Search Results (77)

This paper aims to develop optional semimartingale methods in risk theory to allow for a larger class of risk models. Optional semimartingales are left-continuous with right-limit stochastic processes defined on a probability space where the usual conditions—completeness and right-continuity of the filtration—are not assumed. Three risk models are formulated, accounting for inflation, interest rates, and claim occurrences. The first model extends the martingale approach to calculate ruin probabilities, the second employs the Gerber–Shiu function to evaluate the expected discounted penalty from financial oscillations or jumps, and the third introduces a Gaussian risk model using counting processes to capture premium and claim cash flow jumps in insurance companies. Full article

In this paper, we study a risk model with stochastic premium income and its impact on solvency risk management. It is assumed that both the premium arrival process and the claim arrival process are modelled by homogeneous Poisson processes, and that the premium amounts are modelled by independent and identically distributed random variables. While this model has been studied in the existing literature and certain explicit results are known under more restrictive assumptions, these results are relatively difficult to apply in practice. In this paper, we investigate the factors that differentiate this model and the classical risk model. After reviewing various known results of this model, we derive a simulation approach for obtaining the probability of ultimate ruin based on importance sampling, which does not require specific distributions for the premium and the claim. We demonstrate this approach first with examples where the distribution of the sampling random variable can be identified. We then provide additional examples where we use the fast Fourier transform to obtain an approximation of the sampling random variable. The simulated results are compared with the known results for the probability of ruin. Using the simulation approach, we apply this model to a real-life auto-insurance data set. Differences with the classical model are then discussed. Finally, we comment on the suitability and impact of using this model in the context of solvency risk management. Full article

The paper considers a bidimensional continuous-time risk model with subexponential claims and Brownian perturbations, in which the price processes of the investment portfolio of the two lines of business are two geometric Lévy processes and the two lines of business share a common claim-number process, which is a renewal counting process. The paper mainly considers the claims of each line of business having a dependence structure. When the claims have subexponential distributions, the asymptotics of the finite-time ruin probabilities ${\psi }_{and}(x_1,x_2;T)$ and ${\psi }_{sim}(x_1,x_2;T)$ have been obtained. When the distributions of claims belong to the intersection of long-tailed and dominatedly varying-tailed distribution classes, the asymptotics of the finite-time ruin probability ${\psi }_{or}(x_1,x_2;T)$ is given. Full article

In this paper, we propose a nonparametric estimator of ruin probability in the Wiener–Poisson risk model based on high-frequency data. The estimator is constructed via the Fourier-cosine method and the threshold technique, and the convergence rate is also studied for a large sample size. Finally, we verify the effectiveness of our estimator through some simulation studies. Full article

The purpose of this article is to analyze a sequence of independent bets by modeling it with a convective-diffusion equation (CDE). The approach follows the derivation of the Kelly Criterion (i.e., with a binomial distribution for the numbers of wins and losses in a sequence of bets) and reframes it as a CDE in the limit of many bets. The use of the CDE clarifies the role of steady growth (characterized by a velocity U) and random fluctuations (characterized by a diffusion coefficient D) to predict a probability distribution for the remaining bankroll as a function of time. Whereas the Kelly Criterion selects the investment fraction that maximizes the median bankroll (0.50 quantile), we show that the CDE formulation can readily find an optimum betting fraction f for any quantile. We also consider the effects of “ruin” using an absorbing boundary condition, which describes the termination of the betting sequence when the bankroll becomes too small. We show that the probability of ruin can be expressed by a dimensionless Péclet number characterizing the relative rates of convection and diffusion. Finally, the fractional Kelly heuristic is analyzed to show how it impacts returns and ruin. The reframing of the Kelly approach with the CDE opens new possibilities to use known results from the chemico-physical literature to address sequential betting problems. Full article

This paper deals with the ruin problem of an insurance company investing its capital reserve in a risky asset with the price dynamics given by a conditional geometric Brownian motion whose parameters depend on a Markov process describing random variations in the economic and financial environments. We prove a sufficient condition on the distribution of jumps of the business process ensuring the smoothness of the ruin probability as a function of the initial capital and obtain for this function an integro-differential equation. Full article

In order to deal with complex risk scenarios involving claims, uncertainty, and investments, we consider the ruin problems in a compound Poisson risk model with liquid reserves and proportional investments and study the expected discounted penalty function under threshold dividend strategies. Firstly, the integral differential equation of the expected discounted penalty function is derived. Secondly, since the closed-form solution of the equation cannot be obtained, a sinc method is used to obtain the numerical approximation solution of the equation. Finally, the feasibility and superiority of the sinc method are illustrated by error analysis. In addition, based on a symmetric jump risk market, we discuss the influence of some parameters on the ruin probability with some examples. This study can help actuaries develop more robust risk management strategies and ensure the long-term stability and profitability of insurance companies. It provides a theoretical basis for actuaries to carry out risk management. Full article

To ensure a comfortable post-retirement life and the ability to cover living expenses, it is of utmost importance for individuals to have a clear understanding of how long their pre-retirement savings will last. In this research, we employ a ruin-theory approach to model the inflows and the outflows of retirees’ portfolios. We track all transactions within the portfolios of retired clients sourced by a registered investment provider to Canada’s Financial Wellness Lab at Western University. By utilizing an advanced ruin model, we calculate the mean and the median time it takes for savings to be exhausted, the probabilities of exhaustion of funds within the retirees’ expected remaining lifetime while accounting for the observed withdrawal rates, and the deficit at ruin if a retiree has used up all of their savings. We also account for gender as well as for the risk tolerance of retired clients using a K-Means clustering algorithm. This allows us to compare the financial outcomes for female and male retirees and to enhance some findings in the literature. In the final phase of our study, we compare the results obtained by our methodology to the 4% rule which is a widely used approach for post-retirement spending. Our results show that most retirees can withdraw safely more than they currently do (around 2.5%). A withdrawal rate of about 4.5% is proved to be safe, but it might not provide sufficient income for most retirees since it yields approximately CAD 20,000 per year for male retirees in the highest risk tolerance group who withdraw about 4.5% annually. Full article

We obtain the upper and lower bounds for the ruin probability in the Sparre–Andersen model. These bounds are established under various conditions: when the adjustment coefficient exists, when it does not exist, and when the interarrival distribution belongs to certain aging classes. Additionally, we improve the Lundberg upper bound for the ruin probability. Full article

We consider a two-dimensional risk model with simultaneous Poisson arrivals of claims. Each claim of the first input process is at least as large as the corresponding claim of the second input process. In addition, the two net cumulative claim processes share a common Brownian motion component. For this model we determine the Gerber–Shiu metrics, covering the probability of ruin of each of the two reserve processes before an exponentially distributed time along with the ruin times and the undershoots and overshoots at ruin. Full article

In this article, we study stochastic orders over an interval. Mainly, we focus on orders related to the Laplace transform. The results are then applied to obtain a bound for heavy-tailed distributions and are illustrated by some examples. We also indicate how these ordering relationships can be adapted to the classical risk model in order to derive a moment bound for ruin probability. Finally, we compare it with other existing bounds. Full article

The present work concerns the finite-time ruin probabilities for several bidimensional risk models with constant interest force and correlated Brownian motions. Under the condition that the two Brownian motions $\{B_1\left(t\right),t\ge 0\}$ and $\{B_2\left(t\right),t\ge 0\}$ are correlated, we establish new results for the finite-time ruin probabilities. Our research enriches the development of the ruin theory with heavy tails in unidimensional risk models and the dependence theory of stochastic processes. Full article

In this paper, we propose a new discrete-time risk model of an insurance portfolio with stochastic premiums, in which the temporal dependence among the premium numbers of consecutive periods is fitted by the first-order integer-valued autoregressive (INAR(1)) process and the temporal dependence among the claim numbers of consecutive periods is described by the integer-valued moving average (INMA(1)) process. To measure the risk of the model quantitatively, we study the explicit expression for a function whose solution is defined as the Lundberg adjustment coefficient and give the Lundberg approximation formula for the infinite-time ruin probability. In the case of heavy-tailed claim sizes, we establish the asymptotic formula for the finite-time ruin probability via the large deviations of the aggregate claims. Two numerical examples are provided in order to illustrate our theoretical findings. Full article

The target of this study was the tungsten Regoufe mine, whose exploitation stopped in the 1970s. When the mine closed, an unacceptable legacy constituted of mining waste tailings and the ruins of infrastructure was left behind. This work assessed the soil, plants, and water contamination in the mining area; namely their content in potentially toxic elements (PTEs). The global impact of PTEs in the Regoufe mine surface soil points to a very high to ultrahigh degree of contamination of the area having a serious ecological risk level, mainly related to As and Cd contributions. However, establishing the direct relation between As contamination and the anthropogenic effects caused by the mining process cannot be carried out in a straightforward manner, since the soils were already enriched in metals and metalloids as a result of the geological processes that gave origin to the mineral deposits. The studies performed on the plants revealed that the PTE levels in the plants were lower than in the soil, but site-specific soil concentrations in As and Pb positively influence bioaccumulation in plants. The magnetic studies showed the presence of magnetic technogenic particles concentrated in the magnetic fraction, in the form of magnetic spherules. The magnetic technogenic particles probably result from temperature increases induced by some technological process related to ore processing/mining activity. The PTEs in the surface and groundwater samples were similar and relatively low, being unlikely to pose potential health and environmental risks. Arsenic (As) constituted the exception, with levels above reference for drinking water purposes. Full article

The Devylder–Goovaerts conjecture is probably the oldest conjecture in actuarial mathematics and has received a lot of attention in recent years. It claims that ruin with equalized claim amounts is always less likely than in the classical model. Investigating the validity of this conjecture is important both from a theoretical aspect and a practical point of view, as it suggests that one always underestimates the risk of insolvency by replacing claim amounts with the average claim amount a posteriori. We first state a simplified version of the conjecture in the discrete-time risk model when one equalizes aggregate claim amounts and prove that it holds. We then use properties of the Pareto distribution in risk theory and other ideas to target candidate counterexamples and provide several counterexamples to the original Devylder–Goovaerts conjecture. Full article