Time-consistent investment and reinsurance strategies for mean-variance insurers with relative performance concerns under the Heston model

Finance Research Letters xxx (xxxx) xxx–xxx

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Time-consistent investment and reinsurance strategies for meanvariance insurers with relative performance concerns under the Heston model Huainian Zhua, Ming Cao a b

⁎,b

, Chengke Zhanga

School of Economics & Commence, Guangdong University of Technology, Guangzhou 510520, PR China School of Economics & Trade, Guangdong University of Finance, Guangzhou, Guangzhou 510521, PR China

ARTICLE INFO

ABSTRACT

Keywords: Investment and reinsurance Relative performance Time-consistency Heston model Nash equilibrium Mean-variance

This paper considers the optimal time-consistent investment and reinsurance strategies for two mean-variance insurers subject to the relative performance concerns. Each insurer can purchase a reinsurance protection and invest in a financial market consisted of one risk-free asset and one risky asset. We assume that the price process of risky asset is driven by the Heston model. The main objective of each insurer is to choose a investment and reinsurance strategy such that the mean and variance of his relative terminal wealth with respect to that of his competitor is maximized and minimized, simultaneously. By applying the stochastic control theory, closedform expressions for the equilibrium investment-reinsurance strategies and corresponding value functions are derived. Finally, we provide some numerical studies and draw some economic interpretations.

1. Introduction Investment and reinsurance problem has attracted much attention and became a popular research topic in insurance literature. In the early days, scholars only paid attention to investment problem, for example, Browne (1995) obtained the optimal investment strategies for an insurer who maximizes the expected utility of the terminal wealth or minimizes the ruin probability, where the surplus process of the insurer is modeled by a drifted Brownian motion. Yang and Zhang (2005) studied the optimal investment policies for an insurer who maximizes the expected exponential utility of the terminal wealth or maximizes the survival probability, where surplus process is driven by a jump-diffusion process. Elliott and Siu (2011) discussed a backward stochastic differential equation approach to a risk-based, optimal investment problem of an insurer who minimize the risk described by a convex risk measure of his/her terminal wealth, where the insurer’s risk process is modeled by a diffusion approximation to a compound Poisson risk process. Lin et al. (2012) investigated an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump-diffusion risk model using a game theoretic approach. Liu et al. (2014) investigated an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. In this topic, the investment problem for an insurer can be seen as an asset allocation problem. But the asset allocation problem for an insurer is different from the classic asset allocation problem, the key difference lies in the presence of insurance liabilities, which are mainly due to insurance claims (Zheng et al., 2016). Due to the fact that reinsurance is an effective way to shift risk in the insurance business, optimal reinsurance problems for

⁎

Corresponding author. E-mail address: [email protected] (M. Cao).

https://doi.org/10.1016/j.frl.2018.10.009 Received 9 June 2018; Received in revised form 4 October 2018; Accepted 11 October 2018 1544-6123/ © 2018 Elsevier Inc. All rights reserved.

Please cite this article as: ZHU, H., Finance Research Letters, https://doi.org/10.1016/j.frl.2018.10.009

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insurers with various stochastic investment opportunities have drawn great attention in recent years. Promislow and Young (2005) obtained the optimal reinsurance and investment strategies for an insurer to minimize the ruin probability. Bai and Guo (2008) studied the optimal investment and reinsurance problem with multiple risky assets, who aimed to maximizes the expected exponential utility of the terminal wealth or minimizes the ruin probability. Zhang and Siu (2009), Yi et al. (2015), Gu et al. (2018), and Li et al. (2018) investigated the robust investment-reinsurance strategies for insurers in different situations. Recently, many scholars consider the optimal investment and reinsurance strategies for insurers under the mean-variance criterion proposed by Markowitz (1952). Delong and Gerrard (2007) considered two optimal investment problems for an insurer: one is the classical mean-variance portfolio selection and the other is the mean-variance terminal objective involving a running cost penalizing deviation of the insurer’s wealth from a specified profit-solvency target. They assume that the claim process is a compound Cox process with the intensity described by a drifted Brownian motion and the insurer invests in a financial market consisting of a risk-free asset and a risky asset whose price is driven by a Lévy process. Bai and Zhang (2008) studied the optimal investmentreinsurance policies for an insurer under the mean-variance criterion by the linear quadratic method and the dual method, where they assume that the surplus of the insurer is described by a Cramer–Lundberg model and a diffusion approximation model respectively. Zeng et al. (2010) assumed that the surplus of an insurer is modeled by a jump-diffusion process, and derived the optimal investment policies explicitly under the benchmark and mean-variance criteria by the stochastic maximum principle. More related studies include those of Zeng and Li (2011), Bi et al. (2016) and Li et al. (2017), etc. In the above-mentioned literatures, they only focus on single-agent optimization problems, i.e., they usually take the investment and reinsurance problem as an optimal control problem, thus, they do not consider the strategic interaction among insurers. However, in a competitive economy, firms tend to compare themselves with one another, and relative performance concerns thus play a key role in decision-making (Garcia and Strobl, 2010; Basak and Makarov, 2014). With considering the relative performance concerns, Bensoussan et al. (2014) formulated a nonzero-sum stochastic differential investment and reinsurance game between two insurance companies whose surplus processes were modulated by continuous-time Markov chains. Meng et al. (2015) investigated an optimal reinsurance problem when the two insurers surpluses are subject to quadratic risk controls. Pun and Wong (2016) considered a reinsurance game problem for two ambiguity-averse insurers and obtain equilibrium under a worst-case scenario framework for the exponential utility functions. More studies on reinsurance and/or investment strategies under relative performance concerns can be found in (Pun et al., 2016), (Kwok et al., 2016), (Yan et al., 2017), (Siu et al., 2017), (Hu and Wang, 2018), (Deng et al., 2018) and (Chen et al., 2018). In additional, it is well-known that the prices of risky assets may have different features in the real world and numerous studies have shown that the volatilities of risky assets prices are not deterministic. It is clear that a model with stochastic volatility will be more practical. Heston (1993) used a Cox-Ingersoll–Ross process to characterize the volatility of the risky asset. Since then, the Heston model has been widely used in the field of insurance. Li et al. (2012) applied the Heston model to study the reinsurance and investment problem under the meanvariance criterion. Zhao et al. (2013) discussed the optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model. Yi et al. (2013) investigated the robust optimal reinsurance and investment strategies for an insurer with individual preferences when facing model uncertainty. Actually, the Heston model is classical and very popular for option pricing, and has been recognized as an important feature for asset price models. Meanwhile, the Heston model can be seen as an explanation of many well-known empirical findings, such as the volatility smile, the volatility clustering, and the heavy-tailed nature of return distributions. Therefore, we consider the optimal reinsurance and investment problem for two mean-variance insurers with relative performance concerns under the Heston model. Moreover, it is apparent to all that the mean-variance criterion lacks the iterated-expectation property, which results in that continuous-time mean-variance problems are time-inconsistent. While time-consistence is an essential requirement in decision making, and the question of optimal investment and reinsurance for two mean-variance insurers subject to the relative performance concerns under Heston model has not been investigated, thus, time-consistent investment and reinsurance strategies for meanvariance insurers subject to the relative performance concerns under Heston model worthy to be further explored. In this paper, we extend the work of Li et al. (2012) from single insurer to two competitive insurers. Each insurer aims to maximize his mean-variance utility of his terminal absolute and relative wealth by purchasing reinsurance and investing in a financial market. The financial market consists of one risk-free asset and one risky asset whose price follows the Heston model. By applying stochastic control theory, we establish the extended Hamilton–Jacobi–Bellman (HJB) equation, provide the corresponding verification theorem. Closed-form expressions for the equilibrium investment-reinsurance strategies and corresponding value functions are obtained via complicated analysis. Furthermore, we analyze the properties of the optimal strategy and present a numerical simulation to illustrate our results. The rest of the paper is organized as follows. Section 2 introduces the basic model setup of the two competitive insurers. In Section 3, we derive the extended HJB equation for the case of mean-variance utility function; then, the closed-form expressions for equilibrium time-consistent strategies and the corresponding value functions are obtained. Section 4 provides detailed numerical studies to discuss the impact of model parameters on the equilibrium strategies. Section 5 concludes the paper with some suggestions for future research. 2. Economy with competition In this paper, we aim to extend the work of Li et al. (2012) from one insurance company to two competing insurance companies which are run by two risk-averse managers, for simplicity, referred to as insurer 1 and 2. We assume that trading in the financial market is continuous, no transaction costs or taxes are involved, and short selling is permitted. Let ( , , ) be a complete 2

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probability space equipped with a filtration { t} 0 t T which satisfies the usual conditions, i.e., { t} 0 t T is right-continuous and -complete, where t denotes the information set available until time t. Moreover, we assume that all the stochastic processes described below are well-defined on the probability space ( , , ) and adapted to { t} 0 t T . 2.1. Economic setting We assume that the financial market consists of one risk-free asset (e.g. a bond or a bank account) and one risky asset (e.g. stock); the price dynamics of the risk-free asset, S0(t), is given by the following ordinary differential equation (ODE)

dS0 (t ) = rdt , S0 (0) = s0 > 0, S0 (t )

(1)

where r > 0 is the constant risk-free interest rate; the price dynamics of the risky asset, S(t), is described by the Heston model dS (t ) S (t )

= (r + mL (t )) dt +

dL (t ) =

(

L (t ) dBS (t ), S (0) = s > 0,

L (t )) dt +

L (t ) dBL (t ), L (0) = l0 > 0,

(2)

where m, α, δ and σ are all positive constants, {BS(t)}0 ≤ t ≤ T and {BL(t)}0 ≤ t ≤ T are two standard Brownian motions with [BS (t ) BL (t )] = 1 t . Moreover, we require that 2αδ > σ2 to ensure that L(t) > 0 for all t ≥ 0. In the absent of reinsurance, the surplus process of insurer k ∈ {1, 2} is modeled by a diffusion approximation model

dRk (t ) = µk dt +

(3)

k dBk (t ),

where μk, σk > 0 are the premium return rate and the volatility of the insurer k’s surplus, respectively, and {B1(t)}0 ≤ t ≤ T and {B2(t)}0 ≤ t ≤ T are two standard Brownian motions. In order to consider the correlation of the two insurers’ businesses conveniently, we model the surplus processes of insurer 1 and 2 are interrelated with positive correlation ρ, that is [B1 (t ) B2 (t )] = t . Assume 0 < ρ < 1 which is required to rule out trivial solutions. Moreover, we assume that {B1(t)} and {B2(t)} are independent of {BS(t)} and {BL(t)}. Grandell (1991) and Bensoussan et al. (2014) show that this diffusion approximation model works well for insurance portfolios which are large enough such that each individual claim is relatively small compared to the total reserve. Suppose that the two competing insurers can control their risks by reinsurance. For each t ∈ [0, T], let ak(t) be the reinsurance fraction of the insurer k ∈ {1, 2}. The stochastic process {ak(t)}0 ≤ t ≤ T taking values in + is called the reinsurance strategy of insurer k. When ak(t) > 1, insurer k provides a reinsurance service to a reinsurer or acquires a new business. When ak(t) ∈ [0, 1], insurer k purchases proportional reinsurance protection from the reinsurer, which will cover 1 ak (t ) of the claims and charge a reinsurance premium at the rate of (1 ak (t )) k , where ηk ≥ μk is the premium return rate of the reinsurer; meanwhile, insurer k pays 100a(t)% while the reinsurer pays the rest 100(1 ak (t ))% for each claim occurring at time t. With reinsurance strategy ak(t), the surplus process Rk k (t ) of insurer k ∈ {1, 2} becomes

dRk k (t )

[µk =[

k

(1 ak (t )) k ] dt + k ak (t ) dBk (t ) + k ak (t )] dt + k ak (t ) dBk (t ),

(4)

where k = µk k is the premium difference. Assume that each insurer k ∈ {1, 2} can dynamically provide new insurance business, purchases proportional reinsurance, and adjust investment level in the stock. Let bk(t) be the money amount that insurer k invests in the stock, and k (t ) = (ak (t ), bk (t )) be a reinsurance and investment strategy of insurer k, for k = 1, 2 . Corresponding to a reinsurance and investment strategy πk(t), the wealth dynamics of insurer k, denoted by {Xk k (t )}t 0 , for k = 1, 2, can be expressed as

dXk k (t ) = dRk k (t ) + (Xk k (t ) = [rXk k (t ) +

k

+

bk (t )) k ak (t )

dS0 (t ) S0 (t )

+ bk (t )

dS (t ) S (t )

+ mbk (t ) L (t )] dt +

k ak (t ) dBk (t )

+ bk (t ) L (t ) dBS (t ),

Xk k (0) = xk > 0.

(5)

: = × + and : =[0, T ] × Definition 1. (Admissible Strategy). Let { k (s ) = (ak (s ), bk (s ))}t s T , k ∈ {1, 2}, is said to be admissible if k

.

For

any

fixed

t ∈ [0,

T],

a

strategy

T

(i) ∀s ∈ [t, T], ak(s) and bk(s) are s -progressively measurable and satisfying [ s (|ak ( )|4 + |bk ( )|4 ) d ] < ; , Eq. (5) has a pathwise unique solution {Xk k (s )} s [t , T ] with Xk k (t ) = xk and L (t ) = l ; (ii) (xk , l) (t , x k , l ) , t , xk , l (sups [t , T ] |Xk k (s )|p ) < , where t , xk , l [ · ] is the conditional expectation given (iii) p [1, + ) and k Xk (t ) = xk and L (t ) = l . Denoted the set of all admissible strategies πk by Πk. 2.2. Modeling competition Both of the insurers choose an admissible reinsurance and investment strategy πk to maximize their own terminal wealth. Each 3

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insurer cares about the difference between its terminal wealth and the other’s, and tries to perform better relative to his competitor. Thus, each insurer’s utility not only relies on his own wealth, but also relies on the relative performance. Following (Espinosa and Touzi, 2015), we define the relative performance as the difference of the wealth processes controlled by the two insurers. Each insurer has a mean-variance preference, which is given by

Uk (t , xk , xj , l)

t , x k , xj , l [(1 k

2

=

k ) Xk

k (T )

+

Vart , xk , xj, l [(1

t , x k , xj , l [Xk

k (X k

k ) Xk

k (T )

kXj

j

X j j (T ))]

k (T )

k (T )

+

k (X k k

(T )]

2

k (T )

X j j (T ))]

Vart , xk , xj, l [Xk k (T )

kXj

j

(T )],

where k, j ∈ {1, 2}, k ≠ j. t , xk , xj, l [ · ] and Vart , xk , xj, l [ · ] are the conditional expectation and variance given Xk k (t ) = xk , X j j (t ) = x j and L (t ) = l for (t , xk , xj , l) [0, T ] × × × +, respectively, γk > 0 is the risk-aversion coefficient of insurer k, 1 k (0 < κk < 1) and κk are the weights of absolute wealth and relative wealth. The parameter κk captures the intensity of insurer k’s relative concerns and measures his sensitivity to the performance of his competitor. The greater κk means that he gives more weight to the relative performance and cares more about increasing his relative wealth. Insurer k anticipates that the increasing wealth of the competitor will cause his ranking to drop, trading opportunities to reduce, and insurance market share to decrease. This undoubtedly lowers his satisfaction. Intuitively, the utility of insurer k increases in his own wealth and decreases in the wealth of the other insurer. But this result is not necessarily true under the mean-variance utility due to the adverse impact of the variance component. If the variance is large enough, the opposite result may be true. The mean-variance utility makes each insurer strictly control risk in addition to seeking high profits. Therefore, it is a good measure of preference for the 2 2 insurers. We assume that κ1, κ2 both are positive such that 1 1 2 > 0 and 1 1 1 2 > 0 which are necessary to make equilibrium policies accord with reality. ( , ) j For k = 1, 2, let Zk k j (t ) Xk k (t ) k X j (t ) be the relative wealth process of insurer k, then ( k, j )

dZk

(t ) = dXk k (t ) =

( , ) [rZk k j (t )

+ ( , ) Zk k j (0)

k dX j

+(

j

(t ) k j)

k

k ak (t ) dBk (t )

= xk

+ ( k ak (t )

k j aj (t ) dBj (t )

k j aj (t ))

+ (bk (t )

+ m (bk (t )

k bj (t ))

k bj (t )) L (t )] dt

L (t ) dBS (t ),

k xj ,

(6)

where k ≠ j ∈ {1, 2}. In the following, we will formulate the competition problem under a non-zero-sum stochastic differential game framework, which is studied by Bensoussan et al. (2014), Siu et al. (2017), Deng et al. (2018), etc. For an admissible strategy πk ∈ Πk and any state (t , z , l ) , the insurer k ∈ {1, 2} aims to maximize ( k , *j )

Jk

where j*

(t , z, l)

t , z, l

(aj*, bj*),

Fk (y ) = y

( k , j*)

Zk

k

(T )

( k, *j )

2

k

=

t,z,l

Zk

=

t,z,l

Fk Zk

=

( k, j*) (t ) t , z, l ·|Zk

t,z,l [ · ]

k 2 y ,

[

Gk (y ) =

2

(T )

( k, j*)

( k, j*)

Vart , z , l Zk

2

(T )

t,z,l

+ Gk

(T ) ( k, j*)

Zk

t,z,l

2

(T )

( k, j*)

Zk

t , z, l

(T )

( k , j*)

Zk

2

(T )

,

= z, L (t ) = l], Vart , z, l [ · ] = Vart , z , l [

(7) ( , *) ·|Zk k j (t )

= z , L (t ) = l] and

k 2 y .

2

Problem 1. The classic non-zero-sum stochastic differential game problem between two insurers is to find a Nash equilibrium ( 1*, 2*) 1 × 2 such that, for all π1≜(a1, b1) and π2≜(a2, b2),

J1(

1, 2*) (t ,

* J2( 1 , 2) (t ,

z , l)

J1(

1*, 2*) (t ,

z, l),

z , l)

* * J2( 1 , 2 ) (t ,

z, l).

(8)

The following definition gives the statements of equilibrium reinsurance and investment strategy and equilibrium value function, which is similiar to that in (Bjork and Murgoci, 2010). Definition 2. If the Nash equilibrium strategy ( k*, j*) exists, define a strategy πkτ by k

(s, z˜, l˜)

z˜, l˜), * ( s , z˜, l˜), k k (s ,

t t+

(˜, z l˜) s < T , (˜, z l˜)

s
, ,

(9) 4

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where πk ∈ Πk, τ > 0 and (t , z, l)

lim inf

( *, *) Jk k j (t ,

are arbitrarily chosen. If

( , *) Jk k j (t ,

z, l)

z , l)

0

0

for all πk ∈ Πk and (t , z, l) ( k*, j*)

Jk

. Then for insurer k,

(t , z , l ) .

is called an equilibrium strategy and the equilibrium value function is

k*

3. Nash equilibrium In this section, we solve the time-consistent Nash equilibrium of Problem 1 via stochastic control theory. First, the verification theorem is given, then, closed-form expressions of the time-consistent quilibrium strategies and corresponding value functions are derived. Throughout this paper, if there do not exist relative performance concerns, that is 1 = 2 = 0 and the insurers’ utility just comes from their own wealth, we call the economy as standard economy and the corresponding strategies as standard strategies. 3.1. Verification theorem

, denote by Vk(t, z, l) the equilibrium value function of insurer k ∈ {1, 2}, i.e.

For any fixed (t , z, l) ( k*, j*) (t ,

V k (t , z , l) = Jk

( k, *j )

z, l) = sup Jk k

(t , z , l),

k

, *)

(

where Jk k j (t , z , l) is given by (7) and k* is the equilibrium strategy for insurer k ∈ {1, 2}. We can derive the extended Hamilton–Jocabi–Bellman (HJB) equations for the equilibrium value function and equilibrium n be an open set and O = [0, T ] × O . Denote that strategy. Let O0 0

C1,2 (O) = { (t , z ) | (t , ·)is once continuously differentiable on [0, T ] and (·,z )is twice continuously differentiable on O0}, Dp1,2 (O ) = { (t , z ) | (t , z ) C1,2 (O ), and all once partial derivatives of (·,z ) satisfy the polynomial growth condition on O0}. For any function ( k , j ) k (t ,

×

k (t ,

z , l) = k z (t ,

C1,2 ( ), where k ≠ j ∈ {1, 2}, define the variational operator

z , l) k t (t ,

z, l) + [rz + (

z, l) +

(

l)

k l (t ,

k j)

k

1

z , l) + 2 [

2 k 2 k bj (t ) l] zz (t ,

2 k bk (t ) bj (t ) l +

+ ( k ak (t ) 2 2 k ak (t )

z , l) +

1 2

k ll (t ,

2l

(t , z, l)

k j aj (t ))

2

k

k

z, l) +

+ ml (bk (t )

j ak (t ) aj (t ) + 1

l [bk (t )

2 k

( k , j)

k

k

{

( k , *j ) V k (t ,

z, l)

k

2

( k, *j ) (g k (t ,

z, l))2 +

as follows

k bj (t ))] 2 2 j aj (t )

k bj (t )]

+ bk (t )2l

k zl (t ,

z, l),

.

Theorem 1. (Verification Theorem). For Problem 1, if there exist real value functions Vk(t, z, l), g k (t , z, l) , the following conditions: (t , z, l)

sup

( k , j)

of Zk

kg

k (t ,

z, l)

( k , *j ) g k (t ,

}

z , l) = 0,

Dp1,2 ( ), for k = 1, 2, satisfying

(10)

V k (T , z , l ) = z ,

(11)

( k, *j ) g k (t ,

(12)

g k (T ,

z, l) = 0,

(13)

z , l) = z,

and k*

= arg sup k

then,

( *, *) Jk k j (t ,

k

{

( k , j*) V k (t ,

z, l) = V k (t , z, l),

z , l)

k

2

( k , *j ) (g k (t ,

( k*, j*) (T ) t , z , l Zk

z , l))2 +

kg

k (t ,

z , l)

( k, j*) g k (t ,

}

z, l) ,

(14)

= g k (t , z, l) and k* is the equilibrium strategy.

The proof of the verification theorem can be adapted from Theorem 4.1 of Bjork and Murgoci (2010) and Theorem 1 of Li et al. (2012), here we omit the proof. 3.2. The equilibrium reinsurance and investment strategies In this subsection, we are going to derive the closed-form expressions of the equilibrium reinsurance and investment strategies 5

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and the corresponding value functions. Referring to Theorem 1, the first order condition results in the equilibrium strategy. Differentiating (14) with respect to πk, we obtain k*

k k Vz

= (ak* (t ), bk* (t )), ak* (t ) = k mV zk + 1 V zl

bk* (t ) =

k V zz

k k k 1 gz gl k 2 k ( gz )

+

k 2 k (gz )

k 2 k V zz

j

k

k

aj* (t ),

k b* j (t ).

+

(15)

Substituting (15) into (10) and (12), we have

0 = Vtk +

1 2

l) Vlk +

(

k 2 k (gz )

k 2 k2 V zz

1 2

2 k k Vz

k mV zk + 1 V zl

+ rz + (

k 2 V zz

k k k 1 gz gl l k 2 k (gz )

k V zz

k k k 1 gz gl

l

1

mgzk +

1

gzlk

(

j

j k k

) a* j

V zk

k 2 k (gz ) ],

(16) k

(

j

j

k

) a*

gzk

j

2 ) 2 2 a*2 g k k j j zz

+ 2 (1

2

k

2 ) 2 2 a*2 [V k k j j zz

+ 2 (1

1

k k (g z )

k j)

k

2

k j)

k

k Vk gzz z

gzk

k 2 k ( gz )

k 2 k V zz

2lg k ll

+ rz + (

k 2 k (gz )

k 2 V zz

l) glk +

(

2l (g k ) 2 l

k

2

k mV zk + 1 V zl

2 k 2 k (V z )

0 = gtk +

2lV k ll

k mV zk + 1 V zl k 2 V zz

k k k k 1 gz gl gzz k 2 k (g z )

. (17)

To solve (16) and (17), we try to conjecture the solutions in the following forms

V k (t , z , l ) = A ( t ) z +

B (t )

l+

Ck (t )

,

A (T ) = 1,

B (T ) = Ck (T ) = 0,

B¯ (t ) C¯ (t ) g k (t , z, l) = A¯ (t ) z + l+ k ,

A¯ (T ) = 1,

B¯ (T ) = C¯k (T ) = 0.

k

(18)

k

k

(19)

k

The partial derivatives are

Vtk = A (t ) z +

B (t )

gtk = A¯ (t ) z +

B¯ (t )

k

k

l+ l+

Ck (t )

,

k

C¯ k (t ) k

,

V zk = A (t ), gzk = A¯ (t ),

B (t )

Vlk = glk =

k

B¯ (t ) k

,

.

(20)

Inserting the above partial derivatives back into (15) yields

ak* (t ) =

k A (t ) 2 ¯2 k k A (t )

+

k

j k

aj* (t ), bk* (t ) =

mA (t )

¯ ¯ 1 A (t ) B (t ) + 2 (t ) ¯ A k

k b* j (t ).

(21)

According to (21), we find that each insurer will adjust his reinsurance and investment strategy basing on that of his competitor. Moreover, once the correlation coefficient ρ equals to zero, even though relative performance concerns exist, the strategic interactions on reinsurance business also disappear. As a result, relative performance concerns play a role only in interrelated individuals and ¯ ¯ A (t ) mA (t ) 1 A (t ) B (t ) groups. In the standard economy, insurer 1’s optimal reinsurance and investment strategies are 12 2 and which 2 ¯ 1 1 A (t )

1 D (t )

correspond to the first terms of (21) when k = 1. In contrast, the optimal strategies deviate from the standard levels in the presence of relative performance considerations. The magnitudes of deviation depend on the second terms which represent the hedging demands. Coefficients 1 2 and κ1 measure insurer 1’s sensitivity to insurer 2’s trading actions. Specifically, insurer 2 increases (decreases) his 1 risk retention and investment in the risky asset, which will lead insurer 1 to increase (decrease) his risk retention and investment in the risky asset. This is because reinsurance is expensive relative to insurance; however, investment can bring positive expected profits. Then, if insurer 1 looks forward to outperforming insurer 2, he must invest more money in the risky asset and finance this by purchasing less reinsurance. In this sense, hedging demands protect insurer 1 from the negative effects of relative concerns. The corresponding results also hold for insurer 2. Now, we proceed to derive the closed-form expressions for the equilibrium reinsurance and investment strategies and the corresponding value functions. Inserting (20)–(21) into (16) and (17), we get 6

Finance Research Letters xxx (xxxx) xxx–xxx

H. Zhu et al. B (t )

A (t ) z +

k

B¯ (t )

A¯ (t ) z +

k

2 k A (t ) 2¯ k k A (t )

+

+

k

[m2A (t )

+

l)

2lB¯ (t )2

B (t )

(

l)

m 1 A¯ (t ) B¯ (t )] l ¯ k A (t )

+ rz + (

2 k2

k

2 ¯ ¯ 1 A (t ) B (t )] l 2 k A¯ (t )2

[mA (t )

C¯ k (t )

l+

(

+

k

2 2 k A (t ) 2 k k2 A¯ (t )2

× A (t ) +

+

Ck (t )

l+

B¯ (t )

1 (1 2 k

+ rz + (

k

k j)

k

k

2 ) 2 2 a* (t )2A ¯ (t ) 2 k j j

k j)

k

(

k

(

) a * (t ) j

= 0, j k k

j

j k k

j

(22)

) a * (t )

A¯ (t )

j

= 0.

(23)

For (22) and (23), separating the variables with z and l respectively, we obtain the following system of ODEs:

A (t ) + rA (t ) = 0, B (t )

Ck (t ) + +

2B ¯ (t )2

B (t )

(

k

1 2 (1 2 k

B¯ (t ) +

C¯k (t ) +

= 0,

2A¯ (t ) 2 k j)

k

k

(

) a * (t )

j k k

j

= 0,

B (T ) = 0,

(25)

A (t )

j

Ck (T ) = 0,

(26) (27)

A¯ (T ) = 1, m 1 A¯ (t ) B¯ (t ) = 0, A¯ (t )

m2A (t )

B¯ (t ) +

A¯ (t ) B¯ (t )]2

1

2 ) 2 2 a * ( t ) 2A ¯ (t ) 2 k j j

A¯ (t ) + rA¯ (t ) = 0, B¯ (t )

[mA (t )

+

2

B (t ) +

2 2 k A (t ) 2 k2 A¯ (t )2

(24)

A (T ) = 1,

(

k

k j)

k

B¯ (T ) = 0,

j

k

j

k

k

(28)

aj* (t ) A¯ (t ) +

2 k A (t ) 2 ¯ k A (t )

= 0,

C¯k (T ) = 0,

(29)

Solving the above ODEs, we have

A (t ) = A¯ (t ) = er (T B (t ) =

e2( + m 1 )(t + 2m 1

B¯ (t ) =

m2 +m

+

[1

1

k

(

r 2 k

4r

× [er (T

T)

(2

e(

+ m 1 )(t T )

2) 2 k

(1

k

(

t)

1] +

+

2 k 2 k

m3 1 +m 1

,

+m 1

+ e

(t T )

+

,

(31)

].

(32)

j k k

j

) a* (t ) j

2 2 2r (T t ) j aj* (t ) [e

m2 [e( + m 1 )(t T ) ( + m 1 )2 m2

1 )(t T )

(2 + )(1 e( + m 1 )(t T ) ) m 1 ( +m 1 )

k j)

k

)e( + m m 1

+

(1 e2( + m 1 )(t T ) ) 2( + m 1 )( + 2m 1 )

Ck (t ) =

C¯k (t ) =

(30)

t ).

1] + (T

k

r

1] +

(

+

[er (T 2 k

2 k2

t)

(T

k

e

(t T ) ]

1]

+

k j)

k

[1

(

j

t ),

j k k

(33)

) a * (t ) j

t ),

(34)

where

=

m4 2 (1

2 1)

2( + m 1 )2

,

=

=

m2 2

(

+

),

=

2 + m 1

+ 2m 1

.

(35)

Theorem 2. For Problem 1, the Nash equilibrium reinsurance and investment strategies of insurer 1 and 2 are given by

a1* (t ) =

e r (T t ) 2 1 1 2

b1* (t ) =

e r (T t ) 1 1 2

(

(

1 2 1 1

1 1

1

+

+

2

2 1 2 1 2

) (m

), 1

B¯ (t )) ,

(36) 7

Finance Research Letters xxx (xxxx) xxx–xxx

H. Zhu et al.

a2* (t ) =

e r (T t ) 2 1 1 2

b2* (t ) =

e r (T t ) 1

(

(

1 2

2 2 2 2

1

+

2

2

+

1

1 1 2

) (m

2 1

), 1

B¯ (t )) ;

(37)

and the corresponding equilibrium value functions are given by

V 1 (t , z , l) = z er (T

t)

+

B (t )

l+

C1 (t )

1

V 2 (t , z , l) = z er (T

t)

+

,

(38)

1

B¯ (t )

l+

2

C2 (t )

,

(39)

2

where B(t), B¯ (t ), and Ck(t), k = 1, 2 are given by (31)–(33). Especially, if insurers 1 and 2 do not concern about each other’s performance, that is

a1* (t ) = b1* (t ) =

a2* (t ) = b2* (t ) =

1 r (T t ) , 2e 1 1 ¯ e r (T t ) m 1 B (t )

1

1 2

·

1

2 r (T t ) , 2e 2 2 ¯ e r (T t ) m 1 B (t )

1

1 2

·

2

1

=

2

= 0, the equilibrium strategies degenerate into

,

(40)

,

(41)

which are standard strategies in traditional reinsurance and investment problem. Remark 1. Compared with only single insurer discussed in (Li et al., 2012), the equilibrium reinsurance and investment strategies (a1* (t ), b1* (t )) and (a2* (t ), b2* (t )) of the two insurers with relative performance concerns both include two parts: the first parts are standard components and the second are hedge components coming from relative concerns, and they both do not depend on the relative wealth processes Z1( 1, 2) (t ) and Z2( 1, 2) (t ) . Remark 2. The equilibrium reinsurance strategies a1* (t ) and a2* (t ) increase with respect to reinsurance premiums η1 and η2, this result can be understood that whether insurer1′s reinsurance premium or2′s reinsurance premium rises, both render insurer k ∈ {1, 2} retains more claims risk. Remark 3. Similiar with the results discussed in (Hu and Wang, 2018), the equilibrium reinsurance and investment strategies (a1* (t ), b1* (t )) and (a2* (t ), b2* (t )) are increasing in the degree of relative concerns κ1 and κ2, respectively, and always greater than standard values. 4. Numerical studies In this section, we present some numerical studies on the effect of model parameters on the equilibrium reinsurance and investment strategies. Unless otherwise stated, the following numerical illustrations are based on the model parameters as specified in Table 1. 4.1. Equilibrium reinsurance strategies Fig. 1displays the effect of competition on the equilibrium reinsurance strategies. As parameter κk captures the degree of dependence on the terminal wealth of insurer k’s competitor, a higher κk results in insurer k becoming more concerned with his performance compared with that of his competitor in the terminal period T. While purchasing reinsurance can reduce the risk borne by insurer k, it is nonetheless costly because insurer k needs to pay (1 ak*) k to the reinsurance company for the reinsurance Table 1 Model parameters. Base parameters r 0.05

m 1.5

α 2

μ1 0.2

σ1 1

η1 0.3

μ2 0.2

σ2 1

η2 0.4

δ 0.03 insurer 1 γ1 0.2 insurer 2 γ2 0.5

σ 1.5 κ1 0.7 κ2 0.5

8

ρ1 0.3

ρ 0.5

T 10

Finance Research Letters xxx (xxxx) xxx–xxx

H. Zhu et al. Equilibrium reinsurance strategy of insurer 1 ( 2 =0.5)

2

1 1

1.8

1

=0

2

=0.3

2

1.4

=0.7

2

=0 =0.3 =0.7

1.2

a *2 (t)

1.6

a *1 (t)

Equilibrium reinsurance strategy of insurer 2 ( 1 =0.7)

1.6

1.4

1

1.2

0.8

1

0.6

0.8

0.4 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

t

5

6

7

8

9

10

t

Fig. 1. Effect of κk on the equilibrium reinsurance strategy of insurer k, ak*, for k = 1, 2 .

protection, which decreases his terminal wealth value relative to that of his competitor. In the case, insurer k tends to decrease his expenditure on the reinsurance company, i.e. smaller (1 ak*) k , which increases the dependence parameter κk, which in turn implies increasing ak*. Fig. 2shows the effect of the risk-aversion parameter γk of insurer k on his optimal reinsurance strategy ak* at equilibrium. Observe that ak* is a decreasing function of the risk-aversion parameter γk. This is in agreement with the optimal reinsurance strategy without competition in the existing literature. Because ak* denotes the proportion that insurer k retains upon purchasing a proportional reinsurance protection or accquiring new business, the higher the risk-aversion parameter γk, the greater the risk that insurer k transfers, which in turn implies lower ak*% . Fig. 3depicits a positive correlation between the equilibrium strategies reinsurance strategy ak* and the reinsurance premium ηk. We note that the slopes of the curves become monotonically increasing with respect to t. This can be explained that: as reinsurance premium ηk goes up, this inevitably makes insurer k’s demand drop. The positive relation between the insurers’ risk retention levels results in further reduction of the demand of insurger k. 4.2. Equilibrium investment strategies Fig. 4captures the effect of competition on the equilibrium investment strategies. Contrary to the case of the reinsurance protection, which is an expenditure, investment into the risky asset S has a possibility of generating income and hence both insurers would increase their exposure to the risky asset S at the terminal time T. Fig. 5shows the effect of the mean-reversion rate α on the optimal investment strategy of insurer k at equilibrium. We can find that the equilibrium investment strategy will increase as α increases. α denotes the mean reversion rate of L (i.e., the stock’s volatility); thus, a larger α will cause L to revert more quickly back to its mean δ. This should lead to a more stable return of the stock. As a result, insurer k ∈ {1, 2} will tend to increase its investment in the stock. Fig. 6shows the effect of volatility coefficient σ on the optimal investment strategy of insurer k at equilibrium. it is observing that with the increase of σ, the equilibrium investment strategy will decrease. As σ denotes the volatility of volatility parameter, the larger σ is, the more risk the risky asset S has, thus resulting in the decrease investment in the risky asset, this is also consistent with the intuition.

Equilibrium reinsurance strategy of insurer 1 ( 2 1 1

2

=0.5)

Equilibrium reinsurance strategy of insurer 2 ( 3

=0.2

2

=0.35

2

=0.5 1

2.5

2

1

=0.2)

=0.2 =0.35 =0.5

1.5

a *2 (t)

a *1 (t)

2

1.5 1 1

0.5

0.5 0

1

2

3

4

5

6

7

8

9

10

0

t

1

2

3

4

5

6

7

t

Fig. 2. Effect of γk on the equilibrium reinsurance strategy of insurer k, ak*, for k = 1, 2 . 9

8

9

10

Finance Research Letters xxx (xxxx) xxx–xxx

H. Zhu et al. Equilibrium reinsurance strategy of insurer 1 ( 4.5 1

4

1 1

2

=0.4)

Equilibrium reinsurance strategy of insurer 2 ( 2

=0.3

2

=0.5

1.8

2

=0.7

2

3

1.4

=0.3)

=0.3 =0.5 =0.7

a *2 (t)

1.6

a *1 (t)

3.5

1

2.5

1.2

2

1

1.5

0.8

1

0.6 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

t

5

6

7

8

9

10

t

Fig. 3. Effect of ηk on the equilibrium reinsurance strategy of insurer k, ak*, for k = 1, 2 . Equilibrium investment strategy of insurer 1 ( 2 =0.5)

10

1

9

1 1

Equilibrium investment strategy of insurer 2 ( 1 =0.7)

16

=0

2

=0.3

14

2

=0.7

2

8

=0 =0.3 =0.7

12

7

b *2 (t)

b *1 (t)

10 6

8 5 6

4

4

3 2

2 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

t

5

6

7

8

9

10

9

10

t

Fig. 4. Effect of κk on the equilibrium investment strategy of insurer k, bk*, for k = 1, 2 . Equilibrium investment strategy of insurer 1

13

12

Equilibrium investment strategy of insurer 2

9.5

=0.8 =1.5 =2

=0.8 =1.5 =2

9 8.5

11

8 7.5

b *2 (t)

b *1 (t)

10

9

7 6.5 6

8

5.5 7 5 6

4.5 0

1

2

3

4

5

6

7

8

9

10

0

t

1

2

3

4

5

6

7

8

t

Fig. 5. Effect of α on the equilibrium investment strategy of insurer k, bk*, for k = 1, 2 .

5. Conclusion In this paper, we have investigated the optimal time-consistent investment and reinsurance strategies under a competitive environment, in which both insurers care about relative performances and seek to outperform each other. More specifically, we study the problem in which each insurer has the option of purchasing reinsurance protection and investing in the financial market. By incorporating relative performances concerns into mean-variance utility, we incorporate the concept of competition into the problem. To capture the impact of the volatility risk to the insurers’ strategies, we model the price process of the risky asset by the Heston model. Verification theorem and closed-form expressions for optimal time-consistent reinsurance and investment strategies at equilibrium are provided. Finally, some numerical studies are provided to illustrate the sensitivity of the equilibrium time-consistent strategies. There are some possible extensions of this paper. The first one is the problem with model uncertainty. In fact, model uncertainty is an important issue in all modeling exercises. Thus, how to articulate the optimal reinsurance-investment problem of two competitive 10

Finance Research Letters xxx (xxxx) xxx–xxx

H. Zhu et al. Equilibrium investment strategy of insurer 1

13

12

Equilibrium investment strategy of insurer 2

9.5

=0.5 =1 =1.5

=0.5 =1 =1.5

9 8.5 8

b *2 (t)

b *1 (t)

11

10

7.5 7

9

6.5 6

8 5.5 7

5 0

1

2

3

4

5

6

7

8

9

10

0

t

1

2

3

4

5

6

7

8

9

10

t

Fig. 6. Effect of σ on the equilibrium investment strategy of insurer k, bk*, for k = 1, 2 .

insurers facing uncertainties regarding models for financial and insurance markets deserve further study. The second one is the problem with the regime-switching jump diffusion model, which may be more realistic and interesting. However, it will result in a more sophisticated non-linear partial differential equation and need to apply numerical methods (Jin et al., 2013) to solve it. We shall leave them for a future research. Acknowledgments The authors are very grateful to the anonymous referees and editors for their helpful suggestions. This research was supported by the National Natural Science Foundation of China (71571053, 71673061), National Social Science Fund of China (17BJY045), Humanities and Social Sciences Project of Ministry of Education of China (18YJC790003), Natural Science Foundation of Guangdong Province (2015A030310218, 2016A030313701), Philosophy and Social Sciences Project of Guangzhou city (2017GZQN13, 2018GZQN45, 2018GZYB61), Colleges Innovation Project of Education Commission of Guangdong Province (2017WQNCX107) and Distinguished Innovation Program of Education Commission of Guangdong Province (2015WTSCX014). References Bai, L., Guo, J., 2008. Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insurance: Math. Econ. 42 (3), 968–975. Bai, L., Zhang, H., 2008. Dynamic mean-variance problem with constrained risk control for the insurers. Math. Methods Oper. Res. 68 (1), 181–205. Basak, S., Makarov, D., 2014. Strategic asset allocation in money management. J. Finance 69 (1), 179–217. Bensoussan, A., Siu, C.C., Yam, S.C.P., Yang, H., 2014. A class of non-zero-sum stochastic differential investment and reinsurance games. Automatica 50 (8), 2025–2037. Bi, J., Liang, Z., Xu, F., 2016. Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence. Insurance: Math. Econ. 70, 24–258. Bjork, T., Murgoci, A., 2010. A general theory of Markovian time inconsistent stochastic control problems. Ssrn Electron. J. 18 (3), 545–592. Browne, S., 1995. Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Math.Oper.Res. 24 (4), 937–958. Chen, S., Yang, H., Zeng, Y., 2018. Stochastic differential games between two insurers with generalized mean-variance premium principle. ASTIN Bull. 48 (1), 413–434. Delong, L., Gerrard, R., 2007. Mean-variance portfolio selection for a non-life insurance company. Math. Methods Oper. Res. 66 (2), 339–367. Deng, C., Zeng, X., Zhu, H., 2018. Non-zero-sum stochastic differential reinsurance and investment games with default risk. Eur. J. Oper. Res. 264 (3), 1144–1158. Elliott, R.J., Siu, T.K., 2011. A bsde approach to a risk-based optimal investment of an insurer. Automatica 47 (2), 253–261. Espinosa, G.E., Touzi, N., 2015. Optimal investment under relative performance concerns. Math. Finance 25 (2), 221–257. Garcia, D., Strobl, G., 2010. Relative wealth concerns and complementarities in information acquisition. Rev. Financ. Stud. 24 (1), 169–207. Grandell, J., 1991. Aspects of risk theory. New York, NY: Springer. Gu, A., Viens, F.G., Yao, H., 2018. Optimal robust reinsurance-investment strategies for insurers with mean reversion and mispricing. Insurance: Math. Econ. 80, 93–109. Heston, S.L., 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6 (2), 327–343. Hu, D., Wang, H., 2018. Time-consistent investment and reinsurance under relative performance concerns. Commun. Stat.-Theor. Methods 47 (7), 1693–1717. Jin, Z., Yin, G., Wu, F., 2013. Optimal reinsurance strategies in regime-switching jump diffusion models: stochastic differential game formulation and numerical methods. Insurance: Math. Econ. 53 (3), 733–746. Kwok, K.Y., Chiu, M.C., Wong, H.Y., 2016. Demand for longevity securities under relative performance concerns: stochastic differential games with cointegration. Insurance: Math. Econ. 71, 353–366. Li, D., Li, D., Young, V.R., 2017. Optimality of excess-loss reinsurance under a mean-variance criterion. Insurance: Math. Econ. 75, 82–89. Li, D., Zeng, Y., Yang, H., 2018. Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps. Scand. Actuar. J. 2018 (2), 145–171. Li, Z., Zeng, Y., Lai, Y., 2012. Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s sv model. Insurance: Math. Econ. 51 (1), 191–203. Lin, X., Zhang, C., Siu, T.K., 2012. Stochastic differential portfolio games for an insurer in a jump-diffusion risk process. Math. Methods Oper. Res. 75 (1), 83–100. Liu, J., Yiu, K.F.C., Siu, T.K., 2014. Optimal investment of an insurer with regime-switching and risk constraint. Scand. Actuar. J. 2014 (7), 583–601. Markowitz, H., 1952. Portfolio selection. J. Finance 7 (1), 77–91. Meng, H., Li, S., Jin, Z., 2015. A reinsurance game between two insurance companies with nonlinear risk processes. Insurance: Math. Econ. 62, 91–97. Promislow, S., Young, V.R., 2005. Minimizing the probability of ruin when claims follow Brownian motion with drift. North Am. Actuar. J. 9 (3), 110–128. Pun, C.S., Siu, C.C., Wong, H.Y., 2016. Non-zero-sum reinsurance games subject to ambiguous correlations. Oper. Res. Lett. 44 (5), 578–586.

11

Finance Research Letters xxx (xxxx) xxx–xxx

H. Zhu et al.

Pun, C.S., Wong, H.Y., 2016. Robust non-zero-sum stochastic differential reinsurance game. Insurance: Math. Econ. 68, 169–177. Siu, C.C., Yam, S.C.P., Yang, H., Zhao, H., 2017. A class of nonzero-sum investment and reinsurance games subject to systematic risks. Scand. Actuar. J. 2017 (8), 670–707. Yan, M., Peng, F., Zhang, S., 2017. A reinsurance and investment game between two insurance companies with the different opinions about some extra information. Insurance: Math. Econ. 75, 58–70. Yang, H., Zhang, L., 2005. Optimal investment for insurer with jump-diffusion risk process. Insurance: Math. Econ. 37 (3), 615–634. Yi, B., Li, Z., Viens, F.G., Zeng, Y., 2013. Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model. Insurance: Math. Econ. 53 (3), 601–614. Yi, B., Viens, F., Li, Z., Zeng, Y., 2015. Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria. Scand. Actuar. J. 2015 (8), 725–751. Zeng, Y., Li, Z., 2011. Optimal time-consistent investment and reinsurance policies for mean-variance insurers. Insurance: Math. Econ. 49 (1), 145–154. Zeng, Y., Li, Z., Liu, J., 2010. Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers. J. Indus. Manag. Optim. 6 (3), 483–496. Zhang, X., Siu, T.K., 2009. Optimal investment and reinsurance of an insurer with model uncertainty. Insurance: Math. Econ. 45 (3), 81–88. Zhao, H., Rong, X., Zhao, Y., 2013. Optimal excess-of-loss reinsurance and investment problem for an insurer with jumpdiffusion risk process under the Heston model. Insurance: Math. Econ. 53 (3), 504–514. Zheng, X., Zhou, J., Sun, Z., 2016. Robust optimal portfolio and proportional reinsurance for an insurer under a cev model. Insurance: Math. Econ. 67, 77–87.

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