Optimality of Excess-Loss Reinsurance under a Mean-Variance Criterion
aa r X i v : . [ q -f i n . R M ] M a r Optimality of Excess-Loss Reinsuranceunder a Mean-Variance CriterionREPAIR OF SECTION 3.3
Danping Li ∗ Dongchen Li † Virginia R. Young ‡ September 13, 2018
Abstract
In this paper, we study an insurer’s reinsurance-investment problem under a mean-variancecriterion. We show that excess-loss is the unique equilibrium reinsurance strategy under aspectrally negative L´evy insurance model when the reinsurance premium is computed accordingto the expected value premium principle. Furthermore, we obtain the explicit equilibriumreinsurance-investment strategy by solving the extended Hamilton-Jacobi-Bellman equation.
JEL Codes : C730, G220.
Keywords : Mean-variance criterion; Equilibrium reinsurance-investment strategy; Excess-loss reinsurance; L´evy insurance model.
An integrated reinsurance and investment strategy is commonly employed by an insurer (cedent) toincrease its underwriting capacity, stabilize underwriting results, protect itself against catastrophiclosses, and achieve financial growth. The study of an insurer’s optimal reinsurance-investmentstrategy has received considerable attention in the literature of actuarial science under a varietyof criteria, including minimizing the probability of ruin (see, for example, Promislow and Young[22], Zhang et al. [31], and Chen et al. [9]), maximizing the expected utility of terminal wealth(see, for example, Liu and Ma [18], Bai and Guo [3], Gu et al. [11], and Liang and Bayraktar [15]),and maximizing expected terminal wealth subject to a constraint on the variance, the so-called mean-variance criterion (see, for example, B¨auerle [6] and Zeng and Li [27]).The mean-variance criterion is closely related to maximizing expected utility of terminal wealth.Indeed, Pratt [21] observes that the certainty equivalence for a “small” random gain Y underexpected utility theory approximately equals E ( Y ) − γ Y ) , (1.1) ∗ Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada([email protected]) † Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada([email protected]) ‡ Corresponding author. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA([email protected])
1n which γ is the absolute risk aversion of the utility maximizer. Note that maximizing (1.1) isprecisely the mean-variance criterion. Also, under fairly general conditions, optimal insurance isdeductible insurance for a risk-averse utility maximizer (see, for example, Arrow [1], van Heervar-den [25], and Moore and Young [20]). Thus, when maximizing (1.1) (or solving a related game)with Y equal to terminal wealth of an insurance company, we expect that optimal (or equilibrium)reinsurance will be deductible, or excess-loss, reinsurance, which we prove below in Theorem 3.2.Furthermore, because the risk aversion γ is constant, the deductible is independent of the surplusof the insurer.Under the mean-variance criterion, the reinsurance-investment problem is time-inconsistent inthe sense that Bellman’s optimality principle fails. To tackle the time inconsistency, we formulatethe problem as a non-cooperative game and solve for a subgame perfect Nash equilibrium. Specif-ically, at every time point, the player solves for an equilibrium strategy by treating the problemas a game against all future versions of himself. An equilibrium strategy is, thus, time-consistent.One can trace this approach to Strotz [24], and it has recently been further developed by Bj¨orkand Murgoci [7] for a general class of objective functions in a Markovian framework. Due tothe importance of time consistency for a rational insurer, the approach has already been appliedby many authors to solve for equilibrium strategies in the literature of reinsurance-investmentproblems (see, for example, Zeng et al. [28] and Lin and Qian [17]).Two types of reinsurance policies are most commonly studied in the literature on equilibriumreinsurance and investment under a mean-variance criterion: (1) proportional (quota-share) rein-surance (see, for example, Zeng and Li [27], Shen and Zeng [23], and the two references given atthe end of the previous paragraph) and (2) excess-loss reinsurance (see, for example, Li et al. [14]).Given the rich literature, one question naturally arises which has not received much attention: Which reinsurance policy yields an equilibrium for an insurer under a mean-variance criterionamong all reasonable reinsurance policies ? We show that buying excess-loss reinsurance is theunique equilibrium strategy under this criterion.We model the insurer’s basic surplus process, that is, the surplus process without any reinsurance-investment strategy, by a spectrally negative L´evy process . The model is widely employed in thecontext of risk theory in the actuarial literature (see, for example, Yang and Zhang [26], Chiu andYin [10], Avram et al. [2], and Landriault et al. [13]). It is a generalization of many insurancemodels studied in the context of reinsurance-investment problems, including the Brownian motionmodel (see, for example, Promislow and Young [22]), the classical Cram´er-Lundberg model (see,for example, Zeng et al. [29]), and the jump-diffusion model (see, for example, Zeng et al. [28]).We prove that, when the reinsurance premium is computed according to the expected valuepremium principle, excess-loss reinsurance is the unique equilibrium strategy for a time-consistentinsurer under a mean-variance criterion. As mentioned above, this result is consistent with severalin the literature; specifically, under the expected value premium principle and various objectivefunctions, excess-loss (re)insurance is optimal, including when maximizing the expected utility ofterminal wealth (see, for example, Liang and Guo [16], Zeng and Luo [30]) and when minimizingthe probability of ruin (see, for example, Zhang et al. [31], Meng and Zhang [19], Bai et al. [4],and Zhou and Cai [32]).The remainder of this paper is organized as follows. In Section 2, we formulate our modeland define the equilibrium problem faced by the insurer. In Section 3, we prove that excess-2oss reinsurance is the unique equilibrium strategy, and we obtain explicit expressions for theequilibrium reinsurance-investment strategy and the corresponding equilibrium value function. Wealso discuss two problems closely related to the mean-variance criterion: (1) maximizing expectedexponential utility of terminal wealth, and (2) maximizing the time-0 mean-variance criterionwith commitment. In Section 4, we present some numerical examples to illustrate our findings,and Section 5 concludes the paper.
Let (cid:16) Ω , F , F = {F t } t ≥ , P (cid:17) be a filtered, complete probability space satisfying the usual condi-tions, and let T > U t = c d t + σ d B (1) t − Z ∞ z N (d z, d t ) , U > , in which c > σ > (cid:8) B (1) t (cid:9) t ≥ is an F -adaptedstandard Brownian motion, and N (d z, d t ) is a Poisson random measure representing the numberof insurance claims of size ( z, z +d z ) within the time period ( t, t +d t ). B (1) and N are independent.For more information about L´evy processes, please see Kyprianou [12].Denote the compensated measure of N (d z, d t ) by ˜ N (d z, d t ) = N (d z, d t ) − ν (d z )d t, in which ν is a L´evy measure such that R ∞ z ν (d z ) < ∞ ; ν (d z ) represents the expected number of insuranceclaims of size ( z, z + d z ) within a unit time interval. The insurer’s premium c is determined underthe expected value principle, that is, c = (1 + θ ) R ∞ z ν (d z ), in which θ > { ℓ t } t ∈ [0 ,T ] , with the only restriction 0 ≤ ℓ t ≤ Z t when the claim equals Z t at time t ∈ [0 , T ]. Note that the reinsurer covers the excess loss Z t − ℓ t . We will look for a reinsurancestrategy given in feedback form by ℓ t = ℓ ( Z t , t ), in which we slightly abuse notation by using ℓ on both sides of this equation. Technically, we should assume a priori that the retention strategydepends on surplus, but in Theorem 3.2 below, we will find the equilibrium retention that isindependent of the surplus. Thus, for simplicity, we omit ℓ ’s possible dependency on the surplus.The time- t premium rate of the reinsurance policy is given by(1 + η ) Z ∞ ( z − ℓ ( z, t )) ν (d z ) , determined again under the expected value principle, in which η is the reinsurer’s proportionalsafety loading. It is commonly assumed in the literature that η > θ , indicating that a reinsurancepolicy is more expensive than the primary insurance, and by using this assumption, one generally3voids trivial results. Under the retention ℓ , the dynamics of the surplus process is governed byd R t = d U t − (1 + η ) Z ∞ [ z − ℓ ( z, t )] ν (d z ) d t + Z ∞ [ z − ℓ ( z, t )] N (d z, d t )= (1 + θ ) Z ∞ z ν (d z ) d t + σ d B (1) t − (1 + η ) Z ∞ [ z − ℓ ( z, t )] ν (d z ) d t − Z ∞ z N (d z, d t ) + Z ∞ [ z − ℓ ( z, t )] N (d z, d t )= Z ∞ (( θ − η ) z + ηℓ ( z, t )) ν (d z ) d t + σ d B (1) t − Z ∞ ℓ ( z, t ) ˜ N (d z, d t ) . Furthermore, suppose the insurer invests in a financial market consisting of a risk-free assetwith a constant interest rate r > S t = µ S t d t + σ S t (cid:16) ρ d B (1) t + p − ρ d B (2) t (cid:17) , S > , in which µ > r , σ > ρ ∈ ( − , (cid:8) B (2) t (cid:9) t ≥ is an F -adapted standard Brownian motion,independent of both B (1) and N . Let π t denote the dollar amount of surplus invested in therisky asset at time t , and let { X ut } t ∈ [0 ,T ] denote the corresponding insurance surplus process undera reinsurance-investment strategy u := ( ℓ ( Z t , t ) , π t ) t ∈ [0 ,T ] . The dynamics of the surplus process { X ut } t ∈ [0 ,T ] is, then, given byd X ut = π t d S t S t + ( X ut − π t ) r d t + d R t = (cid:20) rX ut + ( µ − r ) π t + Z ∞ (( θ − η ) z + ηℓ ( z, t )) ν (d z ) (cid:21) d t + q σ + 2 ρσ σ π t + σ π t d B t − Z ∞ ℓ ( z, t ) ˜ N (d z, d t ) , (2.1)in which { B t } t ≥ is an F -adapted standard Brownian motion, independent of N , defined by B t = σ + ρσ π t p σ + 2 ρσ σ π t + σ π t B (1) t + p − ρ σ π t p σ + 2 ρσ σ π t + σ π t B (2) t . For ease of notation, let E x,t [ · ] = E (cid:2) · (cid:12)(cid:12) X ut = x (cid:3) and Var x,t [ · ] = Var (cid:2) · (cid:12)(cid:12) X ut = x (cid:3) . Definition 2.1 ( Admissible strategy ) . A strategy u = ( ℓ ( Z t , t ) , π t ) t ∈ [0 ,T ] is called admissible if itsatisfies the following conditions: ( ) u is F -progressively measurable; ( ) For all t ∈ [0 , T ] and Z t ≥ , ≤ ℓ ( Z t , t ) ≤ Z t ; ( ) For all ( x, t ) ∈ R × [0 , T ] , E x,t hR Tt ( ℓ ( Z s , s ) + π s ) d s i < ∞ with probability one; ( ) For all ( x, t ) ∈ R × [0 , T ] , the stochastic differential equation (2.1) has a unique strong solution. main objective of this paper is to study the reinsurance-investment problem for aninsurer under a mean-variance criterion, that is, one who wishes to maximize J u ( x, t ), in which J u is given by J u ( x, t ) = E x,t [ X uT ] − γ x,t [ X uT ] , ( x, t ) ∈ R × [0 , T ] , (2.2)in which γ > J u ( x, t ) is a time-inconsistent problem in the sense that Bellman’s optimalityprinciple fails. We tackle the problem from a non-cooperative game point of view by defining anequilibrium strategy and its corresponding equilibrium value function; see, for example, Basak andChabakauri [5], Bj¨ork and Murgoci [7], and Bj¨ork et al. [8]. Definition 2.2
For an admissible strategy u ∗ = ( ℓ ∗ ( Z t , t ) , π ∗ t ) t ∈ [0 ,T ] , for ε > , and for t ∈ [0 , T ] ,define the strategy u ε,t by u ε,ts = ( (¯ ℓ ( z, s ) , ¯ π ) , t ≤ s < t + ε,u ∗ s , ≤ s < t or t + ε ≤ s ≤ T, (2.3) in which ¯ ℓ ( z, s ) is an admissible retention strategy and ¯ π is a real constant. If, for all ( x, t ) ∈ R × [0 , T ] , lim inf ε ↓ J u ∗ ( x, t ) − J u ε,t ( x, t ) ε ≥ , then u ∗ is an equilibrium strategy and J u ∗ ( x, t ) is the corresponding equilibrium value function. We first provide a verification theorem whose proof we omit because it is similar to the proof ofthe verification theorem, Theorem 4.1, in Bj¨ork and Murgoci [7]. Also, see the discussion in Bj¨orket al. [8] about applying the verification theorem under the mean-variance criterion.For any admissible retention ℓ and for any constant π ∈ R , we define an integro-differentialoperator A ℓ,π as follows: A ℓ,π φ ( x, t ) := lim ε ↓ E x,t (cid:2) φ ( X ut + ε , t + ε ) (cid:3) − φ ( x, t ) ε = φ t ( x, t ) + (cid:20) rx + ( µ − r ) π + Z ∞ (( θ − η ) z + (1 + η ) ℓ ( z, t )) ν (d z ) (cid:21) φ x ( x, t )+ 12 (cid:0) σ + 2 ρσ σ π + σ π (cid:1) φ xx ( x, t ) + Z ∞ ( φ ( x − ℓ ( z, t ) , t ) − φ ( x, t )) ν (d z ) , (3.1)in which φ ( x, t ) ∈ C , ( R × [0 , T ]). In the first line of (3.1), u = ( ℓ, π ), in which the constant π represents the strategy in which the insurer invests the constant π in the risky asset. Theorem 3.1 ( Verification theorem ) . Suppose there exist V ( x, t ) and g ( x, t ) ∈ C , ( R × [0 , T ]) satisfying the following conditions: ) For all ( x, t ) ∈ R × [0 , T ] , sup ℓ,π n A ℓ,π V ( x, t ) − γ A ℓ,π g ( x, t ) + γ g ( x, t ) A ℓ,π g ( x, t ) o = 0 . (3.2) Let ( ℓ ∗ , π ∗ ) denote the pair that attains the supremum in (3.2) . ( ) For all ( x, t ) ∈ R × [0 , T ] , A ℓ ∗ ,π ∗ g ( x, t ) = 0 . (3.3)( ) For x ∈ R , V ( x, T ) = x and g ( x, T ) = x. (3.4) Then, the equilibrium reinsurance-investment strategy u ∗ is given by u ∗ t = ( ℓ ∗ ( Z t , t ) , π ∗ ( X t , t )) . (3.5) Note that u ∗ is given in feedback form. V ( x, t ) = J u ∗ ( x, t ) is the corresponding equilibrium valuefunction, and g ( x, t ) = E x,t (cid:2) X u ∗ T (cid:3) is the expectation of terminal wealth. One can use Theorem 3.1 directly to obtain an equilibrium strategy. However, we wish to showthat there is only one such equilibrium strategy; to that end, we have the following lemma, whichis similar to Lemma 1 in Basak and Chabakauri [5]. Lemma 3.1
The value function V and expectation of terminal wealth g under the mean-variancecriterion are separable in the surplus x and admit the following representation: (cid:26) V ( x, t ) = e r ( T − t ) x + B ( t ) , B ( T ) = 0 ,g ( x, t ) = e r ( T − t ) x + b ( t ) , b ( T ) = 0 . (3.6) Proof.
From (2.1), we haved (cid:16) e r ( T − t ) X ut (cid:17) = (cid:20) ( µ − r ) π t + Z ∞ (( θ − η ) z + ηℓ ( z, t )) ν (d z ) (cid:21) e r ( T − t ) d t + q σ + 2 ρσ σ π t + σ π t e r ( T − t ) d B t − Z ∞ ℓ ( z, t ) e r ( T − t ) ˜ N (d z, d t )=: G u ( t ) . (3.7)From (3.7) it follows that E x,t [ X uT ] = e r ( T − t ) x + E x,t (cid:20)Z Tt (cid:18) ( µ − r ) π s + Z ∞ (( θ − η ) z + ηℓ ( z, s )) ν (d z ) (cid:19) e r ( T − s ) d s (cid:21) , (3.8)and Var x,t [ X uT ] = Var x,t (cid:20)Z Tt G u ( s ) d s (cid:21) . (3.9)The expressions in (3.8) and (3.9) for the expectation and variance of X uT , respectively, imply thatthe objective and the expectation functions for the mean-variance criterion are separable in thesurplus x , as given in (3.6).In the next theorem, we present the equilibrium strategy and the corresponding equilibriumvalue function. 6 heorem 3.2 The unique equilibrium reinsurance-investment strategy u ∗ = ( ℓ ∗ ( z, t ) , π ∗ ( t )) forthe mean-variance criterion is given by ℓ ∗ ( z, t ) = ηγ e − r ( T − t ) ∧ z,π ∗ ( t ) = µ − rγ σ e − r ( T − t ) − ρ σ σ , (3.10) and the corresponding value function is V ( x, t ) = e r ( T − t ) x + B ( t ) , (3.11) in which B ( t ) = Z Tt ( γ (cid:18) µ − rσ (cid:19) + e r ( T − s ) (cid:20) − ( µ − r ) ρ σ σ + Z ∞ (( θ − η ) z + ηℓ ∗ ( z, s )) ν (d z ) (cid:21) − γ r ( T − s ) (cid:20)(cid:0) − ρ (cid:1) σ + Z ∞ ( ℓ ∗ ( z, s )) ν (d z ) (cid:21)(cid:27) d s. (3.12) Furthermore, E x,t h X u ∗ T i = g ( x, t ) = e r ( T − t ) x + b ( t ) , (3.13) in which b ( t ) = Z Tt ( γ (cid:18) µ − rσ (cid:19) + e r ( T − s ) (cid:20) − ( µ − r ) ρ σ σ + Z ∞ (( θ − η ) z + ηℓ ∗ ( z, s )) ν (d z ) (cid:21)) d s. (3.14) Proof.
We verify that u ∗ , V , and g defined, respectively, in (3.10), (3.11), and (3.13), satisfyconditions (1)–(3) in Theorem 3.1. To that end, from (3.1), we compute A ℓ,π (cid:16) e r ( T − t ) x + β ( t ) (cid:17) = β t + C ℓ,π ( t ) e r ( T − t ) , (3.15)in which C is given by C ℓ,π ( t ) = ( µ − r ) π + Z ∞ (( θ − η ) z + η ℓ ( z, t )) ν (d z ) . Also, from (3.1), we obtain, after simplifying, (cid:16) e r ( T − t ) x + b ( t ) (cid:17) A ℓ,π (cid:16) e r ( T − t ) x + b ( t ) (cid:17) − A ℓ,π (cid:18)(cid:16) e r ( T − t ) x + b ( t ) (cid:17) (cid:19) = −
12 e r ( T − t ) (cid:20)(cid:0) σ + 2 ρσ σ π + σ π (cid:1) + Z ∞ ℓ ( z, t ) ν (d z ) (cid:21) . (3.16)By substituting the expressions for V and g from (3.6) into (3.2) and by using the results of(3.15) and (3.16), we getsup ℓ,π (cid:26) B t + C ℓ,π ( t ) e r ( T − t ) − γ r ( T − t ) (cid:20)(cid:0) σ + 2 ρσ σ π + σ π (cid:1) + Z ∞ ℓ ( z, t ) ν (d z ) (cid:21)(cid:27) = 0 . (3.17)7he expression in (3.17) is concave with respect to π ; thus, we obtain the optimal value of π fromthe first-order condition. Specifically, π ∗ ( t ) = µ − rγ σ e − r ( T − t ) − ρ σ σ . (3.18)Next, consider the terms in ℓ in (3.17), that is, Z ∞ (cid:16) ηℓ ( z, t ) − γ e r ( T − t ) ℓ ( z, t ) (cid:17) ν (d z ) . (3.19)If we maximize the integrand in the integral in (3.19) z -by- z for a given t ∈ [0 , T ], then we willmaximize the integral itself. With respect to ℓ , the graph of f ( ℓ ) := ηℓ − γ e r ( T − t ) ℓ is a concaveparabola that increases through the origin (0 , f (0)) = (0 , f ’s maximizer ℓ ∗ ∈ [0 , z ] is givenby ℓ ∗ ( z, t ) = ηγ e − r ( T − t ) ∧ z. (3.20)If we substitute u ∗ = ( ℓ ∗ , π ∗ ) into (3.17) and solve for B ( t ) (by using the terminal condition B ( T ) = 0), then we obtain the expression in (3.12). Also, if we solve for b ( t ) (with the sameterminal condition b ( T ) = 0) in the equation A ℓ ∗ ,π ∗ (cid:0) e r ( T − t ) x + b ( t ) (cid:1) = 0, then we obtain theexpression in (3.14).Thus, u ∗ , V , and g defined, respectively, in (3.10), (3.11), and (3.13), satisfy conditions (1)–(3)in Theorem 3.1. To complete this proof, note that u ∗ is an admissible strategy, as defined inDefinition 2.2. Remark 3.1
Lemma . and Theorem . prove that excess-loss reinsurance is the unique equi-librium strategy for a time-consistent insurer under the mean-variance criterion; in that sense, weconsider it optimal. Note that the equilibrium strategy is independent of the state variable x . Thisindependence results from the fact that the risk aversion γ is a constant. See (1.1) in which γ in themean-variance approximation of the utility’s certainty equivalence represents the utility’s absoluterisk aversion. Recall that if utility exhibits constant absolute risk aversion, then the form of theutility function is exponential, and decision making under exponential utility invariably results instrategies that are independent of the state variable. See Basak and Chabakauri [5] and Bj¨ork etal. [8] for further discussion.Moreover, the equilibrium excess-loss strategy is independent of the parameters of the risky assetand the safety loading of the insurer, while the equilibrium investment strategy is independent of thesafety loadings of both the insurer and the reinsurer. In other words, the equilibrium reinsurancestrategy is unaffected by the financial market, while the equilibrium investment strategy is unaffectedby the price of reinsurance, and both strategies are unaffected by the price of the primary insurance. The behavior of the equilibrium strategy is given in the following corollary. The proof isstraightforward and hence omitted. We discuss the intuition behind the behavior of the equilibriumstrategy in the numerical analysis in the next section.
Corollary 3.1
The equilibrium retained claim ℓ ∗ ( z, t ) increases in η and t , decreases in r, γ, and T, and is independent of x, θ, ρ, σ , and σ ; the equilibrium amounted invested in the risky asset π ∗ ( t ) increases in µ and t, decreases in r, γ, ρ, and T, and is independent of x and z . .3 Related problems In this section, we compare the equilibrium strategy in Theorem 3.2 with the optimal strategiesfor two related problems.
First, as we observed in the Introduction and in Remark 3.1, the mean-variance criterion is relatedto maximizing expected utility of terminal wealth under constant absolute risk aversion γ . Forthe latter problem, a standard verification theorem states that if we find a classical solution U tosup ℓ,π A ℓ,π U ( x, t ) = 0, with terminal condition U ( x, T ) = − e − γx , then U equals sup ℓ,π E x,t [ u ( X uT )],in which u ( x ) = − e − γx . Furthermore, the optimal strategy is given in feedback form by themaximizer of A ℓ,π U ( x, t ).For the model in this paper, it is straightforward to show that the optimal strategy is ( ℓ u , π u ),in which ℓ u and π u are given by ℓ u ( z, t ) = ln(1 + η ) γ e − r ( T − t ) ∧ z,π u ( t ) = µ − rγ σ e − r ( T − t ) − ρ σ σ . (3.21)Note that, for small values of η , ℓ u is approximately equal to ℓ ∗ , but π u is identically equal to π ∗ .This result further confirms the close relationship between finding the equilibrium strategy for themean-variance criterion with constant risk aversion parameter γ and maximizing expected utilityof terminal wealth with constant absolute risk aversion γ . Second, if the insurer pre-commits to its strategy at time 0 for the entire period [0 , T ] to maximizethe time-0 mean-variance objective function in (2.2), then the optimal investment strategy differsfrom π ∗ , as shown in Basak and Chabakauri [5]. Furthermore, the optimal reinsurance strategyalso differs from ℓ ∗ . We demonstrate the latter statement in this section.The pre-commitment problem is given bysup ℓ,π E x, [ X T ] − γ x, [ X T ] . (3.22)By following the work in Zhou and Li [33], we first solve the following auxiliary problem U ( x, t ) = sup ℓ,π E x,t h αX T − γ X T i , (3.23)with the optimal strategy given in feedback form by (cid:16) ˆ π ( α, X t , t ) , ˆ ℓ ( α, z, X t , t ) (cid:17) . Then, by setting α equal to the solution α ∗ of the following equation α = 1 + γ E x , (cid:16) X ˆ π ( α,X t ,t ) , ˆ ℓ ( α,z,X t ,t ) T (cid:17) , (ˆ π, ˆ ℓ ) with α = α ∗ equals the optimal strategy for the pre-commitment problem in (3.22). Weanticipate the control ℓ will depend on the state variable x and write ℓ = ℓ ( z, x, t ) in feedback9orm. Furthermore, (ˆ π, ˆ ℓ ) clearly depends on x through α ∗ . Note that U in (3.23) is concave withrespect to x because αx − γ x is concave and the surplus is linear with respect to the controls.If we find a classical solution sup ℓ,π A ℓ,π V ( x, t ) = 0, with terminal condition V ( x, T ) = αx − γ x , then V = U , the value function of the auxiliary problem in (3.23). Suppose we have sucha classical solution of this boundary-value problem; without ambiguity, write it as U . Then, theterms in the Hamilton-Jacobi-Bellman equation involving ℓ aremax ℓ Z ∞ ((1 + η ) ℓ U x ( x, t ) + U ( x − ℓ, t )) ν (d z ) . As in the proof of Theorem 3.2, we maximize the integral by maximizing the integrand z -by- z fora fixed value of ( x, t ) over ℓ such that 0 ≤ ℓ ( z, x, t ) ≤ z . Because U is concave with respect to x ,it is straightforward to show that the optimal reinsurance is of the formˆ ℓ ( z, x, t ) = d ( x, t ) ∧ z, (3.24)in which d = d ( x, t ) is given by d ( x, t ) = ℓ c , if ∃ ℓ c ∈ (0 , z ) s . t . (1 + η ) U x ( x, t ) = U x ( x − ℓ c , t ) , ∞ , if (1 + η ) U x ( x, t ) − U x ( x − ℓ c , t ) > , ∀ ℓ > , if U x ( x, t ) ≤ . (3.25)At time T , U ( x, T ) = α ∗ x − γ x , and α ∗ depends on x ; thus,ˆ ℓ ( z, x, T ) = η (cid:18) α ∗ γ − x (cid:19) + ∧ z = ηγ ∧ z = ℓ ∗ ( z, T ) . Thus, the optimal pre-commitment reinsurance strategy differs from the equilibrium reinsurancestrategy for the time-consistent problem. Example 4.1 (Equilibrium strategies)
In this example, we examine the sensitivity of the equi-librium reinsurance-investment strategies given in (3.10) to different parameters. Unless otherwisestated, the parameter values are given by r = 0 . , µ = 0 . , σ = 0 . , σ = 0 . , η = 0 . ,ρ = 0 . , γ = 1 , and T = 9 . Denote the corresponding equilibrium strategy by ( m ∗ , π ∗ ) , in which m ∗ ( t ) = ηγ e − r ( T − t ) . As an aside, note that if we begin the pre-commitment problem at time T − ǫ for ǫ > ǫ → ˆ ℓ ( z, x, T ) = ηγ ∧ z = ℓ ∗ ( z, T ) because lim ǫ → x T − ǫ = x , in which X T = x . This equality makes sensebecause pre-committing over a vanishingly small interval is equivalent to being time-consistent over that interval. m ∗ t = 0 t = 3 t = 6 t = 9 r π ∗ -0.100.10.20.30.40.50.6 t = 0 t = 3 t = 6 t = 9 Figure 1. Impact of r. γ m ∗ t = 0 t = 3 t = 6 t = 9 γ π ∗ -0.500.511.522.5 t = 0 t = 3 t = 6 t = 9 Figure 2. Impact of γ. In Figure 1, we plot the impact of r on the reinsurance-investment strategy for a variety oftimes t . Both m ∗ and π ∗ decrease as the risk-free rate increases, except for m ∗ when t = T , atwhich it is constant. When large claims occur, the insurer might borrow from the risk-free assetto aid in regaining solvency; recall that the amount invested in the risk-free asset equals x − π ∗ ( t ) ,which is negative when the surplus x is negative. Thus, as borrowing money becomes more costly,the insurer retains less insurance risk. Furthermore, it is reasonable for the insurer to decreasethe amount invested in the risky asset as the risk-free asset becomes more attractive.In Figure 2, we plot the impact of γ on the reinsurance-investment strategy. Note that, as theinsurer becomes more risk averse, it assumes less insurance risk and less financial risk. m ∗ t = 0 t = 3 t = 6 t = 9 Figure 3. Impact of η on m ∗ . ρ -0.5 0 0.5 π ∗ t = 0 t = 3 t = 6 t = 9 σ π ∗ -0.2-0.100.10.20.30.4 t = 0 t = 3 t = 6 t = 9 Figure 4. Impact of ρ ( left ) and σ ( right ) on π ∗ .In Figure 3, we plot the impact of η on the retention level m ∗ . Note that m ∗ increases as η increases. In other words, as the reinsurance policy becomes more expensive, the insurer retainsmore insurance risk.In Figure 4, we plot the impact of ρ, σ on the investment strategy. First, we see from the leftpanel that as ρ increases, π ∗ decreases. Second, we see from the right panel that, as the insurancemarket becomes more volatile, the amount invested in the financial market decreases because thereis a positive correlation ( ρ = 0 . between the two markets. xample 4.2 (Proportional vs. excess-loss reinsurance) In this example, we assume thatthe basic surplus process follows the classical Cram´er-Lundberg model d U t = c d t − d N t X i =1 Y i , U = u, in which { Y i } ∞ i =1 is a sequence of independent and identically distributed exponential random vari-ables with common survival function S ( y ) := e − κy for y > representing the amount of indi-vidual claims, and { N t } t ≥ is a Poisson process with intensity λ > representing the number ofclaims, independent of { Y i } . The premium rate c = (1 + θ ) λκ . By applying equation (3.11) with ν (d z ) = λF (d z ) , σ = 0 , and σ = σ, the corresponding value function under the Cram´er-Lundbergmodel is given by V ( x, t ) = e r ( T − t ) x + B ( t ) , ( x, t ) ∈ R × [0 , T ] , in which B ( t ) = Z Tt ( γ (cid:18) µ − rσ (cid:19) + e r ( T − s ) " ( θ − η ) λ E [ Y ] + ηλ Z ηγ e − r ( T − s ) S ( y ) d y − γλ e r ( T − s ) Z ηγ e − r ( T − s ) yS ( y ) d y ) d s. Moreover, we have g ( x, t ) = E x,t h X u ∗ T i = e r ( T − t ) x + b ( t ) , ( x, t ) ∈ R × [0 , T ] , in which b ( t ) = Z Tt ( γ (cid:18) µ − rσ (cid:19) + e r ( T − s ) " ( θ − η ) λ E [ Y ] + ηλ Z ηγ e − r ( T − s ) S ( y ) d y d s, and Var x,t ( X u ∗ T ) = γ ( g ( x, t ) − V ( x, t )) . The other parameter values equal r = 0 . , µ = 0 . ,σ = 0 . , γ = 0 . , T = 3 , θ = 0 . , η = 0 . , λ = 1 , and κ = 0 . .Under this model, we compare the value function V with the value function under the equilib-rium proportional reinsurance V determined in Zeng et al. [28], for example. We see from the toppanel in Figure 5 that V dominates V except at the boundary t = T, where V ( x, T ) = V ( x, T ) = x . In other words, the equilibrium proportional reinsurance policy is demonstrably not optimalwithin a larger class, that is, the broad class of reinsurance policies ℓ for which ≤ ℓ ( Z t , t ) ≤ Z t .We also see from the bottom panel in Figure 5 that when mean and variance are viewed sepa-rately, compared to the equilibrium proportional reinsurance, though the equilibrium excess-losspolicy generates a greater terminal mean, the associated terminal risk is also greater. t x V (t , x ) V V t x E (t , x ) E E t x V a r (t , x ) V ar V ar Figure 5. Excess-loss vs. proportional reinsurance.
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