admissible set of the ceded loss functions:
F , {f (·) : 0 6 f (x) − f (y) 6 x − y, ∀ 0 ≤ y < x ≤ M}.
(1)
where M , esssupX. The ceded loss functions in F are non-decreasing and any increment in
compensation is always less than or equal to the increment in loss, hence potentially reducing moral
hazard from both parties.
In a typical reinsurance treaty, the reinsurer charges a certain amount of premium to the insurer
in exchange of covering the part of loss f (X). Generally, this premium is positive and is a functional
of f (X), say Π(f (X)) for some functional Π. Consequently, the total risk exposure of the insurer in
the presence of reinsurance is given by
Tf = r(X) + Π(f (X)).
In the present paper, the optimal reinsurance is defined as those policies which minimize a certain
distortion risk measure of Tf over the feasible set F . The Value-at-Risk (VaR) and Conditional
Value-at-Risk (CVaR) are two prominent examples in the class of distortion risk measures, and their
definitions are given as follows.
Definition 2.1 The VaR and CVaR of a non-negative random variable Z at a confidence level α
(with 0 < α < 1) are respectively defined as
VaRα (Z) , inf{z ≥ 0 : P(Z > z) ≤ α},
and
1
CVaRα (Z) ,
α
Z
provided that the corresponding integral exists.
4
0
α
VaRs (Z)ds,
To introduce the distortion risk measure, we define the following set of functions:
o
n
G , g : [0, 1] → [0, 1]g(t) is non-decreasing in t, g(0) = 0 and g(1) = 1 .
Definition 2.2 A risk measure is called a distortion risk measure with distortion function g ∈ G,
denoted by ρg , if and only if it admits the following representation
Z 1
g
ρ (Z) ,
VaRs (Z)dg(s)
(2)
0
for random variables Z such that the resulting integral exists.
The following two properties are well known for distortion risk measures:
• Commonotonic Additivity: ρg (X + Y ) = ρg (X) + ρg (Y ) for two commonotonic random
variables X and Y ;
• Translation Invariance: ρg (Z + C) = ρg (Z) + C for any constant C and random variable Z
such that the resulting integral is well defined.
These two properties of the distortion risk measure will be used in our subsequent development.
The distortion risk measure is often alternatively defined in the form of a Choquet integral as
follows
Z ∞
Z 0
ρ(Z) =
g(1 − FZ (t))dt −
[g(FZ (t))] dt.
(3)
0
−∞
It is worth noting that (3) reduces to
g
ρ (Z) =
Z
∞
g(1 − FZ (t))dt
0
for nonnegative random variables Z. The equivalence between (2) and (3) can easily be verified via
a change-of-variable technique.
The distortion risk measure encompasses many interesting risk measures as its special cases. For
example, by setting g(·) = 1(α,1] (·) and g(s) = min{s/α, 1}, the resulting distortion risk measures
become respectively, VaR and CVaR. The family of spectral risk measures, which are defined by
R1
Acerbi (2002) and in a form of 0 VaRs (Z)φ(s)ds for some probability density φ on (0, 1), also
obviously fall into the set of distortion risk measures.
Based on the distortion function, we define the following general distortion premium principle:
Definition 2.3 The distortion premium principle is defined as
Z 1
g
π (Y ) , (1 + θ)
VaRs (Y )dg(s),
0
where the constant θ ≥ 0 and g ∈ G.
5
In the above definition, θ is the safety loading of the reinsurer. When g(x) = x, the distortion
premium principle recovers the expected value premium principle. Furthermore, when the distortion
function is concave with θ = 0, the distortion principle recovers Wang’s premium principle.
We now formulate the two optimal reinsurance models that we will analyze in this paper. These
models are described as follows:
Problem 2.1
(
min
ρg1 (Tf )
s.t.
π g2 (f (X)) ≤ π0 ,
f ∈F
(4)
where g1 and g2 are two distortion functions from G, π0 denotes the reinsurance premium budget. The
presence of the premium constraint signifies that the insurer is willing to seek reinsurance protection
as long as the cost does not exceed π0 . We assume π0 ∈ [0, π g2 (X)], because, from the definition of
F in (1), it is obvious that π g2 (f (X)) ∈ [0, π g2 (X)] for any f ∈ F .
As it will become clear shortly, when the reinsurance premium budget π0 is large enough, the
solution to (2.1) coincides with the following Problem 2.2 where no premium budget constraint is
imposed.
Problem 2.2
min
f ∈F
ρg1 (Tf ).
(5)
In principle, if we could solve Problem 2.1 for a general π0 ∈ [0, π g2 (X)], the solution to Problem
2.2 follows trivially by taking π0 = π g2 (X). Indeed, analyzing Problem 2.1 is much more technically
involved compared to Problem 2.1 due to the existence of the constraint. As pointed out earlier that
both models have been studied by Cui et al. (2013), where the derivation of optimal reinsurance
policy depends on a tedious characterization of the three sets of {t ∈ [0, 1] : g1 (t) < g2 (t)}, {t ∈
[0, 1] : g1 (t) = g2 (t)} and {t ∈ [0, 1] : g1 (t) > g2 (t)}, and a conjecture on the form of optimal ceded
loss function, which are nontrivial and mathematically abstruse. By using the method of MIF, Assa
(2015) provides a more elegant solution to Problem 2.2. We shall generalize Assa (2015)’s method
to study the model with a constraint in the rest of the paper.
3
Marginal Indemnification Formulation (MIF)
The method used in Assa (2015) makes full use of the absolute continuity property of the admissible ceded loss functions, and the procedure for the optimal solution is more readily comprehensible
compared with Cui et al. (2013). The absolute continuity enables one to equivalently represent each
element f ∈ F by
Z x
f (x) =
h(z)dz, x ≥ 0,
(6)
0
6
for some function h ∈ H , {h : R+ 7→ R+ |0 ≤ h ≤ 1 }. Here h(z) is the slope of the ceded loss
function f at z, and it can be interpreted as the “marginal indemnification" from an increase of
the loss X. Thus, the function h is referred to as a “marginal indemnification function (MIF)".
Obviously, two MIF’s only differing each other over a Borel set of zero Lebesgue measure result in
the same ceded loss function f everywhere.
Using the representation of f in (6) and the commonotonic additivity of the distortion risk
measure, a representation result can be developed for ρg (f (X)) for every g ∈ G and f ∈ F , as
asserted in Lemma 3.1.
Lemma 3.1 Given any g ∈ G and f ∈ F , there exists h ∈ H, independent of g, such that
Z x
Z ∞
g
f (x) =
h(z)dz and ρ (f (X)) =
g (1 − FX (z)) h(z)dz.
0
0
Proof: The proof follows from Fubini’s Theorem, in addition to the absolute continuity of f and
the commonotonic additivity of ρg . See Theorem 2 in Assa (2015) for detailed proof.
It follows from Lemma 3.1 that the reinsurance premium π g2 (f (X)) can be represented as
Z ∞
g2
g2
π (f (X)) = (1 + θ)ρ (f (X))) = (1 + θ)
g2 (1 − FX (z)) h(z)dz,
0
and the objective in Problem 2.1 can be rewritten as
ρg (Tf ) = ρg1 (X − f (X) + π g2 (f (X)))
= ρg1 (X) − ρg1 (f (X)) + (1 + θ)ρg2 (f (X)))
Z ∞
Z ∞
g1
= ρ (X) −
g1 (1 − (FX (z))h(z)dz + (1 + θ)
g2 (1 − FX (z))h(z)dz
0
0
Z ∞
g1
= ρ (X) −
ψ(FX (z))h(z)dz,
0
where
ψ(t) = g1 (1 − t) − (1 + θ)g2 (1 − t), t ∈ [0, 1].
(7)
R∞
Since ρg1 (X) is a constant, it suffices to analyze the term 0 ψ(FX (s))h(s)ds for optimal solution.
Using Lemma 3.1, Problems 2.1 and 2.2 can be respectively transformed into a MIF formulation
as follows:
Problem 3.1


 max
h∈H
where π1 = π0 /1 + θ;

 s.t.
Z
∞
U(h) ,
ψ(FX (z))h(z)dz
0
Z ∞
g2 (1 − FX (z)) h(z)dz ≤ π1 ,
0
7
(8)
Problem 3.2
max
h∈H
Z
∞
(9)
ψ(FX (s))h(s)ds.
0
Once we solved the above problems in a MIF formulation and derived an optimal MIF h∗ ∈ H,
Rx
an optimal ceded loss function f ∗ ∈ F can be obtained by f ∗ (x) = 0 h∗ (t)dt, t ≥ 0, in light of the
representation (6). Indeed, the optimal solution to Problem 3.2 can readily be shown to be


1,
if ψ(FX (z)) > 0,



a.e.
h(z) = κ(z),
(10)
if ψ(FX (z)) = 0,



0,
if ψ(FX (z)) < 0,
where κ can be any [0, 1]-valued function on {t ≥ 0 : ψ(FX (t)) = 0}. Thus, function f (x) =
for any h given in (10) solves Problem 2.2. In particular,
solves (9), and
Rx
0
h(z)dz
e
h(s) , 1{ψ(FX (s))>0}
(11)
fe(x) = µ({s ∈ [0, x] |ψ(FX (z)) > 0}), x ≥ 0,
(12)
is one solution to Problem 2.2, where µ denotes the Lebesgue measure.
Remark 3.1 As characterized in Proposition 2.1 in Cui et al. (2013), the three sets {t ∈ [0, 1] :
ψ(t) > 0}, {t ∈ [0, 1] : ψ(t) = 0} and {t ∈ [0, 1] : ψ(t) < 0} involved in (10) can be respectively
represented as the union of countably many disjoint intervals:
(l
) (l
)
1
2
[
[ [
{t : ψ(t) > 0} =
qi− , qi+
qbj− , qbj+ ,
{t : ψ(t) = 0} =
i=1
j=1
(m
[1
) (m
[ [2
+
s−
i , si
(n
1
[
) (n
)
2
[ [
+
b
b+ ,
t−
t−
i , ti
j , tj
i=1
j=1
and
{t : ψ(t) < 0} =
i=1
)
s−
b
s+
,
j ,b
j
j=1
where each of those li , mi and ni , i = 1, 2, is either a positive integer or ∞ depending on how many
points at which the two distortion functions g1 and g2 intersect. The above characterization result
combined with (10) implies that a function f solves Problem 2.2, if it increases at a slope of 1 over
those sets FX−1 (qi− ), FX−1 (qi+ ) , i = 1, . . . , l1 , and FX−1 (b
qj− ), FX−1 (b
qj+ ) , j = 1, . . . , l2 , and increases
−1 +
−1 −
at a slope of any κ(z) ∈ [0, 1] over FX−1 (s−
sj ), FX−1 (b
s+
i ), FX (si ) , i = 1, . . . , l1 , and FX (b
j ) ,
j = 1, . . . , l2 , and keeps constant over the remaining intervals.
8
4
Optimal Solutions under Premium Budget Constraint
In this section, we analyze the solution to Problem 2.1. In view of (6), we focus on Problem 3.1.
Let us now denote
Z ∞
π
e,
g2 (1 − FX (s)) e
h(s)ds,
(13)
0
where e
h is given in (11) as a solution to Problem 3.2. If π1 ≥ π
e, then e
h is obviously an optimal
solution to Problem 3.1, and thus, the ceded loss function fe given in (12) solves Problem 2.1.
In the subsequent analysis we assume π1 < π
e to exclude the known case. We resort to a Lagrangian dual method to solve Problem 3.1. This entails introducing a multiplier λ and the following
auxiliary problem:
Z ∞
Z ∞
max V (λ, h) ,
ψ(FX (s))h(s)ds − λ
g2 (1 − FX (s)) h(s)ds − π1 .
(14)
h∈H
0
0
It follows from (7) that we have V (λ, h) =
R∞
0
ψλ (FX (s))h(s)ds + λπ1 , where
ψλ (t) = g1 (1 − t) − (1 + θ + λ)g2 (1 − t), t ∈ [0, 1].
Thus, the solution to (14) is given by any of the following functions:


1,
if z ∈ Aλ



a.e.
hλ (z) = κλ (z),
if z ∈ Bλ ,



0,
if z ∈ Cλ ,
where κλ is any measurable [0, 1]-valued function, and


 Aλ = {z ≥ 0 |ψλ (FX (z)) > 0 },
Bλ = {z ≥ 0 |ψλ (FX (z)) = 0 },


Cλ = {z ≥ 0 |ψλ (FX (z)) < 0}.
(15)
(16)
(17)
Lemma 4.1 below provides sufficient conditions for a solution of the form (16) to solve Problem
3.1.
Lemma 4.1 Assume that for every λ ≥ 0, hλ solves (14) and there exists a constant λ∗ ≥ 0
satisfying
Z ∞
(18)
g2 (1 − FX (s)) hλ∗ (s)ds = π1 .
0
Then, h∗ , hλ∗ solves Problem 3.1.
Proof: We denote the optimal value of Problem 3.1 by u(π1 ). Then, it follows
u(π1 ) =
max
R∞
0
U(h)
h∈H
g2 (1−FX (s))h(s)ds≤π1
9
≤
max
R∞
0
h∈H
g2 (1−FX (s))h(s)ds≤π1
≤ max U(h) − λ
h∈H
∗
= U(h∗ ) ≤ u(π1 ),
Z
Z
∗
U(h) − λ
∞
g2 (1 − FX (s)) h(s)ds − π1
0
∞
g2 (1 − FX (s)) h(s)ds − π1
0
which implies that h∗ is a solution to Problem 3.1.
The following Proposition 4.1 shows that, when the set {z ≥ 0 |ψ(FX (z)) = 0 } is a µ-null set,
e.
the condition (18) is also necessary for hλ∗ to solve Problem 3.1 in the case of π1 < π
Proposition 4.1 Assume π1 < π
e, and µ{z ≥ 0 |ψ(FX (z)) = 0} = 0. Then, any solution h∗ of (8)
R∞
must satisfy 0 g2 (1 − FX (s)) h∗ (s)ds = π1 .
Proof: By (9) and (10), if µ{z ≥ 0 |ψ(FX (z)) = 0 } = 0, then the optimal solution of Problem 3.2
is unique almost everywhere, and is given by e
h as defined in (11), which satisfies
Z ∞
g2 (1 − FX (s)) e
h(s)ds = π
e > π1 .
0
We prove the desired result by contradiction. Assume that an optimal solution h∗ of Problem 3.1
satisfies
Z ∞
g2 (1 − FX (s)) h∗ (s)ds < π1 .
0
Combining the last two results implies that there exists a constant β ∈ (0, 1) to satisfy
Z ∞
g2 (1 − FX (s)) βh∗ (s) + (1 − β)e
h(s) ds = π1 .
0
We can observe that βh∗ + (1 − β)e
h is also a feasible solution of Problem 3.1. Moreover, since e
h is
the unique optimal solution of (9), U βh∗ + (1 − β)e
h = βU(h∗ ) + (1 − β)U(e
h) > U(h∗ ), which is a
contradiction.
To justify the Lagrangian dual method used for the derivation of optimal solutions to Problem
3.1, we need to show that there indeed exists a constant λ∗ ≥ 0 for a function hλ∗ defined in (16) with
some function κλ∗ that satisfies equation (18). The existence of such λ∗ is shown in the following
Proposition 4.1. To proceed, for λ ≥ 0 and a ∈ [0, 1], we define function
hλ,a (z) = 1Aλ (z) + a1Bλ (z), z ≥ 0.
(19)
Obviously, hλ,a solves (14) for any constant a ∈ [0, 1]. The existence of λ∗ is summarized in part (a)
of Theorem 4.1. Part (b) of the theorem is simply a restatement of the optimality of e
h as we explain
in the beginning of the section, and such a restatement is for presentation convenience in the next
section.
10
Theorem 4.1 (a) Given any π1 < π
e, there exists λ∗ ≥ 0 and a∗ ∈ [0, 1] for hλ∗ ,a∗ to satisfies (18)
and thus, it solves Problem 3.1.
(b) For π1 ≥ π
e, h0,0 = 1A0 solves Problem 3.1.
Proof: See Appendix.
Proposition 4.2 below can be used to verify the uniqueness of the optimal solution.
Proposition 4.2 Assume π1 < π
e, and µ{z ≥ 0 |ψλ (FX (z)) = 0} = 0 for any λ ≥ 0. Then, the
optimal solution to Problem 3.1 is unique almost everywhere.
Proof: Since µ{z ≥ 0 |ψλ (FX (z)) = 0 } = 0, by Theorem 4.1, there exists λ∗ ≥ 0 such that the
R∞
resulting hλ∗ (z) = 1Aλ∗ (z) satisfies 0 g2 (1 − FX (s)) hλ∗ (s)ds = π1 and hλ∗ solves (8). Let b
h be
another solution to (8), which differs from hλ∗ on a Borel set B with µ(B) > 0.
We recall from (17) that Aλ∗ = {z ≥ 0 |ψλ∗ (FX (z))dz > 0 } and Cλ∗ = {z ≥ 0 |ψλ∗ (FX (z))ds < 0 }.
Then, obviously ψ(FX (z)) > λ∗ g1 (1 − FX (z)) on Aλ∗ and ψ(FX (z)) < λ∗ g1 (1 − FX (z)) on Cλ∗ . Moreh = 0−b
h ≥ 0 on Cλ∗ . Consequently,
h = 1−b
h ≥ 0 on Aλ∗ , and hλ∗ − b
over, since hλ∗ = 1Aλ∗ , hλ∗ − b
further noticing the condition of µ{z ≥ 0 |ψλ∗ (FX (z)) = 0 } = 0, we obtain
Z ∞
Z ∞
ψ(FX (s))hλ∗ (s)ds −
ψ(FX (s))b
h(s)ds
0
0
Z
Z
b
=
ψ(FX (s)) hλ∗ (s) − h(s) ds +
h(s) ds
ψ(FX (s)) hλ∗ (s) − b
Cλ∗
Z
ZAλ∗
h(s) ds
ψ(FX (s)) hλ∗ (s) − b
h(s) ds +
ψ(FX (s)) hλ∗ (s) − b
=
Cλ∗ ∩B
Z
ZAλ∗ ∩B
∗
∗
b
b
λ g2 (1 − FX (s)) hλ∗ (s) − h(s) ds
λ g2 (1 − FX (s)) hλ∗ (s) − h(s) ds +
>
Cλ∗ ∩B
Aλ∗ ∩B
Z
Z
∗
b
=
λ g2 (1 − FX (s)) hλ∗ (s) − h(s) ds +
h(s) ds
λ∗ g2 (1 − FX (s)) hλ∗ (s) − b
Aλ ∗
Cλ∗
Z ∞
=
h(s) ds
λ∗ g2 (1 − FX (s)) hλ∗ (s) − b
0
= 0,
which contradicts to the optimality of hλ∗ and therefore, the solution to (8) is unique almost everywhere.
Let λ∗ ≥ 0 and a∗ ∈ [0, 1] be the two constants in Theorem 4.1 such that hλ∗ ,a∗ solves Problem
8. By defining
Aλ∗ ,x = {z ∈ [0, x] : ψλ∗ (FX (z)) > 0} , and Bλ∗ ,x = {z ∈ [0, x] : ψλ∗ (FX (z)) = 0} ,
an optimal solution to Problem 2.1 for a premium budget constraint π1 < π
e is then given by
Z x
fλ∗ ,a∗ (x) =
hλ∗ ,a∗ (z)dz = µ (Aλ∗ ,x ) + a∗ µ (Bλ∗ ,x ) .
0
11
5
CVaR Minimization with ISSD Premium Principle
In this section, we apply the results from the preceding section to study optimal reinsurance
treaties where the distortion risk measure is CVaR and the reinsurance premium is computed by the
Inverse-S Shaped Distortion (ISSD) principle. The ISSD premium principle is a special distortion
premium principle with an inverse-S shaped distortion function. It has interesting economic meaning
in pricing insurance contracts. It can be seen shortly that the optimal solutions under general concave
distortion premium principle can be easily retrieved from those obtained under the ISSD premium
principle.
We will focus on the derivation of the optimal MIF to Problem 3.1, because the optimal indemnification functions to Problem 2.1 can be easily obtained by an integration once we derive the optimal
MIF. Moreover, the MIF can equally tell us the exact shape of the optimal indemnification function
if not better.
5.1
ISSD Premium Principle
The ISSD premium principle is defined as π g = (1 + θ)ρg with a loading factor θ and an ISSD
distortion function satisfying those conditions in Definition 5.1 below.
Definition 5.1 (ISSD Function) A distortion function g is called an ISSD function, if and only if
it satisfies the following conditions:
(1) It is a continuous and strictly increasing mapping from [0,1] onto [0,1] and twice differentiable
in the interior;
(2) There exists b ∈ (0, 1) such that g ′ (·) is strictly decreasing on (0, b) and strictly increasing on
(b, 1);
(3) g ′(0) , limx↓0 g ′ (x) > 1 and g ′ (1) , limx↑1 g ′ (x) > 1.
As proposed by Tversky and Kahneman (1992), a plausible (and popular) ISSD function is of the
form
gγ (x) =
xγ
1
(xγ + (1 − x)γ ) γ
,
where γ is a parameter. Figure 1 displays this inverse-S shaped distortion function for γ = 0.5.
As noted by Rieger and Wang (2006) and Ingersoll (2008), this probability distortion function is
increasing and inverse-S shaped for any γ ∈ (0.279, 1).
To understand the economic meaning of an ISSD premium principle, we apply a simple transform
to derive
Z ∞
g
π (Y ) =
sg ′ (1 − FY (s))dFY (s).
(20)
0
12
1
0.9
0.8
0.7
g(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
c
0.8
1
x
Figure 1: This is an inverse-S shaped probability distortion function which satisfies Definition 5.1.
The point c is explained in Lemma 5.1.
When g is concave, it indicates that the insurance provider puts more weights on great losses (bad
outcomes) than small ones (good ones) in pricing risks. In contrast, when g is SSID, g ′ (0) > 1 and
g ′ (1) > 1 and thus, the insurance provider overweights not only large losses but also small ones,
which is consistent with the empirically observed phenomena in psychological experiments (Quiggin
1982,1992; Tversky and Kahneman 1992; Tversky and Fox 1995; Gonzalez and Wu, 1999).
Moreover, Kaluszka and Krzeszowiec (2012) introduce a premium principle which relies on Cumulative Prospect Theory (CPT). The ISSD premium principle can be viewed as a special CPT
premium principle corresponding to a linear utility function and a zero reference point. Numerical estimates of the probability distortion function are studied in Abdellaoui (2000) and Wu and
Gonzalez (1996).
By introducing η(x; g) , g(x)/x on x ∈ (0, 1] for a given ISSD distortion function, its property,
which is stated in Lemma 5.1 below, will be useful in our subsequent discussion.
Lemma 5.1 If g is an ISSD function, then there exists a unique point c ∈ [b, 1] such that η(·; g) is
strictly decreasing from 0 to c and strictly increasing from c to 1.
Proof: Firstly, η(·; g) is a continuous function on (0, 1]. Moreover, η(·; g) is strictly decreasing on
(0, b] since g ′ (·) is strictly decreasing (or equivalently g(·) is strictly convex) on (0, b]. Then, we take
′
. So, we only need to
the derivative of η(x; g) with respect to x on (b, 1) to get η ′ (x; g) = g (x)x−g(x)
x2
′
′
′′
know the sign of m(x) , g (x)x − g(x). Now, m (x) = g (x)x > 0 for x ∈ (b, 1) which means that
m(x) is a strictly increasing function on x ∈ (b, 1). Thus, the desired result follows.
13
From Definition 5.1, an ISSD distortion function becomes a concave distortion function as b → 1,
and in this case, the point c at which η ′ (x; g) changes its sign as given in Lemma 5.1 is equal to 1. As
we can see shortly, the optimal solutions under the CVaRα and an ISSD premium principle depend
on such point c, and accordingly, the optimal solutions for a concave distortion premium principle
can be obtained by replacing c with 1 in those solutions obtained for an ISSD principle below.
5.2
Optimal Solutions for CVaR Minimization
For presentation convenience, we assume that FX is strictly increasing in the present section,
unless otherwise stated. It is well known that CVaRα is a special distortion risk measure with a
distortion function g1 (s) = min{ αs , 1} so that

1,
if 0 ≤ s ≤ 1 − α,
g1 (1 − s) =
 1 (1 − s),
if 1 − α < s ≤ 1.
α
Assume that g2 in the reinsurance premium principle π g2 is an ISSD function as defined in Definition
5.1. By denoting

1
if 0 ≤ t ≤ 1 − α,
g1 (1 − t)  g2 (1−t) ,
k(t) ,
(21)
=
g2 (1 − t)  1−t ,
if
1
−
α
<
t
<
1,
αg2 (1−t)
and k(1) =
1
,
αg2′ (0+ )
the function ψ defined in (7) is then given by
ψ(t) = g2 (1 − t) k(t) − (1 + θ) , t ∈ [0, 1].
Since FX (z) is strictly increasing, the set {z ≥ 0 |ψ(FX (z)) = 0} has a zero Lebesgue measure, and
thus, according to (10), the optimal MIF to the unconstrained Problem (3.2) is given by e
h(z) =
1{ψ(FX (z))>0} . Moreover, the function ψλ defined in (15) is given by
ψλ (t) = g2 (1 − t) k(t) − (1 + θ + λ) , t ∈ [0, 1].
(22)
We further denote
α
b = min{α, c} and s∗ = VaRαb (X).
Lemmas 5.2 and 5.3 state, respectively, how functions k(t) and k(FX (z)) change their increasing/decreasing patten at the point of 1 − α
b and s∗ . These results are useful in analyzing the optimal
reinsurance contracts.
Lemma 5.2 The function k is strictly increasing on [0, 1 − α
b] and strictly decreasing on [1 − α
b, 1].
Proof: First, we consider the case of α ≤ c so that α
b = α. Since g2 (1 − t) is strictly decreasing on
(1−t)
[0, 1], k is strictly increasing on [0, 1 − α]. Moreover, from Lemma 5.1, g21−t
is strictly increasing
on [1 − α, 1] as α ≤ c, and hence k is strictly decreasing on [1 − α, 1]. Second, we assume α > c so
14
that α
b = c and 1 − α < 1 − c. Since g2 is strictly increasing, k = 1/g2 (1 − t) is strictly increasing
(1−s)
on [0, 1 − α]. Moreover, from Lemma 5.1, 1/k(t) = α g21−s
is strictly decreasing on [1 − α, 1 − c].
(1−s)
is strictly increasing on
Thus, k is strictly increasing on [0, 1 − c]. Further, from Lemma 5.1, g21−s
[1 − c, 1] and thus k is strictly decreasing on [1 − c, 1].
Since FX (s) is strictly increasing and g2 (s) is continuous, Lemma 5.2 and equation (22) imply
that the set {z ≥ 0 |ψλ (FX (z)) = 0} has a zero Lebesgue measure. Therefore, according Proposition
4.2, the solution to Problem 3.1 is unique almost everywhere.
To proceed, recall that M , esssupX.
Lemma 5.3 The function k(FX (s)) is strictly increasing on s ∈ [0, s∗ ) and k(FX (s)) is strictly
decreasing on s ∈ [s∗ , M].
Proof: For s < s∗ = VaRαb (X), we have FX (s) < 1 − α. Thus, by Lemma 5.2 and the assumption
that FX (·) is strictly increasing, we conclude that k(FX (s)) is strictly increasing on s ∈ [0, s∗). For
s > s∗ , we get FX (s) > FX (s∗ ) ≥ 1 − α, and hence, k(FX (s)) is strictly decreasing on s ∈ (s∗ , M]. Generally, the maximum of k(FX (s)) may not be attainable, which occurs when lims↑s∗ k(FX (s)) >
k(FX (s∗ )). In view of Lemma 5.3, let
e
k = max lim∗ k(FX (s)), k(FX (s∗ )) .
s↑s
To derive the optimal solutions to our reinsurance model, the analysis in Section 4 indicates that we
need to consider the set {z ≥ 0 |ψλ (FX (z)) > 0 } for an optimal MIF while (22) additionally suggests
that we need to take into account the value of k(FX (z)). Together with Lemma 5.3, this implies
that it is necessary to determine the optimal reinsurance on a case-by-case basis depending on the
relative magnitude of the following four critical values:
e
k, k0 , k(FX (0)), kM , k(FX (M)), 1 + θ.
Figure 2 enumerates all the cases that we need to consider. For each of these cases, the corresponding
proposition that gives the optimal solution is also provided.
We now analyze the optimal solution on the first case.
Proposition 5.1 If e
k ≤ 1 + θ, an optimal solution to Problem 3.1 is given by h∗ (z) = 0 for
z ∈ [0, M].
Proof: When e
k ≤ 1 + θ, ψλ (t) ≤ 0, ∀ t ∈ [0, 1], and Aλ = ∅, for any λ ≥ 0. The result therefore
follows from Theorem 4.1.
For the case of e
k > 1+θ, we need to discuss a few further subcases based on the relative magnitude
of k0 and kM compared to 1 + θ. If k0 ≤ 1 + θ and kM ≤ 1 + θ, then there exists p ∈ [0, VaRα (X)]
15
e
k, 1 + θ,
k0 , k(FX (0)),
kM , k(FX (M))
e
k ≤1+θ
Proposition 5.1
e
k >1+θ
k0 ≤ 1 + θ
kM ≤ 1 + θ
Proposition 5.2
k0 > 1 + θ
kM ≤ 1 + θ
Proposition 5.4
kM > 1 + θ
Proposition 5.3
kM > 1 + θ
Proposition 5.5
Figure 2: Possible cases for the optimal reinsurance
S
and q ∈ [VaRα (X), M] such that k(FX (s)) ≤ 1 + θ for s ∈ [0, p) [q, M] and k(FX (s)) ≥ 1 + ρ for
s ∈ [p, q).
Proposition 5.2 Assume that e
k > 1 + θ, k0 ≤ 1 + θ and kM ≤ 1 + θ hold.
(i) If π1 ≥ π
e, one optimal solution to Problem 3.1 is given by


0,
if 0 < z < p,



h∗ (z) = 1,
if p < z < q,



0,
if q < z < M.
(ii) If π1 < π
e, the optimal solution to Problem 3.1 is given by


0,
if 0 < z < l,



h∗ (z) = 1,
if l < z < n,



0,
if n < z < M,
S
where l ∈ [p, VaRα (X)], n ∈ [VaRα (X), M] and β are such that k(FX (s)) ≤ β for s ∈ [0, l) [n, M],
Rn
k(FX (s)) ≥ β for s ∈ [l, n) and l g2 (1 − FX (z))dz = π1 .
Proof: The proof of part (i) follows from Theorem 4.1 (b), where A0 = (p, q). For the part (ii),
the existence of l, n and β follows from Lemma 5.3. Denote by λ∗ = β − 1 − θ, it is easy to show
that h∗ in part (ii) satisfies (16) with λ = λ∗ . The residual result follows easily from Theorem 4.1
(a).
16
In the case of e
k > 1 + θ and kM > 1 + θ, there exists p ∈ [0, VaRα (X)] such that k(FX (s)) ≤ 1 + θ
for s ∈ [0, p) and k(FX (s)) ≥ 1 + θ for s ∈ [p, M] and there exists p ∈ [0, VaRα (X)] such that
k(FX (s)) ≤ kM for s ∈ [0, p) and k(FX (s)) ≥ kM for s ∈ [p, M]. Let us denote
π̂ =
Z
M
g2 (1 − FX (z))dz.
p
Proposition 5.3 Assume that e
k > 1 + θ, k0 ≤ 1 + θ and kM > 1 + θ hold.
(i) If π
e ≤ π1 , then the optimal solution to Problem 3.1 is h∗ = e
h, which is given as

0,
if 0 < z < p,
h∗ (z) =
1,
if p < z < M.
(ii) If π̂ ≤ π1 < π
e, then the optimal solution to Problem 3.1 is

0,
if 0 < z < d,
h∗ (z) =
1,
if d < z < M,
RM
where d is such that d g2 (1 − FX (z))dz = π1 .
(iii) If π1 < π̂, then the optimal solution to Problem 3.1 is


0,
if 0 < z < l,



h∗ (z) = 1,
if l < z < n,



0,
if n < z < M,
S
where l ∈ [p, VaRα (X)], n ∈ [VaRα (X), M] and β are such that k(FX (s)) ≤ β for s ∈ [0, l) [n, M],
Rn
k(FX (s)) ≥ β for s ∈ [l, n) and l g2 (1 − FX (z))dz = π1 .
Proof: The proof is similar to that of Proposition 5.2 and hence is omitted. Noted that λ∗ in
parts (ii) and (iii) are k(FX (d)) − 1 − θ and β − 1 − θ respectively.
Remark 5.1 Proposition 5.3 confirms that when the budget is not sufficiently high enough, the
optimal reinsurance policy will change from the stop-loss contract to a one layer contract, i.e., a
contract of stop-loss with an upper limit.
If k0 > 1 + θ and kM ≤ 1 + θ, then there exists p ∈ [V aRα (X), M] such that k(FX (s)) ≥ 1 + θ
for s ∈ [0, q) and k(FX (s)) ≤ 1 + θ for s ∈ [q, M] and there exists q ∈ [V aRα (X), M] such that
Rq
k(FX (s)) ≥ k0 for s ∈ [0, q) and k(FX (s)) ≤ k0 for s ∈ [q, M]. By denoting π̂ = 0 g2 (1 − FX (z))dz,
we have the following proposition.
17
Proposition 5.4 Assume that e
k > 1 + θ, k0 > 1 + θ and kM ≤ 1 + θ hold.
(i) If π
e ≤ π1 , then the optimal solution to Problem 3.1 is h∗ = e
h, which is given as

1,
if 0 < z < p,
∗
h (z) =
0,
if p < z < M.
(ii) If π̂ ≤ π1 < π
e, then the optimal solution to Problem 3.1 is

1,
if 0 < z < d,
∗
h (z) =
0,
if d < z < M,
Rd
where d is such that 0 g2 (1 − FX (z))dz = π1 .
(ii) If π1 < π̂, then the optimal solution to Problem 3.1 is


0,
if 0 < z < l,



h∗ (z) = 1,
if l < z < n,



0,
if n < z < M,
S
where l ∈ [p, VaRα (X)], n ∈ [VaRα (X), M] and β are such that k(FX (s)) ≤ β for s ∈ [0, l) [n, M],
Rn
k(FX (s)) ≥ β for s ∈ [l, n) and l g2 (1 − FX (z))dz = π1 .
Proof: The proof is the same as that of Proposition 5.3 and hence is omitted.
If k0 > 1 + θ and kM > 1 + θ, without loss of generality, we assume that k0 ≤ kM . Then,
then there exists p ∈ [0, V aRα (X)] such that k(FX (s)) ≤ kM for s ∈ [0, p) and k(FX (s)) ≥ kM for
RM
s ∈ [p, M]. By setting nπ̂ = p g2 (1 − FX (z))dz, we obtain the following proposition, where the
proof is the same as above.
Proposition 5.5 Assume that e
k > 1 + θ, k0 > 1 + θ and kM > 1 + θ hold.
(i) If π1 = E[X], then the optimal solution to Problem 3.1 is h∗ (z) = 1 for z ∈ [0, M].
(ii) If π̂ ≤ π1 < E[X], then the optimal solution to Problem 3.1 is

0,
if 0 < z < d,
∗
h (z) =
1,
if d < z < M,
RM
where d is such that d g2 (1 − FX (z))dz = π1 .
(ii) If π1 < π̂, then the optimal solution to Problem 3.1 is


0,
if 0 < z < l,



h∗ (z) = 1,
if l < z < n,



0,
if n < z < M,
S
where l ∈ [p, VaRα (X)], n ∈ [VaRα (X), M] and β are such that k(FX (s)) ≤ β for s ∈ [0, l) [n, M],
Rn
k(FX (s)) ≥ β for s ∈ [l, n) and l g2 (1 − FX (z))dz = π1 .
18
6
Conclusion
The quest for optimal reinsurance has remained a fascinating subject to academicians and practitioners. Over the last few decades many reinsurance models were proposed to provide better guidance
for insurers to manage their risk. This paper contributed to the literature by providing better approach in solving sophisticated optimal reinsurance model whereby the distorted preference of an
insurer’s total risk exposure is minimized under the general distortion risk measure while subject
to a budget constraint on premium. Such optimal reinsurance is quite general. For example, the
distortion risk measure includes VaR, CVaR and spectral risk measure as special cases while the
distortion premium principle includes the expected value principle and Wang’s premium principle
as special cases. We solved this problem explicitly by using marginal indemnification formulation
in conjunction with the method of Lagrange. Compared to the existing method (such as Cui, et al.
2013), our proposed method has the advantages of simplicity and transparent. As another added
advantage of our method, the uniqueness of the optimal reinsurance policy can also be analyzed.
By resorting to a well-specified optimal reinsurance model with CVaR as the risk measure and the
inverse-S shaped distortion function as the premium principle, the proposed method was used to
derive explicitly the optimal solutions.
Acknowledgements
Zhuang acknowledges financial support from the Department of Statistics and Actuarial Science,
University of Waterloo. Weng thanks the financial support from the Natural Sciences and Engineering Research Council of Canada, and Society of Actuaries Centers of Actuarial Excellence Research
Grant. Tan acknowledges the financial support from the Natural Sciences and Engineering Research
Council of Canada (No: 368474), and Society of Actuaries Centers of Actuarial Excellence Research
Grant.
Appendix
R∞
Proof of Proposition 4.1. (a) Let φ(λ, a) , 0 g2 (1 − FX (s)) hλ,a (s)ds, λ ≥ 0 and a ∈
[0, 1], where hλ,a is given in (19). We denote Cp = {z ≥ 0 |g2 (1 − FX (z)) > 0 } and C0 = {z ≥
0 |g2 (1 − FX (z)) = 0 }. It is easy to check
lim µ(Aλ ∩ Cp ) = 0 and
λ→∞
lim µ(Bλ ∩ Cp ).
λ→∞
Further note that g2 (1 − FX (s)) = 0 on C0 . Thus,
Z
Z
φ(λ, a) =
g2 (1 − FX (s))ds + a
g2 (1 − FX (s))ds
Aλ
Bλ
Z
Z
g2 (1 − FX (s))ds
g2 (1 − FX (s))ds + a
=
Bλ ∩Cp
Aλ ∩Cp
19
Combining the above two displays, we get limλ→∞ φ(λ, a) = 0 for any a ∈ [0, 1]. Moreover, for λ = 0
and a = 0, hλ,a = e
h, z ≥ 0, which means φ(0, 0) = π
e.
It is easy to check
lim {z ≥ 0 : ψγ (FX (z)) > 0} = {z ≥ 0 : ψλ (FX (z)) > 0},
γ→λ+
and
lim {z ≥ 0 : ψγ (FX (z)) ≥ 0} = {z ≥ 0 : ψλ (FX (z)) ≥ 0}.
γ→λ−
Thus, we apply the Dominated Convergence Theorem to obtain
lim φ(γ, 0) = φ(λ, 0), and
γ→λ+
lim φ(γ, 1) = φ(λ, 1).
γ→λ−
Define λ∗ = sup Sπe , where
Sπ1 = {λ ≥ 0 : φ(λ, 1) ≥ π1 } .
On one hand, given ǫ > 0, by the superium property of λ∗ , there exists some λ ∈ (λ∗ − ǫ, λ∗ ] and
λ ∈ Sπ1 such that
φ(λ∗ − ǫ, 1) ≥ φ(λ) ≥ π1 ,
where the first inequaility is due to the non-increasing property of φ(λ, a) as a function of λ. On the
other hand, the superium property of λ∗ implies λ∗ + ǫ ∈
/ Sπ and thus,
φ(λ∗ + ǫ, 0) ≤ φ(λ∗ + ǫ, 1) < π1 .
Letting ǫ → 0 in the last two displays yields
φ(λ∗ , 0) ≤ π1 ≤ φ(λ∗ , 1).
(A.1)
Moreover, it is clear from the definition of φ(λ, a) and hλ,a that φ(λ∗ , a) is a continuous function of
a. Thus, it follows form (A.1) and the Intermediate Value Theorem, there exists some a∗ ∈ [0, 1]
such that φ(λ∗ , a∗ ) = π1 , as desired.
(b) The result has been clearly shown at the beginning of Section 4.
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