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Keywords = Premium Calculation Principles

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24 pages, 3233 KiB  
Article
Optimal Reinsurance–Investment Strategy Based on Stochastic Volatility and the Stochastic Interest Rate Model
by Honghan Bei, Qian Wang, Yajie Wang, Wenyang Wang and Roberto Murcio
Axioms 2023, 12(8), 736; https://doi.org/10.3390/axioms12080736 - 27 Jul 2023
Cited by 1 | Viewed by 1278
Abstract
This paper studies insurance companies’ optimal reinsurance–investment strategy under the stochastic interest rate and stochastic volatility model, taking the HARA utility function as the optimal criterion. It uses arithmetic Brownian motion as a diffusion approximation of the insurer’s surplus process and the variance [...] Read more.
This paper studies insurance companies’ optimal reinsurance–investment strategy under the stochastic interest rate and stochastic volatility model, taking the HARA utility function as the optimal criterion. It uses arithmetic Brownian motion as a diffusion approximation of the insurer’s surplus process and the variance premium principle to calculate premiums. In this paper, we assume that insurance companies can invest in risk-free assets, risky assets, and zero-coupon bonds, where the Cox–Ingersoll–Ross model describes the dynamic change in stochastic interest rates and the Heston model describes the price process of risky assets. The analytic solution of the optimal reinsurance–investment strategy is deduced by employing related methods from the stochastic optimal control theory, the stochastic analysis theory, and the dynamic programming principle. Finally, the influence of model parameters on the optimal reinsurance–investment strategy is illustrated using numerical examples. Full article
(This article belongs to the Special Issue Application of Machine Learning and Optimization Methods in Engineering Mathematics)
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Figure 1

Figure 1
<p>The influence of parameter <math display="inline"><semantics><mi>n</mi></semantics></math> on the optimal reinsurance strategy.</p> Full article ">Figure 2
<p>The influence of parameter <math display="inline"><semantics><mi>m</mi></semantics></math> on the optimal reinsurance strategy.</p> Full article ">Figure 3
<p>The influence of parameter <math display="inline"><semantics><mrow><msub><mi>κ</mi><mi>r</mi></msub></mrow></semantics></math> on the optimal reinsurance strategy.</p> Full article ">Figure 4
<p>The influence of parameter <math display="inline"><semantics><mrow><msub><mi>σ</mi><mi>r</mi></msub></mrow></semantics></math> on the optimal reinsurance strategy.</p> Full article ">Figure 5
<p>The influence of parameter <math display="inline"><semantics><mrow><msub><mi>κ</mi><mi>r</mi></msub></mrow></semantics></math> on the optimal investment strategy.</p> Full article ">Figure 6
<p>The influence of parameter <math display="inline"><semantics><mrow><msub><mi>σ</mi><mi>r</mi></msub></mrow></semantics></math> on the optimal investment strategy.</p> Full article ">Figure 7
<p>The influence of parameter <math display="inline"><semantics><mrow><msub><mi>ξ</mi><mi>r</mi></msub></mrow></semantics></math> on the optimal investment strategy.</p> Full article ">Figure 8
<p>The influence of parameter <math display="inline"><semantics><mrow><msub><mi>κ</mi><mi>v</mi></msub></mrow></semantics></math> on the optimal investment strategy.</p> Full article ">Figure 9
<p>The influence of parameter <math display="inline"><semantics><mi>n</mi></semantics></math> on the optimal investment strategy.</p> Full article ">Figure 10
<p>The influence of parameter <math display="inline"><semantics><mi>m</mi></semantics></math> on the optimal investment strategy.</p> Full article ">
26 pages, 479 KiB  
Article
Nash Equilibrium Investment-Reinsurance Strategies for an Insurer and a Reinsurer with Intertemporal Restrictions and Common Interests
by Yanfei Bai, Zhongbao Zhou, Rui Gao and Helu Xiao
Mathematics 2020, 8(1), 139; https://doi.org/10.3390/math8010139 - 19 Jan 2020
Cited by 5 | Viewed by 2530
Abstract
This paper investigates the generalized multi-period mean-variance investment-reinsurance optimization model in a discrete-time framework for a general insurance company that contains a reinsurer and an insurer. The intertemporal restrictions and the common interests of the reinsurer and the insurer are considered. The common [...] Read more.
This paper investigates the generalized multi-period mean-variance investment-reinsurance optimization model in a discrete-time framework for a general insurance company that contains a reinsurer and an insurer. The intertemporal restrictions and the common interests of the reinsurer and the insurer are considered. The common goal of the reinsurer and the insurer is to maximize the expectation of the weighted sum of their wealth processes and minimize the corresponding variance. Based on the game method, we obtain the Nash equilibrium investment-reinsurance strategies for the above-proposed model and find out the equilibrium strategies when unilateral interest is considered. In addition, the Nash equilibrium investment-reinsurance strategies are deduced under two special premium calculated principles (i.e., the expected value premium principle and the variance value premium principle). We theoretically study the effect of the intertemporal restrictions on Nash equilibrium investment-reinsurance strategies and find the effect depends on the value of some parameters, which differs from the previous researches that generally believed that intertemporal restrictions would make investors avoid risks. Finally, we perform corresponding numerical analyses to verify our theoretical results. Full article
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<p>Nash equilibrium investment strategies for (<b>a</b>) the insurer and (<b>b</b>) the reinsurer (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 2
<p>Nash equilibrium investment strategies for (<b>a</b>) the insurer and (<b>b</b>) the reinsurer (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1.01</mn> </mrow> </semantics></math>).</p> Full article ">Figure 3
<p>Nash equilibrium investment strategies for (<b>a</b>) the insurer and (<b>b</b>) the reinsurer (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 4
<p>Nash equilibrium investment strategies for (<b>a</b>) the insurer and (<b>b</b>) the reinsurer (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1.01</mn> </mrow> </semantics></math>).</p> Full article ">Figure 5
<p>Nash equilibrium reinsurance strategy under the expected value premium principle (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 6
<p>Nash equilibrium reinsurance strategy under the expected value premium principle (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>).</p> Full article ">Figure 7
<p>Nash equilibrium reinsurance strategy under the variance value premium principle (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 8
<p>Nash equilibrium reinsurance strategy under the variance value premium principle (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>).</p> Full article ">
18 pages, 544 KiB  
Article
A Generalised CIR Process with Externally-Exciting and Self-Exciting Jumps and Its Applications in Insurance and Finance
by Angelos Dassios, Jiwook Jang and Hongbiao Zhao
Risks 2019, 7(4), 103; https://doi.org/10.3390/risks7040103 - 14 Oct 2019
Cited by 5 | Viewed by 3541
Abstract
In this paper, we study a generalised CIR process with externally-exciting and self-exciting jumps, and focus on the distributional properties and applications of this process and its aggregated process. The aim of the paper is to introduce a more general process that includes [...] Read more.
In this paper, we study a generalised CIR process with externally-exciting and self-exciting jumps, and focus on the distributional properties and applications of this process and its aggregated process. The aim of the paper is to introduce a more general process that includes many models in the literature with self-exciting and external-exciting jumps. The first and second moments of this jump-diffusion process are used to calculate the insurance premium based on mean-variance principle. The Laplace transform of aggregated process is derived, and this leads to an application for pricing default-free bonds which could capture the impacts of both exogenous and endogenous shocks. Illustrative numerical examples and comparisons with other models are also provided. Full article
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Figure 1

Figure 1
<p>Sample paths of simulated process <math display="inline"><semantics> <msub> <mi>S</mi> <mi>t</mi> </msub> </semantics></math> for three different time horizons.</p> Full article ">Figure 2
<p>Bond price <math display="inline"><semantics> <mrow> <mi>B</mi> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> (%) as a function of volatility <math display="inline"><semantics> <mi>σ</mi> </semantics></math>.</p> Full article ">Figure 3
<p>Term structure of bond price <math display="inline"><semantics> <mrow> <mi>B</mi> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics></math> (%).</p> Full article ">
363 KiB  
Article
Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle
by Mi Chen, Wenyuan Wang and Ruixing Ming
Risks 2016, 4(4), 50; https://doi.org/10.3390/risks4040050 - 16 Dec 2016
Cited by 2 | Viewed by 3679
Abstract
In this paper, we study the optimal reinsurance problem where risks of the insurer are measured by general law-invariant risk measures and premiums are calculated under the TVaR premium principle, which extends the work of the expected premium principle. Our objective is to [...] Read more.
In this paper, we study the optimal reinsurance problem where risks of the insurer are measured by general law-invariant risk measures and premiums are calculated under the TVaR premium principle, which extends the work of the expected premium principle. Our objective is to characterize the optimal reinsurance strategy which minimizes the insurer’s risk measure of its total loss. Our calculations show that the optimal reinsurance strategy is of the multi-layer form, i.e., f * ( x ) = x ∧ c * + ( x - d * ) + with c * and d * being constants such that 0 ≤ c * ≤ d * . Full article
(This article belongs to the Special Issue Selected Papers from the 6th Gerber-Shiu Workshop and the 3rd Modeling of Heavy-Tail Phenomena Workshop)
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Figure 1

Figure 1
<p>The horizontal axis represents <span class="html-italic">β</span>, the vertical axis represents <span class="html-italic">α</span>, the lateral axis represents the optimal value given by (<a href="#FD13-risks-04-00050" class="html-disp-formula">13</a>). The Greek <span class="html-italic">θ</span> is set to be 0.1. The distribution of <span class="html-italic">X</span> is exponential: <math display="inline"> <semantics> <mrow> <msub> <mi>S</mi> <mi>X</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>100</mn> </mfrac> <mi>x</mi> </mrow> </msup> </mrow> </semantics> </math>.</p> Full article ">Figure 2
<p>The horizontal axis represents <span class="html-italic">β</span>, the vertical axis represents the optimal value given by (<a href="#FD14-risks-04-00050" class="html-disp-formula">14</a>). The Greek <span class="html-italic">θ</span> is set to be 0.1. The distribution of <span class="html-italic">X</span> is exponential: <math display="inline"> <semantics> <mrow> <msub> <mi>S</mi> <mi>X</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>100</mn> </mfrac> <mi>x</mi> </mrow> </msup> </mrow> </semantics> </math>.</p> Full article ">
1351 KiB  
Article
Demand of Insurance under the Cost-of-Capital Premium Calculation Principle
by Michael Merz and Mario V. Wüthrich
Risks 2014, 2(2), 226-248; https://doi.org/10.3390/risks2020226 - 17 Jun 2014
Cited by 7 | Viewed by 4780
Abstract
We study the optimal insurance design problem. This is a risk sharing problem between an insured and an insurer. The main novelty in this paper is that we study this optimization problem under a risk-adjusted premium calculation principle for the insurance cover. This [...] Read more.
We study the optimal insurance design problem. This is a risk sharing problem between an insured and an insurer. The main novelty in this paper is that we study this optimization problem under a risk-adjusted premium calculation principle for the insurance cover. This risk-adjusted premium calculation principle uses the cost-of-capital approach as it is suggested (and used) by the regulator and the insurance industry. Full article
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Figure 1

Figure 1
<p>Function <math display="inline"> <mrow> <mi>x</mi> <mo>↦</mo> <mi>ψ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math> for parameters <math display="inline"> <mrow> <msub> <mi>r</mi> <mtext>CoC</mtext> </msub> <mo>=</mo> <mn>6</mn> <mo>%</mo> </mrow> </math> and <math display="inline"> <mrow> <mi>ε</mi> <mo>=</mo> <mn>5</mn> <mo>%</mo> </mrow> </math>: lhs: exponential Example 2 with <math display="inline"> <mrow> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>; rhs: Gaussian Example 3 with <math display="inline"> <mrow> <mi>μ</mi> <mo>=</mo> <mn>4</mn> </mrow> </math>.</p> Full article ">Figure 2
<p>Case of large individual risk aversion parameter <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.4</mn> <mo>&gt;</mo> <mi>β</mi> <mo>=</mo> <mn>0.0301</mn> </mrow> </math>: lhs: optimal insurance cover <math display="inline"> <mrow> <mi>x</mi> <mo>↦</mo> <mi>I</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>λ</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </math> for different fixed premiums <math display="inline"> <mrow> <mi>P</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </math>; rhs: optimal expected utility <math display="inline"> <mrow> <mi>P</mi> <mo>↦</mo> <mi>U</mi> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> </math> for different premiums <span class="html-italic">P</span>.</p> Full article ">Figure 3
<p>Case of large individual risk aversion parameter <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.4</mn> <mo>&gt;</mo> <mi>β</mi> <mo>=</mo> <mn>0.0301</mn> </mrow> </math>: the red line illustrates the optimal insurance cover <math display="inline"> <mrow> <mi>x</mi> <mo>↦</mo> <msup> <mi>I</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>I</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>λ</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>P</mi> <mo>*</mo> </msup> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math>.</p> Full article ">Figure 4
<p>Case of small individual risk aversion parameter <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.01</mn> <mo>&lt;</mo> <mi>β</mi> <mo>=</mo> <mn>0.0301</mn> </mrow> </math>: lhs: optimal insurance cover <math display="inline"> <mrow> <mi>x</mi> <mo>↦</mo> <mi>I</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>λ</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </math> for different fixed premium <math display="inline"> <mrow> <mi>P</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </math>; rhs: optimal expected utility <math display="inline"> <mrow> <mi>P</mi> <mo>↦</mo> <mi>U</mi> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> </math> for different premium <span class="html-italic">P</span>.</p> Full article ">Figure 5
<p>Case of small individual risk aversion parameter <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.01</mn> <mo>&lt;</mo> <mi>β</mi> <mo>=</mo> <mn>0.0301</mn> </mrow> </math>: the red line illustrates the optimal insurance cover <math display="inline"> <mrow> <mi>x</mi> <mo>↦</mo> <msup> <mi>I</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>I</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>λ</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>P</mi> <mo>*</mo> </msup> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math>.</p> Full article ">
581 KiB  
Article
Physical Premium Principle: A New Way for Insurance Pricing
by Amir H. Darooneh
Entropy 2005, 7(1), 97-107; https://doi.org/10.3390/e7010097 - 28 Feb 2005
Cited by 8 | Viewed by 8465
Abstract
In our previous work we suggested a way for computing the non-life insurance premium. The probable surplus of the insurer company assumed to be distributed according to the canonical ensemble theory. The Esscher premium principle appeared as its special case. The difference between [...] Read more.
In our previous work we suggested a way for computing the non-life insurance premium. The probable surplus of the insurer company assumed to be distributed according to the canonical ensemble theory. The Esscher premium principle appeared as its special case. The difference between our method and traditional principles for premium calculation was shown by simulation. Here we construct a theoretical foundation for the main assumption in our method, in this respect we present a new (physical) definition for the economic equilibrium. This approach let us to apply the maximum entropy principle in the economic systems. We also extend our method to deal with the problem of premium calculation for correlated risk categories. Like the Buhlman economic premium principle our method considers the effect of the market on the premium but in a different way. Full article
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Figure 1

Figure 1
<p>The loading parameter, (<span class="html-italic">p</span>/<span class="html-italic">p</span><sub>0</sub>) − 1, versus the contract duration for large <span class="html-italic">β</span> parameter. This figure shows the dependence of the premium on the period of insurance contract. It justifies our experience in trading. The premium of a risk category is not change linearly with the contract duration. The long term contract is more advantageous than some short term contraction. The loading parameter is used to get rid of the monetary unit.</p> Full article ">Figure 2
<p>Dependence of the ln <span class="html-italic">βT</span> on the ruin probability, the initial wealth and the mean claim size. The <span class="html-italic">β</span> parameter for apparently is very small for the wealthier insurer. Such a company may offer the insurance with a low price.</p> Full article ">Figure 3
<p>The loading parameter, (<span class="html-italic">p</span>/<span class="html-italic">p</span><sub>0</sub>) − 1, versus the, <span class="html-italic">βT</span>, parameter. The squares display the results of the canonical ensemble theory and the triangles correspond to the Esscher premium principle. The difference between two curves is the result of the market effects.</p> Full article ">
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