Renewable Insurance Research Articles

Inequalities on the ruin probability for light-tailed distributions with some restrictions

When there are some restrictions on the random variables of insurance risk model, it is impossible to calculate the exact value of ruin probabilities. For these cases, even finding a suitable approximation, is very important from a practical point of view. In the present paper, we consider the renewal insurance surplus model with light-tailed claim amount distributions and try to find some inequalities on the infinite time ruin probability depending on the amount of initial reserve using statistical and mathematical approaches if the assumption of net profit does not hold but there exist some other restrictions on the mathematical functions of random variables of model. The assertions depend on the amount of initial reserve, distribution of nonnegative claim occurrences times and successive claim amounts are obtained. Finally, to show the application and effectiveness of given theorems two examples are presented. Through these examples, the infinite time ruin probabilities are estimated using Monte Carlo simulation and give an intuitive way to understand the nature of ruin.

Identification of Potential Valid Clients for a Sustainable Insurance Policy Using an Advanced Mixed Classification Model

Due to the social awareness of risk control, we are witnessing the popularization of the insurance concept and the rapid development of financial insurance. The performance of the insurance industry is highly competitive; thus, in order to develop new and old business from existing clients, information on the renewal of client premiums, purchase of new policies, and new client referrals has become an important research topic in this field. However, based on a review of published literature, few scholars have engaged in relevant research on the above topics by data mining, which motivated the formation of this study, hoping to bridge this gap. We constructed 10 mixed classification prediction models (called Models A–J) using advanced data mining techniques. Moreover, 19 conditional attributes (coded as X1–X19) were selected from the collected insurance client database, plus three different decision attributes (coded as X20–X22): whether to pay the renewal insurance premium, whether to buy a new insurance policy, and whether to introduce new clients. In terms of technical methods, we used two data pretreatment techniques, attribute selection and data discretization, combined with different methods of disassembly in proportion and data cross-validation to conduct data analysis of the collected experimental data set. We also combined and calculated 23 important classification algorithms (or classifiers) in seven different classifications of data mining techniques (i.e., decision tree, Bayes, Function, Lazy, Meta, Mise, and Rule). In terms of the experimental results of insurance data, this study has the following important contributions and findings: (1) finding the best classifier; (2) finding the optimal mixed classification model; (3) determining the best disassembly in proportion; (4) comparing the performance of different disassembly in proportion and data cross-validation methods; (5) determining the important factors influencing the decision attribute “whether to purchase a new insurance policy”, including the time interval to the first purchase, the number of valid policies, the total number of purchased policies, the family salary structure, and gender; and (6) building a knowledge base of decision rules and criteria with the decision tree C4.5 technology, which shall be provided to relevant stakeholders such as insurance dealers and insurance salespeople as a reference for looking for valid clients in the future, and is conducive to the rapid expansion of insurance business. Finally, the important research findings and management implications of this study can serve as a basis for further study of sustainable insurance by academic researchers.

Simple approximation for the ruin probability in renewal risk model under interest force via Laguerre series expansion

Although the ruin probability in a renewal insurance risk model with credit interest may be viewed as a classical research problem, exact solutions are only available in the literature in very special cases when both the claim amounts and the interclaim times follow distributions such as the exponential. This is a long standing problem especially from a computational point of view, and the difficulty lies in the fact that the ruin probability usually satisfies a higher order integro-differential equation and/or an ordinary differential equation with non-constant coefficients. In this paper, for a large class of interclaim time distributions (including a combination of exponentials), we shall develop an approximation for the ruin probability using Laguerre series expansion as a function of the initial surplus level. It is shown that the (approximated) Laguerre coefficients can be solved from a system of linear equations, a procedure that is very easy to implement. A main advantage of our approach is that no specific distributional assumption on the claim amounts is required, apart from some mild differentiability and integrability conditions that can be verified. Numerical examples are provided to illustrate the very good performance of our approximation including both light-tailed and heavy-tailed claims.

Całkowicie prywatna opieka medyczna — próba zarysu

In this paper, we explore what the market for medical services and products could look like if the state completely withdrew from the area of medical care. In section 1, we demonstrate that medical services would be purchased mainly through direct payments and medical insurance. We analyse two models of medical insurance: guaranteed renewable insurance and health-status insurance. Other types of insurance that may emerge on the market are also discussed. In section 2, we exam-ine how the privatisation of the health-care system would affect the prices of medical services. We analyse fundamental problems of the state-run health care and discuss how they contribute to small-er supply and higher prices of medical services. We then describe how the introduction of market mechanisms would allow to solve many of these problems. We argue that internalisation of the costs of medical care in a free market order would create strong economic incentives for individuals to take better care of their health, and we contrast this with the state-run health care in which these costs are externalised. In section 3, we explore how medical services could be obtained by individuals without sufficient funds. In section 4, we discuss how the quality of medical care could be ensured without the help of the state. We argue that competition between service providers would be the main guarantor of quality. We also identify mechanisms that would lead to spontaneous emergence of a system of private medical licencing.

Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps

This paper considers a renewal insurance risk model with two-sided jumps (e.g. Labbé et al., 2011), where downward and upward jumps typically represent claim amounts and random gains respectively. A generalization of the Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) is proposed and analyzed for sample paths leading to ruin. In particular, we shall incorporate the joint moments of the total discounted costs associated with claims and gains until ruin into the Gerber–Shiu function. Because ruin may not occur, the joint moments of the total discounted claim costs and gain costs are also studied upon ultimate survival of the process. General recursive integral equations satisfied by these functions are derived, and our analysis relies on the concept of ‘moment-based discounted densities’ introduced by Cheung (2013). Some explicit solutions are obtained in two examples under different cost functions when the distribution of each claim is exponential or a combination of exponentials (while keeping the distributions of the gains and the inter-arrival times between successive jumps arbitrary). The first example looks at the joint moments of the total discounted amounts of claims and gains whereas the second focuses on the joint moments of the numbers of downward and upward jumps until ruin. Numerical examples including the calculations of covariances between the afore-mentioned quantities are given at the end along with some interpretations.

Some mathematical aspects of price optimisation

Calculation of an optimal tariff is a principal challenge for pricing actuaries. In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff. Our motivation comes from two important actuarial tasks, namely (a) construction of an optimal renewal tariff subject to business and technical constraints, and (b) determination of an optimal allocation of certain premium loadings. We consider both continuous and discrete optimisation and then present several algorithmic suboptimal solutions. Additionally, we explore some simulation techniques. Several illustrative examples show both the complexity and the importance of the optimisation approach.

Analysis of IBNR claims in renewal insurance models

Incurred but not reported (IBNR) claims, which arise naturally in insurance contexts, are of central importance to insurers for risk management and financial reporting purposes. In this paper, we first examine the moments of the total discounted IBNR claim amount at a given time when claim events occur according to a compound renewal process. Under the same claim arrival dynamic, we later consider the joint moments of the total discounted IBNR claim amount and the total incurred and reported claim amount at possibly different time points, a quantity of much interest for claim reserving purposes. In the second part of this article, we examine in more detail properties of the IBNR claim number under specific distributional assumptions for the reporting lags and the interarrival times. Among others, the self-decomposability of the IBNR claim number process is considered when claim events occur according to a compound Poisson process.

Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window

We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival times depending on the claims that arrive within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the bonus-malus (BM) feature of pricing car insurance. We discuss first the asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process, and via an appropriate change of measure, ruin probabilities could be computed to a closed-form formulae. Additionally, we employ the importance sampling simulations to derive ruin probabilities, which further permit an in-depth analysis of a few concrete cases.

Some Mathematical Aspects of Price Optimisation

Calculation of an optimal tariff is a principal challenge for pricing actuaries. In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff. Our motivation comes from two important actuarial tasks, namely a) construction of an optimal renewal tariff subject to business and technical constraints, and b) determination of an optimal allocation of certain premium loadings. We consider both continuous and discrete optimisation and then present several algorithmic sub-optimal solutions. Additionally, we explore some simulation techniques. Several illustrative examples show both the complexity and the importance of the optimisation approach.

Drawdown analysis for the renewal insurance risk process

In this paper, we study some drawdown-related quantities in the context of the renewal insurance risk process with general interarrival times and phase-type distributed jump sizes. We make use of some recent results on the two-sided exit problem for the spectrally negative Markov additive process and a fluid flow analogy between certain queues and risk processes to solve for the two-sided exit problem of the renewal insurance risk process. The two-sided exit quantities are later shown to be central to the analysis of drawdown quantities including the drawdown time, the drawdown size, the running maximum (minimum) at the drawdown time, the last running maximum time prior to drawdown, the number of jumps before drawdown and the number of excursions from running maximum before drawdown. Finally, we consider another application of our methodology for the study of the expected discounted dividend payments until ruin.

Yes, No, Perhaps? Premium Risk and Guaranteed Renewable Insurance Contracts With Heterogeneous Incomplete Private Information

The article shows that heterogeneous incomplete private information can explain the limited existence of guaranteed renewable health insurance (GR) contracts in an otherwise frictionless markets. We derive a unique equilibrium that can be of the form that either only a portion of the population or none will cover themselves against premium risk with a GR contract. Increased risk aversion, increased premium risk, and first-order stochastic improvements of the distribution of private information increase the likelihood of positive take-up. In case GR contracts are in demand, increased risk aversion and first-order stochastic improvements of the distribution of private information lead to more individuals purchasing the GR contract.

After the ACA: Freeing the Market for Health Care

I survey the supply, demand, and market for health care and health insurance. I conclude that a much less regulated system is possible, and necessary. Cost control and technology improvement must come from disruptive competition from new suppliers, as it has in airlines, retail, internet, and other successful industries. People must direct their expenditures at the margin, and feel the benefits and costs of their decisions. Individual, portable, guaranteed renewable insurance can then emerge, addressing the pathologies of today’s insurance markets. I discuss how current law and regulations rather than fundamental market failures are the main reasons a healthy market does not emerge, and why a regulatory approach must fail. I address common objections to market-based health care and insurance.

Exact and Asymptotic Results for Insurance Risk Models with Surplus-dependent Premiums

In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Green's operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cramér-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model.

Asymptotic results for renewal risk models with risky investments

We consider a renewal jump–diffusion process, more specifically a renewal insurance risk model with investments in a stock whose price is modeled by a geometric Brownian motion. Using Laplace transforms and regular variation theory, we introduce a transparent and unifying analytic method for investigating the asymptotic behavior of ruin probabilities and related quantities, in models with light- or heavy-tailed jumps, whenever the distribution of the time between jumps has rational Laplace transform.

Incentive-compatible guaranteed renewable health insurance premiums

Theoretical models of guaranteed renewable insurance display front-loaded premium schedules. Such schedules both cover lifetime total claims of low-risk and high-risk individuals and provide an incentive for those who remain low-risk to continue to purchase the policy. Questions have been raised of whether actual individual insurance markets in the US approximate the behavior predicted by these models, both because young consumers may not be able to “afford” front-loading and because insurers may behave strategically in ways that erode the value of protection against risk reclassification. In this paper, the optimal competitive age-based premium schedule for a benchmark guaranteed renewable health insurance policy is estimated using medical expenditure data. Several factors are shown to reduce the amount of front-loading necessary. Indeed, the resulting optimal premium path increases with age. Actual premium paths exhibited by purchasers of individual insurance are close to the optimal renewable schedule we estimate. Finally, consumer utility associated with the feature is examined.

Lundberg-Type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin Under the Sparre Andersen Model

In this paper we consider the Sparre Andersen insurance risk model. Three cases are discussed: the ordinary renewal risk process, stationary renewal risk process, and s-delayed renewal risk process. In the first part of the paper we study the joint distribution of surplus immediately before and at ruin under the renewal insurance risk model. By constructing an exponential martingale, we obtain Lundberg-type upper bounds for the joint distribution. Consequently we obtain bounds for the distribution of the deficit at ruin and ruin probability. In the second part of the paper, we consider the special case of phase-type claims and rederive the closed-form expression for the distribution of the severity of ruin, obtained by Drekic et al. (2003, 2004). Finally, we present some numerical results to illustrate the tightness of the bounds obtained in this paper.

Approximations for moments of deficit at ruin with exponential and subexponential claims

Consider a renewal insurance risk model with initial surplus u>0 and let A u denote the deficit at the time of ruin. This paper investigates the asymptotic behavior of the moments of A u as u tends to infinity. Under the assumption that the claim size is exponentially or subexponentially distributed, we obtain some asymptotic relationships for the φ-moments of A u , where φ is a non-negative and non-decreasing function satisfying certain conditions.

233017 (E10) Consumer capital market constraints and guaranteed renewable insurance: Frick K.,Journal of Risk and Uncertainty, Volume 16, No 3, July/August 1998

233017 (E10) Consumer capital market constraints and guaranteed renewable insurance: Frick K.,Journal of Risk and Uncertainty, Volume 16, No 3, July/August 1998

Consumer Capital Market Constraints and Guaranteed Renewable Insurance

This paper analyzes a market in which individuals can purchase guaranteed renewable insurance policies to insure against the risk of loss in the current period and the risk of an increased probability of loss. Individuals who cannot borrow will purchase partially guaranteed renewable insurance at most. Relatively impatient individuals may purchase policies with no guaranteed renewability features. Without borrowing, the results are similar regardless of whether consumers can save.

Guaranteed Renewable Disability Insurance

Guaranteed Renewable Disability Insurance