Insurance Risk Process Research Articles

Last passage times for generalized drawdown processes with applications

In recent years, there has been a significant amount of work dedicated to the study of the generalized drawdown process with its extensive applications in insurance and finance. While existing studies have primarily focused on analyzing the associated first passage times, which signal early warnings, the investigation of last passage times should not be overlooked. Last passage times involve knowledge of the future and can thus offer additional insights. This paper aims to fill this gap in the literature by studying the last passage times for the generalized drawdown process with an independent exponential killing and discussing their applications to insurance risk. Our analysis focuses on the Lévy insurance risk processes, for which we derive the Laplace transforms for these random times. Additionally, we obtain new results on the joint distribution of the duration of the drawdown and the surplus level at killing. As applications, we implement our results in the loss-carry-forward tax and dividend models and investigate the valuation of an European digital drawdown option. Detailed numerical examples are presented for illustrative purposes.

On the Ruin Probabilities in a Discrete Time Insurance Risk Process with Capital Injections and Reinsurance

On the Ruin Probabilities in a Discrete Time Insurance Risk Process with Capital Injections and Reinsurance

A refracted Lévy process with delayed dividend pullbacks

ABSTRACT The threshold dividend strategy, under which dividends are paid only when the insurer's surplus exceeds a pre-determined threshold, has received considerable attention in risk theory. However, in practice, it seems rather unlikely that an insurer will immediately pull back the dividend payments as soon as its surplus level drops below the dividend threshold. Hence, in this paper, we propose a refracted Lévy risk model with delayed dividend pullbacks triggered by a certain Poissonian observation scheme. Leveraging the extensive literature on fluctuation identities for spectrally negative Lévy processes, we obtain explicit expressions for two-sided exit identities of the proposed insurance risk process. Also, penalties are incorporated into the analysis of dividend payouts as a mechanism to penalize for the volatility of the dividend policy and account for an investor's typical preference for more stable cash flows. An explicit expression for the expected (discounted) dividend payouts net of penalties is derived. The criterion for the optimal threshold level that maximizes the expected dividend payouts is also discussed. Finally, several numerical examples are considered to assess the impact of dividend delays on ruin-related quantities. We numerically show that dividend strategies with more steady dividend payouts can be preferred (over the well-known threshold dividend strategy) when penalty fee become too onerous.

An Insurance Risk Process With a Generalized Income Process: A Solvency Analysis

An Insurance Risk Process With a Generalized Income Process: A Solvency Analysis

On the formulation of a stochastic model for an accumulated claim amount under renewal risk process

Traditionally an Insurance risk process is characterised by claim process using renewal process assuming claim amount is independent of inter claim time. It is usually modelled as a stochastic process such as Compound Poisson Process. It is also assumed that the premium amount is proportional to the time we refer with each claim. Depending upon the type of portfolio, the insurer can make a variety of different assumptions on the sequence of inter occurrence times and accumulated claim amount as well. In this paper we discuss a stochastic model for Renewal Risk model with different distributions to number of demands and Generalised Exponential distribution to the impact of each demand under insurance claim scenario. Assume that number of cases is independent of severity of each case throughout the model. We present the model when case frequency is Poison or Negative Binomial or Geometric and also present severity of each case with Generalised Exponential distribution.

Some observations on the temporal patterns in the surplus process of an insurer

Abstract In this paper, we explore potential surplus modelling improvements by investigating how well the available models describe an insurance risk process. To this end, we obtain and analyse a real-life data set that is provided by an anonymous insurer. Based on our analysis, we discover that both the purchasing process and the corresponding claim process have seasonal fluctuations. Some special events, such as public holidays, also have impact on these processes. In the existing literature, the seasonality is often stressed in the claim process, while the cash inflow usually assumes simple forms. We further suggest a possible way of modelling the dependence between these two processes. A preliminary analysis of the impact of these patterns on the surplus process is also conducted. As a result, we propose a surplus process model which utilises a non-homogeneous Poisson process for premium counts and a Cox process for claim counts that reflect the specific features of the data.

Parisian excursion with capital injection for drawdown reflected Lévy insurance risk process

ABSTRACT This paper discusses Parisian ruin problem under drawdown with capital injection when the underlying source of randomness of the surplus is modeled by a general Lévy insurance risk process. Capital injection is provided at the first instance the surplus drops below the drawdown level that is a pre-specified function of its current maximum. The capital is continuously injected to keep the surplus above the drawdown level until either it goes above its current maximum or a Parisian type ruin occurs, which is announced at the first time the surplus process has undergone an excursion below its current maximum for an independent exponential period of time consecutively since the most recent drawdown time. Some distributional identities concerning this excursion are presented. Firstly, we give the Parisian ruin probability and the joint Laplace transform (possibly killed at the first passage time above a fixed level for the surplus process) of the ruin time, the surplus position at ruin, and the total capital injection at ruin. Secondly, we obtain the q-potential measure of the surplus process killed at Parisian ruin. Finally, we give the expected present value of the total discounted capitals injected up to the Parisian ruin time. The results are derived using recent developments in fluctuation and excursion theory of spectrally negative Lévy process and are presented semi-explicitly in terms of the scale function of the Lévy process. Some numerical examples are given to facilitate the analysis of the impact of initial surplus and frequency of observation periods to the ruin probability and to the expected total discounted capital injection.

Uncertain insurance risk process with multiple premium processes and single claim process

Uncertain insurance risk process with multiple premium processes and single claim process

A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process

In this paper, we consider a two-dimensional insurance risk model where each business line faces not only stand-alone claims but also common shocks that induce dependent losses to both lines simultaneously. The joint ruin probability is analyzed, and it is shown that under some model assumptions it can be expressed in terms of a bivariate Laguerre series with the initial surplus levels of the two business lines as arguments. Our approach is based on utilizing various attractive properties of Laguerre functions to solve a partial-integro differential equation satisfied by the joint ruin probability, so that continuum operations such as convolutions and partial differentiation are translated to lattice operations on the Laguerre coefficients. For computational purposes, the bivariate Laguerre series needs to be truncated, and the corresponding Laguerre coefficients can be obtained through a system of linear equations. The computational procedure is easy to implement, and a numerical example is provided that illustrates its excellent performance. Finally, the results are also applied to address a related capital allocation problem.

An insurance risk process with a generalized income process: A solvency analysis

In ruin theory, an insurer’s income process is usually assumed to grow at a deterministic rate of c>0 over time. For instance, both the well-known Cramér–Lundberg risk process and the Sparre Andersen risk model have this assumption built in the construction of their respective surplus processes. This assumption is mainly considered for purposes of mathematical tractability, but generally fails to accurately model an insurer’s income dynamics. To better characterize the variability and uncertainty of an insurer’s income process, several papers have studied insurance risk models with random incomes where the main emphasis is placed on carrying the related Gerber–Shiu analysis. However, a systematic and quantitative understanding of how the more volatile income processes impact an insurer’s solvency risk is still lacking. This paper aims to fill this gap in the literature by quantitatively assessing the impact of the choice of income process on some finite-time and infinite-time ruin quantities. To carry this analysis, we consider a generalized Sparre Andersen risk model with a random income process which renews at claim instants. For exponentially distributed claim sizes, we derive explicit expressions for some joint distributions involving the time to ruin and the number of claims until ruin. As special cases of the proposed insurance risk process, we consider income processes modelled by a subordinator or a particular varying premium rate model. Numerical examples are then carried to draw important risk management implications of a solvency nature for the insurer.

Optimal implementation delay of taxation with trade-off for spectrally negative Lévy risk processes

In this paper we consider two cases of optimal implementation delay of taxation with trade-off under spectrally negative Levy insurance risk processes. In the first case, we assume that the insurance company starts to pay tax only when its surplus level reaches a certain level, and at the termination time of the business there is a terminal value incurred to the company. A method is developed to determine the optimal starting-tax surplus level at which the total expected discounted value of all tax payments up to the termination time plus the discounted terminal value is maximized. In the second case, the company still pays tax subject to a starting-tax surplus level, but with capital injections to prevent bankruptcy. The total expected discounted value of tax payments minus the total discounted capital injection costs is maximized to determine the optimal starting-tax surplus level. Numerical examples are given at the end to illustrate the existence of positive optimal starting-tax surplus levels for both cases considered in this paper.

Uncertain insurance risk process with single premium and multiple classes of claims

Traditionally an insurance risk process is considered under the framework of probability theory with a prerequisite that the estimated distribution function is close enough to the real frequency. However, due to economic or technological reasons, sometimes data are unavailable or difficult to obtain, such as when we consider a new insurance product or insurance for valuable weapons. Under these situations, reimbursement policies are based on experts’ belief degree, which has a much wider range than the real frequency. As a result, we should employ uncertain insurance risk models to better deal with human uncertainty in running an insurance company. Noticing the fact that an insurance company pays for different kinds of risks and the uncertainty of the customer’s arrivals and payments, we investigate an uncertain insurance risk process with multiple classes of claims where the premium process follows an uncertain renewal process. Then we derive expressions for the ruin index and the uncertainty distribution of the ruin time. Some numerical examples and a real data example are performed to capture more insights.

Indifference pricing of insurance-linked securities in a multi-period model

Insurance-linked securities (ILS) have recently become an important risk transfer mechanism to help insurers and reinsurers transfer catastrophe risks to the capital market. We employ the utility indifference approach to establish a pricing framework for a representative agent who trades an ILS with payoff linked to an insurance risk process and a reference rate process. The agent, while investing in a financial market composed of traditional financial instruments, discovers her indifference prices of the ILS by weighing the ILS trade on her exponential utility. The problem has been studied extensively, but mainly in one-period models that are best suited for zero-coupon instruments. In view of the prevalence of ILS with interim payments, we extend the study to a multi-period model by working with time 0 equivalent values and solving a multi-period optimization problem. We offer insights into issues such as coherence and time consistency of the ask and bid indifference prices obtained. Finally, we conduct a sensitivity analysis of the prices against certain risk parameters.

An Uncertain Alternating Renewal Insurance Risk Model

The claim process in an insurance risk model with uncertainty is traditionally described by an uncertain renewal reward process. However, the claim process actually includes two processes, which are called the report process and the payment process, respectively. An alternative way is to describe the claim process by an uncertain alternating renewal reward process. Therefore, this paper proposes an insurance risk model under uncertain measure in which the claim process is supposed to be an alternating renewal reward process and the premium process is regarded as a renewal reward process. Then, the paper also gives the inverse uncertainty distribution of the insurance risk process. The expression of ruin index and the uncertainty distribution of the ruin time are derived which both have explicit expressions based on given uncertainty distributions. Finally, several examples are provided to illustrate the modeling ideas.

Fluctuation identities for Omega-killed spectrally negative Markov additive processes and dividend problem

AbstractIn this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of new generalizations of classical scale matrices for MAPs. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the Omega model, where bankruptcy takes place at rate $\omega(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider Markov-modulated Brownian motion (MMBM) as a special case and present analytical and numerical results for a particular choice of piecewise intensity function $\omega(\cdot,\cdot)$ .

Optimal valuation of American callable credit default swaps under drawdown of Lévy insurance risk process

This paper discusses the valuation of credit default swaps, where default is announced when the reference asset price has gone below certain level from the last record maximum, also known as the high-water mark or drawdown. We assume that the protection buyer pays premium at a fixed rate when the asset price is above a pre-specified level and continuously pays whenever the price increases. This payment scheme is in favour of the buyer as she only pays the premium when the market is in good condition for the protection against financial downturn. Under this framework, we look at an embedded option which gives the issuer an opportunity to call back the contract to a new one with reduced premium payment rate and slightly lower default coverage subject to paying a certain cost. We assume that the buyer is risk neutral investor trying to maximize the expected monetary value of the option over a class of stopping time. We discuss optimal solution to the stopping problem when the source of uncertainty of the asset price is modelled by Lévy process with only downward jumps. Using recent development in excursion theory of Lévy process, the results are given explicitly in terms of scale function of the Lévy process. Furthermore, the value function of the stopping problem is shown to satisfy continuous and smooth pasting conditions regardless of regularity of the sample paths of the Lévy process. Optimality and uniqueness of the solution are established using martingale approach for drawdown process and convexity of the scale function under Esscher transform of measure. Some numerical examples are discussed to illustrate the main results.

Expected utility maximization for an insurer with investment and risk control under inside information

This paper studies optimal investment and risk control strategies for an insurer who owns insider information. The insurance risk process is governed by a general jump diffusion process with random parameters and is correlated with the risky asset process in the financial market. We model the inside information by a general random variable related to the insurance risk process and the risky asset process. Under the criterion of expected utility maximization of the terminal wealth, we adopt white noise calculus and BSDE approach to analyze the problem for various utility functions.

Uncertain insurance risk process with multiple classes of claims

Traditionally, an insurance risk process describes an insurance company’s risk through some criteria using the historical data under the framework of probability theory with the prerequisite that the estimated distribution function is close enough to the true frequency. However, because of the complexity and changeability of the world, economical and technological reasons in many cases enough historical data are unavailable and we have to base on belief degrees given by some domain experts, which motivates us to include the human uncertainty in the insurance risk process by regarding interarrival times and claim amounts as uncertain variables using uncertainty theory. Noting the expansion of insurance companies’ operation scale and the increase of businesses with different risk nature, in this paper we extend the uncertain insurance risk process with a single class of claims to that with multiple classes of claims, and derive expressions for the ruin index and the uncertainty distribution of ruin time respectively. As the ruin time can be infinite, we propose a proper uncertain variable and the corresponding proper uncertainty distribution of that. Some numerical examples are documented to illustrate our results. Finally our method is applied to a real-world problem with some satellite insurance data provided by global insurance brokerage MARSH.

Solvency of an Insurance Company in a Dual Risk Model with Investment: Analysis and Numerical Study of Singular Boundary Value Problems

The survival probability of an insurance company in a collective pension insurance model (so-called dual risk model) is investigated in the case when the whole surplus (or its fixed fraction) is invested in risky assets, which are modeled by a geometric Brownian motion. A typical insurance contract for an insurer in this model is a life annuity in exchange for the transfer of the inheritance right to policyholder’s property to the insurance company. The model is treated as dual with respect to the Cramer–Lundberg classical model. In the structure of an insurance risk process, this is expressed by positive random jumps (compound Poisson process) and a linearly decreasing deterministic component corresponding to pension payments. In the case of exponentially distributed jump sizes, it is shown that the survival probability regarded as a function of initial surplus defined on the nonnegative real half-line is a solution of a singular boundary value problem for an integro-differential equation with a non-Volterra integral operator. The existence and uniqueness of a solution to this problem is proved. Asymptotic representations of the survival probability for small and large values of the initial surplus are obtained. An efficient algorithm for the numerical evaluation of the solution is proposed. Numerical results are presented, and their economic interpretation is given. Namely, it is shown that, in pension insurance, investment in risky assets plays an important role in an increase of the company’s solvency for small values of initial surplus.

Continuous-time optimal reinsurance strategy with nontrivial curved structures

This work uses different classes of premium principles (including the expected value premium principle, the variance premium principle and the exponential premium principle) as well as optimal reinsurance problems to minimize the probability of ruin and maximize the expected utility in both a diffusion insurance risk model and a compound Poisson insurance risk model. The optimal reinsurance strategy with a nontrivial structure and its respective optimal value function are obtained. Specifically, the optimal reinsurance strategy has a curved form, which is distinguished from the conventional proportional and excess-of-loss reinsurance strategies. Numerical analyses are provided to illustrate the behaviors of the optimal reinsurance strategies under different objective criteria and different insurance risk processes.