Absolute ruin problems in a compound Poisson risk model with constant dividend barrier and liquid reserves
- Dan Peng^{1}Email author,
- Donghai Liu^{1} and
- Zhenting Hou^{2}
https://doi.org/10.1186/s13662-016-0746-1
© Peng et al. 2016
Received: 2 July 2015
Accepted: 10 January 2016
Published: 10 March 2016
Abstract
In this paper, we consider a compound Poisson surplus model with constant dividend barrier and liquid reserves under absolute ruin. When the surplus is negative, the insurer is allowed to borrow money at a debit interest rate to continue the business; when the surplus is below a fixed level △, the surplus is kept as liquid reserves, which do not earn interest; when the surplus attains the level △, the excess of the surplus over the level receives interest at a constant rate; when the surplus reaches a higher level b, the excess of the surplus above b is all paid out as dividends to shareholders of the insurer. We first derive the integro-differential equations satisfied by the moment-generating function and moment of the discounted dividend payments until absolute ruin. Then, applying these results, we get explicit expressions of them for exponential claims and discuss the impact of the model parameters on the expected dividend payments by numerical examples.
Keywords
MSC
1 Introduction
On the other hand, the surplus of the insurer with a certain dividend strategy has also been receiving more and more attention, including [1, 7–9]. For instance, de Finetti [10] studied the dividend strategy in a discrete process. Lin et al. [9] investigated the classical risk model with constant dividend barrier and analyzed the Gerber-Shiu discounted penalty function at ruin. Albrecher et al. [11] considered the distribution of dividend payments in the Sparre Andersen model with constant dividend barrier. Cai et al. [12] considered a more general model that incorporates the notion of threshold strategy. Based on the model (1.1), they assume that if the surplus continues to surpass a higher level \(b\geq\triangle\), then the excess of the surplus above b is paid out as dividends to the insurer’s shareholders at a constant dividend rate, and no interest is earned on the surplus over the threshold level b, and they discuss the interactions of the liquid reserve level, the interest rate, and the threshold level in the proposed risk model by studying the expected discounted penalty function and the expected present value of dividends paid up to the time of ruin. More specifically, they assume that the portion of the surplus is below a present level △ is liquid, and the amount in excess of this level is invested under a deterministic interest rate. Instead of implementing a threshold in Cai et al. [12], Sendova and Zhang [13] consider a percentage of the current surplus of the insurer and also study the expected discounted penalty function at ruin.
The rest of the paper is organized as follows. In Section 2, we get the integro-differential equations for the moment-generating function and the nth moment of the discounted dividends. In Section 3, we find their explicit expressions for exponential claims and discuss the impact of the model parameters on the expected dividend payments by numerical examples.
2 Integro-differential equations
Theorem 2.1
Proof
When \(u=-\frac{c}{\beta}\), absolute ruin is immediate, namely, no dividend is paid, and we obtain (2.4).
Further, letting \(u\uparrow0\) in (2.1), \(u\downarrow0\) in (2.2), and using (2.6), and then letting \(u\uparrow\triangle\) in (2.2), \(u\downarrow\triangle\) in (2.3), and using (2.6), we get (2.7). □
Remark 2.1
Theorem 2.2
Proof
The proof is obvious and we omit it here. □
Corollary 2.1
3 Explicit expressions for exponential claims
In this section, we assume that \(F(x)=1-e^{-\frac{x}{\mu}}\), \(x>0\), \(\mu>0\), namely, the claim size distribution \(F(x)\) is the exponential distribution with mean μ. We obtain explicit expressions of the moment-generating function and higher moments of the discounted dividends.
We summarize the exact solution for \(V_{n1}(u,b)\), \(V_{n2}(u,b)\), and \(V_{n3}(u,b)\) in the following theorem.
Theorem 3.1
Remark 3.1
At the end of this section, we use the following numerical examples to discuss the impact of the model parameters on the expected dividend payments.
Example 1
Influence of u and r on \(\pmb{V_{13}(u,b)}\) with \(\pmb{b=2.8}\) , \(\pmb{\triangle=1.5}\) , \(\pmb{\alpha=0.03}\) , \(\pmb{\mu=1}\) , \(\pmb{\lambda=1}\) , \(\pmb{c=1.5}\) , \(\pmb{\beta=0.09}\)
u ∖ r | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 |
---|---|---|---|---|---|---|
1.6 | 27.8537 | 28.1717 | 28.4903 | 28.8095 | 29.1294 | 29.4498 |
1.7 | 27.9233 | 28.2420 | 28.5614 | 28.8813 | 29.2018 | 29.5230 |
1.8 | 27.9961 | 28.3155 | 28.6355 | 28.9562 | 29.2744 | 29.5992 |
1.9 | 28.0719 | 28.3920 | 28.7127 | 29.0340 | 29.3559 | 29.6784 |
2.0 | 28.1507 | 28.4714 | 28.7928 | 29.1147 | 29.4372 | 29.7603 |
2.1 | 28.2323 | 28.5537 | 28.8756 | 29.1981 | 29.5212 | 29.8449 |
2.2 | 28.3167 | 28.6386 | 28.9610 | 29.2840 | 29.6077 | 29.9318 |
2.3 | 28.4038 | 28.7261 | 29.0490 | 29.3725 | 29.6965 | 30.0212 |
2.4 | 28.4933 | 28.8161 | 29.1394 | 29.4632 | 29.7877 | 30.1127 |
Example 2
Influence of u and β on \(\pmb{V_{13}(u,b)}\) with \(\pmb{b=2.8}\) , \(\pmb{\triangle=1.5}\) , \(\pmb{\alpha=0.03}\) , \(\pmb{\mu=1}\) , \(\pmb{\lambda=1}\) , \(\pmb{c=1.5}\) , \(\pmb{r=0.04}\)
u ∖ β | 0.09 | 0.10 | 0.11 | 0.12 | 0.13 | 0.14 |
---|---|---|---|---|---|---|
1.6 | 28.1717 | 27.1050 | 26.1146 | 25.1984 | 24.3525 | 23.5719 |
1.7 | 28.2420 | 27.1783 | 26.1906 | 25.2769 | 24.4333 | 23.6549 |
1.8 | 28.3155 | 27.2544 | 26.2691 | 25.3577 | 24.5162 | 23.7396 |
1.9 | 28.3920 | 27.3332 | 26.3501 | 25.4407 | 24.6010 | 23.8262 |
2.0 | 28.4714 | 27.4147 | 26.4334 | 25.5258 | 24.6877 | 23.9144 |
2.1 | 28.5537 | 27.4987 | 26.5191 | 25.6129 | 24.7763 | 24.0042 |
2.2 | 28.6386 | 27.5851 | 26.6069 | 25.7020 | 24.8666 | 24.0956 |
2.3 | 28.7261 | 27.6738 | 29.6968 | 25.7930 | 24.9585 | 24.1884 |
2.4 | 28.8161 | 27.7648 | 26.7887 | 25.8857 | 25.0521 | 24.2827 |
Example 3
Influence of u and △ on \(\pmb{V_{13}(u,b)}\) with \(\pmb{b=2.8}\) , \(\pmb{\alpha=0.03}\) , \(\pmb{\mu=1}\) , \(\pmb{\lambda=1}\) , \(\pmb{c=1.5}\) , \(\pmb{\beta=0.09}\) , \(\pmb{r=0.04}\)
u ∖△ | 0.9 | 1.1 | 1.3 | 1.5 | 1.7 | 1.9 |
---|---|---|---|---|---|---|
1.6 | 20.8956 | 22.6238 | 24.9465 | 28.1717 | 32.8679 | 40.2129 |
1.7 | 20.9881 | 22.7109 | 25.0265 | 28.2420 | 32.9245 | 40.2483 |
1.8 | 21.0815 | 22.7994 | 25.1086 | 28.3155 | 32.9857 | 40.2906 |
1.9 | 21.1756 | 22.8892 | 25.1928 | 28.3920 | 33.0512 | 40.3394 |
2.0 | 21.2706 | 22.9803 | 25.2789 | 28.4714 | 33.1211 | 40.3944 |
2.1 | 21.3662 | 23.0727 | 25.3669 | 28.5537 | 33.1950 | 40.4556 |
2.2 | 21.4625 | 23.1662 | 25.4568 | 28.6386 | 33.2779 | 40.5256 |
2.3 | 21.5594 | 23.2608 | 25.5484 | 28.7261 | 33.3545 | 40.5952 |
2.4 | 21.6570 | 23.3565 | 25.6417 | 28.8161 | 33.4398 | 40.6732 |
Example 4
Influence of u and b on \(\pmb{V_{13}(u,b)}\) with \(\pmb{\triangle=1.5}\) , \(\pmb{\alpha=0.03}\) , \(\pmb{\mu=1}\) , \(\pmb{\lambda=1}\) , \(\pmb{c=1.5}\) , \(\pmb{\beta=0.09}\) , \(\pmb{r=0.04}\)
u ∖ b | 2.5 | 2.6 | 2.7 | 2.8 | 2.9 | 3.0 | 3.1 |
---|---|---|---|---|---|---|---|
1.6 | 30.1186 | 29.4128 | 28.7660 | 28.1717 | 27.6244 | 27.1195 | 26.6526 |
1.7 | 30.1938 | 29.4862 | 28.8378 | 28.2420 | 27.6934 | 27.1872 | 26.7192 |
1.8 | 30.2724 | 29.5630 | 28.9128 | 28.3155 | 27.7655 | 27.2579 | 26.7887 |
1.9 | 30.3542 | 29.6428 | 28.9909 | 28.3920 | 27.8405 | 27.3316 | 26.8611 |
2.0 | 30.4391 | 29.7258 | 29.0720 | 28.4714 | 27.9184 | 27.4080 | 26.9362 |
2.1 | 30.5270 | 29.8116 | 29.1560 | 28.5537 | 27.9990 | 27.4872 | 27.0140 |
2.2 | 30.6178 | 29.9003 | 29.2427 | 28.6386 | 28.0823 | 27.5689 | 27.0943 |
2.3 | 30.7113 | 29.9916 | 29.3321 | 28.7261 | 28.1681 | 27.6532 | 27.1771 |
2.4 | 30.8075 | 30.0856 | 29.4239 | 28.8161 | 28.2563 | 27.7398 | 27.2623 |
Declarations
Acknowledgements
This research is fully supported by a grant from National Natural Science foundation of Hunan (2015JJ6041), by National Natural Science Foundation of China (11501191), by National Natural Science Foundation of China, Tian Yuan Foundation (11426100), by National Social Science Fund (15BTJ028), and by National Social Science Fund (13BGL106).
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.
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