A Perturbation Approach to Optimal Investment, Liability Ratio, and Dividend Strategies
aa r X i v : . [ q -f i n . M F ] D ec A Perturbation Approach toOptimal Investment, Liability Ratio, and Dividend Strategies
Zhuo Jin ∗ Zuo Quan Xu † Bin Zou ‡ This Version: December 6, 2020
Abstract
We study an optimal dividend problem for an insurer who simultaneously controls investmentweights in a financial market, liability ratio in the insurance business, and dividend payout rate.The insurer seeks an optimal strategy to maximize her expected utility of dividend paymentsover an infinite horizon. By applying a perturbation approach, we obtain the optimal strategyand the value function in closed form for log and power utility. We conduct an economic analysisto investigate the impact of various model parameters and risk aversion on the insurer’s optimalstrategy.
Keywords : Jump Diffusion; Optimal Dividend; Reinsurance; Stochastic Control
Stochastic control has enjoyed great success in actuarial science since 1990s; see, e.g., Browne (1995)and Asmussen and Taksar (1997) for early contributions and the monograph Schmidli (2007) for a moresystematic overview. Of particular interest in the application of stochastic control to actuarial problemsis the study of optimal dividend strategies for an insurer. In this paper, we consider an insurer whocontrols not only dividend payments but also investment and liability strategies simultaneously, and applya perturbation approach to solve such an optimization problem under the utility maximization criterion.The optimal dividend problem is a well-studied topic that, generally speaking, seeks an optimal dividendstrategy so as to optimize the insurer’s business objective. In a seminal paper, De Finetti (1957) models ∗ Centre for Actuarial Studies, Department of Economics, University of Melbourne, Australia. Email: [email protected] author is partially supported by the Research Grants Council of the Hong Kong Special Administrative Region (No.17330816). † Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China. Email: [email protected] author is partially supported by by the National Natural Science Foundation of China (No. 11971409) and Hong KongGRF (No. 15204216 and No. 15202817). ‡ Corresponding author. Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009, Storrs, Con-necticut 06269-1009, USA. Email: [email protected]. Phone: +1-860-486-3921. This author is partially supported by astart-up grant from the University of Connecticut. R under the classical Crem´er-Lundberg (CL) setting and considers an insurerwho seeks to maximize the expected discounted dividend payments until the ruin time τ :max Z τ e − δt d e D t , (1.1)where δ > e D = ( e D t ) t ≥ denotes the insurer’s dividend strategy, with e D t representing the cumulative dividend payments up to time t . Note that the ruin time τ in (1.1) dependson the dividend strategy e D ; let us denote X = X e D the insurer’s controlled surplus (wealth) process, then τ = τ e D := inf { t : X e D ≤ } . The optimal dividend strategy e D ∗ to Problem (1.1) is often a band strategyand may further reduce to a barrier strategy in certain cases; see Avanzi (2009) for details. There is a richbody of literature on the optimal dividend problem by now; see Schmidli (2007)[Sections 2.4 and 2.5] fora standard textbook treatment. We refer readers to Albrecher and Thonhauser (2009) and Avanzi (2009)for more comprehensive surveys on this topic. In what follows, we conduct a selective literature review onthis problem by discussing various extensions to the work of De Finetti (1957) from four different angles:the risk model, the type of dividend strategies, additional control components and constraints, and theoptimization objective.First, we discuss extensions to De Finetti (1957) in the modeling of the insurer’s risk and (uncon-trolled) surplus processes. When it comes to the modeling of the insurer’s risk process R , a popular choiceis the diffusion model, which can be seen as a continuous approximation to the classical CL model. Pleasesee Jeanblanc-Picqu´e and Shiryaev (1995), Asmussen and Taksar (1997), Højgaard and Taksar (1999), andChoulli et al. (2003), among many others for investigations of optimal dividend under the diffusion riskmodel. Further generalizations to the standard diffusion model include incorporating mean-reverting (seeCadenillas et al. (2007) and Avanzi and Wong (2012)) and regime switching (see Sotomayor and Cadenillas(2011) and Jin et al. (2013)). Much of the attention in modeling goes to the risk process R , while lessis concerned about the premium process p = ( p t ) t ≥ . In most analyses, p t is a constant (see, e.g.,Taksar and Zhou (1998) and Azcue and Muler (2005)) or further given by a specific premium principle(see, e.g., Asmussen et al. (2000) for the expected value principle). But other studies apply a stochas-tic process (e.g., a diffusion or jump-diffusion process) to model the insurer’s surplus without dividend;see, e.g., Choulli et al. (2003), Gerber and Shiu (2004), Cadenillas et al. (2007), Chen et al. (2014), andZhu (2015). Opposite to the standard risk model is the so-called dual risk model, where p is now theexpense rate and R is interpreted as the positive gains. Optimal dividend in the dual model is studied inAvanzi et al. (2007), Avanzi and Gerber (2008), and Bayraktar et al. (2013, 2014), among others. In this setting, the risk process R = ( R t ) t ≥ is modeled by a compound Poisson process R t = P b N t i =1 Y i , where b N is ahomogeneous Poisson process and ( Y i ) i =1 , , ··· is a series of independently and identically distributed random variables, alsoindependent of b N . Here, b N models the claim frequency and Y i models the severity of the i -th claim. In this setting, the dynamics of the risk process R = ( R t ) t ≥ is given by d R t = α d t + β d W t , where α, β > W = ( W t ) t ≥ is a standard Brownian motion. Here, the parameters α and β can be selected to match the first two momentsof R under the CL model; see Browne (1995). e D = ( e D t ) t ≥ , where e D t is understood as the cumulativedividend payouts up to time t . In the first case, there exists an absolutely continuous and non-negative div-idend rate D = ( D t ) t ≥ such that d e D t = D t d t for all t ≥
0; see Case A in Jeanblanc-Picqu´e and Shiryaev(1995). The optimal dividend problem in this case is a classical control problem. Often an upper bound onthe dividend rate D is imposed (i.e., 0 ≤ D t ≤ m < ∞ ) and, in many cases, the optimal dividend strategyis equal to the upper bound if the controlled surplus is large enough, and 0 otherwise. The study of opti-mal dividend in this case can be found in Asmussen and Taksar (1997), Sotomayor and Cadenillas (2011),Jin et al. (2015), and Zhu (2015), among others. In the second case, the intervention (dividend payout) is not continuous and a dividend strategy consists of a pair of processes ( T i , D i ) i =1 , , ··· , where D i is the amount of the i -th dividend paid at time T i ; see Case B in Jeanblanc-Picqu´e and Shiryaev (1995). The controlchoice in this case is appropriate when paying dividends incurs transaction costs or taxes, and the corre-sponding problem is an impulse control problem. Please refer to Cadenillas et al. (2006), Cadenillas et al.(2007), Bai and Guo (2010), and Yao et al. (2011) for related investigations of this case. In the third case,a dividend strategy e D is a c`adl`ag process (right continuous with left limits) that is non-decreasing andnon-negative; see Case C in Jeanblanc-Picqu´e and Shiryaev (1995). The optimal dividend problem in thissetup often leads to a singular control problem. Jeanblanc-Picqu´e and Shiryaev (1995) show that whenthe upper bound m on the dividend rate goes to infinity in the first case or the fixed transaction costgoes to zero in the second case, the corresponding optimal dividend strategy converges to the one in thethird case. Please consult, e.g., Taksar and Zhou (1998), Azcue and Muler (2005), Albrecher et al. (2005),Bai and Guo (2010), and Sotomayor and Cadenillas (2011) for the related study in this case.Third, we discuss interesting features, mainly in the control components, that have been incorporatedinto the original model of De Finetti (1957). In De Finetti (1957), an insurer only controls the dividendpayout strategy, and neither invests the surplus in a financial market nor resorts to risk control strategies(e.g., reinsurance policies). Much of the following research includes investment and/or risk control strategiesinto the study of optimal dividend; see, e.g., Taksar and Zhou (1998), Asmussen et al. (2000), Choulli et al.(2003), Azcue and Muler (2005), and Cadenillas et al. (2006) for a short list. The insurer’s controls and/oractions are often subject to various restrictions. An important restriction is transaction costs or taxeson dividend strategies; see Cadenillas et al. (2006), Bai and Guo (2010), and Schmidli (2017). Otherfeatures/restrictions proposed in the literature include constraints on risk control (see Choulli et al. (2003)),capital injections (see Kulenko and Schmidli (2008), Schmidli (2017), Albrecher and Ivanovs (2018), andLindensj¨o and Lindskog (2020)), and different credit and debt interest rates (see Zhu (2013)). The majorityof the literature on optimal dividend aggregates the insurer’s entire business together in the study, whileseveral recent works consider the insurer as a multi-line business entity, where capital transfers from onebusiness line to another are possible; see Gu et al. (2018) and Jin et al. (2020).Last, we review the insurer’s optimization objectives in the optimal dividend problem. The most stan-dard choice is given in (1.1), maximizing the expected discounted dividend payments up to ruin. It is3nown that the ruin probability is 1 under the optimal dividend strategy e D ∗ to Problem (1.1), which maybe seen as a “disadvantage for not taking the lifetime of the controlled process into account” as pointed outin Thonhauser and Albrecher (2007). Consequently, Thonhauser and Albrecher (2007) propose to add apenalty term − e − δτ to the objective function in (1.1); such an idea is similar to the bequest penalty used inKaratzas et al. (1986) to deal with bankruptcy in optimal investment and consumption problems. In (1.1),the discounting function is of exponential form, which leads to a time-consistent control problem. How-ever, different discounting functions are proposed in the literature that may cause time-inconsistency; seea piece-wise exponential discounting in Chen et al. (2014), a general discounting (within a discrete frame-work) in Zhou and Jin (2020), and quasi-hyperbolic discounting in Zhu et al. (2020). Another importantalternative objective is rooted from the optimal investment literature (see Merton (1969, 1971)) and aimsto maximize the expected utility of dividend payouts; see, e.g., Cadenillas et al. (2007), Grandits et al.(2007), Thonhauser and Albrecher (2011), Jin et al. (2015), and Xu et al. (2020). In fact, in an early arti-cle Gerber and Shiu (2004)[Sections 9 and 10], the authors point out that “maximizing the expected utilityof the present value of the dividends until ruin is a new and challenging problem.”In this paper, we formulate a stochastic control problem for an insurer who chooses a triplet controlconsisting of investment, liability ratio, and dividend strategies in a combined financial and insurancemarket. We model the insurer’s (unit) risk process R by a diffusion process modulated with jumps, whichis correlated with the risky asset price. In our framework, the insurer can directly control the liability unitsas measured by the liability ratio (defined as the ratio of the total liabilities to the controlled surplus).The goal of the insurer is to seek an optimal investment, liability ratio, and dividend strategy to maximizeher expected utility of dividend payments over an infinite horizon: max E [ R ∞ e − δt U ( D t ) d t ], where δ > D = ( D t ) t ≥ denotes the dividend rate, and the utility function U belongs to thehyperbolic absolute risk aversion (HARA) utility family (including log utility and power utility). To solvesuch a control problem, we apply a perturbation approach, proposed in Herdegen et al. (2020), that isdifferent from the standard Hamilton-Jacobi-Bellman (HJB) and martingale approaches. In the first step,we solve the problem for a restricted class of strategies–constant strategies and obtain the optimal constantstrategy u ∗ c and the corresponding smooth value function V c in closed form. In the second step, for anyadmissible strategy, we perturb it by an ǫ -optimal constant strategy and show that the value function forthe perturbed problem V ǫ ( x ) is equal to V c ( x + ǫ ), and the optimal constant strategy u ∗ c remains optimalamong all the admissible strategies.We next summarize the main contributions of this paper as follows. In the literature, the HJB ap-proach is the dominate choice to optimal dividend, and, to the best of our knowledge, the perturbationapproach developed in Herdegen et al. (2020) has not been applied to solve the optimal dividend problembefore. The perturbation approach is particularly useful in dealing with the HARA utility maximiza-tion criterion adopted in this paper and applies simultaneously for both the cases of 0 < η ≤ η >
1, where η is the relative risk aversion. In comparison, the related papers of Cadenillas et al. (2007), Such a strategy is the same as the optimal constant strategy u ∗ c and starts with an initial wealth ǫ > < η ≤ η > We consider a representative insurer (“she”), who has access to a financial market consisting of a risk-freeasset and a risky asset (e.g., a stock index or a mutual fund). The risk-free asset earns at a constant rate r continuously. The price process S = ( S t ) t ≥ of the risky asset is modeled byd S t = µS t d t + σS t d W (1) t , (2.1)where µ, σ > W (1) is a standard Brownian motion. We assume the underlying financial market isideal and frictionless.The insurer’s main business is to underwrite policies against insurable risks. Following Zou and Cadenillas(2014), we model such risks on a unit basis (e.g., per policy) byd R t = α d t + βρ d W (1) t + β p − ρ d W (2) t + γ d N t , (2.2)where α, β, γ > − < ρ < W (2) is another standard Brownian motion, and N is a homogeneousPoisson process with constant intensity λ >
0. Let p represent the unit premium rate, corresponding tothe unit risk process R in (2.2). We further impose the following assumptions on the model (2.1)-(2.2)5hroughout the rest of this paper: µ > r and p > α + λγ. (2.3)We discuss the economic interpretations of the above assumptions in (2.3) in Remark 2.1. On the technicallevel, we assume the processes W (1) , W (2) , and N are independent under a complete probability space(Ω , F , P ) and the filtration F := ( F t ) t ≥ is generated by these three processes, augmented with P -null sets. Remark 2.1.
As argued in Stein (2012)[Chapter 6], a major mistake in the AIG’s business operationduring the financial crisis of 2007-2008 is ignore or underestimate the negative correlation between thefinancial market and its insurance liabilities. Such a negative correlation can be easily captured by the riskmodel of R in (2.2) by setting ρ ∈ ( − , ; see, e.g., Zou and Cadenillas (2014) and Jin et al. (2015).If we set ρ = γ = 0 , then (2.2) reduces to the standard diffusion model considered in Browne (1995) andAsmussen and Taksar (1997). If we set α = β = 0 and allow γ to be a random variable, then (2.2) becomesthe classical CL model; see Shen and Zou (2020) for more detailed discussions on the risk model (2.2) .The two assumptions in (2.3) both have reasonable economic meanings. The first condition implies thatthe Sharpe ratio of the risky asset, Λ := µ − rσ , is strictly positive, which holds true for all “good” assets(e.g., stock indexes) over long run in the financial market. Since E [d R t ] = ( α + λγ )d t , the second conditionimplies that the insurer should charge the unit premium rate p greater than α + λγ , the so-called “actuarialfair price”. Otherwise (i.e., if p ≤ α + λγ ), the insurer’s ruin probability (without investment) is equal to1. If we apply the expected value principle to calculate insurance premium (i.e., p = (1 + θ ) ( α + λγ ) ), thenassuming p > α + λγ is equivalent to setting θ > , where θ is often called the loading factor. Note thatwe do not assume that p is given by the above expected value principle, which is nevertheless imposed inmany works in order to obtain explicit solutions; see, e.g., Asmussen et al. (2000). The representative insurer chooses a triplet strategy (control) that consists of an investment strategy, aliability ratio strategy, and a dividend strategy in the business operations, as described in what follows. • In the financial market, the insurer chooses a dynamic investment strategy π = ( π t ) t ≥ , where π t denotes the proportion of wealth invested in the risky asset at time t . • In the insurance market, we assume the insurer can directly control the amount of liabilities (numberof units or policies) in the underwriting, denoted by L = ( L t ) t ≥ . Following Stein (2012), we define κ t := L t /X t , where X t is the insurer’s wealth at time t (defined later in (2.4)), and call κ = ( κ t ) t ≥ the insurer’s liability ratio strategy. • The insurer chooses a dividend strategy D = ( D t ) t ≥ to distribute profits to the shareholders, where D t denotes the continuous dividend rate payable at time t . Namely, the dividend payment over[ t, t + d t ] is given by D t d t . 6e denote the insurer’s strategy by u := ( π, κ, D ). We offer some explanations to the insurer’s liabilitystrategy κ and dividend strategy D in the following remark. Remark 2.2.
Motivated by the AIG case during the financial crisis of 2007-2008, Stein (2012) finds thatthe liability ratio provides an early warning signal to the AIG’s failure and sets up a model in which theinsurer directly controls its liability ratio. Such a setup attracts considerable attention in the actuarialscience literature, as evidenced by a series of follow-up works on the optimal control study for an insurer;see Zou and Cadenillas (2014), Jin et al. (2015), and Shen and Zou (2020), among many others. Froman economic point of view, the assumption that the insurer can dynamically control the liability ratio is notunreasonable, as major insurers have monopoly power in the insurance market and may also “discriminate”policy holders. For instance, in the business of health insurance, insurers often reject potential policyapplications based on certain risk factors. Lapham et al. (1996) study a group of participants with geneticdisorders in the family and find that 25% of them believed they were refused life insurance and 22% wererefused health insurance. A recent work of Bernard et al. (2020) confirms that, under certain circumstances,“it may become optimal for the insurer to refuse to sell insurance to some prospects”.The dividend strategy D described above is absolutely continuous with respect to the Lebesgue measure,which is used in many related works; see Asmussen and Taksar (1997), Avanzi and Wong (2012), andJin et al. (2015) for a short list. As such, the corresponding control problem is a classical one, instead of asingular or impulse one. We refer to Jeanblanc-Picqu´e and Shiryaev (1995) and Sotomayor and Cadenillas(2011) for both classical and singular control of optimal dividend problems. As argued in Avanzi and Wong(2012), optimal dividend strategies obtained in the literature are often volatile (e.g., “bang-bang” strategies),which are unlikely to be adopted by managers. In consequence, they consider a setup where dividends arepaid at a constant rate g of the company’s surplus and seek to find the optimal rate g ∗ . Note that the abovedividend strategy D includes the threshold strategies (also called refracting strategies), where D t = constantwhen the wealth (surplus) at time t is greater than a given threshold and D t = 0 otherwise. For analyseson the threshold-type dividend strategies, please see Albrecher and Thonhauser (2009) and Avanzi (2009). Let us denote by X = ( X t ) t ≥ the insurer’s wealth process (also called surplus process) under a tripletstrategy of investment, liability ratio, and dividend u := ( π, κ, D ). The dynamics of X is obtained byd X t = (cid:0) X t − ( r + ( µ − r ) π t + ( p − α ) κ t ) − D t (cid:1) d t + ( σπ t − βρκ t ) X t − d W (1) t (2.4) − β p − ρ κ t X t − d W (2) t − γκ t X t − d N t , X = x > . It is clear from (2.4) that the insurer’s wealth process depends on her initial wealth x and strategy u , andwe write it as X instead of X x,u for notational simplicity.We next define the admissible strategies in Definition 2.3 and close this subsection with some remarks. Definition 2.3 (Admissible Strategies) . A strategy u = ( π, κ, D ) is called admissible, denoted by u ∈ A ,if (1) u is predictable with respect to the filtration F ; (2) for all t ≥ , we have R t π s d s < ∞ , ≤ κ t < γ , nd E [ R t D s d s ] < ∞ with D t ≥ ; (3) there exists a unique strong solution to (2.4) such that X t ≥ forall t ≥ . Remark 2.4.
The constraint ≤ κ t < γ stipulates that the wealth process X will not become negative orzero at the jump times of the Poisson process N . We also mention that the condition on the investmentstrategy π is weaker than the standard square integrability condition E [ R t π s d s ] < ∞ for all t ≥ . Given a triplet strategy u , we define the insurer’s objective functional J by J ( x ; u ) := E (cid:20)Z ∞ e − δt U ( D t ) d t (cid:21) , (2.5)where E denotes taking expectation under the physical measure P , δ > U is a standard utility function (strictly increasing and strictly concave). The concavity of U impliesrisk aversion in the insurer’s decision making. Applying a concave utility function to the dividend paymentsis in fact not unusual in the optimal dividend literature. Cadenillas et al. (2007) provide economic justifica-tions for incorporating risk aversion in the objective (see pp.85 therein); see also Thonhauser and Albrecher(2011), Jin et al. (2015), and Xu et al. (2020). A technical issue with the objective in (2.5) is how to handlethe process after the possible ruin time. To that end, for any admissible strategy u ∈ A , define the ruintime τ u by τ u := inf { t ≥ X ≤ } , u ∈ A , where the insurer’s wealth process X is defined in (2.4). We set U ( D t ) ≡ P for all t ≥ τ u , where P < J in (2.5) as J = E (cid:20)Z τ u e − δt U ( D t ) d t + Z ∞ τ u e − δt U ( D t ) d t (cid:21) = E (cid:20)Z τ u e − δt U ( D t ) d t + Pδ e − δτ u (cid:21) . (2.6)The negative constant P in (2.6) can be interpreted as a free parameter penalizing early ruin. The samepenalty term is also adopted in Thonhauser and Albrecher (2007) in order to take into account the lifetimeof the controlled process; see also Karatzas et al. (1986) for a similar idea. If we let P →
0, the objectivein (2.6) reduces to (1.1), the standard choice in the literature; see Albrecher and Thonhauser (2009) andAvanzi (2009). In other words, early ruin is not penalized in the De Finetti (1957) model. If we let P → −∞ , i.e., a finite ruin is penalized infinitely, then the insurer will adopt a strategy that does notresult in ruin ( τ u = + ∞ ).We now formulate the main optimal investment, liability ratio, and dividend problem of this paper. Problem 2.5.
The insurer seeks an optimal strategy u ∗ = ( π ∗ , κ ∗ , D ∗ ) to maximize the expected discountedutility of dividend over an infinite horizon. Equivalently, the insurer solves the following stochastic controlproblem: V ( x ) := sup u ∈A J ( x ; u ) = J ( x ; u ∗ ) , (2.7) 8 here the admissible set A is defined in Definition 2.3 and the objective functional J is defined in (2.6) . In this work, we consider the utility function U of the hyperbolic absolute risk aversion (HARA) family.In particular, we assume U is given by U ( x ) = 11 − η x − η , η > , (2.8)where the limit case of η = 1 is treated as log utility U ( x ) = ln x . Note that U in (2.8) is well defined forall x >
0. If U is given by log utility ( η = 1) or negative power utility ( η > U (0) = −∞ . Thechoice of power utility is dominant in the optimal investment literature (see the classical papers of Merton(1969, 1971)). We comment that the relative risk aversion η in (2.8) can be any positive number, and isin particular allowed to be greater than 1, arguably the case in real life; see Meyer and Meyer (2005). Incomparison, 0 < η ≤ In this section, we study Problem (2.7) over a restricted set of constant strategies A c , which is defined by A c := { u = ( π, κ, D ) | π t ≡ π c , κ t ≡ κ c , D t ≡ ξ c X t , where π c , κ c , ξ c ∈ R , κ c ∈ [0 , /γ ) , ξ c ≥ } . (3.1)We denote constant strategies in A c by u c := ( π c , κ c , ξ c ), which is slightly different from u = ( π, κ, D ) ∈ A introduced in Section 1. For any u c ∈ A c , there exists a unique strong (positive) solution X to (2.4) and E [ X t ] < ∞ for all t ≥
0; see, e.g., Theorem 1.19 in Øksendal and Sulem (2005). This result, along withthe definition in (3.1), implies that all the conditions in Definition 2.3 are satisfied. Therefore, we concludethat the set of constant strategies A c is a (proper) subset of the set of admissible strategies A .To begin, we solve (2.4) explicitly and obtain the insurer’s wealth X t at time t (for all t ≥
0) by X t = x exp (cid:16)(cid:0) f ( π c , κ c ) − ξ c (cid:1) t + (cid:0) σπ c − βρκ c (cid:1) W (1) t − β p − ρ κ c W (2) t + ln(1 − γκ c ) e N t (cid:17) , (3.2)where e N = ( e N t ) t ≥ , with e N t := N t − λt , is the compensated Poisson process. In addition, the function f is defined over R × [0 , /γ ) by f ( y , y ) := r + ( µ − r ) y + ( p − α ) y − (cid:0) σ y − βρσy y + β y (cid:1) + λ ln(1 − γy ) . (3.3)For future convenience, we define three constants A , B , and C by A := γβ (1 − ρ ) , B := β (1 − ρ ) + γ ( p − α + βρ Λ) , C := p − α + βρ Λ − λγ, (3.4)where Λ is the Sharpe ratio of the risky asset, i.e.,Λ := µ − rσ . (3.5) 9ue to ρ ∈ ( − ,
1) and (2.3), we have
A >
B >
0. If ρ ≥
0, then
C >
0; but if ρ < C may benegative.We first analyze the case of log utility U ( x ) = ln x , corresponding to η = 1 in (2.8). Note that for anadmissible constant strategy u c ∈ A c , the corresponding wealth X t > X (0) = x > J in (2.6), we obtain E (cid:20)Z ∞ e − δt U ( D t ) d t (cid:21) = 1 δ ln x + 1 δ f ( π c , κ c ) + 1 δ ln ξ c − δ ξ c . (3.6)Solving Problem (2.7) over A c when U ( x ) = ln x is now equivalent to maximizing the right hand side of(3.6), which is solved in the proposition below. Proposition 3.1.
Suppose U ( x ) = ln x and the constant C defined in (3.4) is non-negative. The optimalconstant strategy u ∗ c = ( π ∗ c , κ ∗ c , ξ ∗ c ) to Problem (2.7) over A c is given by π ∗ c = µ − rσ + ρβσ κ ∗ c , κ ∗ c = B − √ B − AC A , ξ ∗ c = δ, (3.7) where the constants A , B , and C are defined in (3.4) . Furthermore, the value function V c is obtained by V c ( x ) := sup u c ∈A c E (cid:20)Z ∞ e − δt ln( D t ) d t (cid:21) = J ( x ; u ∗ c ) = 1 δ ln( δx ) + 1 δ ( f ∗ − δ ) , (3.8) where f ∗ := f ( π ∗ c , κ ∗ c ) = max f ( y , y ) , with f defined in (3.3) , and π ∗ c and κ ∗ c derived in (3.7) .Proof. By applying the first-order condition to maximizing the right hand side of (3.6), we obtain that π ∗ c and ξ ∗ c are given by (3.7) and κ ∗ c solves a quadratic equation A y − B y + C = 0, where the constants A , B , and C are defined in (3.4). We compute B − AC = (cid:0) β (1 − ρ ) − γ ( λγ + C ) (cid:1) + 4 λB γ (1 − ρ ) > /γ and the smaller solution is less than 1 /γ . Consequently, κ ∗ c is given by thesmaller solution as shown in (3.7), which is non-negative if and only if C ≥
0. By verifying the second-ordercondition, we confirm that u ∗ c in (3.7) is indeed an optimal constant strategy over A c to Problem (2.7).Plugging u ∗ c back into J in (2.5), after tedious computations, leads to the value function in (3.8).We next study the case of power utility U ( x ) = x − η / (1 − η ), where η > η = 1. For any u c ∈ A c ,we obtain J ( x ; u c ) = E (cid:20)Z ∞ e − δt ( ξ c X t ) − η − η d t (cid:21) = x − η − η ξ − ηc δ − (1 − η ) · g ( π c , κ c ) + (1 − η ) ξ c , (3.9)provided δ − (1 − η ) · g ( π c , κ c ) + (1 − η ) ξ c >
0. Here, the function g is defined over R × [0 , /γ ) by g ( y , y ) := r + ( µ − r ) y + ( p − α ) y − η (cid:0) σ y − βρσy y + β y (cid:1) + λ − η (cid:0) (1 − γy ) − η − (cid:1) . (3.10)We now present the main result for power utility as follows.10 roposition 3.2. Suppose U ( x ) = x − η / (1 − η ) , where η > and η = 1 , and the following two conditionshold C ≥ and ψ := δ − (1 − η ) g ∗ η > , (3.11) where C is defined in (3.4) and g ∗ := max g ( y , y ) , with g defined in (3.10) . The optimal constant strategy u ∗ c = ( π ∗ c , κ ∗ c , ξ ∗ c ) to Problem (2.7) over A c is given by π ∗ c = µ − rησ + ρβσ κ ∗ c , κ ∗ c = unique solution of (3.13) , ξ ∗ c = ψ, (3.12) where κ ∗ c solves the following non-linear equation (1 − γy ) − η + ηAλγ y − Cλγ − . (3.13) Furthermore, the value function V c is obtained by V c ( x ) := sup u c ∈A c E "Z ∞ e − δt D − ηt − η d t = J ( x ; u ∗ c ) = ψ − η − η x − η . (3.14) Proof.
Applying the first-order condition to the maximizing problem of the right hand side of (3.9) leadsto the results of u ∗ c in (3.12) and the non-linear equation of κ ∗ c in (3.13), which has a unique solution in[0 , /γ ) due to the condition C ≥
0. By imposing ψ >
0, we obtain ξ ∗ c = ψ > δ − (1 − η ) g ( π c , κ c ) > π c , κ c ) ∈ R × [0 , /γ ). A straightforward verification process then completes the proof.By (3.11), we have lim η → ψ = δ . We also notice that, when η →
1, the non-linear equation (3.13)reduces to the quadratic equation satisfied by κ ∗ c in the log utility case. As such, Proposition 3.1 for logutility can be seen as the limit result of Proposition 3.2 for power utility. In the literature, a diffusionprocess without jumps is commonly used to approximate the classical Cra ´mer-Lundberg model; see, e.g.,Browne (1995) and Højgaard and Taksar (1998). Such a setup corresponds to setting λ = 0 in the riskprocess (2.2). The results in Proposition 3.2 can be simplified when λ = 0, as shown in the corollary below. Corollary 3.3.
Suppose the utility function U is given by (2.8) and there are no jumps ( λ = 0 ) in the riskprocess (2.2) . Further suppose two technical conditions hold: p − α + βρ Λ > and δ > (1 − η )ˆ g ∗ , (3.15) where the constant ˆ g ∗ is defined by ˆ g ∗ := r + ( p − α + βρ Λ) ηβ (1 − ρ ) + Λ η . (3.16) The optimal constant strategy u ∗ c = ( π ∗ c , κ ∗ c , ξ ∗ c ) to Problem (2.7) over A c is given by π ∗ c = µ − rησ + ρβσ κ ∗ c , κ ∗ c = p − α + βρ Λ ηβ (1 − ρ ) , ξ ∗ c = δ − (1 − η )ˆ g ∗ η . (3.17) 11 roof. Introduce ˆ g := g | λ =0 , where g is defined in (3.10). Maximizing ˆ g and (3.9) leads to the above optimalstrategy (3.17). Note that ˆ g ∗ = max ˆ g ( y , y ) = ˆ g ( π ∗ c , κ ∗ c ), where π ∗ c and κ ∗ c are obtained in (3.17).We end this section by offering some explanations on the technical assumptions C ≥ ψ > C ≥ η → ψ = δ > ψ > C in (3.4), if ρ ≥ ρ is the correlation coefficient between the risky assetand the risk process), we always have C ≥ ρ ≥
0. If ρ < C ≥ C < µ − rσ > p − α − λγ − ρβ , where the left hand side is the Sharpe ratio of the risky asset, andthe numerator of the right hand side measures the insurer’s expected profit (including operation costs)from the insurance business. In practice, on the one hand, insurers are only allowed to invest in “safe”risky assets (with relatively low Sharpe ratio); on the other hand, many insurance businesses are lucrativeand insurers charge sufficient safe loading in premiums for solvency and profitability reasons. Therefore,imposing C ≥ κ ∗ c but also makes economic sense. The othertechnical assumption ψ > ψ ≤ V ( x ) = + ∞ when 0 < η < V ( x ) = −∞ when η > ψ > ⇔ δ > η + (1 − η ) g ∗ , i.e., the subjective discount rate δ should be greaterthan a threshold. Similar conditions are commonly imposed for infinite-horizon control problems; see, e.g.,Jin et al. (2015)[Eq.(4.11)]. We consider the HARA type utility function U , given by (2.8), in the formulation of our main stochasticproblem (see Problem (2.7)). The special scale property of the value function inherited from the HARAutility and the classical results from optimal investment problems (see Merton (1969, 1971)) motivate usto make the following conjecture:The optimal strategy u ∗ over the admissible set A to Problem (2.7) is a constant strategy, andhence coincides with the optimal constant strategy u ∗ c obtained in Section 3.The goal of this section is to verify that the above conjecture is indeed correct when U is given by (2.8). Previously in Sections 2 and 3, we have used simplified notations to make presentation more concise, sincethere is no risk of confusion there. Now we need to introduce notations in a more rigorous way for thegeneral analysis. Given an initial wealth x > u , we denote the insurer’s wealth12t time t by X x,ut , for all t ≥
0, which satisfies the stochastic differential equation (SDE) in (2.4). ByDefinition 2.3, the set of admissible strategies A depends on the insurer’s initial wealth x ( x > A ( x ). However, it is still safe to use A c to denote the set of constant strategies, asits definition in (3.1) is independent of the initial wealth. Introduce X (1) t := X ,u ∗ c t and D (1) t := ξ ∗ c X (1) t for all t ≥
0, where u ∗ c is the optimal constant strategy obtained in (3.12). Namely, X (1) (resp. D (1) ) isthe corresponding wealth process (resp. dividend process) under the unit initial wealth ( x = 1) and theoptimal constant strategy u ∗ c .Let us take two arbitrary admissible controls u = ( π , κ , D ) ∈ A ( x ) and u = ( π , κ , D ) ∈A ( x ), where x , x >
0. We define a new control u = ( π, κ, D ), denoted by u := u ⊕ u , along withthe corresponding wealth X = ( X x,ut ) t ≥ such that the following holds true: x = x + x , π t X x,ut = π ,t X x ,u t + π ,t X x ,u t , κ t X x,ut = κ ,t X x ,u t + κ ,t X x ,u t , and D t = D ,t + D ,t , which together imply X x,ut = X x ,u t + X x ,u t . By the definition of ⊕ and the linearity of (2.4), we have u ⊕ u ∈ A ( x + x )and V ( x ) + V ( x ) ≤ V ( x + x ).For any u ∈ A ( x ) and ǫ >
0, define a perturbed objective functional J ǫ by J ǫ ( x ; u ) := E "Z ∞ e − δt (cid:0) D t + ǫD (1) t (cid:1) − η − η d t = J ( x + ǫ ; u ⊕ u ∗ c ) , (4.1)where the second equality comes from the definitions of the operator ⊕ above and J in (2.6). Let us definethe corresponding value function V ǫ by V ǫ ( x ) := sup u ∈A ( x ) J ǫ ( x ; u ) , (4.2)where J ǫ is defined in (4.1). We now present the main result of this paper in Theorem 4.1.
Theorem 4.1.
Suppose U ( x ) = x − η / (1 − η ) , where η > and η = 1 , C defined in (3.4) is non-negative,and ψ defined in (3.11) is positive. For any ǫ > and x > , we have V ǫ ( x ) = V c ( x + ǫ ) , where V ǫ is defined in (4.2) and V c is obtained in (3.14) .Proof. Using the definition of J ǫ in (4.1), we easily see that J ǫ ( x ; u ∗ c ) = J ( x + ǫ ; u ∗ c ) = V c ( x + ǫ ), where u ∗ c is the optimal constant strategy derived in (3.12). Since u ∗ c ∈ A ( x ), we obtain V ǫ ( x ) ≥ V c ( x + ǫ ). In theremaining of the proof, we aim to show the converse inequality, V ǫ ( x ) ≤ V c ( x + ǫ ), also holds.Let ǫ > x > u ∈ A ( x ). We define a new control u ǫ := u ⊕ u ∗ c ,where the initial wealth associated with the strategy u ∗ c is ǫ , and denote it by u ǫ = ( π ǫ , κ ǫ , D ǫ ). Introduce13he corresponding wealth process by X ǫ = ( X ǫt ) t ≥ , i.e., X ǫt := X x + ǫ,u ǫ t . Recall from (3.2) that the wealthprocess under a constant strategy is always positive, which leads to X ǫt = X x,ut + ǫX (1) t ≥ ǫX (1) t > , ∀ t ≥ . (4.3)Define another new process M ǫ by M ǫt := Z t e − δs U ( D ǫs ) d s + e − δt V c ( X ǫt ) , ∀ t ≥ . (4.4)Recall from (3.14) that V c ( x ) = ψ − η x − η / (1 − η ). Since V c is a smooth function over (0 , ∞ ) and X ǫ > M ǫ . Using the SDE (2.4) of X ǫ , we obtaind M ǫt = e − δt V c ( X ǫt − ) h ( σπ ǫt − βρκ ǫt ) d W (1) t − β p − ρ κ ǫt d W (2) t + (cid:0) (1 − γκ ǫt ) − η − (cid:1) d e N t i +(4.5) e − δt ( − δV c ( X ǫt ) + U ( D ǫt ) + V ′ c ( X ǫt ) (cid:2) ( r + ( µ − r ) π ǫt + ( p − α ) κ ǫt ) X ǫt − D ǫt (cid:3) V ′′ c ( X ǫt ) (cid:0) σ ( π ǫt ) − βρσπ ǫt κ ǫt + β ( κ ǫt ) (cid:1) ( X ǫt ) + λ (cid:0) V c ((1 − γκ ǫt ) X ǫt − ) − V c ( X ǫt − ) (cid:1)) d t. We decompose the d t term in (4.5) into two parts as e − δt ( Y (1) t + Y (2) t )d t , where Y (1) t := U ( D ǫt ) − D ǫt V ′ c ( X ǫt ) − η − η (cid:0) V ′ c ( X ǫt ) (cid:1) − η , and Y (2) t is the remaining part of the d t term in (4.5). Using the first-order condition, we show that theinequality − η y − η − y − η − η ≤ y ≥ y = 1). Bysubstituting D ǫt ( V ′ c ( X ǫt )) /η for y in the above inequality, we obtain that Y (1) t ≤ t ≥
0. Recall theresult of V c in (3.14), from which we get V ′ c ( x ) = ( ψx ) − η and V ′′ c ( x ) = − ηψ − η x − − η . We then analyze Y (2) t as follows Y (2) t := η − η (cid:0) V ′ c ( X ǫt ) (cid:1) − η − δV c ( X ǫt ) + X ǫt V ′ c ( X ǫt ) (cid:0) r + ( µ − r ) π ǫt + ( p − α ) κ ǫt (cid:1) V ′′ c ( X ǫt ) (cid:0) σ ( π ǫt ) − βρσπ ǫt κ ǫt + β ( κ ǫt ) (cid:1) ( X ǫt ) + λ (cid:0) V c ((1 − γκ ǫt ) X ǫt − ) − V c ( X ǫt − ) (cid:1) = ( g ( π ǫt , κ ǫt ) − g ∗ ) · (1 − η ) V c ( X ǫt ) = ( g ( π ǫt , κ ǫt ) − g ∗ ) · ψ − η ( X ǫt ) − η , where we have used the definitions of g in (3.10) and ψ in (3.11) to derive the second equality. Since g ∗ isthe maximum value of the function g over R × [0 , /γ ), we obtain Y (2) t ≤ t ≥
0. In addition, wenotice that Y (1) t = Y (2) t = 0 if and only if u ǫ = u ∗ c .Let us denote L ǫ the local martingale part of M ǫt defined in (4.4). We claim that L ǫ is a supermartingale.To see this result, we define M ǫ similar to that of M ǫ in (4.4), but under x = 0 and u ≡
0. By repeating theabove analysis, we easily show that M ǫ is a uniformly integrable martingale (recall with u ≡ u ǫ = u ∗ c ).14ow using the monotonicity of both U and V c , we deduce that M ǫ ≥ M ǫ , which immediately proves that L ǫ is indeed a supermartingale as claimed. By the previous finding on Y (1) t and Y (2) t , we get E [ M ǫt ] ≤ V c ( x + ǫ ) + E [ L ǫt ] ≤ V c ( x + ǫ ) . (4.6)In the final step, using (4.1), (4.3), (4.4), and the above results, we obtain J ǫ ( x ; u ) = lim t →∞ E (cid:20)Z t e − δs U ( D ǫs ) d s (cid:21) = lim t →∞ E h M ǫt − e − δt V c ( X ǫt ) i ≤ lim sup t →∞ E [ M ǫt ] − lim inf t →∞ E h e − δt V c ( X ǫt ) i ≤ lim sup t →∞ E [ M ǫt ] ≤ V c ( x + ǫ ) , where, to prove the second inequality above, we have used the following resultlim inf t →∞ E h e − δt V c ( X ǫt ) i ≥ ψ − η ǫ − η lim inf t →∞ E " e − δt ( X (1) t ) − η − η = 0 . (4.7)By taking supremum over A ( x ), we obtain V ǫ ( x ) ≤ V c ( x + ǫ ). The proof is now complete.Using Theorem 4.1 and the monotonicity result J ( x ; u ) ≤ J ǫ ( x ; u ), we have the following corollary thateventually verifies the conjecture in the opening of this section. Corollary 4.2.
Under the same assumptions as in Theorem 4.1, we have V ( x ) = V c ( x ) = J ( x, u ∗ c ) , ∀ x > , where u ∗ c and V c are given respectively by (3.12) and (3.14) , and V is the value function to Problem (2.7) . In both Theorem 4.1 and Corollary 4.2, the utility function is of power form, with η = 1. When η = 1(the log utility case), the same result holds, i.e., we still have V ( x ) = V c ( x ), where now V c is given by(3.8) from Proposition 3.1. The analysis leading to this conclusion is similar to the one given above for thepower utility case, and is thus omitted. Corollary 4.3.
Suppose U ( x ) = ln x and the constant C defined in (3.4) is non-negative. We have V ( x ) = V c ( x ) = J ( x, u ∗ c ) , ∀ x > , where u ∗ c and V c are given respectively by (3.7) and (3.8) , and V is the value function to Problem (2.7) . In this subsection, we compare the perturbation approach adopted in this paper (see Herdegen et al. (2020))with the standard HJB approach of solving stochastic control problems (see, e.g., Merton (1969, 1971)).The perturbation approach works brilliantly for the insurer’s problem under HARA (power and log)utility considered in Section 4.2. In addition, the proof to Theorem 4.1 applies to all the cases of the15elative risk aversion η : 0 < η < η = 1, and η >
1. In comparison, the HJB approach encounterstechnical issues when η >
1. For this reason, many related existing works that rely on the HJB approachassume 0 < η ≤
1; see Cadenillas et al. (2007), Thonhauser and Albrecher (2011), Jin et al. (2015), andXu et al. (2020). Below we discuss some of these issues in details. • A key step in the verification theorem under the HJB approach is to guarantee the validness ofinterchanging the order of a limit and a conditional expectation (integral) for a family of processeswhen passing the limit to infinity. A sufficient condition for such an interchange is the uniformintegrability of the processes, which can be achieved by imposing the growth condition on the valuefunction or the utility function. For instance, Capponi and Figueroa-L´opez (2014)[Eq.(4.10)] assumethat U ( x ) ≤ K (1 + x ) holds for the utility function U , where K, x >
0. Alternatively, one may“simply” impose a much stronger condition of uniform boundedness on both the drift and the diffusionterms; see Fleming and Soner (2006)[Section IV.5, Eq.(5.2), p.164]. It is trivial to see that both“solutions” are restrictive and do not apply to Problem (2.7) when η > • The second technical issue we discuss is the possibility of bankruptcy (i.e., X x,ut = 0 at some time t < ∞ ). A full and rigorous treatment of bankruptcy under the standard approaches requires lengthytechnical arguments. Karatzas et al. (1986) introduce a free parameter P as the bequest value atthe bankruptcy time and consider a modified version of the original problem (which has a boundarycondition lim x ↓ V P ( x ) = P ), and take pages to show that the value function V P ( x ) of the modifiedproblem converges to the value function V ( x ) of the original problem as P → −∞ . • Last, a commonly required condition for an infinite horizon problem is the so-called transversalitycondition, which is given bylim t →∞ E (cid:20) e − δt ( X ut ) − η − η (cid:21) = 0 or in a weaker version lim inf t →∞ E (cid:20) e − δt ( X ut ) − η − η (cid:21) ≥ . (4.8)The above transversality condition (4.8) is satisfied automatically if 0 < η < η >
1. Proving (4.8) is certainly not trivial when η >
1, although many applications“simply” assume (4.8) holds to avoid a proof. In fact, (4.8) may not be expected; see Herdegen et al.(2020)[Remark 4.7].Further technical discussions can be found in Herdegen et al. (2020). We remark that another standardapproach, the martingale (duality) method, faces the “dual” side of the technical issues/assumptions inthe HJB approach. For that, we refer readers to Karatzas and Shreve (1998)[Section 3.9] for technicalassumptions imposed on optimal investment problems over an infinite horizon.Having seen some of the technical difficulties under the HJB approach, we now explain why the per-turbation approach proposed in Herdegen et al. (2020) works so well in the proof of Theorem 4.1. First,by (4.3), the perturbed wealth process X ǫ is strictly positive, immediately yielding a significant advantage.16hat is, we no longer need to deal with bankruptcy in the proof of Theorem 4.1. Second, by utilizingthe explicit results from Section 3 and the monotonicity of the utility/value functions, we easily showthat M ǫ is uniformly integrable and M ǫt ≥ M ǫ ,t for all t ≥
0. (Note M ǫ is defined by (4.4) and M ǫ isa special version of M ǫ under x = 0 and u = 0.) There results together verify that the process L ǫ , thelocal martingale part of M ǫ , is a supermartingale, which leads to an essential inequality in (4.6). In otherwords, a carefully-chosen special (optimal) strategy enables us to obtain a lower bound on the process M ǫ ,bypassing the difficulty of proving uniform integrability of ( e − δt V ( X ut )) t ≥ . Note that in the definition of M ǫ in (4.4), the second term is e − δt V c ( X ǫt ), not e − δt V ( X ut ) as in the standard HJB approach. Here theadvantages are at least twofold: (1) V c is already obtained explicitly in (3.14) and is smooth, while thevalue function V is assumed to be smooth and is yet to be solved from the associated HJB equation, and(2) X ǫt > X ut ≥ t . Last, we point out that we derive an inequality in (4.7), which is inthe same spirit to the transversality condition in (4.8). To be precise, replacing X u by X ǫ in (4.8) leadsto (4.7). In our case, proving (4.7) is very easy, by the monotone increasing property of V c and (4.3).However, as already pointed out previously, the transversality condition (4.8) does not hold in general,and even for specific problems when it does hold, proving it is not trivial in the case of η > In this section, we conduct an economic analysis to study how the model parameters and risk aversion affectthe insurer’s optimal strategy u ∗ . Due to the popularity of applying a diffusion process to approximatethe risk model (see Browne (1995) and Højgaard and Taksar (1998)), we assume there are no jumps in therisk process R (setting λ = 0 in (2.2)), unless stated otherwise. Recall that, in the case of no jumps, weobtain the optimal strategy u ∗ = u ∗ c , given by (3.17), in Corollary 3.3. We divide the economic analysisalong two directions, with analytic results in Section 5.1 and numerical results in Section 5.2. In the insurance literature, one often assumes that the insurance market is independent of the financialmarket, i.e., ρ = 0 in (2.2). But as argued in Stein (2012) and many others, ignoring the possible dependencebetween the two markets could lead to catastrophic consequences (e.g., the infamous AIG case in thefinancial crisis of 2007-2008). Hence in the first study, we investigate the impact of ρ on the insurer’s17ptimal strategy. By (3.17), we obtain ∂π ∗ c ∂ρ = βσ κ ∗ c > , ∂κ ∗ c ∂ρ = Λ ηβ + 2 ρκ ∗ c , ∂ξ ∗ c ∂ρ = − − ηη h β Λ κ ∗ c + ρ (cid:0) κ ∗ c (cid:1) i , (5.1)where the Sharpe ratio Λ = ( µ − r ) /σ is defined in (3.5). An interesting result is that ∂π ∗ c /∂ρ >
0, whichis also found in Zou and Cadenillas (2014)[Figures 1-3] and Shen and Zou (2020)[Figure 1]. By (3.17) and(5.1), when ρ >
0, we observe ∂κ ∗ c /∂ρ > ∂ξ ∗ c /∂ρ ) = Sign ( η − π ∗ c >
0. Note thatwhen ρ > R (losses to the insurer) is associated with an increase in the risky asset price S . This observation implies thatthe insurer should long the risky asset, explaining why π ∗ c | ρ> >
0. For the same reason, when the positive correlation becomes stronger, the hedging effect amplifies, or equivalently, the “risky” insurance businessbecomes “less risky”, indicating ∂κ ∗ c ∂ρ | ρ> >
0. Regarding the result of ∂ξ ∗ c /∂ρ in (5.4), observe that theinequality β Λ κ ∗ c + ρ ( κ ∗ c ) > ρ > − β Λ /κ ∗ c , which can be seen as a good indication of theoverall market condition. Next, recall that the power utility function in (2.8) is applied to quantitativelymeasure the welfare of dividend payments (monetary wealth). In similar settings, empirical evidence oftenshows that the relative risk aversion η is greater than 1; see for instance Meyer and Meyer (2005)[Table1]. As a result, it is reasonable for us to assume η >
1, although we do not rule out the possibility of0 < η ≤
1. Now when ρ increases in the region ( − β Λ /κ ∗ c , η >
1) pay dividend at ahigher rate, since they prefer to “cash out” now while the market is still in “good” regime; on the otherhand, insurers with low risk aversion (0 < η <
1) reduce the dividend rate, as the desire to invest morecapital in the risky asset is dominating. We end this study by commenting that our theoretical result in(5.1) offers insight on η > ρ >
0. Then as ρ increases, the overall market improves for the insurerand naturally we expect all the insurer’s strategies π ∗ c , κ ∗ c , and ξ ∗ c to increase at the same time; while (5.1)shows that ξ ∗ c increases only if η > ρ , the insurer’s (relative) risk aversionparameter η is decisive in the comparative statics of the optimal dividend strategy. We next analyze howrisk aversion η affects the insurer’s optimal strategy. To that end, we obtain from (3.17) that ∂π ∗ c ∂η = − η π ∗ c , ∂κ ∗ c ∂η = − η κ ∗ c < , ∂ξ ∗ c ∂η = − η ξ ∗ c + ˆ g ∗ + ( η − rη . (5.2)Due to (3.15), κ ∗ c > ∂π ∗ c /∂η ) = − Sign( π ∗ c ), when the optimal investment strategy is a“buy” strategy (resp. a “short-sell” strategy), the investment proportion in the risky asset reduces (resp. Recall that π ∗ c = µ − rησ + ρβσ κ ∗ c , where all the components are positive except the possibility of ρ being negative. When ρ isnegative, the second term of hedging demand becomes negative and acts on the opposite direction of the first term (Merton’sstrategy) in π ∗ c . When ρ ≤ − β Λ /κ ∗ c , the negative second term dominates the positive first term in π ∗ c , i.e., the insurer needsto “short sell” the risky asset mainly for the hedge purpose, which may lead to excessive risk taking. η increases. Notice that when π ∗ c <
0, an increase in π ∗ c means the absolutevalue of π ∗ c decreases. Therefore, given an increase in the risk aversion η , the insurer always invests lessproportion in absolute values in the risky asset (i.e., less risk taking). Such a result is clearly consistent withthe economic meaning of risk aversion. Despite the analytic result of ∂ξ ∗ c /∂η in (5.2), how risk aversionaffects the dividend strategy is still unclear, since the first term is always negative but the second term isalways positive (note ˆ g ∗ − r > µ and the excess premium ¯ p by¯ µ := µ − r and ¯ p := p − α, (5.3)where ¯ µ > p > µ affects the insurer’s decision, we compute ∂π ∗ c ∂ ¯ µ = 1 ησ (1 − ρ ) > , ∂κ ∗ c ∂ ¯ µ = ρηβσ (1 − ρ ) , ∂ξ ∗ c ∂ ¯ µ = − − ηη ρ ¯ p + β Λ βσ (1 − ρ ) . (5.4)By (5.4), following an increase in ¯ µ , the insurer always invests more in the risky asset but only underwritesmore (resp. less) insurance policies if ρ > ρ < ρ in the above study and the “fact” that the higher the excess return ¯ µ , the more attractivethe risky asset. Note that in our framework the insurer is exposed to both the investment risk from the riskyasset S and the insurable risk R . But according to (5.4), ∂π ∗ c /∂ ¯ µ > ρ . That is, even though we allow a non-zero correlation ρ in our model, the result ∂π ∗ c /∂ ¯ µ > ρ impacts the sensitivity magnitude of π ∗ c with respect to ¯ µ : the stronger the correlation,the more sensitive π ∗ c is to the changes of ¯ µ . Assuming ρ ¯ p + β Λ > as ¯ µ increases, insurers with low riskaversion (0 < η <
1) reduce the dividend rate but insurers with high risk aversion ( η >
1) reacts exactlythe opposite way. But, if ρ ¯ p + β Λ < the converse of the above statement holds true. Another keyparameter in the financial market is the volatility σ of the risky asset, and the related sensitivity resultsare obtained by ∂π ∗ c ∂σ = − (2 − ρ )Λ ησ (1 − ρ ) − ρβσ κ ∗ c , ∂κ ∗ c ∂σ = − ρ ¯ µηβσ (1 − ρ ) , ∂ξ ∗ c ∂σ = 1 − ηη ¯ µσ ρ ¯ p + β Λ β (1 − ρ ) . (5.5)If we set ρ = 0 in (5.5), we have ∂π ∗ c /∂σ = − / ( ησ ) <
0, but such a result is not expected in general,which is different from the standard literature; see Merton (1969, 1971). From (5.5), we observe that ∂π ∗ c /∂σ < ρ ≥ ∂κ ∗ c /∂σ ) = − Sign( ρ ). Consequently, assuming the two marketsare positively correlated, when the volatility σ increases, the insurer invests less in the risky asset andreduces insurance liabilities. Under a negatively correlated condition, the insurer’s reaction to an increase A sufficient condition for this inequality is ρ ≥
0, as ¯ p, β, Λ > Equivalently, ρ < p > − β Λ /ρ >
0. The economic meaning of this condition is that, in an “adverse” market (sincecorrelation is negative), the insurer sets the net insurance premium above a threshold.
19f the volatility σ is to underwrite more policies. However, given ρ <
0, how the insurer should adjust herinvestment strategy to an increase of σ is not easily seen, and will be investigated numerically in the nextsubsection. Comparing (5.3) with (5.5) shows that ∂ξ ∗ c /∂σ = − Λ ∂ξ ∗ c /∂ ¯ µ , and thus the opposite side ofthe previous discussions on ∂ξ ∗ c /∂ ¯ µ applies here.Our last agenda is to study the impact of the insurance market, the drift parameter α and the diffusionparameter β in the risk process (2.2), on the insurer’s optimal strategy. We first focus on the optimalliability strategy κ ∗ c and obtain ∂κ ∗ c ∂α = − ηβ (1 − ρ ) < ∂κ ∗ c ∂β = − pηβ (1 − ρ ) < . (5.6)The sensitivity results in (5.6) fit our intuition perfectly: when α or β increases, the insurable risk R becomes more risky and the insurer should reduce underwriting in response. Regarding the optimal in-vestment strategy π ∗ c , we have ∂π ∗ c ∂α = ρβσ ∂κ ∗ c ∂α and ∂π ∗ c ∂β = ρβσ (cid:18) κ ∗ c β + ∂κ ∗ c ∂β (cid:19) . (5.7)An immediate result is that Sign( ∂π ∗ c /∂α ) = Sign( ρ ), which is consistent with the preceding analysis on ρ . But the story of ∂π ∗ c /∂β in (5.7) is more complex; it not only depends on the correlation ρ but also κ ∗ c (positive by (3.15)) and its derivative with respect to β (negative by (5.6)). Last, we analyze the optimaldividend strategy ξ ∗ c and obtain ∂ξ ∗ c ∂α = 1 − ηη κ ∗ c and ∂ξ ∗ c ∂β = 1 − ηη ¯ p + βρ ¯ p Λ β (1 − ρ ) . (5.8)From (5.8), we easily see that Sign( ∂ξ ∗ c /∂α ) = Sign(1 − η ) and Sign( ∂ξ ∗ c /∂β ) = Sign(1 − η ) if ¯ p > max { , − βρ Λ } . Now if we assume η > p > max { , − βρ Λ } , both derivatives in (5.8) are negative,implying that the insurer should reduce the optimal dividend payment rate when the insurance businessbecomes more risky. This finding may further support that η > In Section 5.1, we analyze the sensitivity of the optimal strategy with respect to various model parameters analytically . Here, in this subsection, we continue the same analysis, but from a numerical point of view.In particular, we focus on the effect of two important parameters: the correlation coefficient ρ and therelative risk aversion η . Table 1: Default Model ParametersParameters r µ σ α β p δ Values 0.01 0.05 0.25 0.1 0.1 0.15 0.15
Note. δ is the subjective discount factor, p is the premium rate, and all other parameters are from the model (2.1) and (2.2). qualitative , we set the base values for the model parameters of (2.1)-(2.2) in Table 1 and allow one parameter to vary over a reasonable range in each study. We compute theinsurer’s optimal investment, liability ratio, and dividend rate strategies u ∗ = ( π ∗ , κ ∗ , ξ ∗ ), where u ∗ = u ∗ c in (3.17), when the correlation coefficient ρ varies from -0.8 to 0.8. We plot the results under four differentrisk aversion levels ( η = 0 . , , ,
5) in Figure 1. Our main findings are summarized as follows: • (Optimal investment π ∗ in the upper panel of Figure 1)We observe that π ∗ is an increasing function of ρ , which verifies the result in (5.1). In consequence,the insurer invests more in the risky asset as ρ increases. There exists a negative threshold ˆ ρ (ˆ ρ isabout -0.32 in the numerical example), below which the optimal investment π ∗ is negative, i.e., shortselling is optimal when ρ < ˆ ρ . We also notice that the absolute investment weight | π ∗ | in the riskyasset is a decreasing function of the relative risk aversion η . Economically, that means insurers withhigher relative risk aversion invest more conservatively than those with lower relative risk aversion,which is obviously consistent with the definition of risk aversion. • (Optimal liability ratio κ ∗ in the middle panel of Figure 1) κ ∗ is not a monotone function of ρ globally. Instead, there exists a negative threshold ˆ ρ (ˆ ρ is around-0.15 in the numerical example) such that κ ∗ is increasing over (ˆ ρ,
1) and decreasing over ( − , ˆ ρ ). Inaddition, the increasing part is “steeper” than the decreasing part, i.e., the same increment of ρ over(ˆ ρ,
1) has a bigger impact on κ ∗ than the one over ( − , ˆ ρ ). On the other hand, we observe that κ ∗ isa decreasing function of the risk aversion parameter η . Therefore, an increase in risk aversion alwaysleads to a decrease in the optimal liability ratio κ ∗ . • (Optimal dividend rate ξ ∗ in the bottom panel of Figure 1)An immediate and also important observation is that the shape of ξ ∗ differs dramatically over differentregions of η . When η = 1 (corresponding to log utility), the optimal dividend rate ξ ∗ is a straight line,equal to δ ( δ = 0 .
15 from Table 1). When 0 < η < ξ ∗ is a concave function of ρ , increasing firstfrom -1 to a negative threshold (around -0.25 in Figure 1) and decreasing afterwards. When η > ξ ∗ is a convex function of ρ , decreasing first and then increasing. Furthermore, when η increases over(1 , ∞ ), the insurer reduces the optimal dividend rate ξ ∗ .Our sensitivity analysis so far is based on the assumption that there are no jumps in the risk process R (2.2). In what follows, we relax this assumption and numerically study how the jump intensity λ affectsthe insurer’s optimal strategy u ∗ , which is now given by (3.12) in Proposition 3.2. Due to the inclusionof jumps, we now set the premium rate by the expected value principle: p = (1 + θ ) × ( α + λγ ), where θ = 50%. We further assume the insurer’s risk aversion is η = 2 > γ = 0 . We use ˆ ρ to denote a genetic threshold constant of ρ , which may differ content by content. Such a rule on premium guarantees that the assumptions in (3.11) hold true. Note that the choice of p = 0 .
15 in Table 1is equivalent to p = (1 + 50%) × ( α + λγ ), with λ = 0 there. ρ and Risk Aversion η on the Optimal Strategies -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-4-202468 Optimal Investment * = 0.8 = 1 = 2 = 5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80510152025
Optimal Liability * = 0.8 = 1 = 2 = 5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.800.050.10.150.20.25
Optimal Dividend * = 0.8 = 1 = 2 = 5
Note. We plot the insurer’s optimal investment π ∗ (upper panel), liability κ ∗ (middle panel), and dividend ξ ∗ (lower), as afunction of the correlation coefficient ρ over ( − . , . η = 0 . η = 1(solid red), η = 2 (dashed brown), and η = 5 (dash-dot black). The parameters are chosen from Table 1. β = 0 . π ∗ , liability ratio κ ∗ , anddividend rate ξ ∗ as a function of λ over (0 . , .
2) in Figure 2. The key results are summarized as follows: • (Optimal investment π ∗ in the upper panel of Figure 2)As anticipated, the results are divided into two regions of the correlation coefficient ρ by a negativethreshold ˆ ρ . When ρ is greater than ˆ ρ , the optimal investment strategy π ∗ is a decreasing functionof the jump intensity λ , and thus the insurer invests less in the risky asset as λ increases. When ρ is less than ˆ ρ , π ∗ becomes an increasing function of λ . In particular, for a very negative ρ (e.g., ρ = − . π ∗ may even change from a negative value to a positive value when λ increases.These results can be explained by the expression of π ∗ in (3.12), which has a positive myopic term,independent of λ , and a hedging term with the same sign as ρ , which depends on λ through κ ∗ . Last,Figure 2 also confirms the finding from Figure 1 that π ∗ is an increasing function of ρ . • (Optimal liability ratio κ ∗ in the middle panel of Figure 2)We observe that κ ∗ is a decreasing function of λ . The economic explanation for such a result is that,as λ increases, the insurable risk becomes more risky, and the insurer’s rational response is to reduceunderwriting. Another observation from the numerical example is that κ ∗ increases as the correlation ρ increases. • (Optimal dividend rate ξ ∗ in the bottom panel of Figure 2)Similar to the result of π ∗ , how the jump intensity λ affects the optimal dividend strategy ξ ∗ alsodepends on the value of ρ . When ρ is greater than a negative threshold ˆ ρ , a negative relation between ξ ∗ and λ is shown by Figure 2. However, for a very negative ρ (e.g., ρ = − . ξ ∗ goes up as λ increases. Following the discussions on ρ in Section 5.1, let us interpret ρ > ˆ ρ asa “preferable” market condition and ρ < ˆ ρ an “adverse” market condition. In a preferable market,although an increase in λ makes the insurance business more risky, the insurer still wants to maintainthe surplus at a relatively high level (since the market is preferable), and to achieve so, the insurerreduces the optimal dividend rate. On the contrary, in an adverse market, the insurer pays dividendat a higher rate when λ increases, that is because keeping a large surplus is no longer attractive whenthe available business opportunities are unfavorable. In other words, when the market is in adversecondition, the “utility” of cashing out dividends now outweighs the “utility” of saving for businessopportunities in future. We introduce a combined financial and insurance market consisting of a risk-free asset, a risky asset, anda risk process R . The risk process R is modeled by a jump-diffusion process, possibly correlated with the23igure 2: Impact of the Jump Intensity λ on the Optimal Strategies Optimal Investment * = -0.6 = -0.2 = 0.2 = 0.6
Optimal Liability * = -0.6 = -0.2 = 0.2 = 0.6
Optimal Dividend * = -0.6 = -0.2 = 0.2 = 0.6
Note. We plot the insurer’s optimal investment π ∗ (upper panel), liability κ ∗ (middle panel), and dividend ξ ∗ (lower), as afunction of the jump intensity λ over (0 . , . ρ = − . ρ = − . ρ = 0 . ρ = 0 . p = 1 . α + λγ ), η = 2, γ = 0 .
3, and the rest by Table 1. constant strategies and then apply a perturbation approach to show that the optimal constantstrategy remains optimal over all admissible strategies. When the insurer’s utility is given by a log or apower utility function, we obtain the optimal strategy and the value function in closed form. We furtherutilize the explicit results to conduct economic analyses, both analytically and numerically, to study theimpact of various market parameters and risk aversion on the insurer’s optimal strategy.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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