JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2012.8.673 MANAGEMENT OPTIMIZATION Volume 8, Number 3, August 2012 pp. 673–690
OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS UNDER MEAN-CAR CRITERIA
Yan Zeng Lingnan (University) College Sun Yat-sen University, Guangzhou 510275, China Zhongfei Li1 Lingnan (University) College/Sun Yat-sen Business School Sun Yat-sen University, Guangzhou 510275, China
(Communicated by Vladimir Veliov)
Abstract. This paper considers an optimal reinsurance-investment problem for an insurer, who aims to minimize the risk measured by Capital-at-Risk (CaR) with the constraint that the expected terminal wealth is not less than a predefined level. The surplus of the insurer is described by a Brownian motion with drift. The insurer can control her/his risk by purchasing pro- portional reinsurance, acquiring new business, and investing her/his surplus in a financial market consisting of one risk-free asset and multiple risky assets. Three mean-CaR models are constructed. By transforming these models into bilevel optimization problems, we derive the explicit expressions of the optimal deterministic rebalance reinsurance-investment strategies and the mean-CaR efficient frontiers. Sensitivity analysis of the results and a numerical example are provided.
1. Introduction. Since reinsurance is an effective way to spread risk and invest- ment is an increasing important element in the insurance business, many optimiza- tion problems have arisen in the last decades. These problems are subject to the control of reinsurance and/or investment, with various objectives in insurance risk management. This topic has attracted a great deal of interest, and has been ex- tensively investigated in literature. For example, Browne [7], Cao and Wan [8], Irgens and Paulsen [16], Liang [20], Xu et al. [34], and Yang and Zhang [35] focus on studying the optimal reinsurance and/or investment strategies to maximize the expected utility from the terminal wealth of insurers. Azcue and Muler [2], Bai and Guo [3], Hipp and Plum [14], Liang and Guo ([21], [22]), Liu and Yang [23], Luo [24], Luo et al. [25], Promislow and Young [29], Schmidli ([30], [31]), and Taksar and
2000 Mathematics Subject Classification. Primary: 90C26; Secondary: 91B28, 49N15. Key words and phrases. Optimal proportional reinsurance-investment strategy, insurers, Capital-at-Risk, Hamilton-Jacobi-Bellman equation. This research was partially supported by the National Science Foundation for Distinguished Young Scholars (No. 70825002), China Postdoctoral Science Foundation funded project (No. 2011M501351), Humanity and Social Science Foundation of Ministry of Education of China (No. 12YJCZH267), the High-level Talent Project of Guangdong“Research on Models and Strategies for Optimal Reinsurance, Investment and Dividend”, Philosophy and Social Science Program- ming Foundation of Guangdong Province (No. GD11YYJ07), and “985 Project” of Sun Yat-sen University. 1Corresponding author.
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Markussen [32] investigate the optimal reinsurance and/or investment strategies in the sense of minimizing the ruin probability of insurers. Bai and Zhang [4], B¨auerle [5], Delong and Gerrard [9], Li et al. [19], Wang et al. [33], Zeng and Li [36], and Zeng et al. [37] concentrate on seeking the optimal reinsurance and/or investment strategies under mean-variance criterion. In the above mentioned papers, three criteria: (1) maximizing the expected utility from the terminal wealth, (2) minimizing the ruin probability, and (3) mean- variance criterion, are used to determine the optimal strategies. However, there are some drawbacks of adopting these three criteria to determine the optimal strategies for insurers. Maximizing the expected utility from the terminal wealth criterion can not illustrate the magnitude of risk. Even for some special utility functions, such as the exponential utility function, the optimal investment strategy is independent of the parameters of the insurance market; thus, the strategy is unrealistic. Min- imizing the ruin probability criterion emphasizes only risk rather than profit, and this focus only on risk does not coincide with the investment intentions of insur- ers. Mean-variance criterion can reveal profit and risk directly, but the variance as a measure of risk has the drawback that it penalizes both upside and downside moments in the portfolio value, which is inconsistent with what insurers expect. To complement the deficiencies, scholars have proposed some alternative risk mea- sures, such as Value-at-Risk (VaR), semivariance, shortfall probability, the limited expected loss, and so on. Among them, VaR is the most notable risk measure, and has been widely accepted by the banks and regulators as the benchmark for control- ling market risk. Recently, Emmer et al. ([10], [11]) define a VaR-based related risk measure called Capital-at-Risk (CaR), and incorporate it in the portfolio optimiza- tion problems. They derive explicitly the optimal investment strategies when the trading strategies are restricted to constant rebalance portfolio (CRP) strategies. Li et al. [18] formulate three mean-CaR models to investigate the continuous-time optimal portfolio selection problems, and derive the corresponding explicit optimal CRP strategies. The CRP strategies mentioned above are the investment strategies which suppose that the proportions of the total wealth invested in each available assets are the same at any time point, regardless of the level of wealth. It is worth pointing out that the CRP strategies are dynamic investment strategies in that they require trading over time, and are also known as the constant mix strategies. These strategies are widely studied in the existing literature. Merton ([26], [27]) indicates that this form of strategies is optimal to the portfolio selection problems of max- imizing expected utility with constant relative risk-aversion. Black and Perold [6] and Perold and Shapre [28] use those strategies in asset allocation practice. Irgens and Paulson [16] investigate the optimal strategies of risk exposure, reinsurance and investments for insurers with the restriction of adopting the constant mix strategies to maximize the expected utility from the terminal wealth. In this paper, we consider an optimal proportional reinsurance-investment prob- lem for an insurer by adopting CaR to measure the risk. Specifically, the surplus of the insurer is described by a diffusion approximation model which has been widely adopted in the existing papers, for example, in Browne [7], Luo [24], Promislow and Young [29], Taksar and Markussen [32], and so on, to investigate optimization problems related to insurers. The insurer in this paper is allowed to purchase pro- portional reinsurance, acquire new business (as a reinsurer of other insurers), and invest her/his surplus in a financial market, which consists of one risk-free asset OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS 675 and multiple risky assets. We construct three mean-CaR models in order to min- imize the risk measured by CaR under the constraint that the expected terminal wealth is not less than a predetermined level. To ensure the tractability of the optimization problems in this paper, we restrict ourselves to adopting deterministic rebalance (DR) strategies. In this paper, a DR strategy is a reinsurance-investment strategy which requires that the risk exposure (the retention level of proportional reinsurance/new business) is proportionate to the total wealth with a proportion- ality factor, which is a deterministic function with respect to (w.r.t.) time t, and the proportions of the total wealth invested in each of available assets are also de- terministic functions w.r.t time t. The DR strategy is an extension of the CRP strategy, and results in much broader trading strategies. Such DR strategy also has the advantages of the CRP strategy, which are discussed in Sections 3.3 and 3.4 of Emmer et al. [11] and Korn [17]. By transforming the original optimization prob- lems into bilevel optimization problems, we derive the explicit expressions of the optimal proportional reinsurance-investment strategies and the mean-CaR efficient frontiers of the three mean-CaR models. We generalize the assumptions of Emmer et al. [11], Irgens and Paulsen [16] and Li et al. [18] in the sense that the market coefficients as well as the trading strategies are time dependent, and generalize the models of Emmer et al. [11] and Li et al. [18] in the sense that the control of risk exposure is introduced. In addition, our results can reduce to the ones of Emmer et al. [11] and Li et al. [18]. In the next section we state the basic assumptions of this paper. Section 3 provides the definition and some properties of CaR. Section 4 solves three mean- CaR models. Section 5 provides sensitivity analysis of our results, and presents a numerical example to illustrate our results. Section 6 concludes this paper.
2. Basic assumptions. To provide a rigorous mathematical formulation of the optimization problems, we start with a filtered probability space (Ω, , F := Ft : 0 t T , P ), where T is a finite and positive constant, representingF the{ time ≤ ≤ } horizon; the filtration Ft represents the information available at time t, and any decision is made based upon this information. All stochastic processes introduced below are assumed to be the adapted processes in this space. 2.1. The surplus process. In this paper, we consider an insurer whose surplus process is described by a diffusion approximation model. For the purpose of moti- vation, we start from the classical Cram´er-Lundberg (C-L) model. In this model, without reinsurance and investment, the surplus process of the insurer is given by
Nt R(t)= x + pt Z , (1) − i i=1 X where x is the initial surplus of the insurer; p is the premium rate; Nt, 0 t T , representing the number of claims occurring in [0,t], is a homogeneous{ ≤ Poisson≤ } process with intensity λ; the claim sizes Zi, i = 1, 2, are assumed to be independent and identically distributed positive{ random variables···} with finite first 2 and second-order moments given by µ∞ and σ∞, respectively. Assume that the premium rate p is calculated via the expected value principle, i.e. p = (1+ η)λµ∞, where η is the relative safety loading of the insurer. Furthermore, we assume that the insurer is allowed to control the risk of the insurance business by purchasing proportional reinsurance or acquiring new business (for example, as a reinsurer of other insurers). The retention level of proportional 676 YANZENGANDZHONGFEILI reinsurance/new business is associated with the value of risk exposure a, where a 0. Assume that the reinsurer uses a safety loading proportional to η with proportional≥ k(a)( 1) depending only on a. If the risk exposure of the insurer a is fixed, when a [0,≥1], it corresponds to a proportional reinsurance cover. The cedent should divert∈ part of the premium to the reinsurer at the rate of (1 + k(a)η)(1 a). Meanwhile, for each claim, the insurer should pay 100a% while the rest 100(1 −a)% is paid by the reinsurer; when a (1, + ), it corresponds to acquiring new business.− Then the surplus process of the∈ insurer∞ with risk exposure a is given by
Nt (a,η) R (t)= x + [(1 + η) (1 + k(a)η)(1 a)]λµ∞t aZ . (2) − − − i i=1 X According to Asmussen et al. [1], Grandell [13] or Iglehart [15], we have
(a,η) 2 ηR (t/η ) µ(a)t + σ0(a)B0(t) in distribution, as η 0 t≥0 → → n o in D[0, + ) (the space of the left-limits and right-continuous functions endowed ∞ 2 with the Skorohod topology), where µ(a) = [1 k(a)(1 a)]λµ∞, σ0(a)= a λσ∞, B (t), 0 t T is a one-dimensional standard− Brownian− motion. So (2) can be 0 p {approximated≤ ≤ by }
(a,η) 2 dR (t)= η[1 k(a)(1 a)]λµ∞dt + a λσ dB (t), R(0) = x. (3) − − ∞ 0 Note that the diffusion approximation model for thep surplus process can be suitable to apply at least for large insurance portfolios (see Asmussen et al. [1], Grandell [13] or Iglehart [15] for more details). Similar to Asmussen et al. [1], throughout this paper we only consider cheap reinsurance in which k(a) = 1, that is, the reinsurer has the same relative safety loading as the insurer. At each time t, the value of risk exposure is selected by the insurer. We denote this value by a(t). Replacing a by a(t) in (3), we have the following diffusion approximation model
dR(t)= a(t)(µdt + σ0dB0(t)), R(0) = x, (4) where µ = ηλµ∞ represents the return (premium) rate of the insurance business; 2 σ0 = λσ∞ measures the risk level of the insurance business. 2.2. Thep financial market. In this paper, we consider a financial market, which consists of one risk-free asset (bond or bank account) and n risky assets (stocks or mutual funds). The price process S0(t) of the risk-free asset is assumed to evolve to the ordinary differential equation
dS0(t)= S0(t)r0(t)dt, 0
3. Capital-at-Risk. Denote by ρ0(α,π,x,T ) the α-quantile of the terminal wealth Xπ(T ) when a real number α (0, 1), initial wealth x, time horizon T , and strategy ∈ 678 YANZENGANDZHONGFEILI
π are chosen. Then ρ0(α,π,x,T ) satisfies P (Xπ(T ) ρ (α,π,x,T )) = α, (11) ≤ 0 where P ( ) is the probability. In addition, using notation ρ0, the expected shortfall or, more precisely,· the conditional tail expectation of Xπ(T ) is defined as ρ (α,π,x,T )= E[Xπ(T ) Xπ(T ) ρ (α,π,x,T )], (12) 1 | ≤ 0 and the conditional tail semi-standard derivation of Xπ(T ) is defined as ρ (α,π,x,T )= E[Xπ(T )2 Xπ(T ) ρ (α,π,x,T )]. (13) 2 | ≤ 0 The following is the definitionp of Capital-at-Risk w.r.t. ρk, (k = 0, 1, 2), which is given in Emmer et al. [10]. In addition, we call the trading strategy π = (δπ(t),βπ(t)) (0, 0), 0 t T the pure bond strategy. Under the pure { ≡ ≤ ≤ } π bond strategy, t [0,T ] the value of risk exposure δπ(t)X (t) 0, and all the wealth is invested∀ in∈ the bond. ≡
Definition 3.1. The CaR of a trading strategy π w.r.t. ρk, (k = 0, 1, 2), initial wealth x and time horizon T is the difference between the terminal wealth of the pure bond strategy and the risk measure ρk, (k =0, 1, 2), i.e., CaR (π)= xξ(T ) ρ (α,π,x,T ). k − k Since Xπ(T ) is lognormal distributed, by using (8) and (11)-(13), the closed-form of the risk measure ρk, (k =0, 1, 2) can be expressed as follows: 1 2 m(T,δπ,βπ)− s(T,δπ ,βπ) +Zαs(T,δπ ,βπ) ρ0(α,π,x,T )= xξ(T )e 2 , (14) Φ(Z s(T,δ ,β )) ρ (α,π,x,T )= xξ(T )em(T,δπ ,βπ) α − π π , (15) 1 α
1 2 m(T,δπ,βπ)+ s(T,δπ ,βπ) Φ(Zα 2s(T,δπ,βπ)) ρ2(α,π,x,T )= xξ(T )e 2 − , (16) r α where Zα is the α-quantile of the standard normal distribution and Φ( ) is the distri- bution function of a standard normal random variable. According to· the definition of CaRk(π), (k =0, 1, 2), we can also express CaRk(π), (k =0, 1, 2) explicitly. Assumption 3.2. To avoid discussion of some (irrelevant) subcases in some of the following results, we suppose α< 0.5 throughout this paper, which leads to Zα < 0.
If α > 0.5, then Zα > 0, and the monotonicity of h( ) in Lemma 4.3 needs to be discussed. Meanwhile, the corresponding discussion also· needs in the following theorems. In practice, we usually only choose α = 0.05, 0.1, and α > 0.5 is quite uncommon. So we make this assumption to avoid discussion of some irrelevant subcases. This assumption is also made in Emmer et al. [11] and Li et al. [18]. Lemma 3.3. If x> 0, α (0, 0.5), then ∈ 2 2 Φ(Zα x) x Φ(Zα 2x) − x +Z x − e 2 − e 2 α . α ≤ r α ≤ Proof. Because
Zα−2x 2 Zα − 2 1 − y 1 − (y 2x) Φ(Zα 2x) = e 2 dy = e 2 dy − −∞ √2π −∞ √2π Z Z Zα 2 −2x2+2Z x 1 − y +2x(y−Z ) −2x2+2Z x = e α e 2 α dy αe α , −∞ √2π ≤ Z OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS 679 we have
2 2 x Φ(Zα 2x) − x +Z x e 2 − e 2 α . r α ≤ In addition,
Zα 2 Zα−2x 2 x2 x2 1 − y 1 − y αe Φ(Zα 2x) = e e 2 dy e 2 dy − √2π √2π Z−∞ Z−∞ Zα−x 2 Zα−x 2 2 1 − (y+x) 1 − y +xy+ x = e 2 dy e 2 2 dy √2π √2π Z−∞ Z−∞ 2 Zα−x 2 1 − y e 2 dy (Schwartz inequality) ≥ √2π Z−∞ ! = (Φ(Z x))2, α − so
Φ(Zα x) x2 Φ(Zα 2x) − e 2 − . α ≤ r α This completes the proof.
Proposition 1. For any trading strategy π Π(x), initial wealth x> 0, and a real number α (0, 0.5), the following results hold:∈ ∈ (i) ρ1(α,π,x,T ) ρ2(α,π,x,T ) ρ0(α,π,x,T ); (ii) CaR (π) CaR≤ (π) CaR (≤π); 0 ≤ 2 ≤ 1 (iii) Supπ∈Π(x)CaRk(π)= xξ(T ), (k =0, 1, 2). Proof. According to (14)-(16) and Lemma 3.3, (i) is obvious. (ii) follows from (i) and Definition 3.1. (iii) follows from (14)-(16) and Definition 3.1. This completes the proof.
4. Mean-CaR reinsurance-investment models. Compared to the Markowitz’s mean-variance model that minimizes the variance of the terminal wealth under a given level of the expected terminal wealth, three optimization models, which we solve in this paper, minimize CaRk, (k =2, 1, 0), respectively, under the constraint that the expected terminal wealth is not less than a predefined level.
4.1. Mean-CaR2 model. In this subsection, we consider the following mean-CaR model associated with CaR2:
min CaR2(π), π∈Π(x) (17) ( s.t. EXπ(T ) C, ≥ where C is the predefined level, which is the minimum that the expected termi- nal wealth EXπ(T ) is allowed to attain. Since the pure bond strategy yields a deterministic wealth xξ(T ), throughout this paper we assume that C satisfies C xξ(T ). (18) ≥ Clearly, this assumption is rational, for the solution of problem (17) under C < xξ(T ) is illogical for insurers. In order to solve the optimization problem (17), we first provide some lemmas. 680 YANZENGANDZHONGFEILI
C T 2 −2 0 −1 Lemma 4.1. Let G(C) = ln xξ(T ) , ΨT = 0 [µ σ0 + r (s)Σ (s)r(s)]ds, εˆ = G(C) 2 2 and Πε = π := (δπ(t),βπ(t)) s(T,δπq,βπ) = ε , m(T,δπ,βπ) G(C), π ΨT { | R ≥ ∈ Π(x) . If ε [0, εˆ), then Π = . } ∈ ε ∅ Proof. Consider the following maximization problem
max m(T,δπ,βπ), π∈Π(x) s.t. s(T,δ ,β )2 = ε2. π π This is a variational problem with an isoperimetric constraint. Its Euler-Lagrangian function is 0 2 2 0 L = µδπ(t)+ βπ(t)r(t)+ τ[σ0 δπ(t)+ βπ(t)Σ(t)βπ(t)], where τ is a constant, representing the Lagrangian multiplier. Applying the cal- culus of variations, we have the first-order condition (Euler equations): ∂L/∂δπ = ˙ ˙ da ∂L/∂δπ, ∂L/∂βπ = ∂L/∂βπ (˙a means dt ), i.e., 2 µ +2τσ δπ(t)=0, 0 (19) ( r(t)+2τΣ(t)βπ(t)=0, and the second-order condition: for each t [0,T ], ∈ σ2 0 2τ 0 0. (20) 0 Σ(t) ≤ 2 From (20) with σ > 0 and Σ(t) θI × , it follows τ < 0. In addition, (19) implies 0 ≥ n n µσ−2 Σ−1(t)r(t) δ (t)= 0 ,β = . (21) π − 2τ π − 2τ Substituting (21) into the constraint results in 1 ε = . (22) 2τ −ΨT Substituting (22) into (21), the unique solution of the maximization problem is µσ−2ε Σ−1(t)r(t)ε ∗ ∗ ∗ ∗ 0 ∗ πε = (δπε (t),βπε (t)),δπε (t)= ,βπε (t)= , (23) ΨT ΨT and
max m(T,δπ,βπ)= εΨT . π∈Π(x) When ε [0, εˆ), εΨ < G(C), which shows Π = . This completes the proof. ∈ T ε ∅ Lemma 4.2. (1/x 1/x3)ϕ(x) < Φ( x) < ϕ(x)/x for any x (0, + ), where ϕ( ) is the density function− of a standard− normal random variable∈. ∞ · Proof. See G¨anssler and Stute [12].
Lemma 4.3. Denote a+ = max a, 0 . Assume that C satisfies { } + C ε˜ := xξ(T )eΨT (ΨT +Zα) . (24) ≥ Then h(ε)=2εΨ + ε2 + ln(Φ(Z 2ε)) attains its maximum on [ˆε, + ) at ε =ε ˆ. T α − ∞ OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS 681
G(C) + Proof. Becauseε ˆ = , (24) impliesε ˆ (ΨT + Zα) ΨT + Zα. In addition, ΨT ≥ ≥ noting that ϕ( x)= ϕ(x) and Lemma 4.2 − 2ϕ(Z 2ε) h(ε)0 = 2Ψ +2ε α − T − Φ(Z 2ε) α − ϕ(2ε Z ) = 2(Ψ + ε − α ) 2(Ψ + Z ε). T − Φ(Z 2ε) ≤ T α − α − Then, h(ε)0 < 0 for ε εˆ when (24) holds, which shows that h(ε) attains its maximum at ε =ε ˆ. This≥ completes the proof. Theorem 4.4. Assume that C satisfies (24). Then the unique optimal strategy of the mean-CaR2 problem (17) is −2 −1 ∗ µσ0 εˆ Σ (t)r(t)ˆε π = (δπ∗ (t),βπ∗ (t)), δπ∗ (t)= , βπ∗ (t)= , (25) ΨT ΨT ∗ the corresponding expected wealth is EXπ (T )= C, and
1 2 ∗ εˆΨT + εˆ Φ(Zα 2ˆε) CaR2(π )= xξ(T ) 1 e 2 − . (26) " − r α #
Proof. With the expressions for (9) and (16), and the definition of CaR2, problem (17) can be equivalently written as
m(T,δ ,β )+ 1 s(T,δ ,β )2 Φ(Zα 2s(T,δπ,βπ)) max e π π 2 π π − , π∈Π(x) r α (27) s.t. m(T,δπ,βπ) G(C). ≥ The feasible set of the problem is Π= π := (δ (t),β (t)) m(T,δ ,β ) G(C), π Π(x) . { π π | π π ≥ ∈ } Given ε 0, the intersection of Π and s(T,δ ,β )2 = ε2 is ≥ π π Π = π := (δ (t),β (t)) s(T,δ ,β )2 = ε2, m(T,δ ,β ) G(C), π Π(x) . ε { π π | π π π π ≥ ∈ } According to Lemma 4.1, Πε = if ε < εˆ. Hence, Π = ε≥εˆΠε. Problem (27) is equivalent to the following bilevel∅ optimization problem ∪
m(T,δ ,β )+ 1 s(T,δ ,β )2 Φ(Zα 2s(T,δπ,βπ)) max max e π π 2 π π − . (28) ε≥εˆ π∈Πε r α Subsequently, we solve the inner-level optimization problem for each fixed ε εˆ ≥ m(T,δ ,β )+ 1 s(T,δ ,β )2 Φ(Zα 2s(T,δπ,βπ)) max e π π 2 π π − , π∈Πε r α or equivalently max m(T,δπ,βπ). π∈Πε The unique optimization solution is µσ−2ε Σ−1(t)r(t)ε ∗ ∗ ∗ 0 ∗ π = (δπε (t),βπε (t)), δπε (t)= , βπε (t)= , ΨT ΨT with maximum m(T,δπ,βπ) = εΨT . Therefore, we obtain the solution of prob- lem(28) by solving the problem
εΨ + 1 ε2 Φ(Zα 2ε) max e T 2 − , (29) ε≥εˆ r α 682 YANZENGANDZHONGFEILI which can be equivalently written as 1 (h(ε)−α) max e 2 . ε≥εˆ According to Lemma 4.3, problem(29)’s objective function attains its maximum at ε =ε ˆ when (24) holds. This completes the proof. Remark 1. The optimal strategy is determined by both the insurance business and the financial market. Remark 2. When µ = 0, the corresponding wealth process of the insurer becomes dXπ(t)= Xπ(t) [r (t)+ β (t)r(t)]dt + σ δ dB (t)+ β (t)σ(t)dB(t) . (30) { 0 π 0 π 0 π } T ¯ ¯ + ¯ 0 −1 ¯ ΨT (ΨT +Zα) Let ΨT = 0 r (s)Σ (s)r(s)ds andε ˇ = G(C)/ΨT . If C xξ(T )e , ∗ −1 ≥ the optimalq strategy of problem (17) is π = 0, Σ (t)r(t)ˇε/Ψ¯ T , the corresponding R ∗ expected wealth is EXπ (T )= C, and
1 2 ∗ εˇΨ¯ T + εˇ Φ(Zα 2ˇε) CaR2(π )= xξ(T ) 1 e 2 − . " − r α # In other words, if the return rate of the insurance business is 0, the optimal risk exposure is 0, which is consistent with our intuition. In the case of µ = 0, if we further choose σ0 = 0 and other parameters are constants in the wealth process (30), problem (17) is the same as that in Li et al. [18], and the optimal investment ∗ strategy, the corresponding expected terminal wealth and the optimal CaR2(π ) are the same as that in Li et al. [18]. In this sense, we extend the results of Li et al. [18]. ∗ Corollary 1. The relation between the optimal CaR2(π ) and the corresponding ∗ expected terminal wealth EXπ (T ) of problem (17) can be described as follows:
∗ 2 G(EXπ (T )) ∗ ∗ π ∗ π 1 ΨT 2G(EX (T )) CaR2(π )= xξ(T ) EX (T )v e Φ Zα , (31) − uα − ΨT u ∗ t where EXπ (T ) ε˜. (31) is known as the efficient frontier of the mean-CaR model ≥ associated with CaR2 in the mean-CaR space.
4.2. Mean-CaR1 model. This subsection is devoted to solving the following mean- CaR model associated with CaR1:
min CaR1(π), π∈Π(x) (32) ( s.t. EXπ(T ) C, ≥ where C is the predefined level, which is the minimum that the expected terminal wealth EXπ(T ) is allowed to attain, and satisfies (18). We can also obtain the explicit solution of problem(32) by adopting a quite similar derivation to that in the proof of Theorem 4.4, which is summarized by the following theorem stated without proof. Theorem 4.5. Assume that C satisfies (24). Then the unique optimal strategy of the mean-CaR1 problem (32) is −2 −1 ∗ µσ0 εˆ Σ (t)r(t)ˆε π = (δπ∗ (t),βπ∗ (t)), δπ∗ (t)= , βπ∗ (t)= , (33) ΨT ΨT OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS 683
∗ the corresponding expected wealth is EXπ (T )= C, and Φ(Z εˆ) CaR (π∗)= xξ(T ) 1 α − eεˆΨT . (34) 1 − α Remark 3. The optimal strategy and the corresponding expected terminal wealth of problem (32) are the same as that of problem (17). In addition, the corresponding results in Remark 2 is also valid for problem (32). ∗ Corollary 2. The relation between the optimal CaR1(π ) and the corresponding ∗ expected terminal wealth EXπ (T ) of problem (32) can be described as follows: ∗ ∗ π π ∗ EX (T ) G(EX (T )) CaR1(π )= xξ(T ) Φ Zα , (35) − α − ΨT ∗ where EXπ (T ) ε˜. (35) is known as the efficient frontier of the mean-CaR model ≥ associated with CaR1 in the mean-CaR space.
4.3. Mean-CaR0 model. In this subsection, we consider the following mean-CaR model associated with CaR0:
min CaR0(π), π∈Π(x) (36) ( s.t. EXπ(T ) C, ≥ where C as that in problem (17) is the predefined level, which is the minimum that the expected terminal wealth EXπ(T ) is allowed to attain, and satisfies (18). We can also obtain the explicit solution of problem (36) by adopting a quit similar derivation to that in the proof of Theorem 4.4, which is summarized by the following theorem stated without proof.
Theorem 4.6. The unique optimal strategy of the mean-CaR0 model (36) is −2 ∗ −1 ∗ ∗ µσ0 ε Σ (t)r(t)ε π = (δπ∗ (t),βπ∗ (t)), δπ∗ (t)= , βπ∗ (t)= , (37) ΨT ΨT the corresponding expected wealth is ∗ EXπ (T )= xξ(T )eεΨT = max C, ε˜ , (38) { } and ∗ ∗ ∗ ε (Ψ +Z )− 1 ε 2 CaR (π )= xξ(T ) 1 e T α 2 . (39) 0 − where ε∗ = max ε,ˆ Ψ + Z . { T α} Remark 4. The optimal strategy and the corresponding expected terminal wealth of problem (36) are the same as that of problem (17) and (32). In addition, the corresponding results in Remark 2 is also valid for problem (36). ∗ Corollary 3. The relation between the optimal CaR0(π ) and the corresponding ∗ expected terminal wealth EXπ (T ) of problem (36) can be described as follows: ∗ ∗ π π 2 G(EX (T )) 1 G(EX (T )) ∗ Zα − ∗ π ΨT 2 ΨT CaR0(π )= xξ(T ) EX (T )e , (40) ∗ − where EXπ (T ) ε˜. (40) is known as the efficient frontier of the mean-CaR model ≥ associated with CaR0 in the mean-CaR space. It is worth to pointing out that when C [xξ(T ), ε˜], the efficient frontier of ∗ ∈ π the mean-CaR0 model degenerates to a single point where EX (T )=˜ε. So the 2 G(˜ε) 1 G(˜ε) Zα Ψ − 2 Ψ efficient frontier of this model starts from point ε,˜ xξ(T ) εe˜ T T .. − 684 YANZENGANDZHONGFEILI
5. Sensitivity analysis and a numerical example. This section analyzes the results derived in the previous section, and presents a numerical example to show ∗ how parameters influence the optimal strategies and CaRk(π ), (k =2, 1, 0).
5.1. Sensitivity analysis. In this subsection, we first analyze the impact of pa- ∗ rameters on the optimal strategies, and then on the optimal CaRk(π ), (k =2, 1, 0). According to Theorem 4.4-4.6 and Corollary 1-3, when predefined level C [˜ε, + ), the optimal reinsurance-investment strategies of the three mean-CaR mod-∈ els are∞ the same and independent of confidence level α, which can be rewritten as
∗ ln C ln xξ(T ) π = (δπ∗ (t),βπ∗ (t)), δπ∗ (t)= − , −1 2 T 0 −1 µT + µ σ0 0 r (s)Σ (s)r(s)ds −1 Σ (t)r(Rt)[ln C ln xξ(T )] βπ∗ (t)= − , (41) 2 −2 T 0 −1 µ σ0 T + 0 r (s)Σ (s)r(s)ds ∗ and the corresponding expected terminal wealth EXR π (T ) are the same and equal ∗ 2 −1 T 0 −1 to predefined level C. Denote µ = σ0 T 0 r (s)Σ (s)r(s)ds. From (41), we have the following results: q R (1) δπ∗ (t) is increasing w.r.t. return rate of the insurance business µ when µ ∗ ∗ ∈ [0,µ ], but decreasing when µ (µ , + ). In addition, δ ∗ (t) is decreasing ∈ ∞ π w.r.t. risk level of the insurance business σ0. (2) βπ∗ (t) is decreasing w.r.t. return rate of the insurance business µ, but increas- ing w.r.t. risk level of the insurance business σ0. (3) δπ∗ (t) and βπ∗ (t) are decreasing w.r.t. time horizon T , but increasing w.r.t. predefined level C. In addition, according to (37), when predefined level C [xξ(T ), ε˜], the optimal ∈ ∗ reinsurance-investment strategy of the mean-CaR0 model π = (δπ∗ (t),βπ∗ (t)) is independent of predefined level C, and δπ∗ (t) and βπ∗ (t) are increasing w.r.t. time horizon T and confidence level α. Assume that predefined level C satisfies (24), by(31), (34), (39), and simple calculation, we have ∗ (1) CaRk(π ), (k =2, 1, 0) is decreasing w.r.t. return rate of the insurance busi- ness µ, but increasing w.r.t. risk level of the insurance business σ0, which ∗ shows that smaller risk measured by CaRk(π ), (k =2, 1, 0) is exposed when µ increases or σ0 decreases. ∗ (2) CaRk(π ), (k = 2, 1, 0) decreases as confidence level α increases, implying ∗ that the risk measured by CaRk(π ), (k = 2, 1, 0) becomes smaller if the insurer improves confidence level α. ∗ (3) CaRk(π ), (k = 2, 1, 0) first increases and then decreases w.r.t. time hori- ∗ zon T . In addition, we find that CaRk(π ), (k = 2, 1, 0) is increasing w.r.t. ∗ EXπ (T ) and CaR (π∗) CaR (π∗) CaR (π∗). 1 ≥ 2 ≥ 0 5.2. Numerical example. In this subsection, we present a numerical example to illustrate the results we derived in the previous section. In the numerical example, we consider a financial market consisting of one risk-free asset and one risky asset which has one risk resource, i.e., n = 1, m = 1. For convenience but without loss of generality, we assume that t [0,T ], the risk-free interest rate is r0(t) r0, the appreciate rate and the volatility∀ ∈ of the risky asset are r (t) r and σ(t)≡ σ, 1 ≡ 1 ≡ OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS 685 respectively. Throughout this section, unless otherwise stated, the basic parameters are given in the following table. Parameter Description Value
x Initial wealth 1 T Time horizon 5 r0 Risk-freeinterestrate 0.05 r1 Appreciate rate of the risky asset 0.1 σ Volatility of the risky asset 0.2 µ Return rate of the insurance business 0.5 σ0 Risk level of the insurance business 1 C Predefinedlevel 3.5 α Confidence level 0.05 Because we assume that the parameters of the financial market are constants in the interval [0,T ], the optimal strategies of the three mean-CaR models are independent of the time t.
0.42 3.5 σ =1 σ =1 0 0 0.4 σ =1.2 σ =1.2 0 3 0 σ =1.4 σ =1.4 0.38 0 0
0.36 2.5
0.34 * * π π 2 δ β 0.32
0.3 1.5
0.28 1 0.26
0.24 0.5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 µ µ
(a) (b)
0.75 2 C=3.5 C=3.5 0.7 C=4 C=4 C=4.5 1.8 C=4.5 0.65
0.6 1.6
0.55 * * π π 1.4 δ β 0.5
0.45 1.2
0.4 1 0.35
0.8 3 3.5 4 4.5 5 3 3.5 4 4.5 5 T T
(c) (d)
Figure 1. The evolution of the optimal reinsurance-investment strategies of the three mean-CaR models when C [˜ε, + ]. ∈ ∞
The impact of parameters on the optimal reinsurance-investment strategies when C [˜ε, + ] is illustrated in Figure 1. Subgraph (a) shows that as µ increases, δ ∗ (t) ∈ ∞ π 686 YANZENGANDZHONGFEILI
firstly increases, then decreases. That is to say, when the return rate of the insur- ance business increases and is less than µ∗, in order to make the expected terminal wealth be not less than predefined level C, the insurer will purchase less reinsurance or acquire more new insurance business; when µ is larger than µ∗ and increases, the insurer will purchase more reinsurance or acquire less new insurance business to decrease the risk while the expected terminal wealth is not less than predefined level C. In addition, this subgraph also illustrates that the larger σ0 is, the smaller δπ∗ (t) is, i.e., if the risk of the insurance business becomes larger, the insurer will purchase more reinsurance or acquire less new insurance business to reduce her/his risk exposure. Subgraph (b) demonstrates that if the return of the insurance busi- ness becomes higher, since the insurer can gain enough return from the insurance business to ensure the expected terminal wealth not less than predefined level C, she/he will invest less in the risky asset to reduce the investment risk. Moreover, in this subgraph we can also find that βπ∗ (t) increases as σ0 increases, i.e., the insurer will invest more in the risky asset when the risk of the insurance business increases. Subgraphs (c) and (d) show that as time horizon T increases or predefined level C decreases, both δπ∗ (t) and βπ∗ (t) become smaller; namely, when time horizon T becomes longer or predefined level C becomes smaller, the insurer will purchase more reinsurance, acquire less new insurance business, and invest less in the risky asset.
0.22 0.55
0.2 0.5
0.18 0.45
0.16 0.4
0.14 0.35 * * π π 0.12 0.3 δ β
0.1 0.25
0.08 0.2
0.06 0.15 α=0.05 α=0.05 α=0.07 α=0.07 0.04 0.1 α=0.1 α=0.1 0.02 0.05 10 11 12 13 14 15 10 11 12 13 14 15 T T
(a) (b)
Figure 2. The evolution of the optimal reinsurance-investment strategy of mean-CaR model when C [xξ(T ), ε˜]. 0 ∈ Figure 2 illustrates the impact of parameters on the optimal reinsurance -investment strategy of the mean-CaR model when C [xξ(T ), ε˜]. From Figure 0 ∈ 2, we can find that both δπ∗ (t) and βπ∗ (t) increase as time horizon T or confidence level α increases. That is, if the time horizon becomes longer or the confidence level increases, the insurer will purchase less reinsurance, acquire more new insurance business, and invest more in the risky asset. ∗ In Figure 3, the impact of parameters on the optimal CaRk(π ), (k = 2, 1, 0) and the efficient frontiers of the three mean-CaR models are demonstrated. The ∗ ∗ ∗ subgraphs all show that CaR1(π ) CaR2(π ) CaR0(π ). Specially, sub- graph (a) illustrates that if the the return≥ rate of the≥ insurance business increases, ∗ CaRk(π ), (k =2, 1, 0) will decreases. The reason is that the insurer will purchase more reinsurance, acquire less new insurance business and invest less in the risky OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS 687
1.4 1.05 CaR CaR 2 2 1 1.3 CaR CaR 1 1 CaR CaR 0 0.95 0 1.2
0.9 1.1 0.85 1 CaR CaR 0.8 0.9 0.75
0.8 0.7
0.7 0.65
0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 µ σ 0
(a) (b)
0.8 1.1 CaR CaR 2 2 0.75 1.05 CaR CaR 1 1 CaR 1 CaR 0.7 0 0 0.95 0.65 0.9 0.6 0.85 CaR 0.55 CaR 0.8 0.5 0.75 0.45 0.7
0.4 0.65
0.35 0.05 0.06 0.07 0.08 0.09 0.1 0 1 2 3 4 5 α T
(c) (d)
1 CaR 2 0.95 CaR 1 CaR 0 0.9
0.85
0.8 CaR
0.75
0.7
0.65
3.5 4 4.5 π* EX (T) (e)
∗ Figure 3. . The evolution of the CaRk(π ). asset. Subgraph (b) reveals that as the risk of the insurance business increases, ∗ CaRk(π ), (k = 2, 1, 0) becomes larger. In this case, the insurer will purchase more reinsurance or acquire less new insurance business to reduce the risk from the insurance business, and invest more in the risky asset to ensure the expected terminal wealth not less than predefined level C. Subgraph (c) shows that as the ∗ insurer improves the confidence level, CaRk(π ), (k = 2, 1, 0) becomes smaller. ∗ Subgraph (d) clarifies that CaRk(π ), (k = 2, 1, 0) is not monotonous w.r.t the time horizon. As time horizon T increases and is large enough, the insurer can 688 YANZENGANDZHONGFEILI
∗ make CaRk(π ), (k = 2, 1, 0) become smaller by purchasing more insurance, ac- quiring less new insurance business and investing more in the risky asset. Subgraph (e) exemplifies the efficient frontiers of the three mean-CaR models. We find that under the optimal strategy, as the profit measured by the expected terminal wealth ∗ π ∗ EX (T ) increases, the risk measured by CaRk(π ), (k =2, 1, 0) will become larger, which is consistent with our intuition.
6. Conclusion and future research. In this paper, we construct three mean- CaR models to investigate the optimal reinsurance-investment problem of an in- surer. The insurer’s surplus is modeled by a diffusion approximation model, and she/he can control her/his risk by purchasing proportional reinsurance, acquiring new business, and investing her/his surplus in a financial market consisting of one risk-free asset and multiple risky assets. By converting these models into bilevel optimization problems, we derive analytically the optimal DR strategies and the efficient frontiers. Moreover, we find that the optimal strategies are the same as the three mean-CaR models, and our results extend the ones of Emmer et al. [11] and Li et al. [18]. In an analogous way, the approach in this paper can also be applied to a safety-first model, a mean-variance model, a mean-VaR model, and so on. In order to ensure tractability of the optimization problems, we restrict ourselves to cheap proportional reinsurance and DR strategies in this paper. However, the DR strategies may not be feedback strategies under general models; therefore, the op- timal DR strategies for our models may not be globally optimal in the set of all dynamic strategies. In future work, we will consider other types of reinsurance, such as noncheap proportional reinsurance and excess-of -loss reinsurance, and replace the DR strate- gies with the general process. Then, the approach in this paper will no longer work, and we have to adopt much more sophisticated techniques, such as Monte Carlo simulation and numerical calculus to solve the corresponding optimization problems.
Acknowledgments. The authors would like to thank the anonymous referee for careful reading of the paper and helpful comments, which contributed to improving the quality of the paper.
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