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Minimizing Lundberg inequality for ruin probability under correlated risk model by investment and reinsurance
- Lin Xu^{1}Email authorView ORCID ID profile,
- Minghan Wang^{1} and
- Bin Zhang^{1}
https://doi.org/10.1186/s13660-018-1838-0
© The Author(s) 2018
- Received: 5 March 2018
- Accepted: 6 September 2018
- Published: 17 September 2018
Abstract
This paper investigates optimal investment and reinsurance policies for an insurance company under a correlated risk model with common Poisson shocks. The goal of the insurance company is to minimize the ultimate ruin probability. By the dynamic programming principle, the Hamilton–Jacobi–Bellman (HJB for short) equation associated with this control problem is obtained. Since there is no explicit solution to the HJB equation, this paper alternates to find the minimal exponential upper bound of the ruin probability. The exponential upper bound of ruin probability is also called Lundberg inequality. Minimizing Lundberg inequality is equal to finding the maximal Lundberg coefficient. It turns out that the optimal investment and reinsurance polices are constant policies. Some numerical examples are given to illustrate the impact of the dependent structure and the investment chance on the upper bound.
Keywords
- Lundberg inequality
- Ruin probability
- Optimal reinsurance
- Optimal investment
- Correlated risk model
1 Introduction
The past two decades have witnessed huge attention on the risk model with dependent structure. The dependent structure is variety. For example, Cossette and Marceau [1], Yuen et al. [2] studied ruin probability under a model with dependent business; Wang and Yuen [3] studied ruin probability when the premiums and claims are thinning dependent; Wang and Yin [4] studied asymptotic ruin probabilities in a dependent discrete risk model. On the other hand, there is also a large amount of literature focus on the model when the claims and claim arrival intervals are correlated; for example, see Denuit et al. [5], Boudreault et al. [6] and the references therein. There is also a lot of literature concentrating on a multivariate risk process, where the components of the multivariate process specify different business of insurance company and they cannot be integrated into a univariate process (or it is meaningless in practice). For details on this aspect, see Weng et al. [7], Anastasiadis and Chukova [8] and the references therein.
Almost every insurance company has investment and reinsurance plans for profit increments and risk exposure control. Due to this fact and regulatory requirements, research on optimal reinsurance and (or) investment for an insurance company has become one of the most important topics in risk theory recently (cf. Browne [9], Fleming and Sheu [10], Taksar and Markussen [11], Yang and Zhang [12], Zhang and Siu [13], Zeng and Li [14], Meng et al. [15]). The aforemention papers did not consider dependent structure or correlated structure. Naturally, it makes sense to consider the optimal investment and reinsurance problem under a correlated risk model. [16] studied optimal dynamic reinsurance with dependent risks and variance premium principle, where the goal of insurance company is to maximize exponential utility at terminal time. Liang and Yuen [17] investigated optimal investment and reinsurance strategies for an insurance company with generalized mean-variance premium principle and no-short selling, the goal of the insurance company wherein is to maximize the expected utility. Other works on this topic can also be found in Landriault et al. [18], Zeng et al. [19], Chiu and Wong [20].
Compared to the study on optimal investment and reinsurance for maximizing expected utility, papers concentrating on minimizing ultimate ruin probability are relatively few. Usually, if one wants to find optimal policies for minimizing the ultimate ruin probability, it is difficult to prove the regularity of the value function. It is also very difficult to obtain an explicit solution to the HJB equation and the optimal policies. This may be one of the reasons that prevent us from progressing the study on minimizing ultimate ruin probability by investment or reinsurance. When the explicit expression of ruin probability is invalid, it is meaningful to find an accurate upper bound estimation of ruin probability. Due to this fact, the upper bound estimation of ruin probability has become one of the three main topics in the classical risk model (cf. Grandell [21]). In the classical risk model, an upper bound for ruin probability is also called Lundberg inequality. Hu and Zhang [22] studied optimal reinsurance for minimizing the upper bound of ultimate ruin probability in a correlated risk model with common shocks. The results in [22] show that the upper bound of the ruin probability in the model with reinsurance chance is less than the one in the model without reinsurance chance. Thus, under the correlated risk model, reinsurance business can really reduce the exposure of the insurance company, and this impact can be quantified by comparing the upper bound. Hu and Zhang [22] only considered reinsurance business, it is natural to take both reinsurance and investment into account. It is well known that if the claim sizes have exponential moments (i.e., the so-called small claim case), the ruin probability decreases exponentially with the increase in the initial surplus (cf. Assussen and Albrecher [23]). However, Kalashnikov and Norberg [24], Frolova et al. [25] found that, even if the claim sizes are small, if the insurance company invests all of its surplus into a risky market, the ruin probability decreases only with some negative power function of the initial surplus. Thus, for large capital, investing more than the surplus into the risky market cannot be optimal. Then, one interesting problem is: What is the minimal ruin probability that it can obtain? Particularly, can it do better than keeping the funds in the bond? And if yes, how much can it do better? Hipp and Plum [26] and Gaier et al. [27] considered this problem when the surplus process is the compound Poisson process and found that when the insurance company can adjust its investment amount to reduce the risk exposure, a constant investment amount is the optimal policy, regardless of the change of surplus of the insurance company. This paper extends the study of [22] by incorporating the investment business into the decision process. But the method applied in this paper relies on the dynamic programming principle and HJB equation. As a result, we also find that the optimal polices for minimizing the upper bound of ruin probability are constant investment amount and constant reinsurance ratio, respectively. Numerical examples show the following: under optimal constant investment policy, an upper bound of the ultimate ruin probability is less than the corresponding one in [22]; when the correlation coefficient of each component risk process increases, the impact of investment on the upper bound of ruin probability inequality is less significant; when the claims are heavy-tailed, reinsurance plays a more important role for the insurance company than the investment.
The rest of this paper is organized as follows. Section 2 presents the model and problem. Section 3 gives the analysis process and the equations satisfying minimal adjustment coefficients. Section 4 gives comparison numerical examples and our conclusions.
2 Model and problem
Remark 1
\(N^{(k)}(t)\), \(k=1,2,\ldots,m\), are Poisson processes with intensity \(\eta_{k}\), \(k = 1,\ldots,m\); \(\alpha_{ik} \circ N^{(k)}(t)\sim B(N^{(k)}(t),\alpha_{ik})\); \(N_{(i)}(t)\), \(i=1,2,\ldots,n\), are Poisson processes with tensity \(\lambda_{i}\), \(i = 1,\ldots,n\).
Lemma 2.1
Remark 2
A natural interpretation of condition (8) is that the insurance company charges more than the net premium of the claims due the expected premium principle. In mathematics, condition (8) guarantees the existence of the root to the Lundberg equation when the claims are “small claims” and thus are of great importance in [22]. However, later we will find that once we take the investment into account, condition (8) is not always necessary. This is one difference between our model and the one of [22].
Assumption 1
- (i)
\(0\leq\delta<\mu\), \(\mathbf{0}< \mathbf{a}\leq\mathbf {1}\), \(\mathbf{0}< \mathbf{M}\), where 0 and 1 are vectors with all the elements being 0 and 1, respectively.
- (ii)
The investment process \(\{A(t)\}_{t\geq0}\) is predictable w.r.t. to \(\mathcal{F}_{t}\) and \(\mathbb {E}[\int_{0}^{t}|A(s)|^{2}\,\mathrm{d}s ]<\infty\), \(\forall t\in[0,\infty)\). This means that the amount of the reinsurance and investment may depend on the information up to time t of the system, but it may not depend on the amount of the claims happened at time t, i.e., \(A(t)\in\mathcal{F}_{t-}\).
3 Minimizing upper bound of ruin probability
Lemma 3.1
If we want to find a solution to the HJB equation of the form (19), then the optimal constant investment policy and optimal reinsurance policy are separately determined.
Proof
Lemma 3.2
Proof
To clarify the impact of a, M, A on the root to Eq. (25), denote the unique positive root by \(R_{\mathbf {a}, \mathbf{M},A}\). Following a standard method, we have the following upper bound estimation for ultimate ruin probability.
Theorem 3.3
Proof
Remark 3
Lemma 3.2 seems similar to Lemma 1 of [22]. However, the sufficient and necessary conditions such that the Lundberg coefficient exists in [22] is the net profit condition (8). By condition (27) we know that, because of the investment chance, the insurance company can derive an exponential upper bound estimation even if the net profit condition (8) does not hold. This indicates that investment offers the insurance company more chance to control its exposure. Then, the remaining problem is how the insurance company should settle the investment position to obtain the minimal upper bound of ruin probability.
- (1)Optimal investment policy Note that by dynamic programming, this is equal to finding the optimizer of Eq. (20). Denotethen$$ h(R):=- \biggl[A_{t}\mu R-\frac{1}{2}A_{t}^{2} \sigma^{2}R^{2} \biggr]-C^{I} R+M_{\tilde{S}(1)}(R), $$(33)is the maximizer of \(h(R)\). Substitute (34) into Eq. (25), the maximum \(R^{*}\) should be the solution to equation$$ A^{*}=\frac{\mu}{\sigma^{2} R} $$(34)Solving Eq. (35) we have an optimal investment amount immediately. To proceed our discussion, let \(H_{\mathbf{a},\mathbf{M},A^{*}}(R):=M_{\tilde{S}(1)}(R)-C^{I}R\) and \(a_{i}'=\frac{\theta_{i}^{R}-\theta_{i}}{\theta_{i}^{R}}\).$$ M_{\tilde{S}(1)}(R)-C^{I}R-\frac{1}{2} \frac{\mu^{2}}{\sigma^{2}}=0. $$(35)
- (2)
Optimal reinsurance policy Suppose that the optimal investment policy is determined, then the optimal reinsurance strategy shall maximize the \(R_{\mathbf{a},\mathbf{M},A^{*}}\) w.r.t \((\mathbf {a},\mathbf{M})\). This is equivalent to determining the optimal quota-share retention levels \(\mathbf{a^{*}}= (a^{*}_{1},\ldots,a^{*}_{n})\), and excess of loss retention limits \(\mathbf {M^{*}}=(M^{*}_{1},\ldots,M^{*}_{n})\) such that \(R_{\mathbf{a},\mathbf {M},A^{*}}\) attains maximum. By Theorem 3.1, the optimal investment amount and optimal reinsurance policies are separately determined, the properties obtained in [22] can be applied directly. In the rest of this paper, we focus on the impact of investment chance on the upper bound of ruin probability by numerical examples.
4 Numerical examples
By previous results, we know that the optimal reinsurance policies and optimal investment amounts are separately determined, and the investment chance implies that the adjustment coefficient in our risk model is less than that in the model of [22]. However, since we have no analytical expression for the solution to the Lundberg equation, we cannot compare the minimal upper bound in our risk model and the one in [22] directly. In this section, we will illustrate some comparative results by numerical examples. For this goal, we choose the same surplus process parameters as those in [22]. To illustrate the effect of investment chance, we compare the Lundberg exponent in [22] and the one in our model. Denote the maximum Lundberg exponent in [22] and the one in our model by \(R_{H}^{*}\) and \(R^{*}\), respectively.
Example 1
Let \(Y^{(1)}\) and \(Y^{(2)}\) be exponentially distributed with means \(\mu_{1} = \mu_{2}=1\), which are typical “small claim”. Let the parameters of the financial market be \(\mu=0.1\), \(\sigma=0.05\), and thus the market price of the market is 2. For any given \((\theta_{1}, \theta_{1}^{R}1)\) and \((\theta_{2}, \theta_{2}^{R})\), by calculating the optimal excess of loss retention levels \((M_{1}^{*},M_{2}^{*})\) with \((\alpha_{11}, \alpha_{12})\) and \((\alpha_{21}, \alpha_{22})\), we have the correspondingly upper bound of ruin probability under different structure.
Comparison of Lundberg exponent: exponential case. \((\theta _{1}, \theta_{1}^{R})=(\theta_{2}, \theta_{2}^{R})= (0.2, 0.4)\), values of Lundberg exponent and upper bound of ruin probability with different dependence parameters for Hu’s model and our model
\((\alpha_{11},\alpha_{12})\) | (0,1) | (0.2,0,8) | (0.4,0.6) | (0.6,0.4) | (0.8,0.2) | (1,0) |
\((\alpha_{21},\alpha_{22})\) | (1,0) | (0.8,0.2) | (0.6,0.4) | (0.4,0.6) | (0.2,0.8) | (0,1) |
\((\lambda_{1},\lambda_{2})\) | (4,2) | (3.6,2.4) | (3.2,2.8) | (2.8,3.2) | (2.4,3.6) | (2,4) |
ρ | 0 | 0.111111 | 0.160714 | 0.160714 | 0.111111 | 0 |
\(M^{*}_{1}\) | 1.48575 | 1.595619 | 1.602443 | 1.542483 | 1.47389 | 1.48575 |
\(M^{*}_{2}\) | 1.48575 | 1.47389 | 1.542483 | 1.602443 | 1.595619 | 1.48575 |
\(R_{H}^{*} \) | 0.226466 | 0.184908 | 0.169814 | 0.169814 | 0.184908 | 0.226466 |
\(e^{-10R_{H}^{*}}\) | 0.1038 | 0.1573 | 0.1830 | 0.1830 | 0.1573 | 0.1038 |
\(R^{*} \) | 0.231466 | 0.194724 | 0.193215 | 0.173001 | 0.194032 | 0.231298 |
\(e^{-10R^{*}}\) | 0.0988 | 0.1427 | 0.1769 | 0.1773 | 0.1437 | 0.0099 |
Comparison of Lundberg exponent: exponential case. \((\theta _{1}, \theta_{1}^{R})=(0.2, 0.4)\) and \((\theta_{2}, \theta_{2}^{R})= (0.25, 0.5)\), values of Lundberg exponent and upper bound of ruin probability with different dependence parameters for Hu’s model and our model
\((\alpha_{11},\alpha_{12})\) | (0,1) | (0.2,0,8) | (0.4,0.6) | (0.6,0.4) | (0.8,0.2) | (1,0) |
\((\alpha_{21},\alpha_{22})\) | (1,0) | (0.8,0.2) | (0.6,0.4) | (0.4,0.6) | (0.2,0.8) | (0,1) |
\((\lambda_{1},\lambda_{2})\) | (4,2) | (3.6,2.4) | (3.2,2.8) | (2.8,3.2) | (2.4,3.6) | (2,4) |
ρ | 0 | 0.111111 | 0.160714 | 0.160714 | 0.111111 | 0 |
\(M^{*}_{1}\) | 1.374693 | 1.426397 | 1.382661 | 1.292643 | 1.22716 | 1.286787 |
\(M^{*}_{2} \) | 1.65657 | 1.679524 | 1.741289 | 1.766061 | 1.70904 | 1.55064 |
\(R_{H}^{*} \) | 0.244762 | 0.201829 | 0.188391 | 0.191707 | 0.211816 | 0.261482 |
\(e^{-10R_{H}^{*}}\) | 0.0865 | 0.1329 | 0.1520 | 0.1470 | 0.1203 | 0.0732 |
\(R^{*} \) | 0.273025 | 0.228736 | 0.203113 | 0.202304 | 0.231029 | 0.271354 |
\(e^{-10R^{*}}\) | 0.0652 | 0.1015 | 0.1312 | 0.1323 | 0.0992 | 0.0663 |
Example 2
Comparison of Lundberg exponent: Pareto distribution. \((\theta_{1}, \theta_{1}^{R})=(\theta_{2}, \theta_{2}^{R})= (0.2, 0.4)\), values of Lundberg exponent and upper bound of ruin probability with different dependence parameters for Hu’s model and our model
\((\alpha_{11},\alpha_{12})\) | (0,1) | (0.2,0,8) | (0.4,0.6) | (0.6,0.4) | (0.8,0.2) | (1,0) |
\((\alpha_{21},\alpha_{22})\) | (1,0) | (0.8,0.2) | (0.6,0.4) | (0.4,0.6) | (0.2,0.8) | (0,1) |
\((\lambda_{1},\lambda_{2})\) | (4,2) | (3.6,2.4) | (3.2,2.8) | (2.8,3.2) | (2.4,3.6) | (2,4) |
ρ | 0 | 0.111111 | 0.160714 | 0.160714 | 0.111111 | 0 |
\(M^{*}_{1}\) | 2.325091 | 2.485092 | 2.521622 | 2.470896 | 2.380495 | 2.325091 |
\(M^{*}_{2} \) | 2.325091 | 2.380495 | 2.470896 | 2.521622 | 2.485092 | 2.325091 |
\(R_{H}^{*} \) | 0.144713 | 0.125136 | 0.117427 | 0.117427 | 0.125136 | 0.144713 |
\(e^{-10R_{H}^{*}} \) | 0.2352 | 0.2861 | 0.3090 | 0.3090 | 0.2861 | 0.2352 |
\(R^{*} \) | 0.160045 | 0.132486 | 0.120324 | 0.120011 | 0.127033 | 0.145129 |
\(e^{-10R^{*}}\) | 0.2018 | 0.2658 | 0.3002 | 0.3012 | 0.2807 | 0.2343 |
Comparison of Lundberg exponent: exponential case. \((\theta _{1}, \theta_{1}^{R})=(0.2, 0.4)\) and \((\theta_{2}, \theta_{2}^{R})= (0.25, 0.5)\), values of Lundberg exponent and upper bound of ruin probability with different dependence parameters for Hu’s model and our model
\((\alpha_{11},\alpha_{12})\) | (0,1) | (0.2,0,8) | (0.4,0.6) | (0.6,0.4) | (0.8,0.2) | (1,0) |
\((\alpha_{21},\alpha_{22})\) | (1,0) | (0.8,0.2) | (0.6,0.4) | (0.4,0.6) | (0.2,0.8) | (0,1) |
\((\lambda_{1},\lambda_{2})\) | (4,2) | (3.6,2.4) | (3.2,2.8) | (2.8,3.2) | (2.4,3.6) | (2,4) |
ρ | 0 | 0.111111 | 0.160714 | 0.160714 | 0.111111 | 0 |
\(M^{*}_{1} \) | 2.151698 | 2.245443 | 2.223645 | 2.134013 | 2.03478 | 2.012961 |
\(M^{*}_{2} \) | 2.592899 | 2.663988 | 2.731697 | 2.735907 | 2.641812 | 2.4257 |
\(R^{*}\) | 0.1564 | 0.1368 | 0.1303 | 0.1323 | 0.1431 | 0.1672 |
\(e^{-10R^{*}} \) | 0.209349 | 0.254557 | 0.271713 | 0.266337 | 0.239386 | 0.18796 |
\(R^{*} \) | 0.162025 | 0.139998 | 0.135537 | 0.137071 | 0.148071 | 0.169951 |
\(e^{-10R^{*}}\) | 0.1978 | 0.2466 | 0.2579 | 0.2539 | 0.2275 | 0.1828 |
Declarations
Acknowledgements
The authors are very grateful to the referees and editors for their constructive advices.
Funding
Lin Xu received a grant from the Humanities and Social Sciences Project of the Ministry Education of China (17YJC910009). Minghan Wang was awarded grants by the NSFC (11301303), the NSSFC(15BJY007), the Taishan Scholars Program of Shandong Province(No.tsqn20161041). Bin Zhang received grants from the Humanities and Social Sciences Project of the Ministry Education of China (16YJC630070), the Natural Science Foundation of Shandong Province (ZR2018MG002), a Project of Shandong Province Higher Educational Science and Technology Program (J15LI03, J15LI53).
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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