CDF Formulation for Solving an Optimal Reinsurance Problem · CDF Formulation for Solving an...
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CDF Formulation for Solving an Optimal Reinsurance Problem
Chengguo Weng∗, and Shengchao Zhuang
Department of Statistics and Actuarial Science
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
July 28, 2015
Abstract
An innovative cumulative distribution function (CDF) based method is proposed for
deriving optimal reinsurance contracts for an insurer to maximize its survival probabil-
ity. The optimal reinsurance model is a non-convex constrained stochastic optimization
problem, and the CDF based method transforms it into a linear problem of determining
an optimal CDF over a corresponding feasible set. Compared to the existing literature,
our proposed CDF formulation provides a more transparent derivation of the optimal solu-
tions, and more interestingly, it enables us to solve a further complex model with an extra
background risk.
Key Words: CDF formulation, Lagrangian dual method, optimal reinsurance, survival prob-
ability maximization, background risk.
∗Corresponding author. Tel: +001(519)888-4567 ext. 31132.
Email: Weng ([email protected]), Zhuang ([email protected]).
1 Model Setup
1.1 Preliminaries
Let (Ω,F,P) be a probability space, where all the random variables in this paper are dened.
We consider a constrained non-convex stochastic optimal decision problem from an insurance
context. The problem is formulated from the perspective of an insurance company (insurer).
We assume that the insurer has an initial capital of W and is subject to a loss X, which is
a nonnegative random variable with a support of [0,M ] for some M ∈ (0,∞]. For the case
of M = ∞, the support of X is interpreted as [0,∞), and throughout the paper, we assume
E[X] <∞.
The stochastic optimization problem considered in the paper is to determine an optimal
reinsurance purchase strategy against the risk X for the insurer to maximize its survival prob-
ability. Mathematically, it is to nd an optimal partition on X into f(X) and r(X) so that
X = f(X) + r(X), where f : [0,M ] → [0,M ] is a measurable function and so is r. In their
economic meanings, f(X) represents the portion of loss that is ceded to a reinsurer, and r(X)
is the residual loss retained by the insurer. In the context of optimal reinsurance, the ceded
loss function f is often restricted to the set C as given below for the solution (Cui, et al., 2013;
Zheng, et al., 2015):
C :=f(·) : 0 6 f(x) 6 x and r(x) := x− f(x) is non-decreasing ∀ x ∈ [0,M ]
. (1)
The non-decreasing assumption on the retained loss function r is imposed to reduce the moral
hazard risk. If the retained function r is not non-decreasing, the insurer may encourage the
policyholders to claim more so as to reduce its own retained loss but increase the ceded loss on
the reinsurer.
In exchange for covering partial risk for the insurer, the reinsurer charges a premium on the
insurer as compensation. Generally, this premium is always positive, and computed according
to a certain premium principle Π so that the reinsurance premium is given by Π(f(X)). In
our work, we consider the expected value principle and compute the reinsurance premium by
Π(f(X)) = (1 + ρ)E[f(X)], where ρ > 0 is the loading factor.
1.2 Optimal Reinsurance Model in Absence of Background Risk
From the preceding subsection, the insurer's net wealth in the presence of a reinsurance is
given by
T := W − r(X)− (1 + ρ)E[f(X)], (2)
and the insurer is subject to a survival probability of
P(W − r(X)− (1 + ρ)E[f(X)] > 0).
Thus, with the reinsurance premium constrained on a level of π > 0, the optimal reinsurance
problem of maximizing the above survival probability is equivalent to the following one:
2
Problem 1.1 maxf∈C
P(W − r(X)− π > 0
),
s.t. (1 + ρ)E[f(X)] = π,(3)
where π is a reinsurance premium budget satisfying 0 < π ≤ (1 + ρ)E[X].
The research of optimal reinsurance has remained a fascinating area since the seminar
papers Borch (1960) and Arrow (1963), where explicit solutions were derived for minimizing the
variance of the insurer's retained loss (or equivalently the variance of T in (2)) and maximizing
the expected utility of its terminal wealth T respectively, subject to a given net reinsurance
premium of E[f(X)]. These two classic results have been extended in a number of important
directions. Just to name a few, Gajek and Zagrodny (2000) and Kaluszka (2001) generalized
Borch's result by considering the standard deviation premium principle and a class of general
premium principles, respectively. Young (1999) generalized Arrow's result by assuming Wang's
premium principle. Among the recent studies on optimal reinsurance, risk measures including
VaR and CVaR have been extensively used; see, for instance, Cai, et al. (2008), Balbás, et al.,
(2009), Tan, et al. (2009), Tan, et al. (2011), Cheung (2010), Chi and Tan (2013), Asimit, et
al. (2013), Chi and Weng (2013), Cheung, et al. (2014).
1.3 Optimal Reinsurance Model in Presence of Background Risk
In the insurance practice, there are some risks which are not insurable and can potentially
occur with other insurable risks. Moreover, the administration fee on the insurer to settle
insurance claims is often not fully deterministic, and not insured. We generally refer to the
aggregate of these risks which occur along with the underlying risk X as background risk, and
model the risk by a nonnegative random variable Y .
The optimal decision problems with background risk in an insurance context have been in-
tensively studied, but most of these are within a utility maximization framework. For example,
Gollier (1996) considered the optimal insurance purchase strategy by maximizing the expected
utility of a policyholder. For another example, Dana and Scarsini (2007) characterized the
optimal risk sharing strategy between two parties, both being expected utility maximizers. In
the literature, it is common to assume that the background risk is statistically independent
of the insurable risk X, because such assumption leads to much more tractable models and it
works as good approximation in the presence of week dependence. For example, Mahul (2000),
Gollier and Pratt (1996), Courbage et al. (2007), and Schlesinger (2013).
Taking the background risk into account, we compute the terminal net wealth of the insurer
as
W − r(X)− Y − (1 + ρ)E[f(X)],
and accordingly the optimal reinsurance model of maximizing the survival probability goes as
follows:
3
Problem 1.2 maxf∈C
P(W − r(X)− Y − π > 0
),
s.t. (1 + ρ)E[f(X)] = π.(4)
1.4 Some Remarks
Problem 1.1 has been studied by Gajek and Zagrodny (2004) and analytical optimal so-
lutions have been derived. However, their results apply only when the reinsurance premium
budget π is small enough and they fail to clearly outline the range of the budget for their results
to apply. Moreover, their approach involves a sophisticated application of the Neyman Pearson
Lemma and is mathematically abstruse. Further, their method can not be readily used to solve
Problem 1.2 where the background risk is additionally considered. For these reasons, in the
present paper, we propose an innovative CDF based method to solve the above two problems.
Our CDF based method rst transforms each of them into a problem of determining an
optimal CDF over a corresponding feasible set, and then combines with a Lagrangian dual
method to derive the optimal solutions in a transparent way. The idea behind our method is to
equivalently reformulate each of the original two problems into a linear functional optimization
problem so as to facilitate the application of a pointwise optimization procedure. Our CDF
based method enables us to derive the optimal solutions of Problem 1.1 for any premium budget
π over the whole range of (0, (1 + ρ)E[X]]. More signicantly, the transparency of the method
enables us to derive analytical optimal reinsurance contracts in the presence of independent
background risk, i.e., the solutions of Problem 1.2, which is otherwise non-tractable.
As one can discover later, our procedure of deriving the optimal solutions can be applied
in parallel if the objective function in Problem 1.1 (or Problem 1.2) were changed to
P(W − r(X)− π > A
)(or P
(W − r(X)− Y − π > A
))
for some some constant A, which represents the net wealth the insurer targets to achieve. Such
an optimal decision problem is typically called goal-reaching model, which was proposed by
Kulldor (1993), and studied extensively in the literature, e.g., Browne (1999, 2000). When
A = 0, the goal-reaching model reduces to the survival probability maximization model.
The rest of the paper proceeds as follows. Section 2 develops the CDF formulation and
some preliminary results from the Lagrangian dual method. Sections 3 and 4 solve Problems
1.1 and 1.2, respectively. Section 5 concludes the paper.
2 CDF Formulation
Hereafter, when it is stated that a functional V (F ) of CDF F is non-decreasing (non-
increasing) in F , we mean that V (F1) ≥ V (F2) if F1(x) ≥ F2(x) (correspondingly F1(x) ≤F2(x)), x ≥ 0. For a given distribution F of a nonnegative random variable, we dene its
inverse as F−1(y) := infz ∈ R+ : F (z) ≥ y
.
4
Denote
R := r(·) : r(x) ≡ x− f(x) ∀x ∈ [0,M ], f ∈ C.
Then, it follows from (1) that
R =r(·) : r(0) = 0, 0 6 r(x) 6 x ∀x ∈ [0,M ], r(x) is non-decreasing on [0,M ]
,
and thus, Problem 1.1 can be equivalently rewritten, in terms of retained loss function r, as
follows: maxr∈R
P(r(X) ≤W − π
),
s.t. E[r(X)] = π,(5)
where π := E[X]−π/(1 +ρ) and 0 ≤ π < E[X] for 0 < π ≤ (1 +ρ)E[X]. From the formulation
of (5), a necessary condition to have a nonempty feasible set for (5) is given by 0 ≤ π ≤ E[X],
and a trivial solution in the case of π = E[X] is given by r(x) = x, x ∈ [0,M ].
We can similarly rewrite Problem 1.2, in terms of retained loss functions, as follows:maxr∈R
P(r(X) + Y ≤W − π
),
s.t. E[r(X)] = π.(6)
Once a solution, say r∗, is obtained for (5) (or (6)), then f∗(x) := x− r∗(x), x ∈ [0,M ], is
a solution to Problem 1.1 (correspondingly Problem 1.2). Thus, we shall focus on studying (5)
and (6) for optimal solutions.
The following two assumptions will be used in the rest of the paper.
Assumption 2.1 Both the left and right derivatives of the CDF FX(x) of X exist and are
strictly positive over [0,M ].
Assumption 2.2 Y has a non-increasing probability density function gY (·), and is statistically
independent of X.
Remark 1 (a) Assumption 2.1 allows the random variable X to have a probability mass of p
at 0, which is a common model for an insurance loss. Assumption 2.1 is certainly satised if
X has a piecewise continuous density function over (0,M ]. The assumption also implies that
FX is continuous and strictly increasing over [0,M ].
(b) Assumption 2.1 also implies, by the Inverse Function Theorem, that both the left and
right derivatives of F−1X exist and are strictly positive, which we respectively denote by
(F−1X )′−(u) and (F−1X )′+(u), u ∈ (0, 1).
(c) Furthermore, it is worth noting that a great number of typical distributions for insurance
loss modelling satisfy Assumption 2.2, such as Pareto distribution, exponential distribution and
so on.
5
Notably, (6) reduces to (5) if we set Y = 0 almost surely. Nevertheless, our CDF method for
solving (6) relies on Assumption 2.2, whereas the condition of Y = 0 almost surely contradicts
to Assumption 2.2. Therefore, the solution of (5) can not be directly retrieved from that of
(6). We need to analyze both problems separately.
To solve (5) and (6), we reformulate each of them into a problem of identifying an optimal
CDF over a feasible set, and recover the optimal ceded loss functions for the original problems
(5) and (6) from those identied optimal CDF's via Lemma 2.2 in the sequel.
To proceed, we dene
F∗ :=F (·) : F (·) is the CDF of r(X), r ∈ R
,
and
F∗∗ :=F (·) : F (·) is a CDF with F (t) ≥ FX(t) ∀ 0 ≤ t ≤M
. (7)
The equivalence between these two sets of F∗ and F∗∗ is shown in Lemma 2.1 below.
Lemma 2.1 Assume that Assumption 2.1 holds, then F∗ = F∗∗.
Proof. On one hand, for every r ∈ R, r(x) ≤ x ∀ x ∈ [0,M ] and thus we must have
P(r(X) ≤ x) ≥ FX(x) ∀ x ∈ [0,M ]. This means F∗ ⊆ F∗∗. On the other hand, for any
F ∈ F∗∗, we dene r(s) = F−1(FX(s)) for s ∈ [0,M ] to get 0 ≤ r(s) ≤ s, where the second
inequality follows from the fact that F (x) ≥ FX(x) ∀ x ∈ [0,M ]. Moreover, Since FX is
continuous and strictly increasing over [0,M ] due to Assumption 2.1, for any t ∈ [0,M ], there
exists y ∈ [0,M ] such that F (t) = FX(y). Hence, we obtain P(r(X) ≤ t
)= P
(F−1(FX(X)) ≤
t)
= P(FX(X) ≤ F (t)
)= P
(FX(X) ≤ FX(y)
)= P(X ≤ y) = F (t) by Assumption 2.1 again.
This means that F ∈ F∗, and thus, F∗ ⊆ F∗∗.
It is obvious that the reinsurance premium budget constraint E[r(X)] = π in problems (5)
and (6) is equivalent to ∫ M
0
(1− Fr(t)
)dt = π,
where Fr denotes the CDF of r(X). Moreover, the objective in (5) is Fr(W − π), which also
merely depends on the CDF of r(X). Such observation motivates us to consider the following
CDF formulation maxF∈F∗∗
U0(F ) := F (W − π),
s.t.∫M0
(1− F (t)
)dt = π.
(8)
For (6) where the background risk is considered, we apply Assumption 2.2 to rewrite its
objective function as follows
P(r(X) + Y ≤W − π
)=
∫ W−π
0P(r(X) ≤W − π − s
)dFY (s)
6
=
∫ W−π
0P(r(X) ≤ t
)gY (W − π − t)dt.
=
∫ W−π
0Fr(t)gY (W − π − t)dt,
which is a linear functional of Fr. Accordingly, regarding (6), we propose the following CDF
formulation: maxF∈F∗∗
U1(F ) :=∫W−π0 F (t)gY (W − π − t)dt,
s.t.∫M0
(1− F (t)
)dt = π.
(9)
For presentation convenience, we refer to (5) and (6) as retained loss function (RLF)
formulation" in contrast to the term of CDF formulation" for (8) and (9). The equivalence
between the RLF formulation and CDF formulation is given in Lemma 2.2 below.
Lemma 2.2 An element F ∗ ∈ F∗∗ solves (8) (or (9)) if and only if r∗, dened by r∗(x) =
(F ∗)−1(FX(x)) for x ∈ [0,M ], solves (5) (correspondingly (6)).
Proof. We only show the relationship between (9) and (6) as the result can be similarly
proved for (8) and (5). We achieve the proof by contradiction. We rst consider the if part.
Assume that F ∗ ∈ F∗∗ is not an optimal solution to (9), i.e., there exists another element, say
F ∈ F∗∗, such that U1(F ) > U1(F∗). Then, we dene r(x) = (F )
−1(FX(x)), ∀x ∈ [0,M ], to
get
P(r(X) + Y ≤W − π
)= U1(F ) > U1(F
∗) = P(r∗(X) + Y ≤W − π
),
which means that r∗ can not be a solution to (6). Thus, if r∗ solves (6), then F ∗ must solve
(9).
To show the only if" part, we assume that r∗ as given is not a solution to (6) so that there
exists another element r ∈ R such that
P(r(X) + Y ≤W − π) > P(r∗(X) + Y ≤W − π).
As we have already seen in the proof of Lemma 2.1, F ∗ is the CDF of r∗(X), and thus, the
last display further implies U1(F ) > U1(F∗), which means that F ∗ is not a solution to (9).
Therefore, if F ∗ solves (9), then r∗ must be a solution to (6).
Remark 2 According to Lemma 2.2, once we solve the CDF formulation (equations (8) or
(9)) and obtain an optimal solution F ∗ ∈ F∗∗, we can construct a solution r∗ to the RLF
formulation (equation (5) or (6) correspondingly) by r∗(x) = (F ∗)−1(FX(x)) for x ∈ [0,M ]
and obtain an optimal ceded loss function f∗(x) = x− r∗(x), x ∈ [0,M ].
7
In view that the objectives and constraints in (8) and (9) are all linear functionals of the
decision variable F and the feasible set F∗∗ is convex, we exploit the Lagrangian dual method
to solve both problems. This entails introducing a Lagrangian multiplier λ and considering the
following auxiliary problem
maxF∈F∗∗
V0(λ, F ) := F (W − π) + λ(∫ M
0
(1− F (t)
)dt− π
)(10)
for (8), and another auxiliary problem
maxF∈F∗∗
V1(λ, F ) :=
∫ W−π
0F (t)gY (W − π − t)dt+ λ
(∫ M
0
(1− F (t)
)dt− π
)(11)
for (9). Once (10) (or (11)) is solved with an optimal solution Fλ(·) for each λ ∈ R, one can
determine λ∗ ∈ R by solving∫ 10 (1 − Fλ∗(t))dt = π. Then, we can show that F ∗ := Fλ∗ is an
optimal solution to (8) (or (9)) as in Lemma 2.3 below.
Lemma 2.3 Assume that for every λ ∈ R, Fλ(·) solves (10) (or (11)) and there exists a
constant λ∗ ∈ R satisfying∫M0 (1 − Fλ∗(t))dt = π. Then, F ∗ := Fλ∗ solves (8) (or (9) corre-
spondingly).
Proof. We generally denote the objective in (8) (or (9)) by U(F ), and let u(π) denotes its
optimal value. Then, it follows
u(π) = maxF∈F∗∗∫M
0 (1−F (t))dt=π
U(F ) = maxF∈F∗∗∫M
0 (1−F (t))dt=π
[U(F ) + λ∗
(∫ M
0(1− F (t))dt− π
)]
≤ maxF∈F∗∗
[U(F ) + λ∗
(∫ M
0(1− F (t))dt− π
)]=U(F ∗) ≤ u(π),
which implies that F ∗ is optimal to (8) (or (9)).
3 Solutions to Problem 1.1
In this section, we study the optimal solutions when there is no background risk, i.e. the
solutions to Problem 1.1. We analyze the solutions in two scenarios separately (W − π ≥ M
and W −π < M), depending on the relative magnitude of W −π and M . When M =∞, only
the case of W − π < M is possible.
We rst consider the case with W − π ≥ M , where for any feasible f ∈ C we have P(W −r(X)− π > 0) ≥ F (r(X) < M) ≥ FX(M) = 1. Therefore, every element in the feasible set of
Problem 1.1 satisfying the constraint is a solution. We summarize this result in Theorem 3.1
below.
8
Theorem 3.1 Assume that Assumption 2.1 and W−π ≥M hold. Then, any f∗ ∈ C satisfying
(1 + ρ)E[f∗(X)] = π is an optimal ceded loss function to Problem 1.1.
In the rest of the section, we assume W − π < M , which is automatically satised for
M =∞, i.e., X has an unbounded support of [0,∞). According to the analysis in the preceding
section, we can solve the CDF formulation (8) and apply Lemma 2.2 to get the optimal retained
loss function and the optimal ceded loss function. By virtue of Lemma 2.3, we can rst analyze
the solution of (10) for each λ. To this end, we note that
F (W − π) ∈ [FX(W − π), 1] for every F ∈ F∗∗
and consider the following sub-problems
maxF∈F∗∗
V0(λ, F ) := u+ λ(∫ M
0
[1− F (t)
]dt− π
), s.t. F (W − π) = u, (12)
indexed by u ∈ [FX(W − π), 1]. If we could derive an optimal solution F ∗λ,u and get optimal
value ξ(u) := V0(λ, F∗λ,u) of (12) for each u ∈∈ [FX(W −π), 1], then F ∗λ := F ∗λ,u∗ is an optimal
solution to (10), where
u∗ = argmaxu∈[FX(W−π),1]
ξ(u).
Let q = F−1X (u) for u ∈ [FX(W − π), 1]. Since W − π > 0 and FX(x) is continuous and strictly
increasing on [0,M ] according to Assumption 2.1, we have u = FX(q) and q ∈ [W − π,M ].
Thus,
u∗ = FX(q∗) with q∗ = argminq∈[W−π,M ]
ξ(FX(q)).
In the subsequent analysis, we follow such a procedure to solve (10) for λ = 0 and λ > 0,
respectively. Its solution for λ < 0 is not relevant when we invoke Lemma 2.3 for the optimal
CDF of (8) and thus omitted.
Case 3.1 λ = 0.
In this case, the objective in (12) is independent of F and ξ(u) = u, and thus,
the solution to (12) F ∗λ can be any F ∈ F∗∗ which satises F (W − π) = 1. (13)
Case 3.2 λ > 0.
In this case, V0(λ, F ) is non-increasing in F , and the pointwise smallest CDF F ∈ F∗∗ whichsatises F (W − π) = u solves (12). Therefore, from the denition of F∗∗ in (7), a solution to
(12) is given by
F ∗λ,u(t) =
FX(t), if 0 ≤ t < W − π,
u, if W − π ≤ t < F−1X (u),
FX(t), if F−1X (u) ≤ t ≤M,
(14)
9
and as a consequence,
ξ(u) = V0(λ, F∗r(X),u)
= u+ λ
∫ W−π
0(1− FX(t))dt+ λ
∫ F−1X (u)
W−π(1− u)dt+ λ
∫ M
F−1X (u)
(1− FX(t))dt− λπ.
By taking the left derivative of ξ(u), we obtain
ξ′−(u) =1 + λ[ ∫ F−1
X (u)
W−π(−1)dt+ (F−1X )′−(u)(1− FX(F−1X (u))
]=− λ(F−1X )′−(u)
[1− FX(F−1X (u))
]=λ[1 + λ(W − π)
λ− F−1X (u)
], u ∈ [FX(W − π), 1]
We can similarly derive its right derivative, and indeed, it is the same as its left derivative
obtained in the above. This means that ξ(u) is dierentiable over [FX(W − π), 1) with
ξ′(u) = λ[1+λ(W−π)
λ − F−1X (u)], which is decreasing in u. This further implies that ξ(u) is a
concave function of u and attains its maximum at
u∗ = max
FX(W − π), FX
(1 + λ(W − π)
λ
)= FX
(1 + λ(W − π)
λ
).
Thus, one solution to (10) is given by F ∗λ = F ∗λ,u∗ as dened in (14).
With the analysis in the above Cases 3.1 and 3.2, we readily apply Lemma 2.3 to analyze
the solutions to the CDF formulation (8). Denote
π0 =
∫ W−π
0
(1− FX(t)
)dt. (15)
Then, depending on the magnitude of π relative to π0, the optimal solution of (8) can be
obtained as summarized in Proposition 3.1 below.
Proposition 3.1 Assume that Assumption 2.1 and W − π < M hold.
(a) If 0 ≤ π ≤ π0, then one optimal CDF to (8) is given by
F ∗(t) =
FX(t), if 0 ≤ t < t0,
1, if t0 ≤ t ≤M,(16)
where t0 ∈ [0,W − π] is such that∫M0
(1− F ∗(t)
)dt = π.
(b) If π0 < π < E[X], then one CDF to (8) is given by
F ∗(t) =
FX(t), if 0 ≤ t < W − π,
FX(t0), if W − π ≤ t < t0,
FX(t), if t0 ≤ t ≤M,
10
where t0 ∈ [W − π,M ] is such that∫M0
(1− F ∗(t)
)dt = π.
Proof. (a) Given π ∈ [0, π0], there obviously exists a constant t0 ∈ [0,W − π] for F ∗(t)
dened in (16) to satisfy∫M0
(1−F ∗(t)
)dt = π, and F ∗ satises the condition (13). Therefore,
by Lemma 2.3, F ∗ as given by (16) is one solution to (8).
(b) For a ∈ [W − π,M ], dene
Fa(t) :=
FX(t), if 0 ≤ t < W − π,
FX(a), if W − π ≤ t < a,
FX(t), if a ≤ t ≤M,
and q(a) :=∫M0
[1 − Fa(t)
]dt. Obviously, q(a) is a continuous and non-increasing function
of a with q(W − π) = E[X] and q(M) = π0. Therefore, for any π ∈ (π0,E[X]), there exists
t0 ∈ (W − π,M ] such that q(t0) = π which means Ft0 solves (10) for t0 = [1 + λ(W − π)]/λ,
i.e., λ = 1/[t0 − (W − π)], as previously shown in Case 3.2. Thus, the desired result follows
from Lemma 2.3.
The optimal ceded loss functions for the case of W − π < M can be consequently obtained
by combining Proposition 3.1 and Lemma 2.2, and we summarize the results in Theorem 3.2
below.
Theorem 3.2 Assume that Assumption 2.1 and W − π < M hold.
(a) If (1 + ρ)(E[X] − π0) ≤ π ≤ (1 + ρ)E[X], then one optimal cede loss function to Problem
1.1 is given by
f∗(t) =
0, if 0 ≤ t < t0,
t− t0, if t0 ≤ t ≤M,
where t0 is such that (1 + ρ)E[f∗(X)] = π.
(b) If 0 < π < (1 + ρ)(E[X]− π0), then one optimal ceded loss function to Problem 1.1 is given
by
f∗(t) =
0, if 0 ≤ t < W − π,
t− (W − π), if W − π ≤ t < t0,
0, if t0 ≤ t ≤M,
where t0 is such that (1 + ρ)E[f∗(X)] = π.
Proof. Since π = E[X]− π/(1 + ρ), the conditions on π given in both parts are equivalent to
those two in terms of π in Proposition 3.1 respectively, whereby the desired results follow from
11
Proposition 3.1 and Lemma 2.2.
Remark 3 Part (b) of Theorem 3.2 indicates that an truncated stop-loss reinsurance is optimal
for reinsurance premium budget smaller than (1 + ρ)(E[X] − π). The optimality of truncated
stop-loss reinsurance has been established in the Corollary 1 of Gajek and Zagrodny (2004) for
the same Problem 1.1. However, as we have commented in section 1.4, their derivation of
the solution involves a complicated application of Neyman Pearson Lemma, and much more
mathematically abstruse than our CDF method. Furthermore, Gajek and Zagrodny (2004) fail
to identify the range of premium budget π for such a solution as clearly as we do in Theorem
3.2. They also fail to discover the optimality of a stop-loss reinsurance for the case of large
premium budget as stipulated in part (a) of Theorem 3.2.
4 Solutions to Problem 1.2
In this section, we study the optimal solutions when the insurer has a background risk, i.e.,
the solutions of Problem 1.2 or equivalently (6). Based on our analysis in section 2, we need to
solve (9) and apply Lemma 2.2 to get the optimal ceded loss functions. To solve (9), we rst
investigate the solutions of (11) and consequently invoke Lemma 2.3 for optimal reinsurance
contracts. Similar to Section 3, the analysis is divided into two cases of W − π ≥ M and
W − π < M .
4.1 Assume W − π ≥M .
In this case, we can rewrite the objective function in (11) as follows
V (λ, F ) =
∫ M
0F (t)
[gY (W − π − t)− λ
]dt+ 1− FY (W − π −M) + λ(M − π).
Hence, (11) reduces to
maxF∈F∗∗
∫ M
0F (t)
[gY (W − π − t)− λ
]dt+ 1− FY (W − π −M) + λ(M − π), (17)
of which one optimal solution follows from (7) as follows
F ∗λ (t) =
FX(t), if 0 ≤ t ≤M and gY (W − π − t)− λ < 0,
any constant, if 0 ≤ t ≤M and gY (W − π − t)− λ = 0,
1, if 0 ≤ t ≤M and gY (W − π − t)− λ > 0,
(18)
provided that F ∗λ is an appropriate CDF in F∗∗.With an aid of Assumption 2.2 and Lemma 2.3, we can obtain a solution to (9) as given in
Proposition 4.1 below.
12
Proposition 4.1 Suppose that Assumptions 2.1 and 2.2 hold and W − π ≥ M is satised.
Then, one optimal solution to (9) is given by
F ∗(t) =
FX(t), if 0 ≤ t < t0,
1, if t0 ≤ t ≤M,(19)
where t0 is such that∫ t00 [1− F ∗(t)]dt = π.
Proof. For a ∈ [0,M ], we dene
Fa(t) :=
FX(t), if 0 ≤ t < a,
1, if a ≤ t ≤M,
and q(a) :=∫M0
[1− Fa(t)
]dt. Obviously, q(a) is a continuous and non-decreasing function of
a with q(0) = 0 and q(M) = E[X]. Thus, for any π ∈ [0, E[X]), we can nd t0 such that
q(t0) = π. Let λ∗ = gY (W −π− t0). Then, by virtue of Lemma 2.3, F ∗ given in (19) solves (9)
if we could show that it is a solution to (17) for λ = λ∗. Indeed, since gY (W − π − t) is a non-
decreasing function of t, we have gY (W−π−t)−λ∗ ≤ 0 for t ∈ [0, t0) and gY (W−π−t)−λ∗ ≥ 0
for t ∈ (t0,M ], whereby it follows from (18) that F ∗ is a solution to (17) for λ = λ∗. Thus, the
proof is complete.
By invoking Lemma 2.2, we can derive an optimal ceded loss function to solve Problem 1.2.
Theorem 4.1 Assume that Assumptions 2.1 and 2.2 hold and W − π ≥ M is satised. One
optimal solution to Problem 1.2 is given by
f∗(x) =
0, if 0 ≤ x < t0,
x− t0, if t0 ≤ x ≤M,(20)
where t0 is determined by (1 + ρ)E[f∗(X)] = π.
Proof. It is a direct consequence of Proposition 4.1 and Lemma 2.2.
4.2 Assume W − π < M .
In this case, the objective function in (11) can be rewritten as follows
V1(λ, F ) =
∫ W−π
0F (t)
[gY (W − π − t)− λ
]dt− λ
∫ M
W−πF (t)dt+ λ(M − π). (21)
13
We follow the same procedure as applied in Section 3 to solve (11). Because F (W − π) ∈[FX(W − π), 1] for every F ∈ F∗∗, we consider the following sub-problems
maxF∈F∗∗
V1(λ, F ), s.t. F (W − π) = u, (22)
indexed by u ∈ [FX(W −π), 1]. If we could derive an optimal solution F ∗λ,u and optimal value
η(u) := V1(λ, F∗λ,u) of (22) for each for each u ∈∈ [FX(W − π), 1], then F ∗λ := F ∗λ,u∗ with
u∗ = argmaxu∈[FX(W−π),1]
η(u) is an optimal solution to (11).
We analyze the solutions of (22) for λ, respectively, in three dierent ranges: λ = 0,
0 < λ ≤ gY ((W − π)−), and gY ((W − π)−) ≤ λ ≤ gY (0+). The case of λ > gY (0+) is not
relevant in order to apply Lemma 2.3 for optimal CDF of (9) and thus omitted.
Case 4.1 λ = 0.
From (21), in this case, V1(λ, F ) =∫W−π0 F (t)gY (W − π − t)dt, which is non-decreasing in
F . Therefore, F ∗λ (t) = 1, t ∈ [0,M ], is an optimal solution to (11).
Case 4.2 0 < λ ≤ gY ((W − π)−).
In this case, gY (W − π − t) ≥ λ for t ∈ [0,W − π]. Therefore, by virtue of (21), a solution
to (22) is given by
F ∗λ,u(t) =
u, if 0 ≤ t < F−1X (u),
FX(t), if F−1X (u) ≤ t ≤M,
which, along with the condition u ∈ [FX(W − π), 1], further implies
η(u) := V1(λ, F∗λ,u)
=
∫ W−π
0u[gY (W − π − t)− λ
]dt− λ
∫ F−1X (u)
W−πudt− λ
∫ M
F−1X (u)
FX(t)dt+ λ(M − π).
We compute the left derivative of η(u) as follows
η′−(u) =
∫ W−π
0
[gY (W − π − t)− λ
]dt− λ
[∫ F−1X (u)
W−πdt+ u · (F−1X )′−(u)
]+ λFX(F−1X (u))(F−1X )′−(u)
=λ[FY (W − π)
λ− F−1X (u)
], u ∈ (FX(W − π), 1).
We can similarly derive its right derivative, and indeed, it is the same as the left derivative we
just obtained. Thus, η(u) is dierentiable over u ∈ (FX(W −π), 1) with η′(u) = λ(FY (W−π)
λ −
F−1X (u)), which is decreasing in u. This implies that η(u) is a concave function for u ∈
[FX(W − π], 1] and attains its maximum over [FX(W − π), 1] at
u∗ = min
max
[FX(W − π), FX
(FY (W − π)
λ
)], 1
.
14
We further note that
FY (W − π) =
∫ W−π
0gY (y)dy ≥
∫ W−π
0gY ((W − π)−)dy = (W − π)gY ((W − π)−),
and thus, FY (W − π)/λ ≥W − π for 0 < λ ≤ gY ((W − π)−). This implies
u∗ = min
FX
(FY (W − π)
λ
), 1
= FX
(min
(FX(W − π)
λ, M
)),
and
F ∗λ (t) = F ∗λ,u∗(t) =
FX(
min(FX(W−π)
λ , M))
, if 0 ≤ t < min(FX(W−π)
λ , M),
FX(t), if min(FX(W−π)
λ , M)≤ t ≤M.
Notably, in the case of M ≤ FX(W − π)/λ, F ∗λ (t) = 1 for t ∈ [0,M ].
Case 4.3 gY ((W − π)−) ≤ λ ≤ gY (0+).
In this case, there exists t0 ∈ [0,W − π] such that gY (W − π− t)− λ ≤ 0 for t ∈ [0, t0) and
gY (W − π − t)− λ ≥ 0 for t ∈ (t0,W − π], where [0, t0) = ∅ for t0 = 0. Hence, in view of (21)
and the fact that F (t) ≥ FX(t), ∀t ∈ [0,M ] and F (W − π) = u for every feasible F in (22), a
solution to (22) is given by
F ∗λ,u(t) =
FX(t), if 0 ≤ t < t0,
u, if t0 ≤ t < F−1X (u),
FX(t), if F−1X (u) ≤ t ≤M.
Accordingly,
η(u) := V1(λ, F∗λ,u)
=
∫ t0
0FX(t)
[gY (W − π − t)− λ
]dt+
∫ W−π
t0
u[gY (W − π − t)− λ
]dt
−λ∫ F−1
X (u)
W−πudt− λ
∫ M
F−1X (u)
FX(t)dt+ λ(M − π).
Similarly to the previous case, we respectively consider the left and right derivatives of η(u)
and nd that they coincide for u ∈ (FX(W − π), 1) and are given by
η′(u) =λ[FY (W − π − t0)
λ+ t0 − F−1X (u)
],
which is non-increasing as a function of u ∈ [FX(W −π), 1]. This implies that η(u) is concave
and attains its maximum over [FX(W − π), 1] at
u∗ = min
max
[FX(W − π), FX
(FY (W − π − t0)λ
+ t0
)], 1
.
15
Since gY (t) is non-increasing in t and gY (W − π − t) ≥ λ for t ∈ (t0,W − π],
FY (W − π − t0)/λ+ t0 =
∫ W−π
t0
gY (W − π − t)λ
dt+ t0 ≥∫ W−π
t0
dt+ t0 = W − π.
Hence,
u∗= min
FX
(FY (W − π − t0)λ
+ t0
), 1
= FX
(min
(FY (W − π − t0)λ
+ t0, M))
=FX (H(t0, λ)) ,
and
F ∗λ (t) = F ∗λ,u∗(t) =
FX(t), if 0 ≤ t < t0,
FX
(H(t0, λ)
), if t0 ≤ t < H(t0, λ),
FX(t), if H(t0, λ) ≤ t ≤M,
(23)
where
H(t, λ) = min
FY (W − π − t)
λ+ t, M
. (24)
With the analysis in the above Cases 4.1, 4.2 and 4.3, Lemma 2.3 can be readily invoked
for the solution of (11). The solution is given in Propositions 4.2 and 4.3, respectively, for a
small π and a large π. To proceed, we dene
d := FY (W − π)/gY ((W − π)−) and φ(F ) :=
∫ M
0[1− F (t)] dt, F ∈ F∗∗. (25)
We further write
π1 :=
∫ d
0[1− FX(t)] dt. (26)
Proposition 4.2 Assume that Assumptions 2.1 and 2.2 hold. For 0 ≤ d ≤M , dene
Fd(t) :=
FX(d), if 0 ≤ t < d,
FX(t), if d ≤ t ≤M.
Then, for each π ∈ [0, π1], there exists a constant t0 ∈[d, M
]to satisfy φ(Ft0) = π and Ft0
solves (9).
Proof. Based on the analysis in Case 4.1, Fd solves (11) with λ = FY (W − π)/d,
d ∈[d, M
]. Moreover, it is easy to check that φ(Fd) is continuous as a function of d with
limd→M φ(Fd) = 0 and limd→d φ(Fd) = φ(F
d) = π1. Thus, by the Intermediate Value Theorem,
16
there exists a constant t0 ∈[d, M
]such that φ(Ft0) = π, and consequently, by Lemma 2.3,
Ft0 is one solution to (9).
To derive the solution to (9) for π larger than π1, some extra notations are necessary. For
each a ∈ [0,W − π], we denote
Λa :=
[limt→a+
gY (W − π − t), limt→a−
gY (W − π − t)].
Further, for each a ∈ [0,W − π] and λa ∈ Λa, we dene
F (a,λa)(t) :=
FX(t), if 0 ≤ t < a,
FX
(H(a, λa)
), if a ≤ t < H(a, λa),
FX(t), if H(a, λa) ≤ t ≤M,
(27)
so that
φ(F (a,λa)) =
∫ a
0[1− FX(t)] dt+ [H(a, λa)− a] · [1− FX(H(a, λa))]
+
∫ M
H(a,λa)[1− FX(t)] dt.
Since gY (t) is non-increasing in t, given any λ ∈ Λa, gY (W −π− t)−λ ≤ 0 for t ∈ [0, a) and
gY (W−π−t)−λ ≥ 0 for t ∈ (a,W−π]. Therefore, according to the analysis in Case 4.3, F(a,λa)
is a solution to (22) for any λa ∈ Λa. Moreover, it is worth noting that Λa = gY (W −π− a),which is a singleton set, at any continuity point a of gY (W − π− a). Therefor, for any interval
(s1, s2) where gY (W − π − t) is continuous, φ(F (a,λa)) is a continuous function of a.
Proposition 4.3 Assume that Assumptions (2.1) and (2.2) hold. For each
π ∈ [π1, E[X]) ,
there exists a constant a ∈ [0,W − π] and λa ∈ Λa such that F (a,λa) solve (9).
Proof. Based on the analysis in Case 4.3 and Lemma 2.3, it is sucient to show the
existence of a ∈ [0,W −π] and λa ∈ Λa to satisfy φ(F (a,λa)) = π for each π ∈ [π1, E[X]). Note
that F (a,λa) = Fdfor a = 0 and λa = limt→0+ gY (W−π−t) = gY ((W−π)−), and F (a,λa) = FX
for a = W − π and any λa ∈ ΛW−π. These two cases lead to φ(F (a,λa)) = φ(Fd) = π1 and
φ(F (a,λa)) = E[X], respectively. Therefore, it is sucient for us to assume π ∈ (π1, E[X]) in
the rest of the proof.
Dene
Sπ :=t ∈ [0,W − π] : ∃λ ∈ Λt such that φ(F (t,λt)) > π
,
17
and t0 := supSπ. If gY (W−π−t) is continuous at t0, then it is continuous over a neighbourhoodof t0, say (t0 − δ, t0 + δ) for some constant δ > 0, because it is a monotone function. In this
case, Λt = g(W − π − t) for each t ∈ (t0 − δ, t0 + δ) and therefore, φ(F (a,λa)) is a continuous
function of a over (t0 − δ, t0 + δ) with λa = g(W − π − a), which in turn implies that, given
any ε > 0,
φ(F (s,λs))− ε ≤ φ(F (t0,λt0 )) ≤ φ(F (s,λs)) + ε,
whenever s ∈ (t0 − κ, t0 + κ) for some constant κ ∈ (0, δ). On one hand, by the supremum
property of t0, there exists s1 ∈ (t0 − δ1, t0) and s1 ∈ Sπ such that
φ(F (t0,λt0 )) ≥ φ(F (s1,λs1 ))− ε ≥ π − ε.
One the other hand, there exists s2 ∈ (t0, t0 + δ2) and s2 /∈ Sπ such that
φ(F (t0,λt0 )) ≤ φ(F (s2,λs2 )) + ε ≤ π + ε.
Letting ε→ 0 in the last two displays, we get φ(F (t0,λt0 )) = π.
Otherwise, we assume that gY (W − π − t) is discontinuous at t0, and dene
g−t0 := limt→t−0
gY (W − π − t), and g+t0 := limt→t+0
gY (W − π − t).
Since gY (W − π − t) is monotone as a function of t, it has at most countably jumps, which
in turn implies that there exists a constant δ > 0 such that gY (W − π − t) is continuous over(t0 − δ, t0) and (t0, t0 + δ). Therefore, φ(F (s,λs)) is a continuous function of s over (t0 − δ, t0)and (t0, t0 + δ) with
lims→t−0
φ(F (s,λs)) = φ(F (t0,g−t0)) and lim
s→t+0φ(F (s,λs)) = φ(F (t0,g
+t0)).
By the supremum property of t0, there exists s1 ∈ (t0 − δ, t0) and s1 ∈ Sπ such that
φ(F (t0,g−t0)) ≥ φ(F (s1,λs1 ))− ε ≥ π − ε.
On the other hand, there exists s2 ∈ (t0, t0 + δ) and s2 /∈ Sπ such that
φ(F (t0,g+t0)) ≤ φ(F (s2,λs2 )) + ε ≤ π + ε.
Letting ε→ 0 in the last two displays, we obtain
φ(F (t0,g+t0)) ≤ π ≤ φ(F (t0,g
−t0)).
φ(F (t0,λ)) with t0 xed is a continuous function of λ and therefore, there must exist some
λt0 ∈ [g+t0 , g−t0
] ≡ Λt0 such that φ(F (t0,λt0 )) = π.
The optimal ceded loss function for Problem 1.2 can be retrieved from the optimal CDF
that is derived in Propositions 4.2 and 4.3 by invoking Lemma 2.2.
18
Theorem 4.2 Assume that Assumptions 2.1 and 2.2 hold.
(a) For each (1 + ρ)(E[X]− π1) ≤ π ≤ (1 + ρ)E[X], one solution to Problem 1.2 is given by
f∗(x) =
0, if 0 ≤ x < t0,
x− t0, if t0 ≤ x ≤M,
where t0 is determined by (1 + ρ)E[f∗(X)] = π.
(b) For each 0 < π < (1 + ρ)(E[X] − π1), there exists a constant a ∈ [0,W − π] and λa ∈ Λa
such that one optimal solution to Problem 1.2 is given by
f (a,λa)(x) =
0, if 0 ≤ x < a,
x− a, if a ≤ x < H(a, λa),
0, if H(a, λa) ≤ x ≤M,
(28)
where (1 + ρ)E[f (a,λa)(X)
]= π.
Proof. Since π = E[X]−π/(1 + ρ), the conditions on π given in both parts are equivalent
to those two in terms of π in Proposition 4.3 respectively. The result of part (a) follows from
Lemma 2.2 and Proposition 4.2 as follows
f∗(x) = x− r∗(x) = x− (Ft0)−1(FX(x)) =
x, if 0 ≤ x < t0,
x− t0, if t0 ≤ x ≤M,
where t0 and Ft0 are as given in Proposition 4.2. The result of part (b) can be proved similarly
by using Lemma 2.2 and Proposition 4.3 as
f∗(x) = x− r∗(x) = x− (F (a,λa))−1(FX(x)) =
0, if 0 ≤ x < a,
x− a, if a ≤ x < H(a, λa),
0, if H(a, λa) ≤ x ≤M,
where a and λa are as given in Proposition 4.3.
Remark 4 It is interesting to compare the solutions between Problems 1.1 and 1.2. For the
case of W −π ≥M , their solutions are given in Theorems 3.1 and 4.1 respectively. In this case,
only a stop-loss reinsurance is shown to be optimal for Problem 1.2, whereas any reinsurance
treaties satisfying the premium budget constraint are optimal to Problem 1.1.
The solutions for the case of W − π < M are given in Theorems 3.2 and 4.2 for the two
problems respectively. For both problems, the optimal reinsurance contract is a stop-loss treaty
for larger premium budget π and a truncated stop-loss treaty for small premium budget π. This
19
means that, in the later case with a small premium budget, the optimal strategy for the insurer
is to entirely sacrice the protection against the occurrence of a large loss. The critical point for
the optimal solution transits from a stop-loss treaty to a truncated stop-loss one diers between
these two problems either. In the presence of Assumption 2.2, gY is non-increasing, and thus,
it follows from (25) that
d = FY (W − π)/gY ((W − π)−) ≥W − π.
This in turn follows form (15) and (26) that π0 ≤ π1, and therefore,
(1 + ρ)(E[X]− π1) ≤ (1 + ρ)(E[X]− π0).
5 Conclusion
In the present paper, we propose an innovative cumulative distribution function (CDF)
based method to solve a constrained and generally non-convex stochastic optimization prob-
lem, which arises from the area of optimal reinsurance, and targets to design the optimal
reinsurance contract for an insurer to maximize its survival probability or for a more gen-
eral goal-reaching model. It is an important decision problem to the insurance companies in
their risk management. Our proposed method reformulates the optimization problem into a
functional linear programming problem of determining an optimal CDF over a corresponding
feasible set. The linearity of the CDF formulation allows us to conduct a pointwise optimiza-
tion procedure, combined with the Lagrangian dual method, to solve the problem. Compared
with the existing literature, our proposed CDF based method is more technically convenient
and transparent in the derivation of optimal solutions. Moreover, our proposed CDF based
method can be readily applied for analytical solutions in the presence of background risk. The
inclusion of background risk leads to a more complex problem, and the analytical solutions are
obtained for the rst time.
Acknowledgements
Weng thanks the nancial support from the Natural Sciences and Engineering Research
Council of Canada, and Society of Actuaries Centers of Actuarial Excellence Research Grant.
Zhuang acknowledges nancial support from the Department of Statistics and Actuarial Science,
University of Waterloo.
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