# Optimal Asset Allocation for CRRA and CARA Insurers under the Vasicek Interest Rate Model

Authors: Hanlei Hu; Shaoyong Lai; Hongjing Chen
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This paper considers the reinsurance-investment problem with interest rate risks under constant relative risk aversion and constant absolute risk aversion preferences, respectively. Stochastic control theory and dynamic programming principle are applied to investigate the optimal proportional reinsurance-investment strategy for an insurer under the Vasicek stochastic interest rate model. Solving the corresponding Hamilton-Jacobi-Bellman equation via the Legendre transform approach, the optimal premium allocation strategies maximizing the expected utilities of terminal wealth are derived. In addition, several sensitivity analyses and numerical illustrations are given to analyze the impacts of different risk preferences and interest rate fluctuation on the optimal strategies. We find that the asset allocation and reinsurance ratio of the insurer are correlated with risk preference coefficient and interest rate fluctuation, and the insurance company may adjust the reinsurance-investment strategy to deal with interest rate risk.


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## Body

## 1. Introduction

As a financial institution, insurance company plays an important role in the modern society, and its reinsurance and investment business is also the focus of the management because reinsurance and investment are effective at dispersing risks and making profits from surplus. Many literature studies have discussed the reinsurance and investment problem from different perspectives, and it is common to convert it into a problem of stochastic optimal control. In the last decades, stochastic control theory has been widely used in risk research. For instance, Browne [1] obtained the optimal investment strategy under the diffusion model through the Hamilton–Jacobi–Bellman (HJB) equation, creating a precedent of combining risk theory with stochastic control theory. Since then, there have been many papers in which the HJB equation was used to solve optimal control problems in insurance. According to the research content, different objective and constraint functions have been studied, such as minimizing ruin probability (Schmidli [2], David Promislow and Young [3], and Bai et al. [4]), maximizing adjustment coefficient (Hald and Schmidli [5] and Liang and Guo [6]), and maximizing expected utility (Irgens and Paulsen [7], Bai and Guo [8], Xu et al. [9], Cao and Wan [10], and Liang et al. [11]). In addition, mean-variance optimization also gets a lot of attention (Bi and Guo [12], Zeng and Li [13], and Wang et al. [14]).In this paper, the objective of the insurer is to maximize the expected utility of terminal wealth in the finite horizon. We suppose that the insurer purchases a proportional reinsurance and is allowed to invest in the financial market. The problem is that the insurer intends to find the optimal strategy to balance the risk and profit. Considering the fact that interest rate is uncertain in the real-world environments, the optimal strategy under stochastic interest rate is more practical. There have been many studies on stochastic interest rate in dynamic portfolio problems; see Li and Wu [15], Noh and Kim [16], Chang [17], Wang and Li [18], and so on. For the reinsurance-investment problem, Liang et al. [19] used an Ornstein-Uhlenbeck process to describe the instantaneous rate of investment return under CRRA utility maximization, and inflation risks are further considered in Guan and Liang [20]. Li et al. [21] obtained the optimal time-consistent reinsurance-investment strategy under the mean-variance criterion. Compared with previous studies, the first contribution of this paper is that we consider the stochastic interest rate in the reinsurance-investment problem, and the stochastic interest rate model and surplus process are different from Guan and Liang [20]. Second, we investigate the optimal reinsurance-investment strategy under two different risk preferences, which may provide the insurer with a more suitable investment strategy. Stochastic dynamic programming is a classical method to solve optimal problems, but the nonlinear partial differential equation generated in it is not easy to solve. Therefore, on the basis of the stochastic control theory, we also use Legendre transformation to obtain the explicit expression of the optimal strategy. For more references on the Legendre transform technique, Jonsson and Sircar [22], Xiao et al. [23], Chang [24], and Hu et al. [25] can be seen. Finally, we analyze the effects of market parameters on the optimal trading strategies.The rest of this paper is organized as follows. Section2 formulates the reinsurance-investment problem with the Vasicek stochastic interest rate. Section 3 derives the explicit expressions of the optimal reinsurance-investment strategies under CRRA and CARA utilities. Section 4 provides several sensitive analyses of market parameters. Section 5 gives conclusions.


## 2. The Model

In this section, we formulate a continuous-time reinsurance-investment model where the insurers can trade in the financial market or the insurance market with no taxes or fees. The framework consists of four parts: the surplus process, the financial market, the wealth process, and the optimization criterion. LetΩ,F,Ft0≤t≤T,P be a complete probability space with filtration Ft,0≤t≤T, where T>0 is the time horizon and P is the probability. All stochastic processes in this paper are supposed to be well defined in this probability space.


### 2.1. Surplus Process

Typically, three types of models are used in the insurance market: the Cramer–Lundberg model, approximating diffusion model, and jump-diffusion model. We adopt the diffusion model to describe the surplus for the insurers. The claim processC is described as(1)dCt=adt−bdW1t,where a and b are positive constants and W1t is a one-dimensional standard Brownian motion. According to the expected value premium principle, the pure premium rate of the insurer is c=1+θa with safety loading θ>0, and the reinsurance premium is paid at the constant rate c1=1+ηa with safety loading η>θ>0. Suppose that the insurer purchases the proportional reinsurance to transfer the underlying risk. For each t∈0,T, the value of risk exposure is denoted by qt∈0,+∞ representing the retention level of reinsurance. When qt∈0,1, it corresponds to a proportional reinsurance cover. Let Y denote the total claim and R:R+⟶R+ denote the reinsurance function. Then, RY=1−qtY, where 1−qt represents the proportion reinsured. The dynamics for the surplus process Rt associated with reinsurance strategy qt is given by(2)dRt=cdt−qtdCt−1−qtc1dt=θ−1−qtηadt+bqtdW1t.


### 2.2. Financial Market

In addition to the reinsurance, we assume that the insurer is allowed to invest its surplus in a financial market consisting of a risk-free asset (i.e., bond) and a risky asset (i.e., stock). The stochastic interest ratert follows the Vasicek model (see [26]).(3)drt=α−λrtdt+βdW0t,r0=r0,where the coefficients α,λ,andβ are positive real constants and W0t is a standard Brownian motion which is independent of W1t.LetS0t denote the price process of the bond, which evolves according to(4)dS0tS0t=rtdt,S00=s0,where r(t) satisfies equation (3).LetSt denote the price process of the risky asset, which follows(5)dStSt=rt+utdt+σdWt,S0=s,where ut>0 is a positive real-valued function, the constant σ>0 denotes the volatility rate of the risk asset, and Wt is another standard Brownian motion, which is independent with W1t, and Wt, W0t satisfy EdW0tWt=ρdt, where ρ∈−1,1 is the correlation coefficient.


### 2.3. Wealth Process

LetXt represent the wealth of the insurer at time t with initial value X0=x0 and πt be the amount of the wealth invested in the risky assets; then, the remainder Xt−πt is invested in the risk-free assets at time t. Since the insurer is allowed to buy reinsurance and invest in the financial market, the trading strategy is a pair of dynamic process which is denoted by Π:=qt,πt, where qt represents the reinsurance strategy and πt denotes the investment strategy. Adopting the reinsurance-investment strategy Π, the corresponding reserve Xt of the insurer is described by(6)dXt=dRt+πtdStSt+Xt−πtdS0tS0t=rtXt+utπt+θa−1−qtηadt+σπtdWt+bqtdW1t.


### 2.4. Optimization Criterion

We focus on maximizing the utility of the insurer’s terminal wealth(7)maxπ,qEUXT,where the utility function U⋅ is typically increasing and concave with constraints (3) and (6). For an admissible strategy Π, the value function HΠt,r,x from state r,x at time t is defined by(8)HΠt,r,x=EUXT|Xt=x,rt=r,and the objective function is(9)Ht,r,x=supπ,qEUXT|Xt=x,rt=r,with boundary condition HT,r,x=Ux. The insurer aims to find a pair of strategy q∗t,π∗t such that HΠ∗t,r,x=Ht,r,x, where q∗t is called the optimal reinsurance strategy and π∗t is called the optimal investment strategy.


## 2.1. Surplus Process

Typically, three types of models are used in the insurance market: the Cramer–Lundberg model, approximating diffusion model, and jump-diffusion model. We adopt the diffusion model to describe the surplus for the insurers. The claim processC is described as(1)dCt=adt−bdW1t,where a and b are positive constants and W1t is a one-dimensional standard Brownian motion. According to the expected value premium principle, the pure premium rate of the insurer is c=1+θa with safety loading θ>0, and the reinsurance premium is paid at the constant rate c1=1+ηa with safety loading η>θ>0. Suppose that the insurer purchases the proportional reinsurance to transfer the underlying risk. For each t∈0,T, the value of risk exposure is denoted by qt∈0,+∞ representing the retention level of reinsurance. When qt∈0,1, it corresponds to a proportional reinsurance cover. Let Y denote the total claim and R:R+⟶R+ denote the reinsurance function. Then, RY=1−qtY, where 1−qt represents the proportion reinsured. The dynamics for the surplus process Rt associated with reinsurance strategy qt is given by(2)dRt=cdt−qtdCt−1−qtc1dt=θ−1−qtηadt+bqtdW1t.


## 2.2. Financial Market

In addition to the reinsurance, we assume that the insurer is allowed to invest its surplus in a financial market consisting of a risk-free asset (i.e., bond) and a risky asset (i.e., stock). The stochastic interest ratert follows the Vasicek model (see [26]).(3)drt=α−λrtdt+βdW0t,r0=r0,where the coefficients α,λ,andβ are positive real constants and W0t is a standard Brownian motion which is independent of W1t.LetS0t denote the price process of the bond, which evolves according to(4)dS0tS0t=rtdt,S00=s0,where r(t) satisfies equation (3).LetSt denote the price process of the risky asset, which follows(5)dStSt=rt+utdt+σdWt,S0=s,where ut>0 is a positive real-valued function, the constant σ>0 denotes the volatility rate of the risk asset, and Wt is another standard Brownian motion, which is independent with W1t, and Wt, W0t satisfy EdW0tWt=ρdt, where ρ∈−1,1 is the correlation coefficient.


## 2.3. Wealth Process

LetXt represent the wealth of the insurer at time t with initial value X0=x0 and πt be the amount of the wealth invested in the risky assets; then, the remainder Xt−πt is invested in the risk-free assets at time t. Since the insurer is allowed to buy reinsurance and invest in the financial market, the trading strategy is a pair of dynamic process which is denoted by Π:=qt,πt, where qt represents the reinsurance strategy and πt denotes the investment strategy. Adopting the reinsurance-investment strategy Π, the corresponding reserve Xt of the insurer is described by(6)dXt=dRt+πtdStSt+Xt−πtdS0tS0t=rtXt+utπt+θa−1−qtηadt+σπtdWt+bqtdW1t.


## 2.4. Optimization Criterion

We focus on maximizing the utility of the insurer’s terminal wealth(7)maxπ,qEUXT,where the utility function U⋅ is typically increasing and concave with constraints (3) and (6). For an admissible strategy Π, the value function HΠt,r,x from state r,x at time t is defined by(8)HΠt,r,x=EUXT|Xt=x,rt=r,and the objective function is(9)Ht,r,x=supπ,qEUXT|Xt=x,rt=r,with boundary condition HT,r,x=Ux. The insurer aims to find a pair of strategy q∗t,π∗t such that HΠ∗t,r,x=Ht,r,x, where q∗t is called the optimal reinsurance strategy and π∗t is called the optimal investment strategy.


## 3. Optimal Reinsurance-Investment Strategy

To solve optimal problem (7), we apply the dynamic programming approach described in Fleming and Soner [27]. Because of the value function Ht,r,x, its partial derivatives Ht,Hr,Hx,Hrr,Hxx, and Hxr are continuous on R+1×R1, and then Ht,r,x satisfies the following Hamilton–Jacobi–Bellman (HJB) equation:(10)Ht+α−λrtHr+rtx+θ−ηaHx+12β2Hrr+supπt>0utπtHx+12σ2π2tHxx+ρβσπtHrx+sup0≤qt≤1ηaqtHx+12b2q2tHxx=0,for t,x∈0,T×R with boundary condition HT,r,x=Ux, where H=Ht,r,x.Differentiating equation (10) with respect to qt and πt and setting their derivatives equal to zero, we have(11)utHx+σ2πtHxx+ρβσHrx=0,ηaHx+b2qtHxx=0.Using the first-order maximizing conditions forqt,πt yields(12)π∗t=−utσ2HxHxx−ρβσHrxHxx,(13)q∗t=−ηab2HxHxx.Note thatq(t) > 0. If qt≤1, then q∗t coincides with equation (13). If q(t) > 1, then we can let qt=1 which means that the proportion of reinsurance is zero. We only consider the case qt≤1.Substituting equations (12) and (13) into the left side of equation (10), we obtain(14)Ht+α−λrtHr+rtx+θ−ηaHx+12β2Hrr−12η2a2b2Hx2Hxx−ρβutσHxHrxHxx−12β2ρ2Hrx2Hxx−12u2tσ2Hx2Hxx=0.Now, the above stochastic control problem has been transformed into solving a partial differential equation for the value functionHt,r,x. In the next step, we shall find the solution to equation (14) with boundary condition HT,r,x=Ux.Definition 1.
(see [23]). Let H:R⟶R be a convex function. For z>0, define the Legendre transform(15)Lz=supx>0Hx−zx.
The functionLz is called the Legendre dual of function Hx.Following the works of Xiao et al. [23], we define a Legendre transform(16)H^t,r,z=supx>0Ht,r,x−zx,(17)gt,r,z=infx>0x|Ht,r,x≥zx+H^t,r,z,where z>0 denotes the dual variable to x. The function H^t,r,z is related to gt,r,z,(18)gt,r,z=−H^zt,r,z.Noting thatHT,r,x=Ux at terminal time T, we have(19)H^T,r,z=supx>0Ux−zx,gT,r,z=infx>0x|Ux≥zx+H^T,r,z,from which we have(20)gT,r,z=U′−1z.Equation (20) implies that gT,r,z is the inverse of marginal utility. From equation (16), we have Hxt,r,x=z, and(21)gt,r,z=x,H^t,r,z=Ht,r,g−zg.Referring to Jonsson and Sircar [22], we have the following transformation rules:(22)Ht=H^t,Hr=H^r,Hxx=−1H^zz,Hrx=−H^rzH^zz,Hrr=H^rr−H^rz2H^zz,where H^=H^t,r,z.Lettingρ2=1 and putting (22) into equation (14), we have(23)H^t+α−λrtH^r+rtx+θ−ηaz+12η2a2b2+u2tσ2z2H^zz−ρβutσzH^rz+12β2H^rr=0.Differentiating equation (23) with respect to z gives the following equation:(24)gt+α−λrtgr−rtg−rtzgz−θ−ηa+K1tz2gzz+2zgz−ρβutσgr+zgrz+12β2grr=0,where K1t=1/2η2a2/b2+u2t/σ2, and the boundary condition gT,r,z=U′−1z.Note that we have transformed the nonlinear partial differential equation (14) into a linear second-order partial differential equation (24). In the following sections, we provide the explicit solutions for equation (14) under CRRA and CARA utilities by the variable change method.


### 3.1. Power Utility

Assume that the insurer takes the power utility function (CRRA)(25)Ux=xpp,p<1.According to equation (20), we have(26)gT,r,z=z1/p−1.We conjecture a solution to equation (26) with the form(27)gt,r,z=ft,rz1/p−1+ht,r,where fT,r and ht,r are suitable functions such that equation (27) is a solution of equation (24), and fT,r=1 and hT,r=0. The derivatives of gt,r,z with respect to the variables t, r, and z are(28)gt=ftt,rz1/p−1+htt,r,gr=frt,rz1/p−1+hrt,r,gz=1p−1ft,rz2−p/p−1,grr=frrt,rz1/p−1+hrrt,r,grz=1p−1frt,rz2−p/p−1,gzz=2−pp−12ft,rz3−2p/p−1.Putting the above derivatives back into equation (24) leads to an equation of ft,r and ht,r,(29)z1/p−1ftt,r+α−λrtfrt,r−rtft,r−ρβutσfrt,r+1p−1frt,r+12β2frrt,r−rtp−1ft,r+K1t2−pp−12ft,r+2p−1ft,r+htt,r+α−λrthrt,r+12β2hrrt,r−ρβutσhrt,r−rtht,r+η−θa=0.To solve equation (27), we decompose it into the following two equations:(30)ftt,r+α−λrt−ρβutσpp−1frt,r+pK1tp−12−prtp−1ft,r+12β2frrt,r=0,with boundary condition fT,r=1, and(31)htt,r+12β2hrrt,r+α−λrt−ρβutσhrt,r−rtht,r+η−θa=0,with boundary condition hT,r=0.Lemma 1.
If a solution of equation (27) is in the form(32)ft,r=AteBtr,with the boundary conditions AT=1andBT=0, then At and Bt are given by(33)At=exp∫tT12β2B2s+α−ρβusσpp−1Bs+pK1sp−12ds,(34)Bt=pλp−1e−λT−t−1.Proof.
Plugging solution (32) into equation (30), we obtain(35)erBtA′t+rAtB′t+12β2AtB2t+α−ρβutσpp−1AtBt+pK1tp−12At−λrAtBt−pp−1rAt=0,where A′tandB′t denote the derivatives with respect to t. In order to eliminate the dependence on r, we decompose equation (35) into the following two equations:(36)rAtB′t−λBt−pp−1=0,(37)A′tAt+12β2B2t+α−ρβutσpp−1Bt+pK1tp−12=0.
Solving the ordinary differential equation (36) with boundary condition B(T) = 0, we obtain equation (34). For equation (37) with A(T) = 1, the solution is given by equation (33).Lemma 2.
If a solution of equation (31) is of the structure(38)ht,r=η−θa∫tTh˜s,rds,then h˜t,r satisfies the following equation:(39)h˜tt,r+12β2h˜rrt,r+α−λrt−ρβutσh˜rt,r−rth˜t,r=0,with the boundary condition h˜T,r=1.Proof.
We define the variational operator∇ on ht,r by(40)∇ht,r=−rtht,r+α−λrt−βutσhrt,r+12β2hrrt,r.
Then, equation (31) is rewritten in the form(41)∂ht,r∂t+∇ht,r+η−θa=0,hT,r=0.
Considering(42)ht,r=η−θa∫tTh˜s,rds,we derive(43)∂ht,r∂t=η−θa∫tT∂h˜s,r∂sds−h˜T,r,(44)∇ht,r=η−θa∫tT∇h˜s,rds.
Substituting equations (43) and (44) into (41), we get(45)η−θa∫tT∂h˜s,r∂s+∇h˜s,rds−h˜T,r+1=0.
Therefore, we obtain(46)∂h˜s,r∂s+∇h˜s,r=0,h˜T,r=1,which completes the proof.Lemma 3.
Assume that(47)h˜t,r=DteEtr,is a solution of equation (39), with boundary conditions D(T) = 1 and E(T) = 0. Then, D(t) and E(t) are given by(48)Dt=exp∫tT12β2E2s+α−ρβusσEsds,(49)Et=1λe−λT−t−1.Proof.
Putting equation (47) into (39) yields(50)DteEtrD′tDt+rE′t+α−λr−ρβutσEt+12β2E2t−r=0.
Eliminating the dependence onr, we decompose equation (50) into the following two equations:(51)rE′t−λEt−1=0,ET=0,(52)D′tDt+α−ρβutσEt+12β2E2t=0,DT=1.
Using the same approach as that of solving equation (36), the solution to equation (51) with E(T) = 0 is given by equation (49). For equation (52) with D(T) = 1, we obtain equation (48).
Note that(53)HxHxx=zgzt,r,z=1p−1x−ht,r,(54)HrxHxx=−grt,r,z=−Btx−ht,r−hrt,r.
Substituting equations (53) and (54) into trading strategies (12) and (13), we get the following theorem.Theorem 1.
Letρ2=1, and assume that the utility is given by a power utility function (25) for the optimal investment-reinsurance problem (7). There exists a solution gt,r,z to the dual Hamilton–Jacobi–Bellman equation (24) with boundary condition gT,r,z=z1/p−1. The corresponding optimal investment πp∗t and proportional strategy qp∗t of problem (7) are given by(55)πp∗t=−utσ2HxHxx−ρβσHrxHxx=utσ211−px−ht,r+ρβσBtx−ht,r+hrt,r,qp∗t=−ηab2HxHxx=ηab211−px−ht,r,where ht,r=η−θa∫tTh˜s,rds is given in Lemmas 2 and 3.


### 3.2. Exponential Utility

Assume that the insurer takes an exponential utility function (CARA)(56)Ux=−1γe−γx,γ>0,where γ represents the absolute risk aversion coefficient. The exponential utility function (56) plays a prominent role in insurance mathematics and actuarial practice.According to terminal condition (20), we have(57)gT,r,z=−lnzγ.We conjecture a solution to equation (24) with the form(58)gt,r,z=−1γkt,rlnz+vt,r+wt,r,with boundary conditions given by kT,r=1,vT,r=0,andwT,r=0.A direct calculation yields the partial derivatives(59)gtt,r,z=−ktt,rγlnz+vt,r−kt,rγvtt,r+wtt,r,grt,r,z=−krt,rγlnz+vt,r−kt,rγvrt,r+wrt,r,gzt,r,z=−kt,rγ1z,gzzt,r,z=kt,rγ1z2,grzt,r,z=−krt,rγ1z,grrt,r,z=−krrt,rγlnz+vt,r−2krt,rvrt,r+kt,rvrrt,rγ+wrrt,r.Introducing the above derivatives back into equation (24), we derive that(60)−lnzγktt,r+α−λrtkrt,r−rtkt,r+12β2krrt,r−ρβutσkrt,r+wtt,r+α−λrtwrt,r−rtwt,r+12β2wrrt,r−ρβutσwrt,r+η−θa−kt,rγvt,rktt,rk+α−λrtvt,rkrt,rk+α−λrtvrt,r−rtvt,r+12β2vt,rkrrt,rk+β2vrt,rkrt,rk−ρβutσvt,rkrt,rk+vrt,r+krt,rk+vtt,r−rt+K1t+12β2vrrt,r=0.Equation (60) is split into the following equations:(61)ktt,r−rtkt,r+α−λrt−ρβutσkrt,r+12β2krrt,r=0,with boundary condition kT,r=1,(62)wtt,r−rtwt,r+α−λrt−ρβutσwrt,r+12β2wrrt,r+η−θa=0,with boundary condition wT,r=0, and(63)vt,rktt,rkt,r+vtt,r−rtvt,r+α−λrt−ρβutσvt,rkrt,rkt,r+α−λrt−ρβutσvrt,r+12β2vt,rkrrt,rkt,r+β2vrt,rkrt,rkt,r−rt+12β2vrrt,r−ρβutσkrt,rkt,r+K1t=0,with boundary condition vT,r=0.Lemma 4.
Assume that a solution of equation (61) is in the form(64)kt,r=eA˜t+B˜tr,with the boundary conditions A˜T=0andB˜T=0. Then, A˜t and B˜t are given by(65)A˜t=∫tTα−ρβusσB˜s+β22B˜2sds,(66)B˜t=1λe−λT−t−1.Proof.
Putting solution (64) into equation (61) yields(67)kt,rA˜′t+rB˜′t−r+α−λr−ρβutσB˜t+β22B˜2t=0.
We separate equation (67) into two equations:(68)A˜′t+α−ρβutσB˜t+β22B˜2t=0,(69)rB˜′t−λB˜t−1=0.
Solving equation (69) with B˜T=0, we obtain equation (66). For equation (68) with A˜T=0, we have equation (67).Lemma 5.
Assume that a solution of equation (62) takes the structure(70)wt,r=η−θa∫tTh˜s,rds.
Then,h˜t,r satisfies equation (39) in Lemma 2.Proof.
Observe that equation (62) has the same solution with equation (31), i.e., wt,r=ht,r. The proof is the same as that of Lemmas 2 and 3; we omit its proof.Lemma 6.
Assume that a solution of equation (63) is in the form(71)vt,r=D˜t+E˜tr,with the boundary conditions D˜T=0andE˜T=0. Then, D˜t and E˜t are given by(72)D˜t=∫tTα−ρβusσ+β2B˜sE˜s+K1s−ρβusσB˜sds,(73)E˜t=1λe−λT−t−1.Proof.
From equation (72), we have krt,r=B˜tkt,r.
Introducing equation (60) and kr(t, r), we simplify equation (61) in the form(74)vtt,r−rt+α−λrt−ρβutσ+β2B˜tvrt,r+12β2vrrt,r−ρβutσB˜t+K1t=0.
Substituting solution (71) into equation (74) yields(75)D˜′t+α−ρβutσ+β2B˜tE˜t+rE˜′t−λE˜t−1+K1t−ρβutσB˜t=0.
Splitting equation (75) into two equations, we have(76)D˜′t+α−ρβutσ+β2B˜tE˜t+K1t−ρβutσB˜t=0,(77)rE˜′t−λE˜t−1=0.
Taking into account the boundary conditionE˜T=0, the solution to equation (77) is given by equation (64). Solving equation (76) with D˜T=0, we obtain equation (63).
Note that(78)HxHxx=zgzt,r,z=−1γkt,r,(79)HrxHxx=−grt,r,z=−B˜tx−wt,r+1γE˜tkt,r−wrt,r.
Substituting equations (78) and (79) into trading strategies (14) and (15), we get the following theorem.Theorem 2.
Letρ2=1, and assume that the utility is given by an exponential utility function (56) for the optimal investment-reinsurance problem (7). There exists a solution gt,r,z to the dual Hamilton–Jacobi–Bellman equation (24) with boundary condition gt,r,z−lnz/γ. The corresponding optimal investment πe∗t and proportional strategy qe∗t of problem (7) are given by(80)πe∗t=−utσ2HxHxx−ρβσHrxHxx=utσ21γkt,r+ρβσB˜tx−wt,r−1γE˜tkt,r+wrt,r,(81)qe∗t=−ηab2HxHxx=ηab21γkt,r,where kt,randwt,r are given in Lemmas 4 and 5.


## 3.1. Power Utility

Assume that the insurer takes the power utility function (CRRA)(25)Ux=xpp,p<1.According to equation (20), we have(26)gT,r,z=z1/p−1.We conjecture a solution to equation (26) with the form(27)gt,r,z=ft,rz1/p−1+ht,r,where fT,r and ht,r are suitable functions such that equation (27) is a solution of equation (24), and fT,r=1 and hT,r=0. The derivatives of gt,r,z with respect to the variables t, r, and z are(28)gt=ftt,rz1/p−1+htt,r,gr=frt,rz1/p−1+hrt,r,gz=1p−1ft,rz2−p/p−1,grr=frrt,rz1/p−1+hrrt,r,grz=1p−1frt,rz2−p/p−1,gzz=2−pp−12ft,rz3−2p/p−1.Putting the above derivatives back into equation (24) leads to an equation of ft,r and ht,r,(29)z1/p−1ftt,r+α−λrtfrt,r−rtft,r−ρβutσfrt,r+1p−1frt,r+12β2frrt,r−rtp−1ft,r+K1t2−pp−12ft,r+2p−1ft,r+htt,r+α−λrthrt,r+12β2hrrt,r−ρβutσhrt,r−rtht,r+η−θa=0.To solve equation (27), we decompose it into the following two equations:(30)ftt,r+α−λrt−ρβutσpp−1frt,r+pK1tp−12−prtp−1ft,r+12β2frrt,r=0,with boundary condition fT,r=1, and(31)htt,r+12β2hrrt,r+α−λrt−ρβutσhrt,r−rtht,r+η−θa=0,with boundary condition hT,r=0.Lemma 1.
If a solution of equation (27) is in the form(32)ft,r=AteBtr,with the boundary conditions AT=1andBT=0, then At and Bt are given by(33)At=exp∫tT12β2B2s+α−ρβusσpp−1Bs+pK1sp−12ds,(34)Bt=pλp−1e−λT−t−1.Proof.
Plugging solution (32) into equation (30), we obtain(35)erBtA′t+rAtB′t+12β2AtB2t+α−ρβutσpp−1AtBt+pK1tp−12At−λrAtBt−pp−1rAt=0,where A′tandB′t denote the derivatives with respect to t. In order to eliminate the dependence on r, we decompose equation (35) into the following two equations:(36)rAtB′t−λBt−pp−1=0,(37)A′tAt+12β2B2t+α−ρβutσpp−1Bt+pK1tp−12=0.
Solving the ordinary differential equation (36) with boundary condition B(T) = 0, we obtain equation (34). For equation (37) with A(T) = 1, the solution is given by equation (33).Lemma 2.
If a solution of equation (31) is of the structure(38)ht,r=η−θa∫tTh˜s,rds,then h˜t,r satisfies the following equation:(39)h˜tt,r+12β2h˜rrt,r+α−λrt−ρβutσh˜rt,r−rth˜t,r=0,with the boundary condition h˜T,r=1.Proof.
We define the variational operator∇ on ht,r by(40)∇ht,r=−rtht,r+α−λrt−βutσhrt,r+12β2hrrt,r.
Then, equation (31) is rewritten in the form(41)∂ht,r∂t+∇ht,r+η−θa=0,hT,r=0.
Considering(42)ht,r=η−θa∫tTh˜s,rds,we derive(43)∂ht,r∂t=η−θa∫tT∂h˜s,r∂sds−h˜T,r,(44)∇ht,r=η−θa∫tT∇h˜s,rds.
Substituting equations (43) and (44) into (41), we get(45)η−θa∫tT∂h˜s,r∂s+∇h˜s,rds−h˜T,r+1=0.
Therefore, we obtain(46)∂h˜s,r∂s+∇h˜s,r=0,h˜T,r=1,which completes the proof.Lemma 3.
Assume that(47)h˜t,r=DteEtr,is a solution of equation (39), with boundary conditions D(T) = 1 and E(T) = 0. Then, D(t) and E(t) are given by(48)Dt=exp∫tT12β2E2s+α−ρβusσEsds,(49)Et=1λe−λT−t−1.Proof.
Putting equation (47) into (39) yields(50)DteEtrD′tDt+rE′t+α−λr−ρβutσEt+12β2E2t−r=0.
Eliminating the dependence onr, we decompose equation (50) into the following two equations:(51)rE′t−λEt−1=0,ET=0,(52)D′tDt+α−ρβutσEt+12β2E2t=0,DT=1.
Using the same approach as that of solving equation (36), the solution to equation (51) with E(T) = 0 is given by equation (49). For equation (52) with D(T) = 1, we obtain equation (48).
Note that(53)HxHxx=zgzt,r,z=1p−1x−ht,r,(54)HrxHxx=−grt,r,z=−Btx−ht,r−hrt,r.
Substituting equations (53) and (54) into trading strategies (12) and (13), we get the following theorem.Theorem 1.
Letρ2=1, and assume that the utility is given by a power utility function (25) for the optimal investment-reinsurance problem (7). There exists a solution gt,r,z to the dual Hamilton–Jacobi–Bellman equation (24) with boundary condition gT,r,z=z1/p−1. The corresponding optimal investment πp∗t and proportional strategy qp∗t of problem (7) are given by(55)πp∗t=−utσ2HxHxx−ρβσHrxHxx=utσ211−px−ht,r+ρβσBtx−ht,r+hrt,r,qp∗t=−ηab2HxHxx=ηab211−px−ht,r,where ht,r=η−θa∫tTh˜s,rds is given in Lemmas 2 and 3.


## 3.2. Exponential Utility

Assume that the insurer takes an exponential utility function (CARA)(56)Ux=−1γe−γx,γ>0,where γ represents the absolute risk aversion coefficient. The exponential utility function (56) plays a prominent role in insurance mathematics and actuarial practice.According to terminal condition (20), we have(57)gT,r,z=−lnzγ.We conjecture a solution to equation (24) with the form(58)gt,r,z=−1γkt,rlnz+vt,r+wt,r,with boundary conditions given by kT,r=1,vT,r=0,andwT,r=0.A direct calculation yields the partial derivatives(59)gtt,r,z=−ktt,rγlnz+vt,r−kt,rγvtt,r+wtt,r,grt,r,z=−krt,rγlnz+vt,r−kt,rγvrt,r+wrt,r,gzt,r,z=−kt,rγ1z,gzzt,r,z=kt,rγ1z2,grzt,r,z=−krt,rγ1z,grrt,r,z=−krrt,rγlnz+vt,r−2krt,rvrt,r+kt,rvrrt,rγ+wrrt,r.Introducing the above derivatives back into equation (24), we derive that(60)−lnzγktt,r+α−λrtkrt,r−rtkt,r+12β2krrt,r−ρβutσkrt,r+wtt,r+α−λrtwrt,r−rtwt,r+12β2wrrt,r−ρβutσwrt,r+η−θa−kt,rγvt,rktt,rk+α−λrtvt,rkrt,rk+α−λrtvrt,r−rtvt,r+12β2vt,rkrrt,rk+β2vrt,rkrt,rk−ρβutσvt,rkrt,rk+vrt,r+krt,rk+vtt,r−rt+K1t+12β2vrrt,r=0.Equation (60) is split into the following equations:(61)ktt,r−rtkt,r+α−λrt−ρβutσkrt,r+12β2krrt,r=0,with boundary condition kT,r=1,(62)wtt,r−rtwt,r+α−λrt−ρβutσwrt,r+12β2wrrt,r+η−θa=0,with boundary condition wT,r=0, and(63)vt,rktt,rkt,r+vtt,r−rtvt,r+α−λrt−ρβutσvt,rkrt,rkt,r+α−λrt−ρβutσvrt,r+12β2vt,rkrrt,rkt,r+β2vrt,rkrt,rkt,r−rt+12β2vrrt,r−ρβutσkrt,rkt,r+K1t=0,with boundary condition vT,r=0.Lemma 4.
Assume that a solution of equation (61) is in the form(64)kt,r=eA˜t+B˜tr,with the boundary conditions A˜T=0andB˜T=0. Then, A˜t and B˜t are given by(65)A˜t=∫tTα−ρβusσB˜s+β22B˜2sds,(66)B˜t=1λe−λT−t−1.Proof.
Putting solution (64) into equation (61) yields(67)kt,rA˜′t+rB˜′t−r+α−λr−ρβutσB˜t+β22B˜2t=0.
We separate equation (67) into two equations:(68)A˜′t+α−ρβutσB˜t+β22B˜2t=0,(69)rB˜′t−λB˜t−1=0.
Solving equation (69) with B˜T=0, we obtain equation (66). For equation (68) with A˜T=0, we have equation (67).Lemma 5.
Assume that a solution of equation (62) takes the structure(70)wt,r=η−θa∫tTh˜s,rds.
Then,h˜t,r satisfies equation (39) in Lemma 2.Proof.
Observe that equation (62) has the same solution with equation (31), i.e., wt,r=ht,r. The proof is the same as that of Lemmas 2 and 3; we omit its proof.Lemma 6.
Assume that a solution of equation (63) is in the form(71)vt,r=D˜t+E˜tr,with the boundary conditions D˜T=0andE˜T=0. Then, D˜t and E˜t are given by(72)D˜t=∫tTα−ρβusσ+β2B˜sE˜s+K1s−ρβusσB˜sds,(73)E˜t=1λe−λT−t−1.Proof.
From equation (72), we have krt,r=B˜tkt,r.
Introducing equation (60) and kr(t, r), we simplify equation (61) in the form(74)vtt,r−rt+α−λrt−ρβutσ+β2B˜tvrt,r+12β2vrrt,r−ρβutσB˜t+K1t=0.
Substituting solution (71) into equation (74) yields(75)D˜′t+α−ρβutσ+β2B˜tE˜t+rE˜′t−λE˜t−1+K1t−ρβutσB˜t=0.
Splitting equation (75) into two equations, we have(76)D˜′t+α−ρβutσ+β2B˜tE˜t+K1t−ρβutσB˜t=0,(77)rE˜′t−λE˜t−1=0.
Taking into account the boundary conditionE˜T=0, the solution to equation (77) is given by equation (64). Solving equation (76) with D˜T=0, we obtain equation (63).
Note that(78)HxHxx=zgzt,r,z=−1γkt,r,(79)HrxHxx=−grt,r,z=−B˜tx−wt,r+1γE˜tkt,r−wrt,r.
Substituting equations (78) and (79) into trading strategies (14) and (15), we get the following theorem.Theorem 2.
Letρ2=1, and assume that the utility is given by an exponential utility function (56) for the optimal investment-reinsurance problem (7). There exists a solution gt,r,z to the dual Hamilton–Jacobi–Bellman equation (24) with boundary condition gt,r,z−lnz/γ. The corresponding optimal investment πe∗t and proportional strategy qe∗t of problem (7) are given by(80)πe∗t=−utσ2HxHxx−ρβσHrxHxx=utσ21γkt,r+ρβσB˜tx−wt,r−1γE˜tkt,r+wrt,r,(81)qe∗t=−ηab2HxHxx=ηab21γkt,r,where kt,randwt,r are given in Lemmas 4 and 5.


## 4. Sensitivity Analyses and Numerical Illustrations

In this section, we analyze the effects of market parameters on the optimal reinsurance-investment strategy, especially the parameters of interest rate and CARA and CRRA utilities, and provide several numerical simulations to illustrate our results. Throughout the numerical analyses, unless otherwise stated, the basic parameters are given byρ=1, a=0.6, b=0.4, θ=0.2, η=0.25, α=0.3,λ=1,σ=0.8,t=0,T=5,andx=1.


### 4.1. Sensitivity Analyses on the Optimal Investment Strategy

From equation (3), we know that the parameter β represents the volatility of short interest rate. It means that the bigger the value of β is, the bigger the volatility resulted from interest rate is. The effect of β on the optimal investment strategy under the CRRA utility is shown in Figure 1, from which we see that πp∗t decreases with the parameter β in the case of ρ=1. Due to the positive correlation between interest rate and stock price dynamics, the volatility of stock price will become larger. It implies that the underlying risk becomes larger when the risk of interest rate becomes larger. Therefore, in order to avoid risks, the investor will decrease the investment in the stocks.Figure 1
The effects ofβ on πp∗t.Figure2 illustrates that the optimal investment strategy under the CARA utility first declines slightly and then increases with β. When interest rate fluctuations are small, the insurer will not change their holdings of risk assets too much. However, the insurer will increase risky investment while interest rate fluctuations become larger.Figure 2
The effects ofβ on πe∗t.For the CRRA utility, the absolute risk aversion coefficientACRRA=−U″x/U′x=1−p/x, which implies that the risk aversion level of the investor decreases as p increases. Therefore, as it is illustrated in Figure 3, πp∗ increases as p becomes larger, and the insurer is willing to invest more money in the financial markets.Figure 3
The effects ofp on πp∗t.For the CARA utility, we obtain the absolute risk aversion coefficientACARA=−U″x/U′x=γ. Thus, πe∗ decreases with γ, which is shown in Figure 4. The larger γ is, the more risk averse the insurer will be and then will reduce the investment in risky assets.Figure 4
The effects ofγ on πe∗t.


### 4.2. Sensitivity Analyses on the Optimal Reinsurance Strategy

For the CRRA utility, the volatility of interest rateβ has a negative effect on the optimal reinsurance strategy πp∗ (see Figure 5). The larger β is, the less the insurer’s retention is. However, for the CARA utility, the effect of β on the optimal reinsurance strategy πe∗ is positive (see Figure 6). Accordingly, the insurer’s reinsurance strategy is influenced by the interest rate risk and the utility function.Figure 5
The effects ofβ on qp∗t.Figure 6
The effects ofβ on qe∗t.From equation (55), we derive that(82)∂qp∗∂p=ηab211−p2x−ht,r>0,which implies that qp∗t increases as p increases as it is shown in Figure 7. The larger p is, the smaller the absolute risk aversion coefficient is, and the insurer would like to take risks on their own and reduce the proportion of reinsurance.Figure 7
The effects ofp on qp∗t.From equation (81), we derive that(83)∂qe∗∂γ=−ηaγ2b2eA˜t+B˜tr<0,which implies that qe∗t decreases with the risk aversion coefficient γ for the CARA utility as it is shown in Figure 8. For larger γ, the insurer is more risk averse and expects to reduce retention and transfer risks.Figure 8
The effects ofγ on qe∗t.


## 4.1. Sensitivity Analyses on the Optimal Investment Strategy

From equation (3), we know that the parameter β represents the volatility of short interest rate. It means that the bigger the value of β is, the bigger the volatility resulted from interest rate is. The effect of β on the optimal investment strategy under the CRRA utility is shown in Figure 1, from which we see that πp∗t decreases with the parameter β in the case of ρ=1. Due to the positive correlation between interest rate and stock price dynamics, the volatility of stock price will become larger. It implies that the underlying risk becomes larger when the risk of interest rate becomes larger. Therefore, in order to avoid risks, the investor will decrease the investment in the stocks.Figure 1
The effects ofβ on πp∗t.Figure2 illustrates that the optimal investment strategy under the CARA utility first declines slightly and then increases with β. When interest rate fluctuations are small, the insurer will not change their holdings of risk assets too much. However, the insurer will increase risky investment while interest rate fluctuations become larger.Figure 2
The effects ofβ on πe∗t.For the CRRA utility, the absolute risk aversion coefficientACRRA=−U″x/U′x=1−p/x, which implies that the risk aversion level of the investor decreases as p increases. Therefore, as it is illustrated in Figure 3, πp∗ increases as p becomes larger, and the insurer is willing to invest more money in the financial markets.Figure 3
The effects ofp on πp∗t.For the CARA utility, we obtain the absolute risk aversion coefficientACARA=−U″x/U′x=γ. Thus, πe∗ decreases with γ, which is shown in Figure 4. The larger γ is, the more risk averse the insurer will be and then will reduce the investment in risky assets.Figure 4
The effects ofγ on πe∗t.


## 4.2. Sensitivity Analyses on the Optimal Reinsurance Strategy

For the CRRA utility, the volatility of interest rateβ has a negative effect on the optimal reinsurance strategy πp∗ (see Figure 5). The larger β is, the less the insurer’s retention is. However, for the CARA utility, the effect of β on the optimal reinsurance strategy πe∗ is positive (see Figure 6). Accordingly, the insurer’s reinsurance strategy is influenced by the interest rate risk and the utility function.Figure 5
The effects ofβ on qp∗t.Figure 6
The effects ofβ on qe∗t.From equation (55), we derive that(82)∂qp∗∂p=ηab211−p2x−ht,r>0,which implies that qp∗t increases as p increases as it is shown in Figure 7. The larger p is, the smaller the absolute risk aversion coefficient is, and the insurer would like to take risks on their own and reduce the proportion of reinsurance.Figure 7
The effects ofp on qp∗t.From equation (81), we derive that(83)∂qe∗∂γ=−ηaγ2b2eA˜t+B˜tr<0,which implies that qe∗t decreases with the risk aversion coefficient γ for the CARA utility as it is shown in Figure 8. For larger γ, the insurer is more risk averse and expects to reduce retention and transfer risks.Figure 8
The effects ofγ on qe∗t.


## 5. Conclusion

This paper investigates the investment-reinsurance problem with stochastic interest rate, in which interest rate is assumed to follow the Vasicek model and be correlated with stock price. The optimal reinsurance-investment strategies for CRRA and CARA utilities are derived by applying the stochastic dynamic programming and Legendre transformation. Through several sensitive analyses of the market parameters, we find that the optimal reinsurance strategy is not only affected by the parameters of reinsurance but also related to the risk preference coefficient and interest rate fluctuation, and the optimal investment strategy is influenced by both financial market and insurance market. For further research, considering the chaos dynamics in the financial market (Vaidyanathan et al. [28] and Sukono et al. [29]), it might be an interesting attempt to study the investment strategy combining the fractional-order financial risk chaotic system.



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