Mean-Variance Investment and Risk Control Strategies -- A Time-Consistent Approach via A Forward Auxiliary Process
aa r X i v : . [ q -f i n . P M ] J a n Mean-Variance Investment and Risk Control Strategies– A Time-Consistent Approach via A Forward Auxiliary Process
Yang Shen ∗ Bin Zou † First Version: July 5, 2020; This Version: November 29, 2020Accepted for publication in
Insurance: Mathematics and Economics
Abstract
We consider an optimal investment and risk control problem for an insurer under the mean-variance(MV) criterion. By introducing a deterministic auxiliary process defined forward in time, we formulatean alternative time-consistent problem related to the original MV problem, and obtain the optimalstrategy and the value function to the new problem in closed-form. We compare our formulation andoptimal strategy to those under the precommitment and game-theoretic framework. Numerical studiesshow that, when the financial market is negatively correlated with the risk process, optimal investmentmay involve short selling the risky asset and, if that happens, a less risk averse insurer short sells morerisky asset.
Keywords : Optimal Reinsurance; Jump Diffusion; Hamilton-Jacobi-Bellman Equation; Time-consistentControl; Precommitment
JEL Code : G11; G22; C61
Investing premiums in financial assets and managing risk from underwriting are two fundamental businessdecisions to an insurer. The close interaction between these two decisions motivates us to model a combinedfinancial and insurance market for an insurer and study them simultaneously. With this in mind, we set upsuch a combined market consisting of one risk-free asset, one risky asset, and one risk process R representingthe liabilities per unit (or per policy). We apply a jump-diffusion model for the risk process R , which is ageneralization of the diffusion approximation model and the classical Cram´er-Lundberg (CL) model. Sucha modeling framework is general in the sense that it covers risk process models used in related literature asspecial cases; see, e.g., Browne (1995) and Højgaard and Taksar (1998) for early works without jumps andZeng et al. (2013) and Zeng et al. (2016) for more recent works with jumps. The setup of the combined ∗ School of Risk and Actuarial Studies, University of New South Wales, Sydney, NSW 2052, Australia. Email:[email protected]. † Corresponding author. Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009, Storrs 06269-1009, USA. Email: [email protected]. Phone: +1-860-486-3921. L times R . One can easily see thatsuch an assumption is equivalent to allowing the insurer to purchase proportional reinsurance to manageher risk exposure from underwriting. Please refer to Højgaard and Taksar (1998) and Schmidli (2001) foran excellent introduction on optimal proportional reinsurance. The insurer in consideration then decidesthe amount π to be invested in the risky asset and the liability units L in the insurance business, with thepurpose to achieve her optimization objective.In the control literature within actuarial science, there are three popular choices for the optimizationobjective: (1) utility maximization, cf. Yang and Zhang (2005), Bai and Guo (2008), and Liang et al.(2011); (2) risk minimization (e.g., ruin probability, VaR, and CVaR), cf. Schmidli (2001), Liu and Yang(2004), and Promislow and Young (2005); and (3) the mean-variance (MV) criterion, with the last one usedin this paper. Please see a recent review article Cai and Chi (2020), the monograph Albrecher et al. (2017),and the references therein for the rich literature on optimal reinsurance under the first two objectives. MVportfolio selection (without risk control or reinsurance) is first studied by Markowitz (1952) and is truly acornerstone of the modern portfolio theory. There are so many works extending Markowitz (1952) alongnumerous directions that it is almost impossible to give credits to all of them even in a review article. Herewe only mention Li and Ng (2000), Zhou and Li (2000), Basak and Chabakauri (2010), Bj¨ork and Murgoci(2010), and Bj¨ork et al. (2014) for a short list. It is well known that a standard dynamic MV problem isa time-inconsistent control problem, in the sense that an optimal strategy obtained at time t may ceaseto be optimal at a later time t . In order for such a strategy to be followed after the initial time, theagent needs to commit herself to it. Hence, this type of optimal strategy is called the precommitment strategy in the literature; see Zhou and Li (2000). However, a rational agent who is aware of such a time-inconsistency issue should search for a time-consistent equilibrium strategy, which she has no incentiveto deviate from once obtained; see Basak and Chabakauri (2010) and Kryger and Steffensen (2010). Fora complete analysis on a more general framework in both discrete and continuous time models, we referreaders to the influential work of Bj¨ork and Murgoci (2010). In the coming paragraph, we shall focus onoptimal MV (proportional) reinsurance problems and provide a selective literature review on the topic.B¨auerle (2005) is among the early contributions to optimal MV reinsurance problems. The authorapplies the embedding technique and the standard Hamilton-Jacobi-Bellman (HJB) approach to obtainthe precommitment proportional reinsurance strategy under the CL model. Bai and Zhang (2008) furtheradd a no short-selling constraint on the investment strategy and solve for the precommitment strategy underboth the CL model and the diffusion model for the risk process. Shen and Zeng (2014) introduce delay intothe controlled portfolio and apply a stochastic maximal principle to handle such a problem. Shen and Zeng(2015) propose an asset model in which both the appreciation and volatility of the risky asset follow non-Markovian processes, and apply a backward stochastic differential equation (BSDE) approach to obtain2he precommitment strategy. On the other hand, time-consistent MV investment-reinsurance problems arefirst studied by Zeng and Li (2011) under a standard Black-Scholes type financial market and a diffusionrisk process. They apply the method from Bj¨ork and Murgoci (2010) and obtain the equilibrium strategyby solving an extend HJB system. Zeng and his collaborators further include jumps in both risky assetsand the risk process in Zeng et al. (2013) and ambiguity aversion in Zeng et al. (2016). Li and Li (2013)allow the risk aversion to be state dependent, similar to that in Bj¨ork et al. (2014). See Li et al. (2015)for extension with stochastic interest rate and inflation risk. A latest paper by Cao et al. (2020) studiesthe problem in a contagious model where the risk process is given by a self-exciting Hawkes process. Theabove mentioned papers consider Markovian, also called feedback or closed-loop, controls in the analysis.Two recent works Wang et al. (2019) and Yan and Wong (2020) consider open-loop controls under a modelwith random parameters and a stochastic volatility model, respectively.This paper lies in the category of time-consistent strategies to MV investment and proportional rein-surance problems. We summarize the main results and contributions of this paper as follows. • We propose an alternative time-consistent formulation to the insurer’s original time-inconsistentMV problem, which is different from both the general approach of Bj¨ork and Murgoci (2010) andthe special approach of Basak and Chabakauri (2010). To be precise, we introduce an auxiliary deterministic process Y defined forward in time and use Y to replace the conditional expectation ofthe insurer’s wealth X in the original objective J , leading to a modified MV objective J (see (2.11)for details). We then consider an alternative MV problem in which the pair ( X, Y ) are taken asthe state processes and J is the optimization objective. Because of the introduction of Y and theenlargement of the state space in the definition of J (which together “kill” the troubling varianceterm in J ), the alternative MV problem is a time-consistent control problem and can be solvedby the standard HJB method. In comparison, Bj¨ork and Murgoci (2010) introduce an auxiliary stochastic process defined backward in time to handle the square term in the MV problems; whileBasak and Chabakauri (2010) take a clever use of the total variance formula to derive a heuristic HJBequation. In terms of applicability to time-inconsistent problems, the approach of Bj¨ork and Murgoci(2010) is the most general one and the approach of Basak and Chabakauri (2010) is likely the mostrestrictive one, as it only applies to the MV problems. The approach of this paper is on the specialside as Basak and Chabakauri (2010), but it can also handle general non-linear term(s) involvingconditional expectation in the objective, other than the square term in the MV objective. • The main approach used in this paper follows from Yang (2020), which considers a standard MVportfolio selection problem without risk control in a Black-Scholes market model. On the technicallevel, we extend the work of Yang (2020) by including an additional risk control strategy for an insurerwhose risk process is modeled by a jump-diffusion process and is correlated with the risky asset inthe financial market. Such an extension leads to many interesting findings along both analytical andnumerical directions. 3
We fully solve both the time-inconsistent and the time-consistent MV investment and risk con-trol problems in the paper, i.e., we obtain both the precommitment and the (time-consistent)optimal strategies, and the efficient frontier and the value function of both problems in closed-form. Utilizing these explicit results, we conduct a comprehensive analysis to compare our optimaland precommitment strategies with those obtained in the standard game-theoretic framework (seeBj¨ork and Murgoci (2010)) and the precommitted framework (see Zhou and Li (2000)). • We also make contributions to the literature in the direction of economic analysis on the insurer’soptimal strategies, which is missing or insufficient in many related works including Yang (2020).We discuss how the model parameters and the insurer’s risk profiles affect the optimal strategiesboth analytically (see Section 3.3) and numerically (see Section 5). In particular, we find that thecorrelation between the financial market and the insurance market (risk process) plays a key rolein the insurer’s optimal decisions. Both analytical results (see Table 1) and numerical findings (seeFigure 1) show that when the correlation coefficient ρ is positive, the insurer increases her investmentin the risky asset and holds more liabilities from underwriting as ρ increases. When ρ is negative, weobtain several interesting findings. First, the optimal liability strategy is no longer monotone withrespect to ρ , but rather has a convex relation, decreasing first and then increasing. Next, when ρ isvery negative (close to -1), the optimal investment involves short selling the risky asset and a lessrisk averse insurer short sells more risky asset.We organize the rest of this paper as follows. In Section 2, we present the market model and statethe insurer’s MV investment and risk control problems. In Section 3, we obtain explicit solutions to theinsurer’s MV problems. We then compare our formulation, approach, and optimal strategy with thoseunder the game-theoretic and precommitted framework in Section 4. Section 5 contains our numericalstudies which focus on the impact of correlation and jumps on the optimal strategy. Our conclusions aresummarized in Section 6. Appendix A collects technical proofs. Let us fix a complete filtered probability space (Ω , F , F = ( F t ) t ∈ [0 ,T ] , P ) over a finite time horizon [0 , T ],with T < ∞ . Here F t contains all the information up to time t and the filtration F satisfies the usualhypotheses. We interpret P as the physical probability measure. We assume all the stochastic processesbelow are well defined and adapted to the given filtration F .We consider an insurer who makes business decisions in a combined financial and insurance market,similar to the one introduced in Zeng and Li (2011) and Zou and Cadenillas (2014). In the financialmarket, there is one risk-free asset and one risky asset (a stock or an index) whose price dynamics are4iven respectively byd S ( t ) = rS ( t ) d t, S (0) = 1 , (2.1) d S ( t ) = S ( t ) (cid:0) µ d t + σ d W ( t ) (cid:1) , S (0) > , (2.2)where r, µ, σ > W is a standard Brownian motion. Here, r is the risk-freeinterest rate, and µ and σ are the appreciation rate and volatility of the risky asset, respectively. Theinsurer can dynamically trade both assets without frictions and taxes. In the insurance market, we modelthe insurer’s unit liabilities (risk) R = ( R ( t )) t ∈ [0 ,T ] by the following jump-diffusion process:d R ( t ) = α d t + β (cid:16) ρ d W ( t ) + p − ρ d W ( t ) (cid:17) + Z R γ ( t, z ) N (d t, d z ) , (2.3)where α, β ≥ ρ ∈ [ − , W is another standard Brownian motion independent of W , N is thePoisson random measure, and γ ( t, z ) > t and z . Here, ρ captures the correlation between thefinancial market and the insurance market.We impose several technical assumptions on the models (2.1)-(2.3) that will be enforced throughoutthe rest of the paper. Suppose the compensated Poisson random measure e N is given by e N (d t, d z ) = N (d t, d z ) − λ d t d F Z ( z ) , (2.4)where λ > F Z is the distribution function of a random variable Z . Inaddition, γ ( t, · ) = γ ( · ) is homogeneous and deterministic, and γ ( Z ) has finite first and second moments,i.e., γ := Z R γ ( z ) d F Z ( z ) ∈ (0 , ∞ ) and γ := Z R γ ( z ) d F Z ( z ) ∈ (0 , ∞ ) . (2.5)We assume W , W , N , and Z are stochastically independent, and the filtration F is generated by themand augmented with P -null sets.In the financial market, the insurer chooses an investment strategy π = ( π ( t )) t ∈ [0 ,T ] , where π ( t ) denotesthe amount of wealth invested in the risky asset at time t . In the insurance market, the insurer chooses arisk control strategy (or a liability strategy) L = ( L ( t )) t ∈ [0 ,T ] , where L ( t ) denotes the amount of liabilitiesin the underwriting at time t . Assume the unit premium rate, corresponding to the unit liabilities (risk) R , is given by p , where p >
0. For any fixed risk control strategy L , the gains from the insurance businessevolve according to L ( t ) (cid:0) p d t − d R ( t ) (cid:1) . Remark 2.1.
In the combined market model (2.1) - (2.3) , we set the model parameters ( r , µ , σ , α , β ,and λ ) to be positive constants. We comment that all the analysis and results in the sequel hold if theseparameters are given by deterministic, bounded, and positive processes, and the volatility process σ isbounded away from zero (i.e., there exists a positive constant K such that σ ( t ) ≥ K > ). Similarly, if thedeterministic function γ ( t, · ) is not time homogeneous, we need to replace γ i in (2.5) by γ i ( t ) , and assume i ( t ) are bounded for all t , where i = 1 , . Given that all the parameters are constants and the function γ ishomogeneous, we set the unit premium rate to be a positive constant p . The above assumptions on modeling,albeit strong, are standard and popular in the related literature; see, e.g, Schmidli (2001), Yang and Zhang(2005), Moore and Young (2006), and Zeng and Li (2011).The risk model (2.3) incorporates several well known models in actuarial science. If we set α = β = 0 ,then the model (2.3) is a generalization of the classical Cram´er-Lundberg (CL) model. Recall the risk process R ( t ) in the CL model is given by R ( t ) = P b N ( t ) i =1 b C i , where b N = ( b N ( t )) t ∈ [0 ,T ] is a homogeneous Poissonprocess with constant intensity b λ and ( b C i ) i =1 , , ··· is a series of independent and identically distributedrandom variables, also independent of b N . Comparing (2.3) with the CL model shows that (i) λ = b λ and(ii) γ ( t, Z ) and b C has the same distribution. If we set λ = 0 , i.e., no jumps in (2.3) , then the model (2.3) can be seen as a diffusion approximation to the CL model; see, e.g., Browne (1995), Højgaard and Taksar(1998), and Moore and Young (2006). In such a case, we have α = b λ E [ b C ] and β = b λ E [ b C ] .In our framework, we interpret L as the amount of liabilities the insurer decides to take in the in-surance business, and p as the premium rate the insurer receives from underwriting the policies againstthe risk R . This modeling choice follows from Stein (2012)[Chapter 6] and its subsequent studies suchas Zou and Cadenillas (2014), Peng and Wang (2016), and Bo and Wang (2017), where the motivationcomes from the AIG case in the financial crisis of 2007-2008 and argues for a negative correlation ρ < .It is clear that our risk control setup is consistent with the model of proportional reinsurance, animportant topic in actuarial science. In the latter case, we should understand R in (2.3) as the risk processof the insurer, and p the reinsurance premium paid by the insurer to the reinsurer. To manage risk, theinsurer chooses proportional insurance, with L denoting the retention proportion. The dynamics of thegains process are then given by L ( t ) d R ( t ) − p (1 − L ( t )) d t . Indeed, our setup (with λ = 0 ) is the same tothat in Zeng and Li (2011) by taking m = 0 and σ to be the negative value there. In the combined market described above, let us introduce u = ( π, L ) as a shorthanded notation forthe insurer’s control or strategy. As usual, we consider self-financing strategies only in the analysis. For afixed strategy u , we write X u = ( X u ( t )) t ∈ [0 ,T ] as the insurer’s wealth process, and obtain the dynamics of X u by d X u ( t ) = (cid:0) rX u ( t ) + ¯ µπ ( t ) + ¯ pL ( t ) (cid:1) d t + (cid:0) σπ ( t ) − ρβL ( t ) (cid:1) d W ( t )(2.6) − β p − ρ L ( t ) d W − L ( t ) Z R γ ( z ) N (d t, d z ) , where the initial wealth X (0) is a (positive) constant, and ¯ µ and ¯ p are defined by¯ µ := µ − r and ¯ p := p − α. (2.7)If we fix an initial state X u ( t ) = x at time t , where t ∈ [0 , T ], and want to emphasize the dependence of X u on the initial state, we use the notation ( X ut,x ( s )) s ∈ [ t,T ] .6o make sure the stochastic differential equation (SDE) (2.6) admits a unique strong solution, we needto impose square integrability conditions on strategies. Given the nature of the Markovian framework,we consider Markov (feedback) controls, π ( t ) = e π ( t, X u ( t )) and L ( t ) = e L ( t, X u ( t )), for some deterministicfunctions e π and e L . We are now ready to state the admissible set of strategies, denoted by A , as follows. Definition 2.2.
A strategy u is called admissible if (1) u is progressively measurable with respect to theunderlying filtration F , (2) E [ R T π ( t ) d t ] < ∞ and E [ R T L ( t ) d t ] < ∞ , (3) u is a feedback control, and(4) X u satisfies the SDE (2.6) . Remark 2.3.
In defining the admissible set, we do not impose L ≥ . In other words, we allow L < ,which corresponds to the case of taking a strategy greater than 1 in the proportional reinsurance and isoften interpreted as acquiring new businesses in the literature; see, e.g., B¨auerle (2005) and Zeng and Li(2011). If we insist on L ≥ , we can impose an extra condition on the model parameters, so that theoptimal liability strategy L ∗ is always non-negative; see Eq.(6) in Zou and Cadenillas (2014). We consider a representative insurer who is a mean-variance (MV) type agent. Namely, the insurer prefershigher mean and lower variance of her terminal wealth. Following the standard literature of Li and Ng(2000) and Zhou and Li (2000), we define the insurer’s objective functional J by J ( t, x ; u ) := E t,x [ X u ( T )] − θ V t,x [ X u ( T )] , (2.8)where θ > X u is given by (2.6), and E t,x (resp. V t,x ) denotes takingconditional expectation (resp. variance) given X u ( t ) = x under the physical measure P . We now state theinsurer’s MV investment and risk control problem as follows. Problem 2.4 (A Time-Inconsistent MV Problem) . The insurer seeks a strategy u pre = ( u pre ( s )) s ∈ [ t,T ] tomaximize the objective functional J defined in (2.8) , i.e., the insurer solves the following MV problem: V ( t, x ) := sup u ∈A J ( t, x ; u ) . (2.9) We call V the value function to Problem (2.9) . As is well known in the literature, a solution to Problem (2.9) is time-inconsistent , and hence theinsurer has the incentive to deviate from such a strategy at a later time s > t . As such, a strategy solvingProblem (2.9) will be followed over the remaining period [ t, T ] only if the insurer commits to it. For thisreason, we call a solution to Problem (2.9) a precommitment strategy with notation u pre . Time-consistentformulations and approaches to the MV problems are then proposed and investigated, while most, if notall, of them follow the influential work of Bj¨ork and Murgoci (2010). However, we will take a differentapproach from Yang (2020) in this paper, which is, although less general, simple and sufficient to handlethe problem under our framework. 7nspired by Yang (2020), for any u = ( π, L ) ∈ A , we introduce an auxiliary process ( Y ut,y ( s )) s ∈ [ t,T ] ,which is defined forward in time byd Y ut,y ( s ) = (cid:0) rY ut,y ( s ) + ¯ µ E t,y (cid:2) π ( s ) (cid:3) + (¯ p − λγ ) E t,y (cid:2) L ( s ) (cid:3)(cid:1) d s, s ∈ [ t, T ] , (2.10)where Y ut,y ( t ) = y is the initial state for arbitrary but fixed t ∈ [0 , T ] and y ∈ R , γ is given by (2.5), and ¯ µ and ¯ p are defined in (2.7). Here, the initial state value y may be different from that of X ut,x ( t ) = x . Using(2.6), we obtain that E t,y [ X ut,y ( s )] = y + E t,y (cid:20)Z st (cid:0) rX ut,y ( v ) + ¯ µπ ( v ) + (¯ p − λγ ) L ( v ) (cid:1) d v (cid:21) + E t,y (cid:20)Z st (cid:0) σπ ( v ) − βρL ( v ) (cid:1) d W ( v ) (cid:21) − E t,y (cid:20)Z st β p − ρ L ( v ) d W ( v ) (cid:21) − E t,y (cid:20)Z st Z R L ( v ) γ ( z ) e N (d v, d z ) (cid:21) = y + Z st (cid:0) r E t,y (cid:2) X ut,y ( v ) (cid:3) + ¯ µ E t,y (cid:2) π ( v ) (cid:3) + (¯ p − λγ ) E t,y (cid:2) L ( v ) (cid:3)(cid:1) d v, ∀ s ∈ [ t, T ] , where X ut,y ( t ) = y , and e N and γ are defined respectively by (2.4) and (2.5). Here, we have used the squareintegrability conditions of π and L from Definition 2.2 and γ = E [ γ ( Z )] < ∞ in (2.5) to conclude thatthe (conditional) expectations of the two Itˆo integrals and the integral with respect to e N are zero. Bycomparing the above result with the dynamics of Y in (2.10), we see that Y ut,y ( s ) = E t,y [ X ut,y ( s )] for all s ∈ [ t, T ] and y ∈ R .We now treat both X and Y as state processes and consider a modified objective functional J definedby J ( t, x, y ; u ) = E t,x,y (cid:20) X ut,x ( T ) − θ (cid:0) X ut,x ( T ) − Y ut,y ( T ) (cid:1) (cid:21) , (2.11)where θ > X u and Y u are given respectively by (2.6) and (2.10), and E t,x,y denotes taking conditional expectation under X ut,x ( t ) = x and Y ut,y ( t ) = y . Notice that there is afundamental difference between J in (2.8) and J in (2.11). That is, we no longer have the “troubling”square term ( E t,x [ X u ( T )]) , which causes time-inconsistency in Problem (2.9). Based on the new objectivefunctional J , we formule a time-consistent version of the original Problem (2.9). Problem 2.5 (A Time-Consistent MV Problem) . The insurer seeks an optimal strategy u ∗ to maximizethe objective functional J defined in (2.11) , i.e., the insurer solves the following MV problem: V ( t, x, y ) = sup u ∈A J ( t, x, y ; u ) . (2.12) We call V the value function to Problem (2.12) . In this section, we first solve the insurer’s time-inconsistent problem (Problem (2.9)) to obtain a precom-mitment strategy in Section 3.1 and then solve the insurer’s time-consistent problem (Problem (2.12)) to8btain an optimal strategy in Section 3.2. We discuss the economic implications based on the results ofProblem (2.12) in Section 3.3.We impose a standing assumption for the subsequent analysis: β (1 − ρ ) + λγ = 0 , (3.1)where γ is given by (2.5). The assumption in (3.1) is rather weak and holds in most conditions. In fact,it only fails when there are (1) no jumps ( λ = 0) and (2) no diffusion term ( β = 0) or perfect correlation( ρ = ± In this subsection, we solve the insurer’s time-inconsistent MV problem, as formulated in Problem (2.9),and obtain explicit solutions in the following theorem.
Theorem 3.1.
A precommitment strategy to Problem (2.9) , denoted by u pre = ( π pre ( s ) , L pre ( s )) s ∈ [ t,T ] , isgiven by π pre ( s ) = − κ (cid:18) X pre ( s ) − x e r ( T − t ) − θ e κ ( T − t ) (cid:19) and L pre ( s ) = − κ κ π pre ( s ) , (3.2) where X pre is the wealth process under the precommitment strategy u pre and the initial state ( t, x ) , and theconstants κ i , i = 1 , , , are given by κ := ¯ µ ( β + λγ ) + ρβσ (¯ p − λγ )( β (1 − ρ ) + λγ ) σ , κ := ρβ ¯ µ + (¯ p − λγ ) σ ( β (1 − ρ ) + λγ ) σ , (3.3) κ := (cid:0) β + λγ (cid:1) ¯ µ + 2 ρβσ ¯ µ (¯ p − λγ ) + (¯ p − λγ ) σ ( β (1 − ρ ) + λγ ) σ = ¯ µ σ + (cid:16) ¯ p − λγ − ρβ ¯ µσ (cid:17) β (1 − ρ ) + λγ , (3.4) with γ and γ defined in (2.5) , and ¯ µ and ¯ p defined in (2.7) .Proof. Please refer to Appendix A for a proof.
Proposition 3.2.
Let X pre be the insurer’s wealth process under the precommitment strategy u pre givenby (3.2) . We have E t,x [ X pre ( T )] = x e r ( T − t ) + e κ ( T − t ) − θ and V t,x [ X pre ( T )] = e κ ( T − t ) − θ , (3.5) where κ is given in (3.4) . The efficient frontier of Problem (2.9) is obtained by V t,x [ X pre ( T )] = (cid:0) E t,x [ X pre ( T )] − x e r ( T − t ) (cid:1) e κ ( T − t ) − , and the value function (mean-variance tradeoff ) of Problem (2.9) is given by V ( t, x ) = E t,x [ X pre ( T )] − θ V t,x [ X pre ( T )] = x e r ( T − t ) + e κ ( T − t ) − θ . (3.6) 9 roof. By plugging u pre in (3.2) back into the SDE (2.6), we obtain the above explicit results. Remark 3.3.
From (3.2) , one can easily see that the strategy u pre = ( π pre , L pre ) strongly depends onthe initial state ( t, x ) , so a more precise but also more cumbersome notation is to replace it by u pre t,x =( π pre t,x , L pre t,x ) . Also it is obvious from (3.2) that u pre is indeed time-consistent as claimed. To see this, let t < t < t < T and X pre t,x ( t ) be the corresponding wealth at time t under the strategy u pre . Using (3.2) ,we have u pre t,x ( t ) = u pre t ,X pre t,x ( t ) ( t ) in general. That means the “best” strategy for a future time t found atthe state ( t, x ) is not the same as the one found at the state ( t , X pre t,x ( t )) . Here, by the “best” strategy,we mean a solution to Problem (2.9) .In Problem (2.9) , θ is a free parameter, called the insurer’s risk aversion parameter, which specifiesthe insurer’s risk attitude towards the mean-variance tradeoff. Since κ > due to (3.4) and θ > bydefinition, we derive from (3.5) that there is a one-to-one relation between the target expected terminalwealth m := E t,x [ X pre ( T )] and the risk aversion parameter θ , for all m > x e r ( T − t ) . The key to solving Problem (2.12) is the standard Hamilton-Jacobi-Bellman (HJB) approach, as presentedin Theorem 3.4. The proof to this theorem is rather standard in the literature, which is based on the flowproperty of SDEs, dynamic programming principle, and a verification theorem. We omit the proof hereand refer readers to Theorems 3.3 and 3.4, and Proposition 3.5 in Chapter 4 of Yong and Zhou (1999) fora standard proof on a more general control problem.
Theorem 3.4.
Suppose there exists a classical solution V to Problem (2.12) . Then V solves the followingHamilton-Jacobi-Bellman (HJB) equation: V t ( t, x, y ) + sup π, L ∈ R (cid:26) ( rx + ¯ µπ + ¯ pL ) V x ( t, x, y ) + 12 (cid:2) ( σπ − ρβL ) + β (1 − ρ ) L (cid:3) V xx ( t, x, y )+( ry + ¯ µπ + (¯ p − λγ ) L ) V y ( t, x, y ) + λ Z R (cid:2) V ( t, x − Lγ ( z ) , y ) − V ( t, x, y ) (cid:3) d F Z ( z ) (cid:27) = 0 , (3.7) for all ( t, x, y ) ∈ [0 , T ) × R × R , and satisfies the terminal condition V ( T, x, y ) = x − θ x − y ) , ∀ x, y ∈ R . (3.8)By applying Theorem 3.4, we obtain explicit solutions to the optimal strategy and the value functionof Problem (2.12), as summarized below. Theorem 3.5.
An optimal strategy to Problem (2.12) , denoted by u ∗ = ( π ∗ ( s ) , L ∗ ( s )) s ∈ [ t,T ] , is given by π ∗ ( s ) = κ θ e − r ( T − s ) and L ∗ ( s ) = κ θ e − r ( T − s ) , s ∈ [ t, T ] , (3.9) where κ and κ are defined in (3.3) . The value function V ( t, x, y ) to Problem (2.12) is given by V ( t, x, y ) = − θ e r ( T − t ) ( x − y ) + e r ( T − t ) x + κ θ ( T − t ) , (3.10) where κ is defined in (3.4) . roof. Please see Appendix A for a proof.
Remark 3.6.
Since both the optimal investment strategy π ∗ and the optimal liability strategy L ∗ in (3.9) are independent of the initial state ( t, x ) , the optimal strategy u ∗ in (3.9) is indeed time-consistent. Wecall u ∗ an optimal strategy, instead of an equilibrium strategy, since the alternative formulated Problem (2.12) is a standard time-consistent stochastic control problem.We next comment on possible generalizations to the model (2.1) - (2.3) . First, as mentioned in Remark2.1, it is straightforward to extend to the case when all model parameters are deterministic and boundedprocesses. Indeed, we simply replace all the constant parameters in κ i by their corresponding (deterministic)process version in κ i ( t ) , where i = 1 , , , and the product with time arguments by an appropriate integral,e.g., we change r ( T − t ) to R Tt r ( s ) d s and κ ( T − t ) to R Tt κ ( s ) d s in Theorems 3.1 and 3.5. Second, weconsider only one risky asset in our model, but the analysis and key results in Theorems 3.1 and 3.5 applyto the case of multiple risky assets in a parallel way once we use appropriate matrix notation. In fact, ourtechnique is adequate to handle an incomplete market with n risky assets driven by d independent Brownianmotions, where n ≤ d . In such a case, we modify the assumption that σ > to σ ( t ) σ ( t ) ⊤ ≥ K I n × n forsome positive K and all t ∈ [0 , T ] , where ⊤ denotes transpose operator and I n × n is an n × n identitymatrix. We next derive the efficient frontier of Problem (2.12) and present the results in the proposition below.
Proposition 3.7.
Let X ∗ be the insurer’s wealth process under the optimal strategy u ∗ given by (3.9) . Weobtain the dynamic efficient frontier of Problem (2.12) by V t,x [ X ∗ ( s )] = (cid:0) E t,x [ X ∗ ( s )] − x e r ( s − t ) (cid:1) κ ( s − t ) , t < s ≤ T, (3.11) and the mean-variance tradeoff by E t,x [ X ∗ ( s )] − θ V t,x [ X ∗ ( s )] = x e r ( s − t ) + κ θ e − r ( T − s ) (cid:18) − e − r ( T − s ) (cid:19) ( s − t ) , t ≤ s ≤ T. (3.12) Proof.
By plugging (3.9) into (2.6), we obtain e r ( T − s ) X ∗ ( s ) = e r ( T − t ) X ∗ ( t ) + ¯ µκ + (¯ p − λγ ) κ θ ( s − t ) + σκ − ρβκ θ (cid:0) W ( s ) − W ( t ) (cid:1) (3.13) − β p − ρ κ θ (cid:0) W ( s ) − W ( t ) (cid:1) − κ θ Z st Z R γ ( z ) e N (d v, d z ) , where κ and κ are defined in (3.3). Taking expectation and variance on (3.13) given the initial state( t, X ∗ ( t ) = x ), we get E t,x [ X ∗ ( s )] = x e r ( s − t ) + ¯ µκ + (¯ p − λγ ) κ θ e − r ( T − s ) ( s − t )(3.14) = x e r ( s − t ) + κ θ e − r ( T − s ) ( s − t ) , t,x [ X ∗ ( s )] = e − r ( T − s ) ( s − t ) θ h ( σκ − ρβκ ) + β (1 − ρ ) κ + λγ κ i (3.15) = κ θ e − r ( T − s ) ( s − t ) , where we have used the following result¯ µκ + (¯ p − λγ ) κ = ( σκ − ρβκ ) + β (1 − ρ ) κ + λγ κ = κ > . (3.16)We obtain (3.16) by recalling the definitions of κ i , i = 1 , ,
3, in (3.3) and (3.4). Combining (3.14), (3.15),and (3.16) leads to the above dynamic efficient frontier of Problem (2.12).With explicit results obtained in (3.14) and (3.15), we can provide a further verification of the optimalinvestment strategy u ∗ in (3.9). To this end, let us recall the definition of Y u in (2.10) and the objectivefunctional J in (2.11). Denote X ∗ and Y ∗ the corresponding processes under the optimal strategy u ∗ . Weobtain J ( t, x, y ; u ∗ ) = E t,x,y (cid:20) X ∗ t,x ( T ) − θ (cid:0) X ∗ t,x ( T ) − Y ∗ t,y ( T ) (cid:1) (cid:21) = E t,x (cid:2) X ∗ t,x ( T ) (cid:3) − θ V t,x (cid:2) X ∗ t,x ( T ) (cid:3) − θ (cid:0) E t,x (cid:2) X ∗ t,x ( T ) (cid:3) − E t,y (cid:2) Y ∗ t,y ( T ) (cid:3)(cid:1) = x e r ( T − t ) + κ θ ( T − t ) − κ θ ( T − t ) − θ (cid:16) x e r ( T − t ) − y e r ( T − t ) (cid:17) = V ( t, x, y ) derived in (3.10) . (3.17)Hence, u ∗ given by (3.9) is optimal to Problem (2.12). In this subsection, we present economic discussions on the explicit results of Problem (2.12) in Theorem3.5 and Proposition 3.7.We first derive two corollaries when the insurer has no access to the financial market or decides notto take any insurance business. To this purpose, we directly follow the same arguments in the proof ofTheorem 3.5 by setting π ≡ L ≡
0) and obtain the optimal strategy and the value function in eachcase. We skip the cumbersome computations and report the results below.
Corollary 3.8.
If the insurer has no access to the financial market ( π ( s ) ≡ for all s ∈ [ t, T ] ), the optimalrisk control strategy L ∗ to Problem (2.12) is given by L ∗ ( s ) (cid:12)(cid:12)(cid:12) π ≡ = ¯ p − λγ ( β + λγ ) θ e − r ( T − s ) , ∀ s ∈ [ t, T ] , and the value function is given by V ( t, x, y ) (cid:12)(cid:12)(cid:12) π ≡ = − θ e r ( T − t ) ( x − y ) + e r ( T − t ) x + κ θ ( T − t ) , where κ = (¯ p − λγ ) β + λγ . he loss due to no access to investing in the financial market is measured by V ( t, x, y ) − V ( t, x, y ) (cid:12)(cid:12)(cid:12) π ≡ = (cid:0) ¯ µ ( β + λγ ) + ρβσ (¯ p − λγ ) (cid:1) ( β + λγ )( β (1 − ρ ) + λγ ) · T − t θ > . Corollary 3.9.
If the insurer sets L ( s ) ≡ for all s ∈ [ t, T ] , the optimal investment strategy π ∗ to Problem (2.12) is given by π ∗ ( s ) (cid:12)(cid:12)(cid:12) L ≡ = ¯ µθσ e − r ( T − s ) , ∀ s ∈ [ t, T ] , (3.18) and the value function is given by V ( t, x, y ) (cid:12)(cid:12)(cid:12) L ≡ = − θ e r ( T − t ) ( x − y ) + e r ( T − t ) x + ¯ µ θσ ( T − t ) . The loss due to not taking insurance business is measured by V ( t, x, y ) − V ( t, x, y ) (cid:12)(cid:12)(cid:12) L ≡ = (cid:16) ¯ p − λγ − ρβ ¯ µσ (cid:17) β (1 − ρ ) + λγ · T − t θ > . Several important remarks and explanations are due regarding Theorem 3.5 and Corollaries 3.8 and3.9. First, the optimal strategy depends on both the financial market and the insurance market. Namely, π ∗ also depends on the risk model (2.3) and π ∗ = 0 in general, while L ∗ depends on the price models(2.1)-(2.2) and L ∗ = 0 either. This observation testifies the importance of considering a combined financialand insurance market in the risk management study for an insurer. Corollary 3.8 further shows thatthe loss in the value function is strictly positive for all t ∈ [0 , T ); not having the access to the financialmarket makes the insurer worse off. On the other hand, when the insurer does not engage in the insurancebusiness (i.e., L = 0), the insurer invests as if she were a utility maximizer equipped with an exponentialutility U ( x ) = − e − θx . In this case, the loss in the value function is also strictly positive, as shown inCorollary 3.9, even when the insurance policy is “cheap”, i.e., when ¯ p = λγ ( p d t = E [d R ( t )]). Along thisdiscussion, given ¯ p ≤ λγ (insurance policies are underpriced), the optimal risk control strategy L ∗ maystill be positive, if ρβ ¯ µ > ρ >
0, given ¯ µ >
0. Thisfinding is certainly interesting, as the insurer takes positive shares in a “losing” insurance business, whichseems an irrational decision at first thought. But a second thought reveals that such a business provides anatural hedge to the risky asset, making it still desirable to hold positive shares in the portfolio. Anotherplausible explanation is that the insurer sees underwriting policies as a financing tool to raise capital forinvestment purpose. If the gain from investing premiums in the financial market outweighs the shortfalldue to underpricing policies, the insurer indeed has the incentive to sell underpriced insurance policies.Next, we investigate the impact of the correlation between the two markets on the optimal strategy.In the extreme case of ρ = 0, i.e., when the two markets are independent, we have π ∗ ( s ) (cid:12)(cid:12)(cid:12) ρ =0 = ¯ µθσ e − r ( T − s ) and L ∗ ( s ) (cid:12)(cid:12)(cid:12) ρ =0 = ¯ p − λγ θ ( β + λγ ) e − r ( T − s ) , ∀ s ∈ [ t, T ] , Please see Remark 1 in Basak and Chabakauri (2010) for further discussions on the connection with exponential utilitymaximization. π ∗ | ρ =0 = π ∗ | L =0 is the same as the optimalstrategy in the standard Merton’s problem with an exponential utility U ( x ) = − e − θx . For the given riskprocess R in (2.3), if p ≤ α + λγ (i.e., ¯ p ≤ λγ ), ruin occurs for sure to the insurer. With that in mind,let us suppose the following conditions hold for the rest of the section:¯ p − λγ > µ = µ − r > , (3.19)where the second inequality means the risky asset has higher return than the risk-free asset. Let us denote π ∗ | ρ> the optimal investment in the risky asset under a positive correlation ρ . We rewrite the optimalstrategy in (3.9) as follows π ∗ ( s ) = β + λγ β (1 − ρ ) + λγ · π ∗ ( s ) (cid:12)(cid:12)(cid:12) ρ =0 + ρ β (¯ p − λγ )( β (1 − ρ ) + λγ ) σ θ e − r ( T − s ) , (3.20)from which we deduce π ∗ ( s ) | ρ> > π ∗ ( s ) | ρ =0 . The economic meaning is that an insurer, with risk positivelycorrelated with the risky asset, invests more aggressively in the risky asset, as if she were less risk averse.This makes perfect sense, since, with ρ >
0, a decrease in the price of the risky asset is accompanied by adecrease in the liabilities to be paid out, making the risky asset less risky to the insurer. However, whenthese two markets are negatively correlated, less can be said, as although the factor in the first term in(3.20) is still greater than 1, the second term in (3.20) is negative. In consequence, we expect differentparameter values lead to different monotonicity results. One could carry out the same analysis on theoptimal risk control strategy L ∗ , and the findings are the same. We point out that the numerical analysisin Zou and Cadenillas (2014) shows π ∗ is increasing with respect to ρ ∈ ( − , L ∗ seems to exhibit a“smile” shape (decreasing first and then increasing) as a function of negative ρ .We continue to study the impact of other model parameters on the optimal strategy. After carefulcomputations and analysis, we summarize the impact of all the model parameters on the optimal strategyand the value function in Table 1. Note the results are obtained under the additional conditions in (3.19).As the excess return ¯ µ (recall ¯ µ = µ − r ) increases, the optimal investment strategy π ∗ increases, and theoptimal liability strategy L ∗ increases (resp. decreases) if and only if ρ > ρ < p (recall ¯ p = p − α ) on the optimal strategy is exactly the opposite, comparing to thatof ¯ µ . However, how the rest of parameters affect the optimal strategy is less clear, with only partial resultsas presented in Table 1. As already discussed in details in the proceeding paragraph, when ρ >
0, thefinancial and insurance markets provide a natural hedge to each other, which allows us to gain full insighton the optimal strategy. When the two markets are negatively correlated, argued in Stein (2012) to be amain contributor to the failure of AIG during the financial crisis, the change of a parameter (e.g., β and λ )results in opposite reactions from the two markets. To better explain this result, let us consider the impactof the asset volatility σ on the optimal investment π ∗ . From (3.20), it is clear that, as σ increases, themyopic component ¯ µ/ ( θσ ) decreases, but the second hedging component increases if ρ <
0. In summary,14he case of negative correlation ρ < π ∗ Optimal Liability L ∗ Value Function V correlation ρ ∂π ∗ /∂ρ > ρ > ∂L ∗ /∂ρ > ρ > ∂ V /∂ρ < ρ < µ ∂π ∗ /∂ ¯ µ > ∂L ∗ /∂ ¯ µ = sign ( ρ ) ∂ V /∂ ¯ µ > ρ < σ ∂π ∗ /∂σ < ρ > ∂L ∗ /∂σ = sign ( ρ ) ∂ V /∂σ > ρ < p ∂π ∗ /∂ ¯ p = sign ( ρ ) ∂L ∗ /∂ ¯ p > ∂ V /∂ ¯ p < ρ < β ∂π ∗ /∂β > ρ > ∂L ∗ /∂β > ρ = 0 ∂ V /∂β < ρ < λ ∂π ∗ /∂λ < ρ > ∂L ∗ /∂λ < ρ > ∂ V /∂λ < ρ > γ ∂π ∗ /∂γ < ρ > ∂L ∗ /∂γ < ρ > ∂ V /∂γ < ρ > Note. In the last row of “jump size γ ”, we derive the results assuming γ ( · ) ≡ γ is a positive constant. From the efficient frontier (3.11) of Problem (2.12), one immediate observation is the so-called securitymarket line (SML), E t,x [ X ∗ ( s )] = x e r ( s − t ) + p κ ( s − t ) × q V t,x [ X ∗ ( s )] . Since the dependence of the value function V ( t, x, y ) on the model parameters (except r ) is only throughthe coefficient κ , the last column of Table 1 offers some monotonic results regarding the slope of the SML.For instance, when ρ <
0, the slope of the SML is increasing with respect to (w.r.t.) the excess return ¯ µ and asset volatility σ , and decreasing w.r.t. the correlation coefficient ρ , the excess premium ¯ p , and therisk volatility β .Finally, according to Theorem 3.5, both the optimal investment strategy π ∗ and the optimal liabilitystrategy L ∗ are independent of the wealth, but are increasing at an exponential rate with respect to thetime variable. Setting y = x , we have V x ( t, x, x ) > V t ( t, x, x ) <
0. That is, with higher initial wealth x or a longer investment horizon T − t , the insurer is able to derive a higher expected utility, which fitsour intuition perfectly. The goal of this section is to compare our approach and optimal strategy in Section 3 with those underthe game-theoretic and precommitted framework. 15 .1 Comparison Analysis with Game-Theoretic Strategies
If we consider a standard MV portfolio selection problem without risk control strategies, the optimalinvestment π ∗ | L ≡ is obtained in (3.18), which is the same as that in Basak and Chabakauri (2010) (with-out stochastic factor), Bj¨ork and Murgoci (2010), Bj¨ork et al. (2014) (under constant risk aversion), andKryger et al. (2020). But as witnessed in Section 3, we arrive at the same result via a different analysis.Next, let us compare our optimal strategy in (3.9) to those in the time-consistent MV investment-reinsurance literature. If we set ρ = 0 and γ ≡
0, i.e., the financial market is independent of the insurancemarket and the risk process R is a diffusion approximation process, then we recover the same results as inZeng and Li (2011); see Theorem 2 therein. If we set ρ = 0, then our optimal strategy is consistent withthat in Zeng et al. (2013) and Zeng et al. (2016) without ambiguity aversion. There is a key difference between our optimal strategy and those discussed above. Under (3.19), theoptimal strategy in the above papers is always positive (excluding the wealth-dependent risk aversion casein Bj¨ork et al. (2014)) and does not involve short selling. In our model, the optimal strategy is positive if ρ ≥
0. However, if ρ <
0, it is possible that κ and κ are negative at the same time, or they have differentsigns. In other words, our optimal strategy may involve short selling when ρ <
0; see Figure 1.Our optimal strategy is independent of the wealth process X , since the risk aversion parameter θ is takento be a constant in the analysis. For the same problem but under a wealth-dependent risk aversion (e.g., θ ( x ) = constant /x ), the optimal strategy is likely to be proportional to the wealth level; see, Bj¨ork et al.(2014), Dai et al. (2020), and Kryger et al. (2020). A standard approach to tackle the time-inconsistency issue of Problem (2.9) is to formulate the problemunder a game-theoretic framework. That is, one sets up a game between the current self of an MV-typeagent and her future incarnations (selfs). A strategy b u is “optimal”, more often called equilibrium (usedhereafter in this subsection), if all her future incarnations living in [ t + ǫ, T ] will follow this strategy andthere is no gain for her to choose a different strategy u at time t (lasting from t to t + ǫ for an infinitesimalperiod ǫ ). The above “informal” definition of an equilibrium strategy b u is indeed rigorous in a discrete-timemodel. To see this, suppose b u = ( b u ( s )) s ∈ [ t,T ] is an equilibrium strategy and consider a “perturbed” strategy u = ( u ( s )) s ∈ [ t,T ] , where u ( t ) = u = b u ( t ) and u ( s ) = b u ( s ) for all s = t + 1 , · · · , T . By the definition ofan equilibrium strategy, we have J ( t, x ; b u ) ≥ J ( t, x ; u ) for any F t -measurable u in the admissible domain.However, extending the same idea to a continuous-time model becomes problematic, since a deviation from b u at time t is only effective for an infinitesimal period of time (a set with Lebesgue measure zero) andhence its impact on the objective functional is negligible in most important problems (including the MV There is slight difference in the setup of risk management between ours and those in the works of Zeng and his collaborators’;see Remark 2.1. For instance, using our notation, m and θ in Zeng and Li (2011) are 0 and ¯ p − λγ . b u is called an equilibrium strategy iflim inf ǫ ↓ J ( t, x ; b u ) − J ( t, x ; u ) ǫ ≥ , (4.1)holds for all perturbed strategies u within the admissible set, where we assume the goal is to maximize theobjective J in the problem. The definition (4.1) is proposed by Ekeland and Lazrak (2006) to study anoptimization problem with a non-exponential discounting function. The equilibrium definition (4.1) is alsofundamental and used in almost all the subsequent works on time-consistent MV and MV-reinsurance prob-lems; see, e.g., Bj¨ork and Murgoci (2010), Kryger and Steffensen (2010), Zeng and Li (2011), Zeng et al.(2013), Bj¨ork et al. (2014), and many others.However, the equilibrium condition (4.1) is only a necessary condition, not a sufficient condition. Thatbrings an immediate problem: a strategy b u satisfying (4.1) with an equality may fail to dominate anotheradmissible strategy (See Remark 7.1 in Bj¨ork and Murgoci (2010)). Huang and Zhou (2020) construct acounterexample (see Example 4.3 therein) in which an equilibrium strategy b u satisfying (4.1) is strictlyworse off than a different admissible strategy u ′ , i.e., J ( t, x ; u ′ ) > J ( t, x ; b u ). Namely, there exist caseswhere an agent has the incentive to deviate from an equilibrium strategy b u , as characterized by (4.1),which contradicts the very definition of equilibrium.In comparison, the problem considered in this paper, Problem (2.12), is a standard time-consistentcontrol problem. Namely, once a solution u ∗ is obtained to Problem (2.12) at time t , the insurer will followthe strategy u ∗ over [ t, T ] and the optimality of u ∗ holds trivially by definition, i.e., J ( t, x, y ; u ∗ ) ≥ J ( t, x, y ; u ) ∀ u ∈ A . (4.2)In fact, our analysis in Section 3 does find an optimal strategy u ∗ such that J ( t, x, y ; u ∗ ) = sup u ∈A J ( t, x, y );see Theorem 3.5 and (3.17). The key differences between the equilibrium definition of Bj¨ork and Murgoci(2010) in (4.1) and the optimality definition in (4.2) are as follows: (1) we introduce an auxiliary process Y (with initial value y ) and consider a modified objective J , in which Y replaces the conditional expectationin the original objective J ; and (2) we seek an optimal strategy u ∗ that maximizes J over all admissiblestrategies. Because of these differences, our alternative formulation in (2.12) always leads to a well-definedoptimal strategy (for a modified objective J ), while the same cannot be said in general under the definition(4.1). We discuss two differences between our approach and those in Bj¨ork and Murgoci (2010) and its followingworks, such as Zeng and Li (2011) and Bj¨ork et al. (2014). The first one comes from the auxiliary process,while the second one comes from the HJB equation.17o facilitate the presentation, let us restate the MV problem considered in Bj¨ork and Murgoci (2010)and Bj¨ork et al. (2014) as follows: b V ( t, x ) = sup u ∈A b J ( t, x ; u ) = sup u ∈A (cid:26) E t,x [ X u ( T )] − θ V t,x [ X u ( T )] (cid:27) , θ > . In Bj¨ork and Murgoci (2010), an essential technique to handle the “troubling” square term is to introducean auxiliary process g u defined by g u ( t, x ) = E t,x [ X u ( T )] , t < T and g u ( T, x ) = x. (4.3)Notice that g u in (4.3) is defined in a backward way and its dynamics g u ( t, X u ( t )) are stochastic , while ourauxiliary process Y u is defined in a forward way and its dynamics Y u ( t ) are deterministic ; see (2.10). Dueto the deterministic nature of process Y u , we do not have V yy and V xy terms in our HJB equation (3.7).For any admissible control u ∈ A , define the associated Dynkin operator of process X u in (2.6) by L u . As u ∈ A , X u is the unique strong solution to the SDE (2.6) and X ( T ) is square integrable. It thenfollows from (4.3) that g u ( t, X u ( t )) is a martingale and we have L u g u ( t, x ) = 0 for all u ∈ A by Dynkin’sformula. Assume an equilibrium strategy ˆ u as defined in (4.1) exists, Bj¨ork and Murgoci (2010) derive anextended system of HJB equations satisfied by the pair ( b V ( t, x ) , g ˆ u ( t, x )) and solve the system to obtain ˆ u ;see Theorems 2.1 and 7.4 therein. In our formulation, Problem (2.12) is a standard control problem, withtwo state variables, and the HJB equation (3.7) is about the value function V ( t, x, y ) only. To summarize,the approach of Bj¨ork and Murgoci (2010) ends up with solving a system of two one-dimensional partialdifferential equations (PDEs), while our approach leads to a two-dimensional PDE. In Section 3, we obtain the precommitment strategy u pre to Problem (2.9) in (3.2) and the optimal strategy u ∗ to Problem (2.12) in (3.9). Since u pre and u ∗ are solutions to two different stochastic control problems, adirect side-by-side comparison has little mathematical meaning. However, Problem (2.12) is an alternativetime-consistent formulation to Problem (2.9), and both u pre and u ∗ are available investment and riskcontrol strategies to the insurer, comparing the end results of these strategies makes economic sense.First, we fix the same risk aversion parameter θ for both Problems (2.9) and (2.12), and investigatethree important results: the mean E t,x [ X u ( T )], the variance V t,x [ X u ( T )], and the objective J definedin (2.8). Let X pre and X ∗ denote the insurer’s wealth process (2.6) under the precommitment strategy u pre and the optimal strategy u ∗ , respectively. By comparing (3.5)-(3.6) with (3.14)-(3.15) and (3.12), weobtain: E t,x [ X pre ( T )] > E t,x [ X ∗ ( T )] , V t,x [ X pre ( T )] > V t,x [ X ∗ ( T )] , J ( t, x ; u pre ) > J ( t, x ; u ∗ ) . (4.4)From (4.4), we conclude that the optimal strategy u ∗ is more conservative than the precommitment strategy u pre , leading to a smaller risk under the compromise of performance (mean). Note that to derive (4.4), we18ake the parameter θ to be the same for Problems (2.9) and (2.12). However, the results in (4.4) clearlyreveal different risk attitudes in terms of both mean and variance. As such, in the next step, we fix thesame target E t,x [ X u ( T )] for the insurer and study how the two different formulations achieve the sametarget.Second, let us fix m := E t,x [ X u ( T )] > xe r ( T − t ) for the insurer, and re-consider Problems (2.9) and(2.12) under a constrained admissible set A m , defined by A m := { u ∈ A : E t,x [ X u ( T )] = m } . Denote the corresponding solutions by u pre m and u ∗ m . We apply Theorems 3.1 and 3.5 to obtain u pre m and u ∗ m in the next two corollaries and then compare them. Corollary 4.1.
Let a target level m := E t,x [ X u ( T )] > xe r ( T − t ) be given. A precommitment strategy,denoted by u pre m = ( π pre m ( s ) , L pre m ( s )) s ∈ [ t,T ) , to Problem (2.9) over the admissible set A m is given by π pre m ( s ) = − κ X pre m ( s ) − m − x e ( r − κ )( T − t ) − e − κ ( T − t ) e − r ( T − s ) ! and L pre m ( s ) = − κ κ π pre m ( s ) , (4.5) where X pre m denotes the wealth process under the precommitment strategy u pre m .Proof. Recall E t,x [ X pre ( T )] is obtained in (3.5). By equating E t,x [ X pre ( T )] = m , we get θ pre ( m ) = e κ ( T − t ) − m − xe r ( T − t ) > . Substituting the free parameter θ by the above θ pre ( m ) in (3.2) leads to the desired results in (4.5). Corollary 4.2.
Let a target level m := E t,x [ X u ( T )] > xe r ( T − t ) be given. A precommitment strategy, u ∗ m = ( π ∗ m ( s ) , L ∗ m ( s )) s ∈ [ t,T ) , to Problem (2.12) over the admissible set A m is given by π ∗ m ( s ) = κ (cid:0) m − x e r ( T − t ) (cid:1) κ ( T − t ) e − r ( T − s ) and L ∗ m ( s ) = κ (cid:0) m − x e r ( T − t ) (cid:1) κ ( T − t ) e − r ( T − s ) . (4.6) Proof.
Recall E t,x [ X ∗ ( s )] is obtained in (3.14). By equating E t,x [ X ∗ ( T )] = m , we get θ ∗ ( m ) = κ ( T − t ) m − xe r ( T − t ) > . Substituting the free parameter θ by the above θ ∗ ( m ) in (3.9) leads to the desired results in (4.6).Despite successfully obtaining π pre m and π ∗ m in close-form, we cannot directly compare them at anarbitrary time s ( t ≤ s < T ). However, when s = t , by using (4.5) and (4.6), and e κ ( T − t ) − > κ ( T − t ),we obtain π pre m ( t ) > π ∗ m ( t ) and L pre m ( t ) > L ∗ m ( t ) . (4.7) 194.7) implies that, to achieve the same target m , the precommitment framework yields a more risky strategyin both investment and liability at the initial time t than the alternative time-consistent formulation.Note that we call u ∗ m in Corollary 4.1 a precommitment strategy, since both π ∗ m and L ∗ m depend onthe initial state ( t, x ) as seen in (4.6). That means, by restricting the admissible set from A to A m , thealternative time-consistent formulation in Problem (2.12) becomes time-inconsistent. In fact, u ∗ m being aprecommitment strategy is not surprising at all, because the restriction of A m leads to a state-dependentrisk aversion θ ∗ ( m ), which is a well-known contributor to time-inconsistent problems (see Bj¨ork et al.(2014)). Given the explicit results on both the optimal strategy and the value function in Theorem 3.5, we discusstheir analytical properties with respect to the model parameters, and summarize the main findings in Table1. However, most findings there are partial and limited to the assumption of ρ >
0. That motivates us toconduct numerical studies in this section to further investigate how the model parameters and risk aversionaffect the insurer’s decisions.We follow Zou and Cadenillas (2014) to initiate the default model parameters in (2.1)-(2.3), and presentthem in Table 2, where we assume γ ( t, z ) ≡ γ for all t ∈ [0 , T ] and z ∈ R . Note that we leave both ρ (correlation coefficient) and θ (risk aversion) unspecified in the default setting, as they are the mostsignificant factors of the optimal strategy. r µ σ α β λ γ p ρ (correlation coefficient) and θ (risk aversion) on the optimal strategy.We plot the optimal investment strategy π ∗ ( t ) and the optimal liability strategy L ∗ ( t ) against all ρ ∈ [ − , θ = 1 , , ,
10 in Figure 1. Several important observations andexplanations are due as follows. • The optimal investment in the risky asset may be negative (i.e., shorting selling may be optimal),but only when ρ is close to -1. We further observe that a less risk averse insurer short sells more ina combined market with extreme negative correlation. To understand these results, we look at theextreme case of ρ = −
1, in which the movements from the Brownian motion W cause the risky asset S and the risk process R to affect the insurer in the exactly same direction, if she holds positivepositions. For instance, a decrease of W leads to a lower price of S and more liabilities fromunderwriting policies. In turn, the volatility due to W is amplified and results in a more volatilemarket in the insurer’s view. A strategy to counter such an amplification effect is to short sell the20 Optimal Investment Strategy * = 1 = 2 = 5 = 10 -1 -0.5 0 0.5 101234567 Optimal Liability Strategy L * = 1 = 2 = 5 = 10 Figure 1: Impact of ρ and θ on the Optimal Strategies π ∗ (left) and L ∗ (right) Notes. We plot π ∗ ( t ) and L ∗ ( t ) under the default model parameters in Table 2, x = 1, and T − t = 1. risky asset. A less risk averse insurer is less prone to the amplification effect, and then short sellsmore risky asset when ρ is close to -1. • The optimal investment π ∗ is an increasing function of ρ , and shares the same intersection point at π ∗ = 0 (recall π ∗ = 0 iff κ = 0 by (3.9) and κ is independent of θ by (3.3)). The optimal liability L ∗ first decreases and then increases with respect to (w.r.t.) ρ , revealing a convex relation. However,if ρ > L ∗ is an increasing function of ρ , as already shown in Table 1. Under the given parametersin Table 2, we calculate that L ∗ takes minimum values at ρ = − . ρ >
0, the financialmarket and the insurance market provide a natural hedge to each other, making the combined marketless volatile to the insurer, and hence, both π ∗ and L ∗ increase w.r.t. ρ when ρ > • As risk aversion θ increases, we observe that the optimal liability strategy L ∗ taken by the insurerdecreases, and is almost insensitive to ρ when θ is large enough. Similar statement holds true for theoptimal investment π ∗ (in the absolute value).21 Optimal Investment Strategy * = -0.5 = 0 = 0.5 Optimal Liability Strategy L * = -0.5 = 0 = 0.5 Figure 2: Impact of λ on the Optimal Strategies π ∗ (left) and L ∗ (right) Notes. We calculate the premium p by the expected value principle with loading η = 40%. We set θ = 2, x = 1, T − t = 1, and other parameters as in Table 2. We next study how the jump intensity λ of the risk process R affects the insurer’s optimal strategy. Tothis end, we allow λ to vary in [0 , . p = 0 .
15 as now λ is changing, but instead applythe expected value principle with loading η = 40% to calculate the premium p by p = (1 + η ) × ( α + λγ ).We set θ = 2 and consider three correlation levels ρ = − . , , .
5. The remaining model parameters arethe same as in Table 2. We then plot π ∗ ( t ) and L ∗ ( t ), with x = 1 and T − t = 1, against λ ∈ [0 , . λ in Table 1. When ρ = 0, the flat solid line (in black) in the left panel of Figure 2 represents the Merton ratio. The impactof λ on L ∗ is more direct and easy to understand. As λ increases, the liabilities per unit increase and theinsurer responds by taking less units (policies) in the insurance business. However, the impact of λ on π ∗ isindirect and more complex. We notice a decreasing relation when ρ = 0 . ρ = − .
5. In a positively correlated combined market, the risk process R acts as a hedge to the risky asset.But, as λ increases, this hedging effect weakens; or putting it differently, the hedging tool itself becomestoo volatile to serve its hedging purpose. Upon understanding that, the insurer reacts by reducing risky22nvestment. Less can be said in general regarding the relation between π ∗ and λ when ρ is negative. Weknow that, with ρ <
0, the increase of λ leads to a shrinking effect on the risky asset’s volatility σ , makingthe insurer willing to hold more risky asset, but at the same time a bigger λ makes the insurance businessmore volatile, causing the insurer to be cautious of making risky investment. The former effect outweighsthe latter one, leading to a positive relation found in Figure 2. We remark that the opposite may happenif a different set of model parameters is taken. We study an optimal investment and risk control optimization problem for an insurer with mean-variance(MV) preference. The insurer has access to a combined market with financial assets and insurance businessopportunities. The financial market is given by a standard Black-Scholes model, and the risk process(liabilities) per unit is given by a jump-diffusion process. The insurer seeks an optimal investment and riskcontrol strategy under the MV preference, which is a well-known time-inconsistent problem. We introducea deterministic auxiliary process to replicate the conditional expectation of the insurer’s wealth and extendit as a second state process. We then formulate an alternative time-consistent problem which is not onlyintimately related to the original problem but also can be solved by the standard dynamic programmingmethod. We obtain the optimal strategy, the efficient frontier, and the value function, all in closed-form, tothe alternative problem. We conduct analytical studies to compare our formulation and optimal strategywith those under the game-theoretic and precommitted framework. Among many findings, we emphasizethat the correlation between the financial market and the risk process plays a key role in the optimalstrategy. When the correlation coefficient ρ is positive, both the optimal investment strategy π ∗ and theoptimal liability strategy L ∗ are increasing functions of ρ . However, when ρ is negative, L ∗ first decreasesand then increases with respect to ρ , showing a convex U-shaped relation. As ρ moves towards -1, π ∗ becomes negative, suggesting the optimal investment decision is to short sell the risky asset. Lastly, wereport that the jump intensity of the risk process also has a major impact on the optimal strategy. Acknowledgments
We would like to thank anonymous referees and editor for their careful reading and many insightful com-ments that help us improve the quality of an early version of this paper. Bin Zou is partially supported bya start-up grant from the University of Connecticut. Yang Shen is partially supported by the DiscoveryEarly Career Researcher Award (Grant No. DE200101266) from the Australian Research Council.Declarations of interest: none. 23
Proofs
Problem (2.9) is a standard MV type control problem that is well studied in the literature; see, e.g.,Markowitz (1952) and Li and Ng (2000) in discrete time, and Zhou and Li (2000) and the book Yong and Zhou(1999) in continuous time. There are various approaches to finding the precommitment strategy, and mostof them rely on the so-called embedding technique, first used in Li and Ng (2000) and Zhou and Li (2000).We denote a constrained admissible set A m by A m := { u ∈ A : E t,x [ X u ( T )] = m } , where m > x e r ( T − t ) , and explain such a technique using the (equivalence) relations as follows:sup u ∈A J ( t, x ; u ) ⇔ sup m ∈ R sup u ∈A m J ( t, x ; u ) = m − θ E t,x h ( X u ( T ) − m ) i ⇔ inf u ∈A m V t,x [ X u ( T )] ⇔ inf u ∈A V t,x [ X u ( T )] − w ( E t,x [ X u ( T )] − m ) ⇔ inf u ∈A E t,x h ( X u ( T ) − ξ ) i − ( ξ − m ) , ξ = m + w (Lagrange multiplier) . (A.1)The above relation shows that the key to solving Problem (2.9) is finding an optimal solution to thefollowing problem: W ( t, x ; ξ ) := inf u ∈A E t,x h ( X u ( T ) − ξ ) i , where ξ ∈ R . (A.2) Theorem A.1.
Suppose β (1 − ρ ) + λγ = 0 . An optimal solution to Problem (A.2) is given by π pre ( s ; ξ ) = − κ (cid:16) X pre ( s ; ξ ) − ξ e − r ( T − s ) (cid:17) and L pre ( s ; ξ ) = − κ (cid:16) X pre ( s ; ξ ) − ξ e − r ( T − s ) (cid:17) , (A.3) where κ and κ are defined in (3.3) and X pre ( · ; ξ ) is the corresponding wealth process under the initialstate ( t, x ) . The value function to Problem (A.2) is given by W ( t, x ; ξ ) = (cid:16) x e r ( T − t ) − ξ (cid:17) e − κ ( T − t ) , (A.4) where κ is defined in (3.4) .Proof. Since Problem (A.2) is a standard control problem, by using the dynamic programming principle, weobtain that W solves the following HJB equation (assuming W satisfies the needed regularity conditions) W t ( t, x ; ξ ) + sup ( π,L ) ∈ R (cid:26) ( rx + ¯ µπ + ¯ pL ) W x ( t, x ; ξ ) + 12 (cid:0) σ π − ρβσπL + β L (cid:1) W xx ( t, x ; ξ )+ λ Z R (cid:0) W ( t, x − Lγ ( z ); ξ ) − W ( t, x ; ξ ) (cid:1) d F Z ( z ) (cid:27) = 0along with the terminal condition W ( T, x ; ξ ) = ( x − ξ ) . We then make an educated guess for the valuefunction in the form of W ( t, x ; ξ ) = (cid:16) x e r ( T − t ) − ξ (cid:17) f ( t ) , f ( T ) = 1 . Straightforward computations and verification then yield the desired results in (A.3) and (A.4).24e next apply the above theorem and the road map in (A.1) to solve Problem (2.9).
Proof of Theorem 3.1.
Using (A.4), we consider the problem below c W ( t, x ; m ) = sup ξ ∈ R W ( t, x ; ξ ) − ( ξ − m ) and easily obtain the optimal solution and the value function, under the given expectation level m , by ξ ∗ ( m ) = m − x e ( r − κ )( T − t ) − e − κ ( T − t ) and c W ( t, x ; m ) = (cid:0) m − x e r ( T − t ) (cid:1) e κ ( T − t ) − . (A.5)Next, we solve sup m ∈ R m − θ c W ( t, x ; m )and obtain the optimal target level m ∗ by m ∗ = x e r ( T − t ) + e κ ( T − t ) − θ . (A.6)Finally, we substitute ξ in (A.3) by ξ ∗ ( m ∗ ), where ξ ∗ ( · ) and m ∗ are given in (A.5) and (A.6), and obtainthe desired results after tedious computations. Proof to Theorem 3.5.
From the terminal condition (3.8) and the nature of stochastic linear-quadraticproblems, we guess an ansatz to the value function in the form of V ( t, x, y ) = A ( t ) ( x − y ) + B ( t ) x + C ( t ) , where A , B and C are yet to be determined and A ( t ) < t . Note that with A ( t ) <
0, the above V is concave in both x and y arguments. It is clear from (3.8) that A ( T ) = − θ , B ( T ) = 1 , C ( T ) = 0 . Using the above ansatz and the HJB equation (3.7), we derive that an (candidate of) optimal strategy u ∗ = ( π ∗ , L ∗ ) should solve the following system of equations σ · π ∗ − ρβσ · L ∗ = − ¯ µ B ( t )2 A ( t ) ρβσ · π ∗ − (cid:0) β + λγ (cid:1) · L ∗ = (¯ p − λγ ) B ( t )2 A ( t )which, under the assumption β (1 − ρ ) + λγ = 0 in (3.1), admits a unique solution π ∗ = − κ B ( t )2 A ( t ) and L ∗ = − κ B ( t )2 A ( t ) , where κ and κ are defined in (3.3). 25e plug the above ( π ∗ , L ∗ ) into the HJB equation (3.7) and simplify to get( x − y ) A ′ ( t ) + xB ′ ( t ) + C ′ ( t ) + 2 r ( x − y ) A ( t ) + rxB ( t ) − κ B ( t )4 A ( t ) = 0 , where κ is defined by (3.4). Since the above equation holds for all x, y ∈ R , we obtain A ′ ( t ) + 2 r A ( t ) = 0 , A ( T ) = − θ ,B ′ ( t ) + r B ( t ) = 0 , B ( T ) = 1 ,C ′ ( t ) − κ B ( t )4 A ( t ) = 0 , C ( T ) = 0 , which leads to the solutions A ( t ) = − θ e r ( T − t ) < , B ( t ) = e r ( T − t ) > , C ( t ) = κ θ ( T − t ) > . It is straightforward to verify that V given by (3.10) is smooth ( V ∈ C , , ) and, by construction, satisfiesthe HJB equation (3.7). As a result, the value function to Problem (2.12) is indeed given by (3.10). Thestrategy ( π ∗ , L ∗ ) given by (3.9) solves the supremum problem uniquely in the HJB equation (3.7), whichis guaranteed by V xx ( t, x, y ) = 2 A ( t ) <
0. By Definition 2.2, ( π ∗ , L ∗ ) is admissible. Hence, we concludethat the strategy given by (3.9) is optimal to Problem (2.12). The proof is now complete. References
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