scholarly journals Minimizing an Insurer's Ultimate Ruin Probability by Noncheap Proportional Reinsurance Arrangements and Investments

2019 ◽  
Author(s):  
Christian Kasumo
Keyword(s):  
Integral Equation ◽  
Ruin Probability ◽  
Stock Price ◽  
Risk Model ◽  
Vital Role ◽  
Insurance Companies ◽  
Proportional Reinsurance ◽  
Investment Return ◽  
Heavy Tailed ◽  
Return Process

In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard Black-Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton-Jacobi-Bellman approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

scholarly journals Minimizing an Insurer's Ultimate Ruin Probability by Noncheap Proportional Reinsurance Arrangements and Investments

2019 ◽  
Author(s):  
Christian Kasumo
Keyword(s):  
Integral Equation ◽  
Ruin Probability ◽  
Stock Price ◽  
Risk Model ◽  
Vital Role ◽  
Insurance Companies ◽  
Proportional Reinsurance ◽  
Investment Return ◽  
Heavy Tailed ◽  
Return Process

In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard Black-Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton-Jacobi-Bellman approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

scholarly journals Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments

2019 ◽  
Vol 24 (1) ◽  
pp. 21 ◽  
Author(s):  
Christian Kasumo
Keyword(s):  
Integral Equation ◽  
Ruin Probability ◽  
Stock Price ◽  
Risk Model ◽  
Vital Role ◽  
Insurance Companies ◽  
Claim Amount ◽  
Proportional Reinsurance ◽  
Investment Return ◽  
Return Process

In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black–Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton–Jacobi–Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed individual claim amount distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

scholarly journals Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Yan Li ◽  
Guoxin Liu
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Optimal Strategies ◽  
Proportional Reinsurance ◽  
Two Dimensional ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Optimal Value ◽  
Hamilton Jacobi Bellman ◽  
Poisson Risk Model

We consider the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. The optimization in the sense of minimizing the ruin probability which is defined by the sum of subportfolio is being ruined. Via the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal value function and prove the verification theorem. In addition, we obtain the Lundberg bounds and the Cramér-Lundberg approximation for the ruin probability and show that as the capital tends to infinity, the optimal strategies converge to the asymptotically optimal constant strategies. The asymptotic value can be found by maximizing the adjustment coefficient.

scholarly journals Numerical Ruin Probability in the Dual Risk Model with Risk-Free Investments

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 110 ◽  
Author(s):  
Sooie-Hoe Loke ◽  
Enrique Thomann
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Ruin Probability ◽  
Risk Model ◽  
Constant Force ◽  
Time Of Ruin ◽  
Dual Risk Model ◽  
The Laplace Transform ◽  
Force Of Interest ◽  
The Time Of Ruin

In this paper, a dual risk model under constant force of interest is considered. The ruin probability in this model is shown to satisfy an integro-differential equation, which can then be written as an integral equation. Using the collocation method, the ruin probability can be well approximated for any gain distributions. Examples involving exponential, uniform, Pareto and discrete gains are considered. Finally, the same numerical method is applied to the Laplace transform of the time of ruin.

scholarly journals Estimates for the Finite-Time Ruin Probability of a Time-Dependent Risk Model with a Brownian Perturbation

2020 ◽  
Vol 2020 ◽  
pp. 1-5 ◽  
Author(s):  
Kaiyong Wang ◽  
Yongfang Cui ◽  
Yanzhu Mao
Keyword(s):  
Finite Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Time Dependent ◽  
Dependence Structure ◽  
Finite Time Ruin Probability ◽  
Dependent Model ◽  
Heavy Tailed ◽  
Time Dependent Model ◽  
Asymptotic Upper Bound

In this paper, we consider a time-dependent risk model with a Brownian perturbation. In this model, there is a dependence structure between the claim sizes and their corresponding interarrival times. Assuming the claim sizes have subexponential distributions, we obtain the asymptotic lower bound of the finite-time ruin probability. When the claim sizes have distributions from the class L∩D, the asymptotic upper bound of the finite-time ruin probability has been presented. These results confirm that when the claim sizes are heavy-tailed, the asymptotics of the finite-time ruin probability of this time-dependent model are insensitive to the Brownian perturbation.

scholarly journals Asymptotic Ruin Probability of a Bidimensional Risk Model Based on Entrance Processes with Constant Interest Rate

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 135
Author(s):  
Hongmin Xiao ◽  
Lin Xie
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Dependence Structure ◽  
Arrival Times ◽  
Interest Force ◽  
Heavy Tailed Distribution ◽  
Constant Interest Force ◽  
Heavy Tailed ◽  
Constant Interest Rate ◽  
Constant Interest

In this paper, the risk model with constant interest based on an entrance process is investigated. Under the assumptions that the entrance process is a renewal process and the claims sizes satisfy a certain dependence structure, which belong to the different heavy-tailed distribution classes, the finite-time asymptotic estimate of the bidimensional risk model with constant interest force is obtained. Particularly, when inter-arrival times also satisfy a certain dependence structure, these formulas still hold.

scholarly journals On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Christian Kasumo ◽  
Juma Kasozi ◽  
Dmitry Kuznetsov
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Insurance Company ◽  
Block Method ◽  
Numerical Examples ◽  
Volterra Integrodifferential Equation ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Hamilton Jacobi Bellman ◽  
And Diffusion

We consider an insurance company whose reserves dynamics follow a diffusion-perturbed risk model. To reduce its risk, the company chooses to reinsure using proportional or excess-of-loss reinsurance. Using the Hamilton-Jacobi-Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation (VIDE) which we transform into a linear Volterra integral equation (VIE) of the second kind. We then proceed to solve this linear VIE numerically using the block-by-block method for the optimal reinsurance policy that minimizes the ultimate ruin probability for the chosen parameters. Numerical examples with both light- and heavy-tailed distributions are given. The results show that proportional reinsurance increases the survival of the company in both light- and heavy-tailed distributions for the Cramér-Lundberg and diffusion-perturbed models.

scholarly journals Asymptotics for ultimate ruin probability in a by-claim risk model

2021 ◽  
Vol 26 (2) ◽  
pp. 259-270
Author(s):  
Aili Zhang ◽  
Shuang Liu ◽  
Yang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Asymptotic Formulas ◽  
Dependence Structure ◽  
Simulation Studies ◽  
Main Claim ◽  
Heavy Tailed ◽  
Constant Interest Rate ◽  
Constant Interest ◽  
Theoretical Results

This paper considers a by-claim risk model with constant interest rate in which the main claim and by-claim random vectors form a sequence of independent and identically distributed random pairs with each pair obeying some certain dependence or arbitrary dependence structure. Under the assumption of heavy-tailed claims, we derive some asymptotic formulas for ultimate ruin probability. Some simulation studies are also performed to check the accuracy of the obtained theoretical results via the crude Monte Carlo method.

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