A continuous-time theory of reinsurance chains

Lv Chen,∗ Yang Shen,† Jianxi Su‡ §

January 29, 2020

Abstract

A chain of reinsurance is a hierarchical system formed by the subsequent interactions among multiple (re)insurance

agents, which is quite often encountered in practice. This paper proposes a novel continuous-time framework for studying

the optimal reinsurance strategies within a chain of reinsurance. The transactions between reinsurance buyers and sellers

are formulated by means of Stackelberg games, in order to reflect the conflicting interests and unequal negotiation powers

in the bargaining process. Assuming the variance premium principle and the mean-variance criterion on the surplus

processes, we solve the time-consistent optimal reinsurance demands and pricing strategies in explicit forms, which are

surprisingly plain.

Based on the proposed reinsurance chain models, our in-depth theoretical analysis shows that: a.) it is optimal to

situate more (resp. less) risk averse reinsurers to the latter (resp. former) positions in a chain of reinsurance; b.) adding

new reinsurers will lower the reinsurance prices at all levels in a chain of reinsurance, promoting the existing agents to

rationally control their respective risk exposures; and essentially c.) alleviate the systemic risk in the chain structure.

Keywords: Stackelberg games, variance principle, mean-variance optimization, time inconsistency, diversification, sys-

temic risk.

JEL classifications: C61, G11, G22.

∗Academy of Statistics and Interdisciplinary Sciences, East China Normal University, Shanghai 200062, China. Email: chenlvhero@sina.com;†School of Risk and Actuarial Studies, University of New South Wales, Sydney, NSW 2052, Australia. Email: y.shen@unsw.edu.au;‡Department of Statistics, Purdue University, West Lafayette, IN, 47906, United States. Email: jianxi@purdue.edu;§Corresponding author; postal address: 150 N. University Street, West Lafayette, IN, 47906.

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1 Introduction

Reinsurance is an integral component of the insurance sector. It has been a common practice that insurance companies

opt to cede parts of their risk exposures to reinsurers, in order to limit the earning volatility. In fact, the use of reinsurance

reaches far beyond risk transfer. Reinsurance further contributes to the portfolio diversification and capital management

of the insurance conglomerates, underlying how risk and profit are two sides of the same coin.

Along with the rapid development in the global insurance market, it is the emerging concept of reinsurance networks.

For instance, traditional insurers may engage in reinsurance business, and reinsurers purchase insurance from other

counterparties in the insurance market. This present article focuses on the study of reinsurance chains which are one

of the most fundamental structures in the more complex concept of reinsurance networks. To be specific, a chain of

reinsurance is a hierarchical system formed by the interactions among multiple (re)insurance agents. Each agent in the

reinsurance chain is in direct contact with only two others: the one to which (re)insurance coverage is sold, and the one

from which coverage is bought. One exception is the end-position agent who purchases no further reinsurance and thus

has to bear all residual losses.

Before placing the present paper into perspective, here is a coarse overview of the current literature. The first in-

vestigation on reinsurance chains can date back to the seminal work Gerber (1984) in which the optimal strategies on

reinsurance demand and pricing were obtained based on the normally distributed insurance claims. Soon after Gerber’s

paper was published, the study of reinsurance chains had been extended along several directions. For example, d’Ursel

and Lauwers (1985) considered the same reinsurance chain problem using another equilibrium concept that is of Cournot

type (Cournot, 1838). Lemaire and Quairiere (1986) generalized the results in Gerber (1984) by applying Borch’s theorem

(Borch, 1974). In contrast to the non-cooperative equilibria used in the aforementioned works, d’Ursel and Lauwers (1986)

studied the Pareto optimality of reinsurance chains.

Despite that reinsurance chains are often encountered in practice, related research received rather litter attention after

the earlier studies initiated in 1980s (e.g., d’Ursel and Lauwers, 1985, 1986; Gerber, 1984; Lemaire and Quairiere, 1986).

Mathematical investigations of reinsurance chains have not been revisited for over decades. In particular, all the existing

studies of reinsurance chains are based on single-period models which cannot reflect the changes in the claim process

and the associated (re)insurance transactions over time. Thus, the use of single-period models may not be sufficient for

the managerial needs in the context of dynamic insurance risk management in the long run. Through stochastic control

theory which has been a set of mature toolkits applied in numerous directions of economics research, we strive to put

forth to a continuous-time framework for studying the optimal strategies within a reinsurance chain structure. To the

best of our knowledge, this is the first attempt in the economic dynamics literature. Our primary contribution consists

of developing a rigorously formulated yet operational continuous-time paradigm for modeling the dynamic transactions

within reinsurance chains under uncertainty. We hope that our endeavor in this present paper may serve as the stepping

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stone towards future investigations on the more intricate object of reinsurance networks.

To briefly preview the proposed framework, for a given chain of reinsurance, we use the classic Cramer-Lundberg

risk process to model the occurrences of insurance claims which will be indemnified subsequently by the agents in the

reinsurance chain. Each agent in the reinsurance chain aims to decide the optimal level of risk to acquire and to transfer.

Thus the interactions among the agents form a system of bargaining rules. Motivated by the empirical finding that the

dominance between the reinsurance buyer and seller is unequal (Kanno, 2016; Plantin, 2006), we formulate the bargaining

rules in terms of leader-follower games (a.k.a., Stackelberg games; also see Chen and Shen, 2018, 2019; Gerber, 1984,

for a similar consideration). Indeed, the framework of Stackelberg games has been widely used in economics research

(see Jørgensen et al., 1989; Øksendal et al., 2013, for other applications besides reinsurance). To determine the optimal

reinsurance demands and prices in the reinsurance chain, we concern about both the performance and stability of the

surplus processes of individual agents. The mean-variance criterion merges naturally in the optimization problem as a

systematic treatment of the dilimma that each agent faces: the conflicting objectives of “high profit” versus “low risk”.

One major challenge in the corresponding analysis is that the Bellman optimality principle does not hold for dynamic

mean-variance problems, which unfavorably results in an time-inconsistency nature (Bjork et al., 2014). To properly

address the just-mentioned issue, we simultaneously work with two large sets of embedded games with one set describing

the bargaining processes and the other set accounting for the time-inconsistency. We manage to solve the complex control

problem in a recursive manner. The solutions are explicit and surprisingly plain, which allow us to derive a collection of

economics implications.

In addition to the methodological contributions mentioned above, the present study also contains practical contributions

that help to explain the economics underpinnings behind the form of reinsurance chains. Namely, capitalizing on the

proposed reinsurance chain models, we will develop fairly complete theoretical answers to the following set of practical

questions which to the best of our knowledge, have not been investigated exclusively before, at least so far as a continuous-

time framework is concerned.

Q1. What is the optimal structure for a chain of reinsurance?

Q2. How do extra reinsurers influence the existing agents’ strategies in a chain of reinsurance?

Q3. How do extra reinsurers alleviate the systemic risk when a chain of reinsurance is viewed as a whole?

We believe that the study of these questions is needfully important since they offer, on the one hand, a general guidance

for regulatory authorities to oversee the rationality of the insurance market (i.e., via the answer to Q1), and on the other

hand, sound theoretical evidences to justify the economics benefits by promoting an open market in the insurance sector

(i.e., via the answers to Q2 and Q3).

The rest of the article will be proceeded as follows. Setting up the reinsurance chain model and the associated

optimization problem in Section 2, we derive the optimal solutions in Section 3. In Section 4, we provide the theoretical

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answers to Questions Q1-Q3 outlined above. Section 5 concludes the paper. To facilitate the reading, a summary of the

notation system is provided in Appendix A, and all the technical proofs are relegated to Appendix B.

2 Statement of the problem

The subject matter of this section is to put forth a mathematical formulation for the reinsurance chain problem of interest.

2.1 The reinsurance chain model

We are going to propose a “minimalist” setup that allows us to study the questions posed in Section 1. At the outset, we

shall prepare ourselves with the terminology and notation about the insurance claims to be disbursed along the reinsurance

chain. We fix a filtered probability space(Ω,F ,F := F(t)t∈[0,T ],P

)satisfying the usual conditions of completeness and

right continuity. Here, T > 0 is the terminal time of decision making, P is a real-world probability measure, and F is a

right-continuous and P-complete filtration generated by the insurance claim process which will be introduced next.

For a given hypothetical primary insurer, we view the insurance claims from the pool of all customers as a whole and

use a compound Poisson process to model the aggregate claim over time. More precisely, the frequency distribution of

insurance claims is modeled by a homogeneous Poisson process, notationally N(t)t∈[0,T ] with intensity λ ∈ R+, which

counts the accumulative number of claims up to time t. The severity random variables (RV’s), Yj ∈ R+, j ∈ N are used

to model the distribution of individual claims, and they are assumed to be independently and identically distributed with

a common distribution function F : R+ → [0, 1]. By convention, we further assume N(t)t∈[0,T ] and Yjj∈N to be

independent, and the first two raw moments of F exist, that is, for j ∈ N,

µ := µ′1 = E[Yj ] =

∫R+

y dF (y) <∞ and µ′2 = E[Y 2j ] =

∫R+

y2 dF (y) <∞.

Denote by M(·, ·) the Poisson random measure associated with N(t)t∈[0,T ] and Yii∈N. The corresponding compensator

is given by ν(dy)ds = λ dF (y)ds. The aggregate claim RV up to time t for the primary insurer, can be expressed as

L(t) :=

N(t)∑i=1

Yi =

∫ t

0

∫R+

y M(dy, ds). (1)

The expectation and variance of the aggregate claim RV can be computed via

E[L(t)] =

∫ t

0

∫R+

y ν(dy)ds and Var(L(t)

)=

∫ t

0

∫R+

y2 ν(dy)ds.

Lastly, we assume the primary insurer receives an instantaneous premium at rate c > 0 for insuring aggregate loss L in

Equation (1).

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Net premium c0 − c1 c1 − c2 cn−1 − cn cn

Premium payment c0 c1 c2 cn−1 cn

Customers −−→←−− PI −−→

←−− RI1 −−→←−− · · · −−→

←−− RIn−1 −−→←−− RIn

Indemnity l0 l1 l2 ln−1 ln

Retained loss l0 − l1 l1 − l2 ln−1 − ln ln

Table 1: Premium and indemnity flows in the reinsurance chain with “PI” and “RIi”, i = 1 . . . , n, representing the primaryinsurer and the i-th reinsurer, respectively.

No industry is immune to failures. Although the comparative frequency of insurer insolvencies has so far been relatively

low, the US alone witnessed approximately 700 such failures between 1969 and 1998 history (Massey et al., 2003). A natural

way with which the dangers of insolvency can be ameliorated, is to secure an appropriate level of reinsurance. Namely

the primary insurer will give up a part of the premiums collected from the customers, and pay to a reinsurer in exchange

for the protection against excessive insurance losses. Similarly, in order to manage the excessive risk of the insurance

portfolio, the reinsurer will further seek reinsurance from another party. This risk management activity will be repeated

sequentially, and as a consequence, a chain of reinsurance will be formed.

We proceed to suggest a mathematical formulation for the reinsurance chain described above. Denote by N0 the zero

augmented set of natural numbers, and throughout the rest of this article, let n ∈ N0. Without loss of generality, we

assume that there are(n+ 1

)participants in the reinsurance chain, of which one is the primary insurer and the remaining

participants are reinsurers. Thus, the reinsurance chain contains (n+1) levels of interactions. At each level, the (re)insurer

is in direct contact with only two other members in the chain: the one to which coverage is sold, and the one from which

coverage is bought. The first interaction in the chain is between the customers and the primary insurer; the position of

the primary insurer in the reinsurance chain is indexed by “0”. It is then followed by the interaction between the primary

insurer and the level 1 reinsurer (indexed by “1” in the chain), etc., through the n-th level of reinsurer. The flowchart

displayed in Table 1 schematically illuminates the hierarchical structure underlying the reinsurance chain of interest.

Some notations for describing the reinsurance demands and prices are also needed. We first introduce a set of loss

acquisition functions li : R2+ → R+, i = 0, . . . , n with l0 ≤ l1 ≤ · · · ≤ ln, to denote the loss acquired by the i-th

level (re)insurer from the lower level of the adjacent agent in the reinsurance chain. Given that a loss amounted to y

occurs at time t, the quantity l0(t, y) = y is the insurance indemnity paid by the primary insurer to the customers. In a

similar fashion, li(t, y), i = 1, . . . , n represents the reinsurance indemnity paid by the i-th level reinsurer to the (i− 1)-th

level (re)insurer. Equivalently, the function li describes the demand of reinsurance for the (i − 1)-th level (re)insurer,

i = 1, . . . , n. The loss borne by the i-th level (re)insurer is thus hi(t, y) := li(t, y) − li+1(t, y) for i = 0, . . . , (n − 1). The

n-th level reinsurer who is at the end position of the reinsurance chain, bears the residual loss hn(t, y) := ln(t, y).

Obviously, the demands for reinsurance are related to the reinsurance prices. For i = 1, . . . , n, denote by ci(t) > 0 the

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instantaneous reinsurance cost diverted from the (i− 1)-th level (re)insurer to the i-th level reinsurer for covering the risk

li(t, ·) at time t ∈ [0, T ]. The function c0(t) > 0 can be used to represent the instantaneous premium that the primary

insurer gathers from the pool of insureds, which is equal to c as we mentioned earlier.

Appealing to the language of actuarial mathematics, a premium principle is a functional H : X → [0,∞], where X

denotes the set of RV’s representing insurance losses. The literature on premium principle is vast and growing quickly.

Our article is normative and operating under the variance principle which has earned an unprecedented popularity among

both practitioners and theoreticians, particularly in the context of optimal reinsurance research (see, e.g., Chen and

Shen, 2018; Guerra and Centeno, 2010; Liang and Yuen, 2016). Formally, let L ∈ X and assume E[L2] <∞, the variance

principle is formulated as

H(L) = E[L] + ξVar(L), (2)

where ξ > 0 denotes the security loading parameter determined subjectively based on the insurance company’s risk

preference (Kaas et al., 2008). The choice of variance principle in our paper guarantees a desired level of mathematical

tractability for the optimization problem of interest, and as such explicit solutions are possible to obtain. Based on the

loss model specified in Equation (1) and the variance premium principle in Equation (2), the reinsurance cost function

associated with the loss proportion acquired by the i-th level reinsurer can be computed via

ci(t; (li, ξi)

)=

∫R+

li(t, y) ν(dy) + ξi(t)

∫R+

l2i (t, y) ν(dy), i = 1, . . . , n, (3)

where ξi(t) > 0 denotes the security loading factor set by the i-th level reinsurer at time t ∈ [0, T ]. With a bit abuse of

notation, we set ξ0(t) = ξ0 for all t ∈ [0, T ] such that c0(t; (l0, ξ0)

)= λ(µ+ ξ0 µ

′2) = c, or equivalently ξ0 = (c/λ− µ)/µ′2.

As can be seen from the cost function in Equation (3), for a given level of loss acquisition li, there is a one-to-one

correspondence between ξi and ci, i = 1, . . . , n. Hence, under our setup, the security loading factors ξ = (ξ1, . . . , ξn) are

the determinants of the reinsurance pricing strategies.

Collectively, for i = 1, . . . , n, we have that li reflects the demand of reinsurance for the (i− 1)-th level reinsurer, and ξi

describes the pricing strategy of the i-th reinsurer (i.e., ξi determines the cost ci) for insuring the risk li. We remark that

both li and ξi, i = 1, . . . , n are allowed to be time dependent. This implies that the (re)insurers in the reinsurance chain

can continuously adjust the reinsurance strategies based on the evolution of the claim processes and the interactions with

the other members.

In optimal reinsurance research, the study of the (re)insurer’s surplus process often plays a critical role (see, e.g., Chen

and Shen, 2018, 2019; Li et al., 2016, and references therein). In what follows, we report formally the stochastic models

underlying the surplus processes of the reinsurance chain’s agents. For notational convenience, denote by u = (u1, . . . ,un)

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with ui =(li, ξi

), i = 1, . . . , n, the set of reinsurance strategies for all the (re)insurers. Associated with u, let Ri(t ;u),

i = 0, . . . , n be the surplus process of the i-th level (re)insurer at time t ∈ [0, T ] and ρ > 0 be the exogenously given

interest rate credited/debited on the surplus. Based on the insurance model established thus far, we have

dRi(t ;u) =[ρRi(t ;u)︸ ︷︷ ︸

1

+ ci(t;ui)− ci+1(t;ui+1)︸ ︷︷ ︸2

]dt−

∫R+

[li(t, y)− li+1(t, y)︸ ︷︷ ︸

3

]M(dy, dt) (4)

for i = 0, . . . , n− 1, and

dRn(t ;u) =[ρRn(t ;u)︸ ︷︷ ︸

1

+ cn(t;un)︸ ︷︷ ︸2

]dt−

∫R+

ln(t, y)︸ ︷︷ ︸3

M(dy, dt). (5)

Table 2 summaries the descriptions for the components in surplus process (4)-(5). A visual illustration of the notation

system proposed for describing the reinsurance chain of interest can be found in Table 1.

Number Description

1 Interest credited/debited on surplus.

2 Net premium earned.

3 Retained loss proportion, given that an insurance claim occurs.

Table 2: Descriptions of the components in the surplus dynamics.

2.2 The optimization problem

With no doubt, evaluating the optimal level of risk to retain is a vital task for any insurance companies in practice. In the

context of reinsurance chains, the interactions between the neighbouring (re)insurers form a system of bargaining rules in

which each agent aims to determine the optimal price for selling reinsurance to the lower level (re)insurer and the optimal

demand for further reinsurance from the higher level reinsurer. Identifying the optimal reinsurance strategies which consist

of the optimal reinsurance demands and prices, is one of the major objectives in our study. Before proceeding forward,

we define formally the set of all admissible reinsurance strategies.

Definition 1. A set of strategies u(· , ·) = (u1(· , ·), . . . ,un(· , ·)) with ui(· , ·) =(li(· , ·), ξi(·)

)for i = 1, . . . , n is said to

be admissible if

(i) the reinsurance acquisition processes li(· , ·), i = 1, . . . , n are F-predictable and satisfy 0 ≤ ln(· , ·) ≤ · · · ≤ l1(· , ·) ≤ y;

(ii) the reinsurance premium processes ξi(·), i = 1, . . . , n are F-predictable and such that the associated reinsurance cost

functions satisfy 0 ≤ cn(· ;un) ≤ · · · ≤ c1(· ;u1) ≤ c;

(iii) associated with u(· , ·), the surplus equations (4) and (5) have unique strong solutions Ri(· ;u), i = 0, . . . , n such that

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they are cadlag, F-adapted processes satisfying

E[

supt∈[0,T ]

|Ri(t ;u)|2]<∞, i = 0, . . . , n.

Denote by Al and Aξ respectively the spaces of all admissible strategies associated with the reinsurance acquisition and

premium, l(·, ·) = (l1(·, ·), . . . , ln(·, ·)) and ξ(·) = (ξ1(·), . . . , ξn(·)), then A := Al × Aξ is the space of all admissible

reinsurance strategies.

Remark 1. The first set of inequalities about the loss acquisition functions in condition (i) of Definition 1 assures that

short-selling of insurance risk is prohibited. The second set of inequalities about the cost functions in condition (ii) tells

that the (re)insurance premium earned from selling coverage to the lower level agent should at least suffice to cover the

cost for further reinsuring. This is a natural condition, since otherwise, there is no financial incentive for the agents to

enter the reinsurance chain.

In order to study the optimal strategies in the reinsurance chain, we need to lay down a meaningful groundwork for

formulating the optimality properly. At the outset, we remind the reader that there is a negative relationship between

reinsurance prices and demands. Namely, a cheaper reinsurance policy offered by the upper level reinsurers will boost the

reinsurance demands from the lower level (re)insurers, and versa vice. This reveals the presence of conflicting interests in

the bargainning processes between the neighboring agents, which motivates us to study the optimization problem using a

game theoretic approach.

In the insurance economics literature, game theory is one of the most prevailing techniques for studying optimal

reinsurance problems. Depending on the nature of the problems at hand, a variety of game equilibrium concepts can

be employed. For instance, concerning about the non-cooperative conflicts between reinsurer and insurer, Aase (2009)

and Boonen et al. (2016) proposed to use the Nash equilibrium concept to price reinsurance contracts. In a monopolistic

reinsurance market, the principal-agent framework can be used to capture the information asymmetry presented between

reinsurer and insurer (see, e.g., Doherty and Smetters, 2005; Horst and Moreno-Bromberg, 2008; Laffont and Martimort,

2002, and references therein). Another important equilibrium concept which is of great relevance to the study of the present

optimal reinsurance problem, is the Stackelberg equilibrium meaning that one player of the game (a.k.a. the “leader”)

moves first, and the other player (a.k.a. the “follower”) moves afterward (von Stackelberg, 1934; also see, Simaan and

Cruz, 1973, for the multi-period extension). Assuming that the leader may anticipate the follower’s rational reaction, the

leader will announce the policy that optimizes her performance criterion, and then the follower would pick an optimal

strategy according to the leader’s announced strategy. For recent applications of Stackelberg game in insurance studies,

we refer the reader to, e.g., Chen and Shen (2018, 2019), Dutang et al. (2013) and Lin et al. (2012)

In this paper, we will focus on the Stackelberg equilibrium and consider agents in the reinsurance chain as the players of a

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set of stochastic Stackelberg differential games. The choice of Stackelberg equilibrium is suitable for our problem because

empirical studies have shown that reinsurance sellers often possess a dominating role in the bargaining of reinsurance

contracts (Kanno, 2016; Plantin, 2006). It is natural to view the reinsurance seller (resp. buyer) as the leader (resp.

follower) of the Stackelberg games. To be more concrete, consider the bargaining between the (i − 1)-th and i-th level

(re)insurers, i = 1, . . . , n. Under the Stackelberg terminology, for a given security loading ξi set by the i-th level reinsurer

(thus determining the reinsurance cost ci), the optimal reinsurance demand from the (i−1)-th level (re)insurer is assumed

to be predictable and denoted by l∗i (ξi). In anticipating of l∗i (ξi), the i-th level reinsurer determines the optimal pricing

strategy ξ∗i in order to achieve the best reward at l∗i := l∗i (ξ∗i ). The set of optimal strategies u∗ :=

(l∗i , ξ∗i i=1,...,n

)∈ A

are the Stackelberg equilibria that we aim to establish.

Finally, we need an optimality criterion to make our optimization problem complete. Among the set of admissible

reinsurance strategies, we assume that the (re)insurers concern about the performance of the resulting surplus processes.

The (re)insurers aim to maximize the expected terminal values of the surplus processes while minimize the associated

variabilities. Let t ∈ [0, T ] be an initial time, and suppose that x = (x0, . . . , xn) ∈ Rn+1 is a vector of initial surpluses of

all the (re)insurers in the chain, that is, R(t) = (R0(t), R1(t), . . . , Rn(t)) = x. Shorthand the conditional expectation and

variance of the terminal surplus by Et,x[Ri(T )] = E[Ri(T ) | Ft, R(t) = x

]and Vart,x(Ri(T )) = Var

(Ri(T ) | Ft, R(t) = x

),

i = 0, . . . , n, respectively. Also, for i = 0, . . . , n, we introduce the risk aversion parameter γi > 0 to describe the level of

risk tolerance for the i-th (re)insurer. Under the mean-variance criterion, it follows that the objective function associated

with the reinsurance strategies u ∈ A for the i-th level (re)insurer is given by

Ji(t,x;u) = Et,x[Ri(T ;u)

]− γi

2Vart,x

(Ri(T ;u)

), i = 0, . . . , n, (6)

for x ∈ Rn+1, t ∈ [0, T ] and Ri(· ;u) is as per Equations (4)-(5). Our optimization problem is then to determine a set of

Stackelberg equilibria u∗ = (u∗1, . . . ,u∗n) ∈ A, such that the objective functions (6) are maximized.

Remark 2. It is clear from Equations (4) and (5) that each surplus process Ri(·) depends explicitly only on (ui,ui+1) for

i = 0, . . . , n− 1 and un for i = n. For simplicity, we include the entire strategy matrix u in the surplus process notation

(i.e., Ri(· ;u), i = 0, . . . , n) whenever there is no risk of confusion. The same terminology is also applied to the notations

for the objective functions in Equation (6). In Definition 2, we will highlight that each objective function depends on

explicitly only on (ui,ui+1) for i = 0, . . . , n− 1 and un for i = n.

We conclude this current section by noting that objective functions (6) are separable in the surplus processes. Namely,

each objective function Ji, i = 0, . . . , n is solely dependent on the surplus process of the i-th (re)insurer, but independent

of the others’. Thereby, we can use Ji(t, xi;u), i = 0, . . . , n to denote the objective function of the the i-th (re)insurer in

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the rest of the paper. To see the reason, for t ∈ [0, T ], let us first define the generalized risk aversion coefficients to be

γi(t) = γi expρ (T − t), i = 0, . . . , n. (7)

Recall that hi(·, ·) = li(·, ·) − li+1(·, ·), i = 0, . . . , n − 1, and hn(·, ·) = ln(·, ·) denote the risk retained by the individual

reinsurers. With some elementary algebraic manipulations, we obtain

J0(t, x0;u) = x0 eρ(T−t) + Et

∫ T

t

∫R+

eρ(T−s)(c

λ− y − ξ1(s) l21(s, y)

)ν(dy)ds

×[1 +

∫ T

t

∫R+

γ0(s)h0(s, y) M(dy, ds)

]− 1

2γ0

∫ T

t

∫R+

(γ0(s)h0(s, y)

)2ν(dy)ds

(8)

and

Ji(t, xi;u) = xi eρ(T−t) + Et

∫ T

t

∫R+

eρ(T−s)(ξi(s) l

2i (s, y)− ξi+1(s) l2i+1(s, y)

)ν(dy)ds

×[1 +

∫ T

t

∫R+

γi(s)hi(s, y) M(dy, ds)

]− 1

2γi

∫ T

t

∫R+

(γi(s)hi(s, y)

)2ν(dy)ds

(9)

for i = 1, . . . , n− 1, and

Jn(t, xn;u) = xn eρ(T−t) + Et

∫ T

t

∫R+

eρ(T−s) ξn(s) l2n(s, y) ν(dy)ds

×[1 +

∫ T

t

∫R+

γn(s)hn(s, y) M(dy, ds)

]− 1

2γn

∫ T

t

∫R+

(γn(s)hn(s, y)

)2ν(dy)ds

, (10)

where M(dy, ds) := M(dy, ds)− ν(dy)ds is a compensated random measure.

Remark 3. We note that if the reinsurnace strategies l(·, ·) and ξ(·) are deterministic functions, then there is only one

random component in Equations (8)-(10) with expectation

Et[ ∫ T

t

∫R+

γi(s)hi(s, y) M(dy, ds)

]= 0, i = 0, . . . , n, (11)

for all y ∈ R+ and t ∈ [0, T ]. As we will see latter in Theorem 1, it is exactly the case for the solution of our optimization

problem.

What is more, the control variables that describe the reinsurance strategies (i.e., li and ξi, i = 1, . . . , n), only appear

inside the expectation terms of Equations (8)-(10) which are independent of x. Hence, we can further conclude that the

optimal trajectories of li and ξi, i = 1, . . . , n must be independent of the present surpluses. The aforementioned separation

and state-independence properties are of central importance in the present study, as they will pave a mathematically

10

convenient route for us to attack the associated control problem agent by agent along the reinsurance chain.

3 Time-consistent optimal solutions

In this section, with an eye towards the related economics implications, we derive the optimal reinsurance rules for the

agents of the reinsurance chain described in Section 2. Though straightforward, we recall the elementary formula for

computing variance:

Var(X) = E[X2]−(E[X]

)2,

given that the expectations exist. Due to the non-linear function of the expected terminal surplus presented in objective

function (6), the law of iterated expectations does not hold in our control problem. This fact unfavorably results in a

time-inconsistency nature. One way of handling time-inconsistent control problems is pre-commitment, meaning that

the (re)insurers will disregard the time-inconsistency feature of the control problem and set their optimal reinsurance

strategies in stone at the initial time point (see, e.g., Bauerle, 2005; Chiu and Wong, 2011, and references therein). While

pre-commitment may be more convenient to work with from the mathematical perspective, the major drawback of this

approach is that the pre-committed rules are only optimal at present, but will no longer be optimal at future time points.

Thus, under the pre-commitment formulation, the implication of optimality is “conceptually unclear” (Bjork et al., 2014).

The limitations involved in the aforementioned pre-commitment approach have been frequently noticed and reported in

the economics literature (e.g., Chen and Shen, 2019; Chen et al., 2014; Cui et al., 2017; Wang and Forsyth, 2011).

In this article, we endeavour to derive the time-consistent solutions for the reinsurance chain problem such that the

optimality can hold over time. Toward this aim, we resort to the renewed notion of equilibrium control (see, Strotz, 1955,

for the earliest work; Bjork et al., 2014, for the recent development), which has found fruitful applications in the recent

economic dynamics literature. Specifically, for a given agent in the reinsurance chain, we consider another game played

by a continuum of completely different individuals, each of which represents the future incarnation of the (re)insurer at

the time point t ∈ [0, T ]. The Nash equilibria of the game then realize the time-consistent optimal reinsurance strategies

for the given (re)insurer. As was mentioned in Section 2, the interactions among the agents in the reinsurance chain

are also formulated using the game theoretic framework. Thereby, in order to derive the time-consistent solution, we

are actually dealing with a control problem consisting of two sets of embedded games. One set of Stackelberg games

describes the conflicting interests among all pairs of neighboring (re)insurers in the reinsurance chain, while the other

set of non-cooperative games is used to capture the time-inconsistency feature of the control problem. Table 3 depicts a

discrete-time illustration of the two-sets embedded games mechanism underlying the control problem. A rigorous definition

of the control problem is given in the sequel. Beforehand, let us recall that l = (l1, . . . , ln) ∈ Al and ξ = (ξ1, . . . , ξn) ∈ Aξ

11

t1 t2 · · · tm

PI A NCm game for the time-inconsistency of PI.

mRI1 A NCm game for the time-inconsistency of RI1.

A set of n Stackelberg games for cater-ing the interactions of (re)insurers.

m...

......

mRIn A NCm game for the time-inconsistency of RIn.

Table 3: A discrete-time illustration of the two-sets embedded games control problem of interest with 0 ≤ t1 < · · · < tm ≤T , m ∈ N. In this table, PI and RIi, i = 1, . . . , n are short for the primary insurer and i-th level reinsurer, respectively,and NCm denotes a non-corporative game with m players.

represent respectively, a set of given reinsurance acquisition and pricing rules in the reinsurance chain. Moveover, let

u∗i = (l∗i , ξ∗i ) be the time-consistent optimal trajectory associated with ui = (li, ξi), i = 1, . . . , n. With a bit abuse of

notation, we set u∗0 = u0 = (l0, ξ0). To make our presentation accurate, we highlight the dependence of the objective

functions Ji on ui and ui+1, for i = 0, . . . , n − 1, and Jn on un, rather than on the collective strategy u, and write

Ji(t, xi;ui,ui+1), for i = 0, . . . , n− 1, and Jn(t, xn;un).

Definition 2. For any fixed i ∈ 1, . . . , n, suppose that u∗i−1 is known. Given any admissible ξi, the loss acquisition rule

l∗i (ξi) is said to be a time-consistent Stackelberg follower action of the (i− 1)-th level reinsurer if, for xi−1 ∈ R, t ∈ [0, T ],

lim infε→0+

Ji−1(t, xi−1;u∗i−1,

(l∗i (ξi), ξi

))− Ji−1

(t, xi−1;u∗i−1,

(lεi (ξi), ξi

))ε

≥ 0 (12)

holds for any admissible li such that

lεi(s, y; ξi(·)

)=

li(y) for s ∈ [t, t+ ε)

l∗i(s, y; ξi(·)

)for s ∈ [t+ ε, T ]

,

where y, ε > 0.

Furthermore, assume l∗i (ξi) is known ex ante, the reinsurance premium rule ξ∗i is said to be a time-consistent Stackelberg

leader action if, for xi ∈ R, t ∈ [0, T ],

lim infε→0+

Ji(t, xi;

(l∗i (ξ

∗i ), ξ∗i

),ui+1

)− Ji

(t, xi;

(l∗i (ξ

εi ), ξ

εi

),ui+1

)ε

≥ 0, for i = 1, . . . , n− 1

lim infε→0+

Jn(t, xn;

(l∗n(ξ∗n), ξ∗n

))− Jn

(t, xn;

(l∗n(ξεn), ξεn

))ε

≥ 0, for i = n

(13)

12

holds for any admissible ξi such that

ξεi (s) =

ξi for s ∈ [t, t+ ε)

ξ∗i(s)

for s ∈ [t+ ε, T ]

,

where ε > 0.

Collectively, for i = 1, . . . , n, the pair of actions u∗i = (l∗i (ξ∗i ), ξ∗i ) are the time-consistent Stackelberg reinsurance

strategies between the neighbouring (i−1)-th and i-th reinsurers in the reinsurance chain described in Section 2. The value

functions associated with the aforementioned time-consistent Stackelberg reinsurance strategies are given by

Vi(t, xi) =

Ji(t, xi;u

∗i ,u∗i+1), for i = 0, . . . , n− 1,

Jn(t, xn;u∗n), for i = n.

(14)

We stress that Definition 2 contains two sets of optimal control problems. The limit inferior signs operating on the

objective functions Ji, i = 0, . . . , n describe the subgame perfect Nash equilibria of non-cooperative games accounting for

the time-inconsistency feature, and the two-steps sequential movements for determining optimal acquisition and pricing

strategies correspond to the Stackelberg games among the reinsurance chain’s agents. Importantly, Definition 2 character-

izes our optimal control problem in a recursive manner. To be specific, given that the insurance portfolio of the primary

insurer, u∗0 = (l0, ξ0), which is exogenously given, the first level reinsurer predicts the optimal reinsurance demand of the

primary insurer and designs the corresponding optimal premium strategy. Sequentially, based on the optimal strategies

appointed by the former reinsurers, the same decision-making process is repeated by the second, third, etc., till the last

reinsurer.

In order to derive the optimal strategies, we establish and solve a system of extended Hamilton-Jacobi-Bellman (HJB)

equations corresponding to the course of actions mentioned above. The detailed derivations are rather tedious and thus

are relegated to Appendix B. The succeeding theorem outlines the time-consistent optimal reinsurance strategies.

Theorem 1. Assume the reinsurance chain is described in Section 2. For the i-th level reinsurer, the optimal reinsurance

premium strategy is

ξ∗i (t) =

[2i−1∑j=0

(γj(t)

)−1]−1+

[(γi(t)

)−1+(2 ξ∗i+1(t)

)−1]−1, if i = 1, . . . , n− 1

[2n−1∑j=0

(γj(t)

)−1]−1+ γn(t), if i = n

(15)

13

which is calculated recursively in a backward manner. Furthermore, the optimal risk acquisition strategy is given by

l∗i (y) := l∗i (t, y; ξ∗i ) =

[(1/2)i

1/2 + ζ∗ii−1∑j=0

γ−1j

]× y, i = 1, . . . , n, (16)

where ζ∗i = ξ∗i (t) e−ρ(T−t) is time independent.

Proof. See Appendix B.

The optimal reinsurance strategies reported in the assertion above are surprisingly plain. What is more, both the

optimal reinsurance premium and acquisition strategies are independent of the claim process’s parameters. Therefore,

there is no risk of parameters miss-calibration, and the optimal solutions are likely to hold across a wide range of real

world situations, as long as the reinsurance arrangements comply with the chain structure described in Section 2.

A few economics implications can be immediately gleaned from Equations (15)-(16). We begin with the discussion

on the optimal reinsurance premium strategies. First, fixing the risk aversion coefficients γ = (γ0, . . . , γn) and the

instantaneous interest rate ρ, we observe that the optimal security loading ξ∗i (t), i = 1, . . . , n, is decreasing in time

t ∈ [0, T ]. Speaking bluntly, that is, advancing time leaves less uncertainty inherent in the decision making process, so

the reinsurers rationally choose to decrease the premium rate in order to attract additional business. Second, as will be

shown in Proposition 2 below, the optimal premium strategies track closely the risk and time preferences of the agents in

the reinsurance chain.

Proposition 2. For any fixed k ∈ 1, . . . , n, if any one of the elements in γ = (γ0, . . . , γn) or ρ increases, then the

optimal reinsurance pricing rule ξ∗k(·) increases.

Proof. See Appendix B.

As a result, even if only one of the agents shifts its risk preference and turns to be more risk averse, due to the inter-

dependencies in the reinsurance chain, the change will influence the pricing strategies of all other agents and reinsurance

will become more expensive along the chain. High interest rate affects the optimal pricing strategies in the same manner

as high risk aversion. This is because high return on risk-free investment eliminates the speculation demand of reinsurers,

consequently pushing up the reinsurance prices.

Next we turn to discuss the optimal acquisition strategies in Equation (16). First and perhaps formost, the optimal

amount of risk acquisition is time independent and is proportional to the loss occurred. Appealing to the language of

actuarial mathematics, this form of reinsurance arrangement is known as the proportional reinsurance. Another important

note on the optimal acquisition strategies is about their relationships with the risk aversion parameters. In the same vein

as the optimal premium strategies, each optimal acquisition strategy depends on the risk profiles of all the agents in

14

the reinsurance chain. However, various agents may influence the optimal acquisition strategies in a strikingly different

manner. This interesting observation is summarized in the succeeding assertion.

Proposition 3. For any fixed k ∈ 1, . . . , n, the optimal reinsurance acquisition rule l∗k(·) is increasing in γi for all

i = 0, . . . , k − 1, while decreasing in γi for all i = k, . . . , n

Proof. See Appendix B.

The result of Proposition 3 is practically meaningful. Fix k ∈ 1, . . . , n, if the k-th level reinsurer becomes more risk

averse, then it will rationally reduce the amount of risk to acquire from the lower level adjacent (re)insurer while at the

same time, demand extra reinsurance from the higher level adjacent reinsurer. Due to the interplays of the agents, the

impacts will spread out along the reinsurance chain with a domino effect.

As hitherto, we have established the optimal reinsurance strategies for the reinsurance chain of interest. The rest of

this current section is devoted to studying several ordering properties of these optimal rules. Recall that the definition

of admissible reinsurance strategies requires 0 ≤ ln(y, t) ≤ · · · ≤ l1(y, t) ≤ y to hold for any l = (l1, . . . , ln) ∈ Al (see,

Definition 1). For notational convenience, shorthand the reinsurance proportion by

p∗i = (1/2)i[1/2 + ζ∗i

i−1∑j=0

γ−1j

]−1, (17)

such that the optimal reinsurance acquisition strategies in Equation (16) can be expressed as l∗i (y) = p∗i y for y ∈ R+ and

i = 1, . . . , n. We note that

p∗1 =1/2

1/2 + ζ∗1 / γ0≤ 1,

thus l∗1(y) ≤ y for all y ∈ R+. Moreover, for any i = 1, . . . , n−1, evoking the recursive relationship in Equation (15) yields

p∗i = (1/2)i[1 +

[γ−1i +

(2 ζ∗i+1

)−1]−1 i−1∑j=0

γ−1j

]−1

= (1/2)(i+1)

[1/2 + ζ∗i+1

i∑j=0

γ−1j

]−1(1 +

2 ζ∗i+1

γi

)= p∗i+1

(1 +

2 ζ∗i+1

γi

)≥ p∗i+1. (18)

Hence, we can conclude 0 ≤ l∗n(y) ≤ · · · ≤ l∗1(y) ≤ y.

There is unfortunately, no ordering property can be established for the optimal security loadings. Two relevant

counterexamples are indicated. For both examples, without loss of generality, let γ0 = · · · = γn−2 = 1 and ρ = 0.

In the first example, further set γn−1 < ∞ and γn = ∞, then it is elementary to check that ∞ = ξ∗n(·) > ξ∗n−1(·)

where ξ∗n−1(·) is finite. In the other example, consider the case when γn−1 = ∞ and γn < ∞, then we readily have

ξ∗n−1(·) = 2ξ∗n(·) + (n− 2)/2 > ξ∗n(·), which stands in contrast to the inequality implied by the previous example.

15

Based on the variance premium principle, the reinsurance cost functions are increasing in both the acquisition rules

and security loadings (see, Equation (3)). However, it is noteworthy that there is a negative relationship between the

optimal risk acquisition and premium strategies (see, Equation (16)). At this point, it is a natural question as to the

ordering property for the reinsurance cost functions associated with the optimal strategies in Theorem 1. To answer this

question, let ω∗i (t) = ξ∗i (t)(p∗i)2, with i = 1, . . . , n and t ∈ [0, T ]. By evoking the recursive formula for computing ξ∗ as

well as that for p∗ = (p∗1, . . . , p∗n) established in Equation (18), we get

ω∗i (t) =

[2

i−1∑j=0

(γj(t)

)−1]−1+

[(γi(t)

)−1+(2 ξ∗i+1(t)

)−1]−1(1 +

2 ζ∗i+1

γi

)2 (p∗i+1

)2≥(

1 +2 ζ∗i+1

γi

)ξ∗i+1(t)

(p∗i+1

)2 ≥ ω∗i+1(t),

for t ∈ [0, T ], i = 1, . . . , n− 1. So under the optimal strategies in Theorem 1, the reinsurance cost functions satisfy

c∗i (t) := ci(t; (l∗i , ξ∗i )) = p∗i

∫R+

y ν(dy) + ω∗i (t)

∫R+

y2 ν(dy)

≥ p∗i+1

∫R+

y ν(dy) + ω∗i+1(t)

∫R+

y2 ν(dy) = c∗i+1(t).

In summary, we have shown that both the optimal reinsurance acquisition rules and the associated reinsurance cost

functions decline with the position in the reinsurance chain. So the inequalities with regard to the admissible acquisition

processes and cost functions in Definition 1 can be indeed satisfied by the optimal reinsurance strategies (15) and (16).

At the end of this section, let us study a monotonicity property for the value functions in the reinsurance chain. The

property shows that, for a specific agent, the value function is decreasing in the security loading set by the next level

reinsurer. In other words, high reinsurance premium hampers the value function of the reinsurance buyer. The result is

of auxiliary importance in the study of the next section.

Proposition 4. Assume the reinsurance chain is described in Section 2. Fix k ∈ 0, . . . , n−1, there is a one-to-one and

decreasing relationship between the value function of the k-th agent, i.e., Vk(t, xk), and the security loading factor set by

the (k + 1)-th agent, i.e., ξ∗k+1(t), for any t ∈ [0, T ] and xk ∈ R.

Proof. See Appendix B.

Remark 4. Thus far, we have assumed all the agents in the reinsurance chain share the same interest rate ρ > 0. On

the one hand, this is a fairly reasonable assumption. It is particularly true if all the agents are operating under the same

market environement. On the other hand, the common interest rate is not a necessary assumption for our mathematical

derivation and is made only for the sake of brevity. In fact, if heterogenous interest rates are considered such that the i-th

agent possesses interest rate ρi > 0, i = 0, . . . , n, then all the reported results in this section remain unchanged except the

16

optimal loss acquisition laws become

l∗i (t, y; ξ∗i ) =

[(1/2)i

1/2 + ξ∗i (t)i−1∑j=0

(γj(t)

)−1]× y,

where y ∈ R+, t ∈ [0, T ] and i = 1, . . . , n. To see the reason, we refer the reader to the proof of Theorem 1 which is based

on this slightly more general case. It is worth mentioning that, under this more general case, the optimal loss acquisition

laws still admit the proportional reinsurance arrangement but are time dependent in general.

4 Economics implications

Assume the companies in the reinsurance chain are rational and their reinsurance strategies are set according to the

optimal rules in Theorem 1. We are in the position to answer the three questions posted in the introduction section.

4.1 Optimal structure of reinsurance chains

The reinsurance chain that we have investigated so far assumes the positions of the given reinsurers to be fixed ahead

of time. This restriction can be relaxed so as to study the optimal order for the lineup of reinsurers in the reinsurance

chain. To this end, let us begin with a pool of n rational reinsurance companies with risk aversion parameters βi > 0,

i = 1, . . . , n. The reinsurance companies are heterogeneous in the sense that all the elements of β = (β1, . . . , βn) are

distinct. Note that the risk aversion coefficients γi and βi correspond respectively to the i-th company, i = 1, . . . , n, in the

reinsurance chain and the candidate reinsurers pool, thus they must not be identical. Independent of their placement in

the reinsurance chain, we assume that these companies are always willing to underwrite reinsurance businesses with any

other parties.

Here is how to formulate the optimal placement problem of interest. Starting from the primary insurer, each company

in the reinsurance chain may have the option to select the following agent for purchasing reinsurance in order to maximize

its value function. However, it is important for us to point out that not every company in the reinsurance chain has the

same number of potential reinsurers to choose. In fact, the number of candidate reinsurers depends on the position in the

reinsurance chain. For the i-th level agent where i ∈ 1, . . . , n, there are (n − i) available reinsurers for it to consider,

since the other companies have been previously selected and already placed in the former positions. Now, the task is to

study the optimal selection of the succeeding reinsurer for each company in the reinsurance chain.

In order to tackle the task, consider two reinsurance chains, both are as per the description in Section 2. Assume

that the interest rate, denoted by ρ, is identical between the two reinsurance chains. For i ∈ 1, 2 and j ∈ 0, . . . , n,

let γij be the risk aversion parameters of the j-th agent in the i-th reinsurance chain. Furthermore, set γ1j = γ2j = γj

17

for j ∈ 0, . . . , k − 2 ∪ k, . . . , n, k = 2, . . . , n, while γ1(k−1) = γ2k = γk−1 and γ1k = γ2(k−1) = γk. Plainly speaking,

the only difference between the two aforementioned reinsurance chains is that the positions of the (k − 1)-th and k-th

level reinsurers are interchanged. Denote by ξ∗ij the optimal pricing strategy associated with the j-th agent of the i-th

reinsurance chain. The following assertion indicates that reinsurance is cheaper when the reinsurance chain’s agents are

placed in an increasing order according to the risk aversion coefficients.

Proposition 5. Fix k ∈ 2, . . . , n, it holds that ξ∗1j(·) < ξ∗2j(·) for j = 1, . . . , k − 1, if and only if γk−1(·) < γk(·).

Proof. See Appendix B.

We now proceed to derive the optimal structure of a reinsurance chain. Our plan for solving the problem is as follows.

We begin with the primary insurer who is the first agent to select the succeeding reinsurer. There are n candidate

companies for the primary insurer to choose for purchasing reinsurance. Fix t ∈ [0, T ] and x0 ∈ R, we know from

Proposition 4 that maximizing the value function V0(t, x0) is equivalent to minimizing the security loading ξ∗1(t). Namely,

assuming that the future agents will adopt the same optimal selection strategy, the primary insurer will rationally purchase

reinsurance from the candidate company that asks for the lowest price, or equivalently by Proposition 5, has the lowest

risk aversion coefficient. It is noteworthy that, as we discussed earlier in Proposition 2, the determinants of the optimal

security loadings involve the risk profiles of all the agents in the reinsurance chain. That being said, when making the

optimal selection of succeeding reinsurer, the primary insurer has to be also mindful about the selections made by the

latter agents in the chain. We shall verify theoretically that the latter agents will indeed follow the blueprint conceived

by the primary insurer in order to maximize the respective value functions. Consequently, the optimal structure of the

reinsurance chain can be established. We believe that this is a very interesting result which will be reported formally in

the following assertion.

For j = 1, . . . , n, let β(j) be the j-th smallest element among β =(β1, . . . , βn

).

Theorem 6. Assume the reinsurance chain is described in Section 2. The optimal structure that maximizes the value

functions of all the agents in the reinsurance chain, is such that γj = β(j), for j = 1, . . . , n, or equivalently

γ1 < γ2 < · · · < γn. (19)

Proof. See Appendix B.

At the first glance, it may be somewhat suspicious to conclude that it is optimal to situate the most risk averse company

to the end position of the reinsurance chain. However, this result by no mean represents a flaw with our model or the

theoretical derivations. The finding can be explained from two perspectives. On the one hand, the optimal selection of

the succeeding agent depends on the price at which reinsurance is offered. Since the more risk averse companies tend to

18

ask for higher reinsurance premiums, thus it is more likely that they will be placed at the latter chain positions. On the

other hand, under the variance premium principle, the optimal reinsurance arrangement for the reinsurance chain is the

proportional reinsurance (see, Theorem 1). Hence, it is not necessarily true that the latter positions in the reinsurance

chain represent a more daunting situation. When there is an insurance claim occurs, the risk is spread out and absorbed

proportionally by all members of the reinsurance chain. Indeed, as we will show formally in Proposition 7, the variances

of the surplus processes are decreasing in accordance with the positions in the reinsurance chain. Because of the mean-

variance criterion that we adopted to define the optimization problem, the value functions of the reinsurers with higher

risk aversion coefficients will be penalized harsher by the variance terms. Therefore, more risk averse agents will rationally

choose to be placed at the latter positions of the reinsurance chain in order to avoid the excessively volatile surplus

processes. In light of the discussion above, we think that the optimal order reported in Theorem 6 makes a perfect sense.

Proposition 7. Suppose that the reinsurance chain of interest follows the optimal structure according to Theorem 6.

Associated with the optimal rules u∗ = (l∗, ξ∗) outlined in Theorem 1, let R∗i (t) := Ri(t;u∗) be the surplus process of the

i-th reinsurance agent, i = 0, . . . , n, t > 0. Then for any t ∈ [0, T ] and T > 0, the conditional variances of the surplus

processes given the filtration up to time t, satisfy

Vart(R∗0(T )

)≥ Vart

(R∗1(T )

)≥ · · · ≥ Vart

(R∗n(T )

).

Proof. See Appendix B.

This part of Section 4 answers question Q1 posted in the introduction.

4.2 Impacts of involving additional participants

In this subsection, we study the impacts on the reinsurance strategies of the existing agents when a new participant is

added to the reinsurance chain. To this end, some extra notations are needed. Let γa > 0 be the risk aversion coefficient

for the newly added reinsurer. In order to articulate the optimal reinsurance strategies after the new reinsurer is added,

we put a hat above the related notations such that i(·) in the expanded chain is associated with i(·) in the original

chain, where ∈ ξ∗, ζ∗, l∗, p∗, V , i = 1, . . . , n. The newly added reinsurer is indexed by “a”. Assuming that the new

reinsurer is added between the (k − 1)-th and k-th agents in the original chain, where k ∈ 1, . . . , n, Figure 1 helps to

elucidate the notions proposed herein. As a side note, due to the added reinsurer, i(·) corresponds to the i-th agent in

the expanded reinsurance chain for i = 1, . . . , k − 1, but corresponds to the (i+ 1)-th agent in the expanded reinsurance

chain for i = k, . . . , n.

At first, we remark that, depending on the risk profile of the new reinsurer, its impacts on the reinsurance strategies of

the existing agents are mixed. For illustration, consider two kinds of rather extremal yet illuminating risk profiles for the

19

Customers PI 1RI 1RIk RIk RIn

RIa

… …

Original chain 0 1 1k k n

Expanded chain 0 1 1ˆ k ˆ a ˆ k ˆ n

… …

… …

: Premium flows;

: Indemnity flows.

Infinitely risk averse hypothetical reinsurer

Figure 1: Illustration of the original and expanded chains with “PI” and “RIi”, i = 1 . . . , n, representing the primaryinsurer and the i-th reinsurer, respectively, and “” being one of the strategies ξ∗, ζ∗, l∗, p∗, h∗. The infinitely risk aversehypothetical reinsurer is used in the proof of Theorem 8.

new reinsurer added between the (k − 1)-th and k-th agents in the original reinsurance chain, where k ∈ 1, . . . , n. One

kind is risk neutral (i.e., γa → 0) and the other kind is infinitely risk averse (i.e., γa →∞). On the one hand, if the added

reinsurer is risk neutral, then according to the optimal risk acquisition strategies in Theorem 1, we have l∗i (·) = 0 for all

i ≥ k. Stating bluntly, this means that the added reinsurer will rationally keep all the residual risk so as to maximize

the expected final surplus without worrying about the associated variability. The reinsurance chain will be ceased at the

position of the added reinsurer. On the other hand, if the added reinsurer is infinitely risk averse, then l∗a(·) ≡ l∗k(·).

Namely, the added reinsurer will transfer all the risk acquired from the former agent to the succeeding agent. In this case,

the added reinsurer induces no impact on the reinsurance demands in the reinsurance chain.

In order to avoid the trivial situations discussed above and make the comparison actionable, we confine ourselves to

the case where both the original and expanded reinsurance chains admit the optimal structure such that Equation (19)

holds. Also, we assume that the new agent is added to the reinsurance chain as a reinsurer, and it cannot replace the role

of the primary insurer. Our assumption is described formally below.

Assumption N. Consider the reinsurance chain described in Section 2, and assume that the optimal structure in Theorem

6 holds. Consider a new reinsurer having risk aversion coefficient γa 6= γi, i = 1, . . . , n. If γa < γ1, then the new reinsurer

is added between the primary insurer and the first level reinsurer. If γ1 < γa < γn, then the new reinsurer is added between

the (k − 1)-th and k-th agents of the original reinsurance chain such that γk−1 < γa < γk for some k ∈ 2, . . . , n. If

γa > γn, then the new reinsurer is added after the n-th (i.e., last) agent of the original reinsurance chain.

The main result of this current subsection is summarized in the following assertion.

Theorem 8. Suppose that Assumption N holds, then the expanded reinsurance chain still admits the optimal structure.

20

Moreover, we have ξ∗i (·) > ξ∗i (·) for i = 1, . . . , n, as well as

l∗i (·) < l∗i (·) and Vi(·, ·) < Vi(·, ·), for i = 1, . . . , k − 1;

l∗i (·) > l∗i (·) and Vi(·, ·) > Vi(·, ·), for i = k, . . . , n.

Proof. See Appendix B.

Here is how we should interpret the results in Theorem 8. We have shown that, if the reinsurers are rationale and so

the structure of the reinsurance chain is optimal, then the involvement of an extra reinsurer lowers the reinsurance prices

at all levels. However, depending on the positions of the agents in the reinsurance chain, the impacts of the extra reinsurer

on the optimal reinsurance demands and the associated value functions can be very different. The extra reinsurer boosts

the reinsurance demands for the agents who are in the former positions while weakens the reinsurance demands for those

in the latter positions. This finding is in accordance with our intuition. On the one hand, because the extra reinsurer

makes reinsurance cheaper in the reinsurance chain, the reinsurance demands increase for all the former agents. On the

other hand, the new reinsurer will rationally keep a part of losses to maximize its reward. Consequently, the residual

risk that can be acquired by the latter agents becomes less. Due to this reason, the extra reinsurer will benefit the value

functions of the former agents but harm those of the latter agents.

We remark that although the changes in the optimal reinsurance acquisition strategies due to the extra reinsurer have

been specified in Theorem 8, the impacts on the retained losses of the individual agents remain unclear. This question

will be answered in the succeeding assertion. In what follows, let

h∗i (·) =

l∗i (·)− l∗i+1(·), if i ∈ 0, . . . , n− 1

l∗n(·), if i = n

(20)

be the optimal volume of loss retained by i-th agent in the original reinsurance chain (also see, Table 1 for illustration).

Also, let h∗i (·), i = 0, . . . , n, be the respective counterparts of h∗i (·) in the expanded reinsurance chain, which are defined

in the same manner except h∗k−1(·) = l∗k−1(·)− l∗a(·), while h∗k−1(·) = l∗k−1(·)− l∗k(·). The optimal volume of loss retained

by the new reinsurer is h∗a(·) = l∗a(·)− l∗k(·).

Theorem 9. Consider the reinsurance chain structure described in Section 2 and suppose that Assumption N holds. The

involvement of the extra reinsurer lowers the retained loss at all levels of the reinsurance chain. Namely, for the i-th agent

in the original reinsurance chain, we have h∗i (·) ≤ h∗i (·), for all i = 0, . . . , n.

Proof. See Appendix B.

This part of Section 4 answers question Q2 posted in the introduction.

21

4.3 Implications on risk management

Risk management plays a ubiquitously pivotal role in maintaining the stability of the insurance sector. At the heart of

risk management is diversification. In the context of the reinsurance chain of interest, diversification can be enhanced by

bringing in additional participants. That said, systemic risk may opposingly emerge at the same time. Namely, devastating

losses hitting the reinsurance chain may result in the insolvency of certain agents, which will further affect the solvency of

all the former agents due to the hieratical structure of the reinsurance chain. As a consequence, it is of central importance

for both the insurance industry and regulators to be aware of the potential limitation of diversification and to understand

the mechanism that may hamper it.

In the previous subsection, we studied the impacts of an extra reinsuer on the individual strategies of the existing

agents. This subsection examines a similar question but approaches it from the angle of risk management. It is noteworthy

that, if there is no restriction imposed on the chain structure, then an overly aggressive reinsurer (for an extremal instance,

a risk neutral reinsurer) added in the middle of the reinsurance chain may keep excessive proportion of the insurance risk.

Hence, the solvency of the reinsurance chain may be deteriorated. In a similar vein to the discussion in Section 4.2, the

optimal structure of the reinsurance chain must be assumed in order to manage the build up of systemic risk due to the

involvement of the extra reinsurer.

Throughout the rest of this subsection, we again assume the reinsurance chain admits the optimal structure described

in Theorem 6. Intuitively speaking, if the reinsurance chain admits the optimal structure, then adding an extra reinsurer

lowers the risk exposures of all the existing agents (see, Theorem 9), and thus the reinsurance chain becomes safer. We

verify this conjecture formally in what follows. To be consistent with how risks are discussed in the present optimization

framework, in the succeeding investigation, we will use the volatility of surplus processes as the primary measure for

studying the solvency risks associated with the individual reinsurance agents and the reinsurance chain as a whole.

Similarly to the notation used in Section 4.2, we index the new reinsurer by i = a and denote by R∗i (·) the i-th

existing agent’s surplus process under the optimal reinsurance strategies after the extra reinsurer is added. Recall that

the conditional variance given the filtration till time t > 0 is denoted by Vart(·) = Var(·|Ft). The conditional covariance

Covt(·, ·) is defined similarly by using the conditional expectation. The main result of the present subsection is summarized

in the next assertion.

Theorem 10. Consider a situation in which an extra agent is added to the reinsurance chain. Suppose that Assumption

N holds. For any t ∈ [0, T ] with terminal time T > 0, we have

(i) Covt(R∗i (T ), R∗j (T )

)≤ Covt

(R∗i (T ), R∗j (T )

)for any i, j ∈ 0, . . . , n;

(ii) maxi∈1,...,n∪aVart(R∗i (T )

)≤ maxi∈1,...,nVart

(R∗i (T )

).

Proof. See Appendix B.

22

Remark 5. Set i = j ∈ 0, . . . , n, inequality (i) in Theorem 10 readily implies

Vart(R∗i (T )

)≤ Vart

(R∗i (T )

), for all i ∈ 0, . . . , n.

Theorem 10 reveals the following implications on the reinsurance chain’s risk management. First, the involvement of an

extra reinsurer to the reinsurance chain admitting the optimal structure lowers the volatilities of the surplus processes for

all the existing agents. This is due to the diversification benefit induced by the extra reinsurer who lowers the reinsurance

prices in the reinsurance chain and promotes the existing agents to rationally control their respective risk exposures.

Second, the involvement of the extra reinsurer mitigates the contagion risk in the reinsurance chain if the dependencies

of the surplus processes are measured by covariances. Third, as was discussed earlier, the extra reinsurer may take over

excessive amount of insurance risk, which threatens the stability of the reinsurance chain as a whole. Since the insolvency

of a certain succeeding agent in the reinsurance chain may essentially lead to the insolvency of the primary insurer to the

customers, the maximal variance among the individual agents’ surpluses naturally emerges as a meaningful quantity for

measuring the system’s stability. In this respect, the inequality in (ii) of Theorem 10 indicates that the involvement of

the extra reinsurer will strengthen the system’s stability if the reinsurance chain admits the optimal structure.

This part of Section 4 answers question Q3 posted in the introduction.

5 Conclusions

In this paper, we established a continuous-time framework for studying the optimal reinsurnace strategies within a reinsur-

ance chain structure having arbitrary number of agents. Each agent has to decide the optimal amount of risks to acquire

and to transfer. Due to the unequal bargaining powers carried between reinsurance buyers and sellers, we formulated the

transactions in the reinsurance chain based on the Stackelberg games terminology. In the construction of the optimization

problem, we applied the mean-variance criterion on the individual surplus processes in order to address the conflicting

objectives of “high profit” versus “low risk” faced by the reinsurance chain’s agents. By dealing with two large systems of

embedded games, we successfully obtained the time-consistent optimal reinsurance strategies in explicit forms.

What is more, our paper offered a parsimonious theory to answer three open questions that are of great importance in

the study of insurance economics. Namely, under the proposed construction of reinsurnace chain which is fairly general and

realistic, we showed formally: a.) it is optimal to situate more (resp. less) risk averse reinsurers to the latter (resp. former)

positions of the reinsurance chain; b.) adding a new reinsurer will lower the reinsurance prices at all levels while promote

the existing agents to rationally control their respective risk exposures; and c.) the involvement of the new reinsurer will

alleviate the systemic risk in the reinsurance chain structure. Consequently, we believe that our endeavors in this present

paper offer sound theoretical arguments which help persuade the regulatory authorities of emerging countries to withdraw

23

the barriers on the (re)insurance businesses, so that a global-wise open insurance marketplace can be ultimately achieved.

In future research, it will be interesting to extend the proposed framework to the study of reinsurance network in

which a given agent may simultaneously conduct reinsurance businesses with multiple other agents in the system. The

formulation of reinsurance network will become considerably more complex than that of the reinsurance chain studied in

this present paper. In this respect, we think that the notion of graphical models may be helpful for catering the intricate

network structure. The continuous-time framework established in this present paper will serve as the building block for

identifying the optimal reinsurance strategies in reinsurance networks. Understanding the economics benefits induced by

the form of reinsurance networks will be the major objective in this future research direction.

References

Aase, K. K. (2009). The Nash bargaining solution vs. equilibrium in a reinsurance syndicate. Scandinavian Actuarial

Journal, 2009(3):219–238.

Bauerle, N. (2005). Benchmark and mean-variance problems for insurers. Mathematical Methods of Operations Research,

62(1):159–165.

Bjork, T., Murgoci, A., and Zhou, X. Y. (2014). Mean-variance portfolio optimization with state-dependent risk aversion.

Mathematical Finance, 24(1):1–24.

Boonen, T. J., Tan, K. S., and Zhuang, S. C. (2016). Pricing in reinsurance bargaining with comonotonic additive utility

functions. ASTIN Bulletin, 46(2):507–530.

Borch, K. H. (1974). The Mathematical Theory of Insurance. Lexington Books, Lexington.

Chen, L. and Shen, Y. (2018). On a new paradigm of optimal reinsurance: A stochastic Stackelberg differential game

between an insurer and a reinsurer. ASTIN Bulletin, 48(02):905–960.

Chen, L. and Shen, Y. (2019). Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance

framework. Insurance: Mathematics and Economics, 88:120–137.

Chen, S., Li, Z., and Zeng, Y. (2014). Optimal dividend strategies with time-inconsistent preferences. Journal of Economic

Dynamics and Control, 46:150–172.

Chiu, M. C. and Wong, H. Y. (2011). Mean-variance portfolio selection of cointegrated assets. Journal of Economic

Dynamics and Control, 35(8):1369–1385.

Cournot, A. (1838). Recherches sur les Principes Mathematiques de la Theorie des Richesses. Hachette, Paris.

24

Cui, X., Li, D., and Shi, Y. (2017). Self-coordination in time inconsistent stochastic decision problems: A planner–doer

game framework. Journal of Economic Dynamics and Control, 75:91–113.

Doherty, N. and Smetters, K. (2005). Moral hazard in reinsurance markets. Journal of Risk and Insurance, 72(3):375–391.

d’Ursel, L. and Lauwers, M. (1985). Chains of reinsurance: Non-cooperative equilibria and Pareto optimality. Insurance:

Mathematics and Economics, 4(4):279–285.

d’Ursel, L. and Lauwers, M. (1986). Pareto optimal chains of reinsurance. Economics Letters, 20(4):307–310.

Dutang, C., Albrecher, H., and Loisel, S. (2013). Competition among non-life insurers under solvency constraints: A

game-theoretic approach. European Journal of Operational Research, 231(3):702–711.

Gerber, H. U. (1984). Chains of reinsurance. Insurance: Mathematics and Economics, 3(1):43–48.

Guerra, M. and Centeno, M. d. L. (2010). Optimal reinsurance for variance related premium calculation principles. ASTIN

Bulletin, 40(1):97–121.

Horst, U. and Moreno-Bromberg, S. (2008). Risk minimization and optimal derivative design in a principal agent game.

Mathematics and Financial Economics, 2(1):1–27.

Jørgensen, S., Kort, P. M., and van Schijndel, G. J. C. (1989). Optimal investment, financing, and dividends: A Stackelberg

differential game. Journal of Economic Dynamics and Control, 13(3):339–377.

Kaas, R., Goovaerts, M., Dhaene, J., and Denuit, M. (2008). Modern Actuarial Risk Theory. Springer, Heidelberg.

Kanno, M. (2016). The network structure and systemic risk in the global non-life insurance market. Insurance: Mathe-

matics and Economics, 67:38–53.

Laffont, J.-J. and Martimort, D. (2002). The Theory of Incentives: The Principal-Agent Model. Princeton University

Press, New Jersey.

Lemaire, J. and Quairiere, J.-P. (1986). Chains of reinsurance revisited. ASTIN Bulletin, 16(2):77–88.

Li, B., Li, D., and Xiong, D. (2016). Alpha-robust mean-variance reinsurance-investment strategy. Journal of Economic

Dynamics and Control, 70:101–123.

Liang, Z. and Yuen, K. C. (2016). Optimal dynamic reinsurance with dependent risks: Variance premium principle.

Scandinavian Actuarial Journal, 2016(1):18–36.

Lin, X., Zhang, C., and Siu, T. K. (2012). Stochastic differential portfolio games for an insurer in a jump-diffusion risk

process. Mathematical Methods of Operations Research, 75(1):83–100.

25

Massey, R., Hart, D., Widdows, J., Law, D., Bhattacharya, K., Hawes, W., and Shaw, R. (2003). Insurance company

failure. Technical report, Institute and Faculty of Actuaries, London.

Øksendal, B., Sandal, L., and Ubøe, J. (2013). Stochastic Stackelberg equilibria with applications to time-dependent

newsvendor models. Journal of Economic Dynamics and Control, 37(7):1284–1299.

Plantin, G. (2006). Does reinsurance need reinsurers? Journal of Risk and Insurance, 73(1):153–168.

Simaan, M. and Cruz, J. B. (1973). On the Stackelberg strategy in nonzero-sum games. Journal of Optimization Theory

and Applications, 11(5):533–555.

Strotz, R. H. (1955). Myopia and inconsistency in dynamic utility maximization. The Review of Economic Studies,

23(3):165.

von Stackelberg, H. (1934). Marktform und Gleichgewicht. Springer, Vienna.

Wang, J. and Forsyth, P. A. (2011). Continuous time mean variance asset allocation: A time-consistent strategy. European

Journal of Operational Research, 209(2):184–201.

26

Appendix A Summary of the notation system

We summarize the notation system used in this present article herein.

Notation Description

ρ Risk-free interest rate parameter.

γi for i = 0, . . . , n Risk aversion coefficient of the i-th agent.

γi for i = 0, . . . , n Generalized risk aversion coefficient of the i-th agent; see, Equation (7).

Ri, R∗i for i = 0, . . . , n Surplus process of the i-th agent (under the optimal strategies).

Ji for i = 0, . . . , n The objective function of the i-th agent.

li, l∗i for i = 1, . . . , n The (optimal) risk acquisition strategy of the i-th agent.

hi, h∗i for i = 1, . . . , n The (optimal) risk retention of the i-th agent; see, e.g., Equation (20).

ξi, ξ∗i for i = 1, . . . , n The (optimal) pricing strategy of the i-th agent.

ζ∗i for i = 1, . . . , n The interest adjusted pricing rule of the i-th agent; see, Theorem 1.

ci, c∗i for i = 1, . . . , n The reinsurance cost of the i-th agent (under the optimal strategies).

p∗i for i = 1, . . . , n The optimal reinsurance proportion of the i-th agent; see, Equation (17).

Vi for i = 0, . . . , n The value function of the i-th agent.

with ∈ ξ∗, ζ∗, l∗, p∗, h∗, V The optimal strategies and value function in the expanded reinsurnace chain.

Table 4: Summary of the notation system.

Appendix B Technical proofs

Proof of Theorem 1. In order to facilitate the discussion in Remark 4, we consider a slightly more general version of the

theorem in which the interest rates may vary among the agents in the reinsurance chain. The interest rate of the i-th

agent is denoted by ρi > 0, i = 0, . . . , n. The interest rates in the surplus processes and objective functions are adjusted

accordingly. The expressions in Theorem 1 can be obtained by setting ρ0 = · · · = ρn = ρ.

We begin the proof by introducing some additional notations. Define a set of partial differential generators acting on

the function ϕ ∈ C1([0, T ]× R;R), such that

L(ui,ui+1)i

(ϕ(t, xi)

):= ϕt(t, xi) +

(ρixi + ci(t;ui)−ci+1(t;ui+1)

)ϕx(t, xi)

+

∫R+

[ϕ(t, xi − li(t, y) + li+1(t, y)

)− ϕ(t, xi)

]ν(dy)

27

for i = 0, . . . , n− 1, and

L(un)n

(ϕ(t, xn)

):= ϕt(t, xn) +

(ρnxn + cn(t;un)

)ϕx(t, xn) +

∫R+

[ϕ(t, xn − ln(t, y)

)− ϕ(t, xn)

]ν(dy).

Moreover, for some function gi ∈ C1([0, T ]× R;R), i = 0, . . . , n, let

G(ui,ui+1)i (t, xi) := L(ui,ui+1)

i

(Vi(t, xi)

)− γi

2L(ui,ui+1)i

(g2i (t, xi)

)+ γi gi(t, xi)L(ui,ui+1)

i

(gi(t, xi)

).

Whenever there is no risk of confusion, we simply write “supl1

”, “ supξi,li+1

” for i = 1, . . . , n − 1, and “supξn

” in the following

HJB equations, which should be understood as choosing the optimal reinsurance premium/acquisition strategies from the

corresponding sets of admissible strategies.

Now we consider the interaction between the primary insurer and the first level reinsurer. The associated reinsurance

strategies is u1 = (l1, ξ1). Also recall that we set u∗0 = u0 = (l0, ξ0). For a given admissible premium strategy ξ1, evoking

the verification theorem in Bjork et al. (2014) yields the time-consistent Stackelberg follower action for the primary insurer

(see, Definition 2) must satisfy the following system of extended HJB equations:

supl1

G

(u∗0 ,(l1,ξ1))

0 (t, x0)

= 0, t ∈ [0, T ], x0 ∈ R, (21)

L(u∗0 ,(l

∗1 ,ξ1))

0

(g0(t, x0)

)= 0, t ∈ [0, T ], x0 ∈ R,

V0(T, x0) = x0, x0 ∈ R,

g0(T, x0) = x0, x0 ∈ R,

where l∗1 realizes the supremum in Equation (21).

We conjecture the pair of ansatze, for t ∈ [0, T ], x0 ∈ R,

V0(t, x0) = eρ0(T−t)x0 +B0(t) and g0(t, x0) = eρ0(T−t)x0 + b0(t)

is a solution to the extended HJB equation (21). Substitution gives

∂B0(t)

∂t+ eρ0(T−t) sup

l1

∫R+

[ cλ− y − ξ1(t) l21(t, y)− γ0(t)

2

(y − l1(t, y)

)2]ν(dy)

= 0.

By the first order condition of the equation above, we readily obtain

l∗1(t, y; ξ1) =γ0(t)

2ξ1(t) + γ0(t)y. (22)

28

Next we turn to determine the time-consistent Stackelberg actions that are related to the first reinsurer, namely ξ∗1 and

l∗2. For any given admissible strategy ξ2, the extended HJB equation which we need to solve is

supξ1,l2

G

((l∗1 ,ξ1),(l2,ξ2))1 (t, x1)

= 0, t ∈ [0, T ], x1 ∈ R, (23)

L((l∗1 ,ξ∗1 ),(l

∗2 ,ξ2))

1

(g1(t, x)

)= 0, t ∈ [0, T ], x1 ∈ R,

V1(T, x1) = x1, x1 ∈ R,

g1(T, x1) = x1, x1 ∈ R,

where ξ∗1 and l∗2 realize the supremum in Equation (23), and l∗1 has already been computed in Equation (22). In this case,

we try

V1(t, x1) = eρ1(T−t)x1 +B1(t) and g1(t, x1) = eρ1(T−t)x1 + b1(t),

for t ∈ [0, T ], x1 ∈ R, and substitute these ansatze into Equation (23). We obtain

∂B1(t)

∂t+ eρ1(T−t) sup

ξ1,l2

∫R+

[ξ1(t)

(l∗1(t, y)

)2 − ξ2(t)(l2(t, y)

)2 − γ1(t)

2

(l∗1(t, y)− l2(t, y)

)2]ν(dy)

= 0.

Application of the first order condition with respect to ξ1 and l2 yields the system of equations:

γ0(t) y − 4 ξ1(t) l∗1(t, y) + 2 γ1(t)(l∗1(t, y)− l2(t, y)

)= 0;

γ1(t)(l∗1(t, y)− l2(t, y)

)− 2 ξ2(t) l2(t, y) = 0.

Hence we have

ξ∗1(t) =γ0(t)

2+

[(γ1(t)

)−1+(2 ξ2(t)

)−1]−1,

and

l∗2(t, y; ξ2) =

[2 + 4 ξ2(t)

(1

γ0(t)+

1

γ1(t)

)]−1y.

The same procedure as above can be repeated sequentially to derive the optimal strategies for the remaining reinsurers.

29

By mathematical induction, we get for t ∈ [0, T ] and y ∈ R+,

l∗i (t, y) =

[(1/2)i

1/2 + ξ∗i (t)i−1∑j=0

(γj(t)

)−1]y, i = 1, . . . , n, (24)

and

ξ∗i (t) =

[2

i−1∑j=0

(γj(t)

)−1]−1+

[(γi(t)

)−1+(2 ξ∗i+1(t)

)−1]−1, i = 1, . . . , n− 1.

Now, it only remains to establish the expression for ξ∗n. In this case, we have to solve

supξn

L(l∗n,ξn)n

(Vn(t, xn)

)− γn

2L(l∗n,ξn)n

(g2n(t, xn)

)+ γn gn(t, xn)L(l∗n,ξn)

n

(gn(t, xn)

)= 0, t ∈ [0, T ], xn ∈ R, (25)

L(l∗n,ξ∗n)

n

(gn(t, xn)

)= 0, t ∈ [0, T ], xn ∈ R,

Vn(T, xn) = xn, xn ∈ R,

gn(T, xn) = xn, xn ∈ R,

where ξ∗n realizes the supremum in Equation (25), and l∗n is given in Equation (24). Substituting the ansatze

Vn(t, xn) = eρn(T−t)xn +Bn(t) and gn(t, xn) = eρn(T−t)xn + bn(t)

to Equation (25), we get

∂Bn(t)

∂t+ eρn(T−t) sup

ξn

∫R+

[ξn(t)

(l∗n(t, y)

)2 − γn2

(l∗n(t, y)

)2]ν(dy)

= 0.

Applying the first order condition and evoking the expression for l∗n established in Equation (24), we readily obtain

ξ∗n(t) =

[2

n−1∑j=0

(γj(t)

)−1]−1+ γn(t).

This completes the proof of the theorem.

Proof of Proposition 2. From the recursive formula in Equation (15), it can be seen that each ξ∗k(t), k = 1, . . . , n, increases

with all the elements of γ(t) =(γ0(t), . . . , γn(t)

)for any fixed t ∈ [0, T ]. Note that γi(·) is increasing in both γi and ρ for

i = 1, . . . , n, and so is ξ∗(·) =(ξ∗1(·), . . . , ξ∗n(·)

).

30

Proof of Proposition 3. It is evident that γi given in Equation (7) is increasing in γi, i = 0, 1, . . . , n. So the proof boils

down to examining the increasing and decreasing properties of the functions

fk(γ) := ξ∗k

k−1∑j=0

(γj)−1

, k = 1, . . . , n, (26)

where we have temporarily suppressed the time variable for brevity. It is expected that the following statement is true.

Statement 1. Fix k ∈ 1, . . . , n, the function fk(γ) is decreasing in γi for i = 0, . . . , k − 1 while is increasing in γi for

i = k, . . . , n.

We next proceed by induction. Consider k = n, and we have

fn(γ) = 1/2 + γn

n−1∑j=0

(γj)−1

.

Clearly, fn(γ) is decreasing inγii=0,...,n−1, while is increasing in γn. So Statement 1 holds when k = n.

Assuming Statement 1 to hold for k, k + 1, . . . , n, we prove it for k − 1. Recall that ξ∗k−1 is increasing in γi for all

i = 0, . . . , n, according to Proposition 2. It is straightforward to check that fk−1(γ) increases with γi for i = k− 1, . . . , n.

Now, rewrite Equation (26) as

fk−1(γ) = 1/2 +

[γk−1

k−2∑j=0

(γj)−1]−1

+

[2 ξ∗k

k−2∑j=0

(γj)−1]−1−1

= 1/2 +

[γk−1

k−2∑j=0

(γj)−1]−1

+1

2

[fk(γ)− ξ∗k γk−1

]−1−1,

which is decreasing inγii=0,...,k−2. By mathematical induction, we have proved Statement 1 which is equivalent to the

statement in the proposition. The proof is now finished.

Proof of Proposition 4. Fix t ∈ [0, T ] and xk ∈ R, k = 0, . . . , n− 1, throughout the rest of the proof. Let us first consider

the value function for the primary insurer:

V0(t, x0) = J0(t, x0;u∗) = x0 eρ (T−t) +

∫ T

t

eρ (T−s)c− λµ− λµ′2

[(ξ∗1(s)

)−1+ 2

(γ0(s)

)−1]−1ds,

which is decreasing in ξ∗1 .

For the k-th agent with k = 1, . . . , n− 1, the value function can be computed via

Vk(t, xk) = Jk(t, xk;u∗) = xk eρ (T−t) +

∫ T

t

Bk(s) ds,

31

where for s ∈ [0, T ],

Bk(s) =

∫R+

ξ∗k(s)(l∗k(y)

)2 − ξ∗k+1(s)(l∗k+1(y)

)2 − 1

2γk(s)

(l∗k(y)− l∗k+1(y)

)2ν(dy)

=

∫R+

ξ∗k(s)

[2ξ∗k+1(s)

γk(s)l∗k+1(y) + l∗k+1(y)

]2− ξ∗k+1(s)

(l∗k+1(s)

)2 − γk(s)

2

(2ξ∗k+1(s)

γk(s)l∗k+1(y)

)2

ν(dy)

=

[2 k−1∑

j=0

(γj(s)

)−1]−1+

[(γk(s)

)−1+(2 ξ∗k+1(s)

)−1]−1(2ξ∗k+1(s)

γk(s)+ 1

)2

− ξ∗k+1(s)−2(ξ∗k+1(s)

)2γk(s)

×

[(1/2)k+1

1/2 + ξ∗k+1(s)k∑j=0

(γj(s)

)−1]2λµ′2.

For simplicity, let us suppress the time variable in the expression above. Meanwhile, introduce a shorthand notation

Si =∑ij=0

(γj)−1

, i = 0, . . . , n. Then we can write

4k Bkλµ′2

=

[1

2Sk−1+

2γkξ∗k+1

γk + 2ξ∗k+1

](2ξ∗k+1

γk+ 1

)2

− ξ∗k+1 −2(ξ∗k+1

)2γk

(1 + 2Sk ξ

∗k+1

)−2=

[2

(1

Sk−1γ2k+

1

γk

)(ξ∗k+1

)2+

(2

Sk−1γk+ 1

)ξ∗k+1 +

1

2Sk−1

] (1 + 2Sk ξ

∗k+1

)−2=

1

2Sk−1(1 + Sk−1γk

) +

[ξ∗k+1 +

γk2(1 + Sk−1γk)

] (1 + 2Sk ξ

∗k+1

)−2=

1

2

(1 + Sk−1γk

)−1 [S−1k−1 + γ2k

(γk + 2

(1 + Sk−1γk

)ξ∗k+1

)−1].

From the above expression, it is evident that Bk, and thus Jk, is decreasing in ξ∗k+1 when the coefficientsγii=0,...,k

are

fixed.

The proof is completed.

Proof of Proposition 5. For ease of presentation, we again suppress the time variable in the succeeding expressions. Recall-

ing the recursive formula for deriving the optimal pricing strategies in Equation (15), we can readily check that ξ∗1j = ξ∗2j

for j = k + 1, . . . , n, k ∈ 2, . . . , n. Let Si =∑ij=0

(γj)−1

, i = 1, . . . , n, and consider

Dk =1

2

[(ξ∗1k)−1 − (ξ∗2k)−1] =

1

2

1

2

[Sk−2 +

(γk−1

)−1]−1+[(γk)−1

+(2 ξ∗k+1

)−1]−1−1

−1

2

1

2

[Sk−2 +

(γk)−1]−1

+[(γk−1

)−1+(2 ξ∗k+1

)−1]−1−1,

32

where ξ∗k+1 = ξ∗1(k+1) = ξ∗2(k+1). In order to simplify the expression for Dk, define Ak = Sk−2 + Sk +(2ξ∗k+1

)−1and

Ck(x, y) =(Sk−2 + x−1

) [(2ξ∗k+1

)−1+ y−1

], (x, y) ∈ R2

+.

Moreover, let Ck = Ck(γk−1, γk

)and C ′k = Ck

(γk, γk−1

). Some algebraic manipulations lead to

Dk =Ck

Ak +(γk−1

)−1 − C ′k

Ak +(γk)−1 .

We are now ready to prove the inequalities in the proposition. First, let us focus on the respective (k − 1)-th agent in

the two reinsurance chains. Notice that comparing the size between

ξ∗1(k−1) =(2Sk−2

)−1+[(γk−1

)−1+(2 ξ∗1k

)−1]−1and ξ∗2(k−1) =

(2Sk−2

)−1+[(γk)−1

+(2 ξ∗2k

)−1]−1is equivalent to studying the sign of

Dk +(γk−1

)−1−(γk)−1 =

[Ck(Ak + γ−1k

)− C ′k

(Ak + γ−1k−1

)]γk−1 γk + (γk − γk−1)

∏kl=k−1

(Ak + γ−1l

)∏kl=k−1 γl

(Ak + γ−1l

) .

Evidently, the denominator of the expression above is positive. We now check the numerator:

[Ck(Ak + γ−1k

)− C ′k

(Ak + γ−1k−1

)]γk−1 γk + (γk − γk−1)

k∏l=k−1

(Ak + γ−1l

)= Ak

[(Ck − C ′k

)γk−1 γk +

(γk − γk−1

)(Ak + γ−1k−1 + γ−1k

)]+ γk−1 Ck − γk C ′k + γ−1k−1 − γ

−1k

= γk

Ak[Sk−2 +

(ξ∗k+1

)−1+ 2 γ−1k−1

]− C ′k

− γk−1

Ak[Sk−2 +

(ξ∗k+1

)−1+ 2 γ−1k

]− Ck

+(γ−1k−1 − γ

−1k

)= γk

[Ak + (ξ∗k+1)−1 + γ−1k−1

][Sk−2 + (2ξ∗k+1)−1 + γ−1k−1

]− γk−1

[Ak + (ξ∗k+1)−1 + γ−1k

][Sk−2 + (2ξ∗k+1)−1 + γ−1k

]+(γ−1k−1 − γ

−1k

)which is positive if γk−1 < γk, or equivalently γk−1 < γk. Therefore, we have ξ∗1(k−1) < ξ∗2(k−1). Finally, according to the

recursive formula in Equation (15), the aforementioned inequality holds for all the agents that are in front of the k-th

position (i.e., ξ∗1j > ξ∗2j for all j = 1, . . . , k − 1).

When γk−1 > γk, the aforementioned relationship is reversed. The proposition is proved.

Proof of Theorem 6. We conjuncture that the optimal reinsurance chain structure for the primary insurer is such that

33

γj = β(j), for j = 1, . . . , n, or equivalently

γ1 < γ2 < · · · < γn. (27)

We prove our conjuncture by contradiction. Assuming that there exists an optimal order other than the one in Equation

(27), such that the value function of the primary insurer is maximized. Then, there must exist at least one pair of

neighboring agents satisfying γk−1 > γk for k ∈ 2, . . . , n. According to Proposition 5, by interchanging the agents in the

(k− 1)-th and k-th position, the optimal security loading ξ∗1(·) becomes smaller, and thus V0(·, ·) becomes larger. Clearly,

this is a contradiction to the assumption that the order is optimal.

We have shown that the optimal order of placement for the primary insurer is given by Equation (27). Since the primary

insurer is only able to select the first level reinsurer, it remains to confirm that the latter agents will still adopt the same

ordering strategy. Let us now consider the optimal selection for k-th agent in the insurance chain, k ∈ 1, . . . , n − 1.

Assume that the selection of reinsurer for the former positions has been determined and thus γi, i = 0, . . . , k are fixed.

The k-th agent faces an optimal selection problem with (n− k) candidate companies which have risk aversion coefficients

βk+1, . . . , βn. By the repeated application of Proposition 4 to the k-th agent and using the same argument as for the

primary insurer, we are able to conclude that the optimal order of placement for the k-th agent is such that γi = β(i),

i = k + 1, . . . , n.

The proof is now completed.

Proof of Proposition 7. Recall that p∗i , i = 0, . . . , n, denotes the optimal reinsurance proportion given in Equation (17).

Let O∗i = p∗i − p∗i+1 for i = 0, . . . , n− 1 and O∗n = p∗n. From Equations (4) and (5), we readily know

R∗i (T ) = eρ (T−t)Ri(t) +

∫ T

t

eρ (T−s)[c∗i (s)− c∗i+1(s)

]ds−O∗i

∫ T

t

∫R+

eρ (T−s) yM(dy, ds), i = 0, . . . , n− 1,

and

R∗n(T ) = eρ (T−t)Rn(t) +

∫ T

t

eρ (T−s)c∗n(s)ds−O∗n∫ T

t

∫R+

eρ (T−s) yM(dy, ds).

It follows that

Vart(R∗i (T )

)= (O∗i )2 µ′2

(e2ρ(T−t) − 1

2ρ

), i = 0, . . . , n.

Next, we aim to compare the sizes of O∗i , i = 0, . . . , n. Some elementary algebraic manipulations on the recursive

34

relationship (18) yield, for i = 0, . . . , n− 2,

O∗i =2 ζ∗i+1

γip∗i+1

=2 ζ∗i+1

γi

(1 +

2 ζ∗i+2

γi+1

)p∗i+2

=2 ζ∗i+1

γi

(1 +

γi+1

2 ζ∗i+2

)O∗i+1

=(2γi+1

γi+

1 + γi+1 (2ζ∗i+2)−1

γi∑nj=0 γ

−1j

)O∗i+1

≥ O∗i+1,

where the last inequality holds because of the optimal structure assumption according to (19). Moreover,

O∗n−1 =2 ζ∗nγn−1

p∗n =[(

2 γn−1

n−1∑j=0

γ−1j)−1

+2γnγn−1

]O∗n ≥ O∗n.

Collectively, we have proved

Vart(R∗0(T )

)≥ Vart

(R∗1(T )

)≥ · · · ≥ Vart

(R∗n(T )

).

The proof is now completed.

Proof of Theorem 8. Suppose that Assumption N holds. Without loss of generality, in this proof, we assume γa < γn.

This means the extra reinsurer is added at the middle of any two consecutive agents in the original reinsurance chain.

The proof for the end-position case (i.e., γa > γn) is the same as the middle-position case, but with another hypothetical

infinitely risk averse agent added to the original insurance chain (see, Figure 1 for a graphical exposition).

First, we study the impact of the extra reinsurer on the optimal pricing strategies. For any t ∈ [0, T ],

ξ∗n(t) =1

2

[ n−1∑j=0

(γj(t)

)−1]−1+ γn(t) >

1

2

[ n−1∑j=0

(γj(t)

)−1+(γa(t)

)−1]−1+ γn(t) = ξ∗n(t).

Consider two cases in which k = n and k < n, respectively. If k = n, then

ξ∗n(t) =1

2

[ n−1∑j=0

(γj(t)

)−1]−1+ γn(t) >

1

2

[ n−1∑j=0

(γj(t)

)−1]−1+

[(γa(t)

)−1+(2 ξ∗n(t)

)−1]−1= ξ∗a(t). (28)

35

If k < n, then based on the recursive formula in (15), it is elementary to check that the inequality

ξ∗i (t) =1

2

[ i−1∑j=0

(γj(t)

)−1]−1+

[(γi(t)

)−1+(2 ξ∗i+1(t)

)−1]−1

>1

2

[ i−1∑j=0

(γj(t)

)−1+(γa(t)

)−1]−1+

[(γi(t)

)−1+(2 ξ∗i+1(t)

)−1]−1= ξ∗i (t),

holds for i = k, . . . , n− 1. Moreover, note that

ξ∗n(t) =1

2

[ n−1∑j=0

(γj(t)

)−1]−1+ γn(t) >

1

2

[ n−2∑j=0

(γj(t)

)−1+(γa(t)

)−1]−1+

[(γn−1(t)

)−1+(2 ξ∗n(t)

)−1]−1= ξ∗n−1(t).

Applying again the recursive formula in (15) yields

ξ∗i+1(t) =1

2

[ i∑j=0

(γj(t)

)−1]−1+

[(γi+1(t)

)−1+(2 ξ∗i+2(t)

)−1]−1

>1

2

[ i−1∑j=0

(γj(t)

)−1+(γa(t)

)−1]−1+

[(γi(t)

)−1+(2 ξ∗i+1(t)

)−1]−1= ξ∗i (t),

for i = k, . . . , n− 2. So,

ξ∗k(t) =1

2

[ k−1∑j=0

(γj(t)

)−1]−1+

[(γk(t)

)−1+(2 ξ∗k+1(t)

)−1]−1

>1

2

[ k−1∑j=0

(γj(t)

)−1]−1+

[(γa(t)

)−1+(2 ξ∗k(t)

)−1]−1= ξ∗a(t). (29)

Collectively, what we have established thus far are ξ∗i (t) > ξ∗i (t) and ξ∗k(t) > ξ∗a(t) for all i = k, . . . , n and any

k ∈ 1, . . . , n. By the repeated application of the recursive formula in (15), we arrive at the conclusion that ξ∗i (t) ≥ ξ∗i (t)

holds also for all i = 1, . . . , k − 1.

Now, we turn to consider the impact on the optimal risk acquisition strategies. Recall the optimal reinsurance

proportions defined in (17). For the last agent in the reinsurance chain, its risk acquisition strategy satisfies

p∗n = (1/2)n[1 + γn

n−1∑j=0

γ−1j

]−1> (1/2)n

[1 + γn

( n−1∑j=0

γ−1j + γ−1a

)]−1= p∗n.

Based on the recursive relationship derived in Equation (18), we get

p∗i = p∗i+1

(1 +

2 ζ∗i+1

γi

)> p∗i+1

(1 +

2 ζ∗i+1

γi

)= p∗i ,

36

for i = k, . . . , n − 1. However, the inequality is reversed for all the agents that are in front of the extra reinsurer. To be

specific, we have for i = 1, . . . , k − 1,

p∗i = (1/2)i[1/2 + ζ∗i

i−1∑j=0

γ−1j

]−1< (1/2)i

[1/2 + ζ∗i

i−1∑j=0

γ−1j

]−1= p∗i .

So, it can be concluded that

l∗i (·) < l∗i (·), for i = 1, . . . , k − 1;

l∗i (·) > l∗i (·), for i = k, . . . , n.

Finally, we study the impact of the additional reinsurer on the value functions of the existing agents. Continue with

the shorthand notations used in Propositions 4 and 5. Fix t ∈ [0, T ] and xn ∈ R, according to Equation (10), the value

function for the last agent in the reinsurance chain can be computed via

Vn(t, xn) = xn eρ (T−t) + λµ′2

∫ T

t

eρ (T−s)(p∗n)2 [

ξ∗n(s)− γn(s)

2

]ds

> xn eρ (T−t) + λµ′2

∫ T

t

eρ (T−s)(p∗n)2 [

ξ∗n(s)− γn(s)

2

]ds = Vn(t, x),

where the inequality holds due to p∗n > p∗n and ξ∗n(s) > ξ∗n(s) > γn(s)2 established in the earlier part of the proof.

Suppressing the time variable for a moment, we have already known from the proof of Proposition 4 that

4iBiλµ′2

=1

2Si−1(1 + Si−1 γi

) +

[ξ∗i+1 +

γi2(1 + Si−1 γi)

]p∗i+1

>1

2Si−1(1 + Si−1 γi

) +

[ξ∗i+1 +

γi

2(1 + Si−1 γi)

]p∗i+1 =

4i Biλµ′2

,

where Si =∑ij=0 γ

−1j + γ−1a , i = k, . . . , n− 1. So, for fixed t ∈ [0, T ] and xi ∈ R,

Vi(t, xi) = xi eρ (T−t) +

∫ T

t

Bi(s) ds > xi eρ (T−t) +

∫ T

t

Bi(s) ds = Vi(t, xi), i = k, . . . , n− 1.

Consider the agents that are in front of the k-th position. Since ξ∗i (·) > ξ∗i (·) for i = 1, . . . , k, application of Proposition

4 leads to the conclusion that Vi(t, xi) < Vi(t, xi) for i = 1, . . . , k − 1.

The proof is now completed.

Proof of Theorem 9. Suppose that Assumption N holds. Furthermore, we assume γa < γn. If γa > γn, then the same

argument as that in the proof of Theorem 8 can be used to obtain the desired result.

37

Since the optimal reinsurance arrangement is the proportional reinsurance, it suffices to study the optimal proportions

in Equation (18). We know from the proof of Proposition 7 that

p∗i − p∗i+1 =(1/2)i ζ∗i+1

γi

(1/2 + ζ∗i+1

i∑j=0

γ−1j

) , i = 0, . . . , n− 1.

Assume that the reinsurance chain admits the optimal structure according to Equation (19), and the extra reinsurer

with risk aversion coefficient γa > 0 is added between the (k− 1)-th and k-th agents such that γk−1 < γa < γk holds for a

given k ∈ 1, . . . , n. We have shown in Theorem 8 that the involvement of the extra reinsurer will lower the reinsurance

premiums at all levels of the reinsurance chain. Namely, ξ∗i (·) ≤ ξ∗i (·) or equivelenatly, ζ∗i ≤ ζ∗i , i = 0, . . . , n. For the

reinsurance agents that are at the positions before the newly added reinsurer, it holds that

p∗i − p∗i+1 =(1/2)i ζ∗i+1

γi

(1/2 + ζ∗i+1

i∑j=0

γ−1j

) ≤ p∗i − p∗i+1, i = 0, . . . , k − 2.

For the (k − 1)-th agent who is at the position right before the extra reinsurer, recall that ξ∗k(·) > ξ∗a(·) (see, Equations

(28) and (29)), thus p∗k−1 − p∗a ≤ p∗k−1 − p∗k.

For those who are at the positions after the added reinsurer, we have

p∗i − p∗i+1 =(1/2)i+1 ζ∗i+1

γi

[1/2 + ζ∗i+1

( i∑j=0

γ−1j + γ−1a)] < (1/2)i ζ∗i+1

γi

(1/2 + ζ∗i+1

i∑j=0

γ−1j

) ≤ p∗i − p∗i+1, i = k, . . . , n− 1.

For the last reinsurance agent, note that

ζ∗n =

(2

n−1∑j=0

γ−1j + γa

)−1+ γn.

It holds that

p∗n =(1/2)n+1

1/2 + ζ∗n( n−1∑j=0

γ−1j + γ−1a) =

(1/2)n+1

1 + γn( n−1∑j=0

γ−1j + γ−1a) ≤ (1/2)n

1 + γnn−1∑j=0

γ−1j

= p∗n.

Collectively, we have already obtained that h∗i (·) ≤ h∗i (·) for all i = 0, . . . , n. The proof is now completed.

Proof of Theorem 10. Suppose that Assumption N holds. Throughout the proof, let us fix t ∈ [0, T ] with T > 0. First,

consider the inequality in (i). For i = 1, . . . , n, recall that h∗i (·) denotes the optimal level of loss retained by the i-th agent

38

in the original reinsurance chain and h∗i (·) is the counterpart after adding the extra reinsurer. Recall that, from Equations

(4) and (5), we readily know

R∗i (T ) = eρ (T−t)Ri(t) +

∫ T

t

eρ (T−s)[c∗i (s)− c∗i+1(s)

]ds−

∫ T

t

∫R+

eρ (T−s) h∗i (y)M(dy, ds).

For any y > 0 and i = 0, . . . , n, recall O∗i = h∗i (y) · y−1 used in Proposition 7, and let O∗i = h∗i (y) · y−1 which are

independent of y. Then, for any i, j ∈ 0, . . . , n,

Covt(R∗i (T ), R∗j (T )

)= Covt

(∫ T

t

∫R+

eρ (T−s) h∗i (y)M(dy, ds),

∫ T

t

∫R+

eρ (T−s) h∗j (y)M(dy, ds))

= O∗i O∗j Vart

(∫ T

t

∫R+

eρ (T−s) yM(dy, ds))

(1)

≥ O∗i O∗j Vart

(∫ T

t

∫R+

eρ (T−s) yM(dy, ds))

= Covt

(∫ T

t

∫R+

eρ (T−s) h∗i (y)M(dy, ds),

∫ T

t

∫R+

eρ (T−s) h∗j (y)M(dy, ds))

= Covt(R∗i (T ), R∗j (T )

),

where inequality “(1)

≥” holds because of Theorem 9.

Then, application of Proposition 7 leads to the inequality in (ii). The proof is finished.

39