Search Results (24)

This article provides an overview of the current state-of-the-art in cyber risk and cyber risk management, focusing on the mathematical models that have been created to help with risk quantification and insurance pricing. We discuss the main ways that cyber risk is measured, starting with vulnerability functions that show how systems react to threats and going all the way up to more complex stochastic and dynamic models that show how cyber attacks change over time. Next, we examine cyber insurance, including the structure and main features of the cyber insurance market, as well as the growing role of cyber reinsurance in strategies for transferring risk. Finally, we review the mathematical models that have been proposed in the literature for setting the prices of cyber insurance premiums and structuring reinsurance contracts, analysing their advantages, limitations, and potential applications for more effective risk management. The aim of this article is to provide researchers and professionals with a clear picture of the main quantitative tools available and to point out areas that need further research by summarising these contributions. Full article

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

In this study, we design an algorithm to work on gate-based quantum computers. Based on the algorithm, we construct a quantum circuit that represents the surplus process of a cedant under a reinsurance agreement. This circuit takes into account a variety of factors: initial reserve, insurance premium, reinsurance premium, and specific amounts related to claims, retention, and deductibles for two different non-proportional reinsurance contracts. Additionally, we demonstrate how to perturb the actuarial stochastic process using Hadamard gates to account for unpredictable damage. We conclude by presenting graphs and numerical results to validate our capital modelling approach. Full article

This paper studies insurance companies’ optimal reinsurance–investment strategy under the stochastic interest rate and stochastic volatility model, taking the HARA utility function as the optimal criterion. It uses arithmetic Brownian motion as a diffusion approximation of the insurer’s surplus process and the variance premium principle to calculate premiums. In this paper, we assume that insurance companies can invest in risk-free assets, risky assets, and zero-coupon bonds, where the Cox–Ingersoll–Ross model describes the dynamic change in stochastic interest rates and the Heston model describes the price process of risky assets. The analytic solution of the optimal reinsurance–investment strategy is deduced by employing related methods from the stochastic optimal control theory, the stochastic analysis theory, and the dynamic programming principle. Finally, the influence of model parameters on the optimal reinsurance–investment strategy is illustrated using numerical examples. Full article

This paper investigates the optimal private health insurance contract design problem, considering the joint interests of a policyholder and an insurer. Both the policyholder and the insurer jointly determine the premium of private health insurance. In order to better reflect reality, the illness expenditure is modelled by an extended compound Poisson process depending on health status. Under the mean–variance criterion and by applying dynamic programming, control theory, and leader–follower game techniques, analytically time-consistent private health insurance strategies are derived, optimal private health insurance contracts are designed, and their implications toward insurance are analysed. Finally, we perform numerical experiments assuming that the policyholder and the insurer calculate their wealth every year and they deposit their disposable income into the Bank of China with the interest rate being $r=0.021$. The values of other model parameters are set by referring to the data in the related literature. We find that the worse the policyholder’s health, the higher the premium that they pay for private health insurance, and buying private health insurance can effectively reduce the policyholder’s economic losses caused by illnesses. 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

This paper investigates an optimal reinsurance policy using a risk model with dependent claim and insurance premium by assuming that the insurance premium is random. Their dependence structure is modeled using Sarmanov’s bivariate exponential distribution and the Farlie–Gumbel–Morgenstern (FGM) copula-based bivariate exponential distribution. The reinsurance premium paid by the insurer to the reinsurer is fixed and is charged by the expected value premium principle (EVPP) and standard deviation premium principle (SDPP). The main objective of this paper is to determine the proportion and retention limit of the optimal combination of proportional and stop-loss reinsurance for the insurer. Specifically, with a constrained reinsurance premium, we use the minimization of the Value-at-Risk (VaR) of the insurer’s net cost. When determining the optimal proportion and retention limit, we provide some numerical examples to illustrate the theoretical results. We show that the dependence parameter, the probability of claim occurrence, and the confidence level have effects on the optimal VaR of the insurer’s net cost. Full article

Credit Default Swap (CDS) spread is a realistic measure of credit risk. Changes in the spreads showcase changes in the underlying uncertainty or credit volatility regarding the credit risk, associated with the asset class. We use Multifractal Detrended Fluctuation Analysis (MF-DFA) to further investigate the presence of asymmetries and the difference between Greece and G7 countries in terms of credit risk. We have considered 2587 daily observations for each of the 48 CDS spreads. Hence, a total of 124,176 data points were under consideration across six yearly CDS categories of Greece and most of the G7 countries (Germany, USA, UK, Canada, Japan). The tenure of these CDS were 1 year, 2 years, 3 years, 5 years, 7 years, 10 years, 20 years, and 30 years. We have found that the Greek CDS spread movement is purely stochastic and anti-persistent, having practically no predictability at all. On the other hand, the remaining countries’ CDSs were highly predictable, showing a consistent long memory or long-range dependence, having embedded the bubble caused by herding. This is reflected in terms of flight-to-quality behavior and in estimates of CDS premiums for insurance against a default on government bonds. Full article

This paper examines the role of trade credit insurance in a supply chain consisting of a capital-constrained supplier and a capital-constrained retailer. The retailer faces stochastic market demand and seeks trade credit from the supplier. The supplier, who is the Stackelberg game leader, decides the production quantity and the insurance coverage rate. We find that when the supplier’s initial capital is not sufficient, the use of trade credit insurance may reduce the trade quantity and the expected profit of the retailer. However, when the initial capital of the supplier is sufficient, the use of trade credit insurance will always increase the trade quantity. In the extension, we assume the supplier will face a potential financing cost if the net income is lower than the threshold. We find that if the insurance company has to keep its expected return positive and has no way to invest the insurance premium, the supplier will never buy the trade credit insurance no matter how much the marginal financing cost is when threshold is outside a certain range. Both the results and the methods in this paper can help businesses achieve a balance of funds and the logistics of the supply chain and risks, thereby improving the effectiveness of the supply chain operation. Full article

Water scarcity is an increasingly recurring problem for irrigated agriculture in Mediterranean regions. It is, therefore, necessary to establish technical and financial measures to enable irrigators to deal with this problem. This study presents a new index-based drought insurance scheme in an irrigation district in the Jucar river basin in Spain, a highly regulated water system. Three insurance scheme options were evaluated and, the values of the fair risk premiums, the maximum compensation, and the deductible franchise were established. These insurance schemes were designed in agreement with the preexisting drought system operating rules to reduce moral hazard and adverse selection. Risk-reducing and effective evaluation methods were used to determine the insurance coverage’s viability for irrigators: standard deviation gross margin, minimum gross margin, and RMSL. The proposed insurances were also evaluated using synthetic hydrological time series generated with a stochastic ARMA model through a basin-wide water resource simulation model developed in the DSS Shell AQUATOOL. Financial indicators, such as the basis risk and claim ratio were applied to analyze the economic feasibility for insurance companies. The results show that a suitable and efficient option is an early-bird contract combined with a trigger of emergency or alert state in a multi-year contract. This type of specialized insurance helps to fill the existing gap in traditional insurance schemes for irrigated crops and offered additional coverage to farmers under drought and water scarcity conditions. Full article

The present paper introduces a theoretical framework through which the degree of risk aversion with respect uncertain prices can be measured through the context of the indirect utility function (IUF) using a lab experiment. First, the paper introduces the main elements of the duality theory (DT) in economics. Next, it proposes the context of IUFs as a suitable framework for measuring price risk aversion through varying prices as opposed to varying payoffs, which has been common practice in the mainstream of experimental economics. Indeed, the DT in modern microeconomics indicates that the direct utility function (DUF) and the IUF are dual to each other, implicitly suggesting that the degree of risk aversion (or risk seeking) that a given rational subject exhibits in the context of the DUF must be equivalent to the degree of risk aversion (or risk seeking) elicited through the context of the IUF. This paper tests the accuracy of this theoretical prediction through a lab experiment using a series of relevant statistical tests. This study uses the multiple price list (MPL) method, which has been one of the most popular sets of elicitation procedures in experimental economics to study risk preferences in the experimental laboratory using non-interactive settings. The key findings of this study indicate that price risk aversion (PrRA) is statistically significantly greater than payoff risk aversion (PaRA). Additionally, it is shown that the risk preferences elicited under the expected utility theory (EUT) are somewhat subject to context. Other findings imply that the risk premium (RP), as a measure of willingness to pay for insuring an uncertain situation, is statistically significantly greater for stochastic prices compared to that for stochastic payoffs. These results are robust across different MPL designs and various statistical tests that are utilized. Full article

This work uses classic stochastic dynamic programming techniques to determine the equivalence premium that each of two extraction agents of a non-renewable natural resource must pay to an insurer to cover the risk that the extraction pore explodes. We use statistical and geological methods to calibrate the time-until-failure distribution of extraction status for each agent and couple a simple approximation scheme with the actuarial standard of Bühlmann’s recommendations to charge the extracting agents a variance premium, while the insurer earns a return on its investment at risk. We test our analytical results through Monte Carlo simulations to verify that the probability of ruin does not exceed a certain predetermined level. Full article

We introduce here a diffusion-type approximation of the ruin probability both in finite and infinite time for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. This type of process is often called the insurer–reinsurer model. We assume that the flow of claims is governed by a general renewal process. A simple ruin probability formula for the model is known only in infinite time for the special case of the Poisson process and exponentially distributed claims. Therefore, there is a need for simple analytical approximations. In the literature, in the infinite-time case, for the Poisson process, a De Vylder-type approximation has already been introduced. The idea of the diffusion approximation presented here is based on the weak convergence of stochastic processes, which enables one to replace the original risk process with a Brownian motion with drift. By applying this idea to the insurer–reinsurer model, we obtain simple ruin probability approximations for both finite and infinite time. We check the usefulness of the approximations by studying several claim amount distributions and comparing the results with the De Vylder-type approximation and Monte Carlo simulations. All the results show that the proposed approximations are promising and often yield small relative errors. Full article

Currently, insurance companies exclude the buildings’ envelope of their policies since they lack reliable information about the risks and degradation models and are unable to estimate the probabilities of intervention and corresponding costs. This study intends to overcome the existing gap, proposing property maintenance insurance policies developed based on condition-based maintenance plans, using stochastic information regarding the degradation process of the buildings’ envelope elements in the definition of insurance policies. To perform this work, external thermal insulation composite systems (ETICS) are used as case study, for the definition of an insurance policy. This approach allows reducing the uncertainty associated with the degradation of ETICS even when subject to scheduled maintenance actions. Several insurance policies are analysed, with different insurance premiums, evaluating different risks accepted by the owners when adopting a certain maintenance plan. For owners, the main advantages of acquiring this insurance product are: (i) changing the nature of the risk, transferring the risk to the insurer; and (ii) increasing the asset’s equity value, reducing the risk associated with the degradation of ETICS and the uncertainty of maintenance costs over time. Full article