Insurance Risk Model Research Articles

How Might Model Uncertainty and Transaction Costs Impact Retained Earning & Dividend Strategies? An Examination Through a Classical Insurance Risk Model

How Might Model Uncertainty and Transaction Costs Impact Retained Earning & Dividend Strategies? An Examination Through a Classical Insurance Risk Model

Utility of classical insurance risk models for measuring the risks of cyber incidents

We demonstrate that the classical insurance risk models yield significant advantages in the context of cyber risk analysis. This model exhibits commendable attributes in terms of both computational efficiency and predictive capabilities. Utilizing several compound point risk models, we derive the conditional Value-at-Risk and Tail Value-at-Risk predictions for the cumulative breach size within specified time intervals. To verify the reliability of our method, we conduct backtesting exercises, comparing our predictions with actual breach sizes.

Uniform asymptotics for a risk model with constant force of interest and a random number of delayed claims

Consider an insurance risk model with constant force of interest under the assumption that an immediate claim is accompanied by a random number of delayed claims. In the presence of heavy tails on claim distributions and dependence structures among modelling components, we study the uniform asymptotics for an insurer's ruin-related quantities, where the uniformity holds for all time horizons varying in a relevant interval.

Multivariate regularly varying insurance and financial risks in multidimensional risk models

AbstractMultivariate regular variation is a key concept that has been applied in finance, insurance, and risk management. This paper proposes a new dependence assumption via a framework of multivariate regular variation. Under the condition that financial and insurance risks satisfy our assumption, we conduct asymptotic analyses for multidimensional ruin probabilities in the discrete-time and continuous-time cases. Also, we present a two-dimensional numerical example satisfying our assumption, through which we show the accuracy of the asymptotic result for the discrete-time multidimensional insurance risk model.

Asymptotics for ruin probabilities of a dependent delayed-claim risk model with general investment returns and diffusion

In this paper, we study a delayed-claim insurance risk model perturbed by diffusion with general investment returns, in which each main claim may induce a delayed claim. Assume that the main claim sizes follow a one-sided linear process with independent and identically distributed step sizes. Furthermore, we assume that the step sizes and the inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a bivariate Sarmanov distribution, and so do the delayed claim sizes and corresponding delayed times. In the presence of heavy tails, asymptotic upper and lower bounds for ruin probabilities are obtained. Finally, in order to verify the accuracy of our results, we conduct numerical simulations by Crude Monte Carlo (CMC) method.

Supermodular and directionally convex comparison results for general factor models

This paper provides comparison results for general factor models with respect to the supermodular and directionally convex order. These results extend and strengthen previous ordering results from the literature concerning certain classes of mixture models as mixtures of multivariate normals, multivariate elliptic and exchangeable models to general factor models. For the main results, we first strengthen some known orthant ordering results for the multivariate ▪ -product of the specifications, which represents the copula of the factor model, to the stronger notion of the supermodular ordering. The stronger comparison results are based on classical rearrangement results and in particular are rendered possible by some involved constructions of transfers as arising from mass transfer theory. The ordering results for ▪ -products are then extended to factor models with general conditional dependencies. As a consequence of the ordering results, we derive worst case scenarios in relevant classes of factor models allowing, in particular, interesting applications to deriving sharp bounds in financial and insurance risk models. The results and methods of this paper are a further indication of the particular effectiveness of Sklar‘s copula notion.

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.

Asymptotics for a diffusion-perturbed risk model with dependence structures, constant interest force, and a random number of delayed claims

During extreme events such as natural or man-made disasters, it is likely that after immediate claims directly caused by the events, there are other delayed claims occurring in a certain period of future. In the classical risk model, each immediate claim is accompanied with a delayed one, but in reality each immediate claim can lead to a random number of delayed ones. To this end, we consider an insurance risk model with a random number of delayed claims as well as Brownian perturbation, dependence structures and constant interest force, and explore asymptotic estimations for an insurer’s ruin-related risks exposed to the extreme events. This article provides a novel insight to model and analyze the potential risks by fully considering the ongoing impacts of heavy-tailed claims, and thus accurately assesses the insurer’s operation risk in the long run.

Estimating the Gerber–Shiu Function in the Two-Sided Jumps Risk Model by Laguerre Series Expansion

In this paper, we consider an insurance risk model with two-sided jumps, where downward and upward jumps typically represent claim amounts and random gains, respectively. We use the Laguerre series to expand the Gerber–Shiu function and estimate it based on observed information. Moreover, we show that the estimator is easily computed and has a fast convergence rate. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

A scale function based approach for solving integral-differential equations in insurance risk models

In risk theory, the resolutions of many interesting problems are reduced to solving some integro-differential equations (IDE), see [4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 21, 22, 23, 24, 25, 26, 27, 29, 31], and references therein. Meanwhile, due to the recent advances made on Lévy processes, explicit analytical expressions of the scale functions associated with Lévy processes become on offer (see [1, 2, 11, 16, 17], Chapter 8 of [19], [20], etc). This paper aims at bridging together the scale functions and the IDEs by presenting a unified scale function based approach for solving IDEs that arise in risk theory. In particular, to demonstrate the effectiveness of this approach, a dividend and capital injection problem is considered under a jump-diffusion risk model. We first derive the IDEs satisfied by the expected accumulated discounted difference between the net dividends and the costs of capital injections, and then solve the IDEs with its solution being expressed in compact and transparent forms.

Uniform Asymptotic Estimate for the Ruin Probability in a Renewal Risk Model with Cox–Ingersoll–Ross Returns

Consider an insurance risk model with arbitrary dependence structures between the claim sizes. Suppose that the risky investment in the insurer can be established by the Cox–Ingersoll–Ross model. When the claim-size distribution is heavy-tailed, a uniform asymptotic formula for ruin probability is obtained.

Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary

In this article we present a space–time epidemic-type aftershock sequence (ETAS) model for the area of Hungary, motivated by the goal of its application in insurance risk models. High-quality recent instrumental data from the period 1996–2021 are used for model parameterization, including data from the recent nearby Zagreb and Petrinja event sequences. In the earthquake-triggering equations of our ETAS model, we replace the commonly used modified Omori law with the more recently proposed stretched exponential time response form, and a Gaussian space response function is applied with a variance add-on for epicenter error. After this model was tested against the observations, an appropriate overall fit for magnitudes M≥3.0 was found, which is sufficient for insurance applications, although the tests also show deviations at the M=2.5 threshold. Since the data used for parameterization are dominated by Croatian earthquake sequences, we also downscale the model to regional zones via parameter adjustments. In the downscaling older historical data are incorporated for a better representation of the key events within Hungary itself. Comparison of long-term large event numbers in simulated catalogues versus historical data shows that the model fit by zone is improved by the downscaling.

A decomposition for Lévy processes inspected at Poisson moments

AbstractWe consider a Lévy process Y(t) that is not continuously observed, but rather inspected at Poisson( $\omega$ ) moments only, over an exponentially distributed time $T_\beta$ with parameter $\beta$ . The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to $T_\beta$ , denoted by $Y_{\beta,\omega}$ . Our main result is a decomposition: we derive a remarkable distributional equality that contains $Y_{\beta,\omega}$ as well as the running maximum process $\bar Y(t)$ at the exponentially distributed times $T_\beta$ and $T_{\beta+\omega}$ . Concretely, $\overline{Y}(T_\beta)$ can be written as the sum of two independent random variables that are distributed as $Y_{\beta,\omega}$ and $\overline{Y}(T_{\beta+\omega})$ . The distribution of $Y_{\beta,\omega}$ can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative Lévy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cramér–Lundberg insurance risk model.

Queueing and risk models with dependencies

This paper analyzes various stochastic recursions that arise in queueing and insurance risk models with a ‘semi-linear’ dependence structure. For example, an interarrival time depends on the workload, or the capital, immediately after the previous arrival; or the service time of a customer depends on her waiting time. In each case, we derive and solve a fixed-point equation for the Laplace–Stieltjes transform of a key performance measure of the model, like waiting time or ruin time.

Joint moments of discounted claims and discounted perturbation until ruin in the compound Poisson risk model with diffusion

This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.

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.

Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite.

Cascade Sensitivity Measures.

In risk analysis, sensitivity measures quantify the extent to which the probability distribution of a model output is affected by changes (stresses) in individual random input factors. For input factors that are statistically dependent, we argue that a stress on one input should also precipitate stresses in other input factors. We introduce a novel sensitivity measure, termed cascade sensitivity, defined as a derivative of a risk measure applied on the output, in the direction of an input factor. The derivative is taken after suitably transforming the random vector of inputs, thus explicitly capturing the direct impact of the stressed input factor, as well as indirect effects via other inputs. Furthermore, alternative representations of the cascade sensitivity measure are derived, allowing us to address practical issues, such as incomplete specification of the model and high computational costs. The applicability of the methodology is illustrated through the analysis of a commercially used insurance risk model.

Poissonian occupation times of spectrally negative Lévy processes with applications

In this paper, we introduce the concept of Poissonian occupation times below level 0 of a spectrally negative Lévy process. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known continuously observed quantities involving occupation times of spectrally negative Lévy processes. As an application, we establish a link between Poissonian occupation times and insurance risk models with Parisian implementation delays.

Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.