scholarly journals Discrete-Time Risk Models Based on Time Series for Count Random Variables

2010 ◽  
Vol 40 (1) ◽  
pp. 123-150 ◽  
Author(s):  
Hélène Cossette ◽  
Etienne Marceau ◽  
Véronique Maume-Deschamps
Keyword(s):  
Time Series ◽  
Discrete Time ◽  
Regime Switching ◽  
Risk Model ◽  
Dependence Structure ◽  
Serial Dependence ◽  
Bernoulli Process ◽  
Temporal Dependence ◽  
Risk Models ◽  
Numerical Examples

AbstractIn this paper, we consider various specifications of the general discrete-time risk model in which a serial dependence structure is introduced between the claim numbers for each period. We consider risk models based on compound distributions assuming several examples of discrete variate time series as specific temporal dependence structures: Poisson MA(1) process, Poisson AR(1) process, Markov Bernoulli process and Markov regime-switching process. In these models, we derive expressions for a function that allow us to find the Lundberg coefficient. Specific cases for which an explicit expression can be found for the Lundberg coefficient are also presented. Numerical examples are provided to illustrate different topics discussed in the paper.

scholarly journals Ruin Probabilities of a Discrete-time Multi-risk Model

2015 ◽  
Vol 44 (4) ◽  
pp. 367-379 ◽  
Author(s):  
Andrius Grigutis ◽  
Agneška Korvel ◽  
Jonas Šiaulys
Keyword(s):  
Discrete Time ◽  
Ruin Probability ◽  
Arrival Time ◽  
Risk Model ◽  
Recursive Formula ◽  
Ruin Probabilities ◽  
Numerical Examples ◽  
Premium Rate ◽  
Insurance Business ◽  
Independent Series

In this work,  we investigate a  multi-risk model describing insurance business with  two or more independent series of claim amounts. Each series of claim amounts consists of independent nonnegative random variables. Claims of each series occur periodically with some fixed   inter-arrival time. Claim amounts occur until they   can be compensated by a common premium rate and the initial insurer's surplus.  In this article, wederive a recursive formula for calculation of finite-time ruin probabilities. In the case of bi-risk model, we present a procedure to calculate the ultimate ruin probability. We add several numerical examples illustrating application  of the derived formulas.DOI: http://dx.doi.org/10.5755/j01.itc.44.4.8635

scholarly journals Perturbed MAP Risk Models with Dividend Barrier Strategies

2009 ◽  
Vol 46 (2) ◽  
pp. 521-541 ◽  
Author(s):  
Eric C. K. Cheung ◽  
David Landriault
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Arrival Process ◽  
Risk Models ◽  
Expected Discounted Penalty Function ◽  
Numerical Examples ◽  
Dividend Payments ◽  
Discounted Penalty Function ◽  
Markov Modulated ◽  
Expected Discounted Penalty

In the context of a dividend barrier strategy (see, e.g. Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identity of Gerber, Lin and Yang (2006) can be established for the class of perturbed MAP surplus processes, extending in the process some results of Li and Lu (2008) for the Markov-modulated risk model. Also, we revisit the same ruin-related quantities in an identical MAP risk model with the only exception that the barrier level effective at time t depends on the state of the underlying environment at this time. Similar relationships are investigated and derived. Numerical examples are also considered.

scholarly journals Perturbed MAP Risk Models with Dividend Barrier Strategies

2009 ◽  
Vol 46 (02) ◽  
pp. 521-541 ◽  
Author(s):  
Eric C. K. Cheung ◽  
David Landriault
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Arrival Process ◽  
Risk Models ◽  
Expected Discounted Penalty Function ◽  
Numerical Examples ◽  
Dividend Payments ◽  
Discounted Penalty Function ◽  
Markov Modulated ◽  
Expected Discounted Penalty

In the context of a dividend barrier strategy (see, e.g. Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identity of Gerber, Lin and Yang (2006) can be established for the class of perturbed MAP surplus processes, extending in the process some results of Li and Lu (2008) for the Markov-modulated risk model. Also, we revisit the same ruin-related quantities in an identical MAP risk model with the only exception that the barrier level effective at time t depends on the state of the underlying environment at this time. Similar relationships are investigated and derived. Numerical examples are also considered.

scholarly journals Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment

Risks ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 26
Author(s):  
Dhiti Osatakul ◽  
Xueyuan Wu
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Insurance Industry ◽  
External Environment ◽  
Ruin Probabilities ◽  
Risk Models ◽  
Markovian Environment ◽  
Life Insurance Industry ◽  
Claim Frequency ◽  
The Impact

In this paper we consider a discrete-time risk model, which allows the premium to be adjusted according to claims experience. This model is inspired by the well-known bonus-malus system in the non-life insurance industry. Two strategies of adjusting periodic premiums are considered: aggregate claims or claim frequency. Recursive formulae are derived to compute the finite-time ruin probabilities, and Lundberg-type upper bounds are also derived to evaluate the ultimate-time ruin probabilities. In addition, we extend the risk model by considering an external Markovian environment in which the claims distributions are governed by an external Markov process so that the periodic premium adjustments vary when the external environment state changes. We then study the joint distribution of premium level and environment state at ruin given ruin occurs. Two numerical examples are provided at the end of this paper to illustrate the impact of the initial external environment state, the initial premium level and the initial surplus on the ruin probability.

scholarly journals Ruin Problems of Multidimensional Risk Models under Constant Interest Rates and Dependent Risks with Heavy Tails

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Xinmei Shen ◽  
Meng Yuan ◽  
Dawei Lu
Keyword(s):  
Interest Rates ◽  
Ruin Probability ◽  
Heavy Tails ◽  
Optimal Allocation ◽  
Risk Model ◽  
Dependence Structure ◽  
Asymptotic Estimates ◽  
Risk Models ◽  
Dependent Risks ◽  
Constant Interest

Consider a discrete-time multidimensional risk model with constant interest rates where capital transfers between lines are partially allowed over each period. By assuming a large initial capital and regularly varying distributions for the losses, we derive asymptotic estimates for the ruin probability under some dependence structure and study the optimal allocation of the initial reserve. Some numerical simulations are provided to illuminate our main results.

scholarly journals On a discrete-time risk model with claim correlated premiums

2015 ◽  
Vol 9 (2) ◽  
pp. 322-342 ◽  
Author(s):  
Xueyuan Wu ◽  
Mi Chen ◽  
Junyi Guo ◽  
Can Jin
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Insurance Industry ◽  
Joint Probability ◽  
Ruin Probabilities ◽  
Deficit At Ruin ◽  
Probability Of Ruin ◽  
Numerical Examples ◽  
Car Insurance ◽  
The Impact

AbstractThis paper proposes a discrete-time risk model that has a certain type of correlation between premiums and claim amounts. It is motivated by the well-known bonus-malus system (also known as the no claims discount) in the car insurance industry. Such a system penalises policyholders at fault in accidents by surcharges, and rewards claim-free years by discounts. For simplicity, only up to three levels of premium are considered in this paper and recursive formulae are derived to calculate the ultimate ruin probabilities. Explicit expressions of ruin probabilities are obtained in a simplified case. The impact of the proposed correlation between premiums and claims on ruin probabilities is examined through numerical examples. In the end, the joint probability of ruin and deficit at ruin is also considered.

scholarly journals Asymptotic Tail Probability of the Discounted Aggregate Claims under Homogeneous, Non-Homogeneous and Mixed Poisson Risk Model

Risks ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 122
Author(s):  
Franck Adékambi ◽  
Kokou Essiomle
Keyword(s):  
Risk Model ◽  
Tail Probability ◽  
Closed Form Expression ◽  
Dependence Structure ◽  
Risk Models ◽  
Occurrence Time ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Force Of Interest ◽  
Asymptotic Tail Probability

In this paper, we derive a closed-form expression of the tail probability of the aggregate discounted claims under homogeneous, non-homogeneous and mixed Poisson risk models with constant force of interest by using a general dependence structure between the inter-occurrence time and the claim sizes. This dependence structure is relevant since it is well known that under catastrophic or extreme events the inter-occurrence time and the claim severities are dependent.

scholarly journals A Discrete-Time Approach to Evaluate Path-Dependent Derivatives in a Regime-Switching Risk Model

Risks ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 9
Author(s):  
Emilio Russo
Keyword(s):  
Discrete Time ◽  
Regime Switching ◽  
Risk Model ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Fixed Number ◽  
Regime Changes ◽  
Quadratic Interpolation ◽  
Asset Value ◽  
Path Dependent

This paper provides a discrete-time approach for evaluating financial and actuarial products characterized by path-dependent features in a regime-switching risk model. In each regime, a binomial discretization of the asset value is obtained by modifying the parameters used to generate the lattice in the highest-volatility regime, thus allowing a simultaneous asset description in all the regimes. The path-dependent feature is treated by computing representative values of the path-dependent function on a fixed number of effective trajectories reaching each lattice node. The prices of the analyzed products are calculated as the expected values of their payoffs registered over the lattice branches, invoking a quadratic interpolation technique if the regime changes, and capturing the switches among regimes by using a transition probability matrix. Some numerical applications are provided to support the model, which is also useful to accurately capture the market risk concerning path-dependent financial and actuarial instruments.

scholarly journals Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 147 ◽  
Author(s):  
Andrius Grigutis ◽  
Jonas Šiaulys
Keyword(s):  
Discrete Time ◽  
Survival Probability ◽  
Risk Model ◽  
Time Model ◽  
Numerical Examples ◽  
Discrete Time Model ◽  
Survival Probabilities ◽  
Renewal Model ◽  
Recursive Formulas

In this paper, we prove recursive formulas for ultimate time survival probability when three random claims X , Y , Z in the discrete time risk model occur in a special way. Namely, we suppose that claim X occurs at each moment of time t ∈ { 1 , 2 , … } , claim Y additionally occurs at even moments of time t ∈ { 2 , 4 , … } and claim Z additionally occurs at every moment of time, which is a multiple of three t ∈ { 3 , 6 , … } . Under such assumptions, the model that is obtained is called the three-risk discrete time model. Such a model is a particular case of a nonhomogeneous risk renewal model. The sequence of claims has the form { X , X + Y , X + Z , X + Y , X , X + Y + Z , … } . Using the recursive formulas, algorithms were developed to calculate the exact values of survival probabilities for the three-risk discrete time model. The running of algorithms is illustrated via numerical examples.

scholarly journals A Class of Copula-Based Bivariate Poisson Time Series Models with Applications

Computation ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 108
Author(s):  
Mohammed Alqawba ◽  
Dimuthu Fernando ◽  
Norou Diawara
Keyword(s):  
Time Series ◽  
Transition Probabilities ◽  
Real Life ◽  
Dependence Structure ◽  
Time Series Models ◽  
Serial Dependence ◽  
Complicated Structure ◽  
Copula Functions ◽  
Copula Theory ◽  
T Copula

A class of bivariate integer-valued time series models was constructed via copula theory. Each series follows a Markov chain with the serial dependence captured using copula-based transition probabilities from the Poisson and the zero-inflated Poisson (ZIP) margins. The copula theory was also used again to capture the dependence between the two series using either the bivariate Gaussian or “t-copula” functions. Such a method provides a flexible dependence structure that allows for positive and negative correlation, as well. In addition, the use of a copula permits applying different margins with a complicated structure such as the ZIP distribution. Likelihood-based inference was used to estimate the models’ parameters with the bivariate integrals of the Gaussian or t-copula functions being evaluated using standard randomized Monte Carlo methods. To evaluate the proposed class of models, a comprehensive simulated study was conducted. Then, two sets of real-life examples were analyzed assuming the Poisson and the ZIP marginals, respectively. The results showed the superiority of the proposed class of models.

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