Surplus Process Research Articles

Periodic threshold-type dividends for a perturbed dual risk model with a random time horizon

In this paper, we investigate the dividend problem where the surplus process is inscribed by a perturbed dual risk model with a random time horizon, where a periodic threshold-type dividend strategy is applied to the surplus process. Under such a dividend strategy, if the surplus level is bigger than the maximum of threshold b and the last observed level (post-dividend) at dividend-decision times, then a fraction of the excess is paid off as a lump sum dividend. In addition, we assume that ruin can only occur at dividend-decision times. When the individual gains density has a rational Laplace transform, explicit expressions for the expected value of the present dividends before ruin are provided. Finally, numerical examples are presented to explore the effect of the parameters in the model on the dividend function.

Computational Methods of Ruin Probability: Actuarial Comparison of De-Vylder and Tijim’s Models

The underwriting operation of insurance firms is to assume the risk of the insured in return of premium received. In order to shield itself against extreme losses and avoid the risk of insolvency, it becomes necessary to examine how the portfolio is expected to perform over a long-time horizon. The surplus process is connected with the excess of the premium received over claims outgo of the insurer’s portfolio to enable it predict the level at which the insurer could survive. When the surplus approaches a defined lower limit irrespective of the initial reserve, then the insurer is ruined. The probability of ruin apparently defines the volatility embedded in underwriting process as a useful tool of risk measurement in long range planning of premium rate. This paper is anchored on the following objectives: solve the adjustment co-efficient using the moment generating function, compute the De-Vylder’s ruin, compute the Tijim’s ruin approximation and then compare the two. Although it was verified that as the initial capital increases, the probability of ruin decreases under the two models, computational evidence from our results in tables 2 and 3 shows that the Tijim’s approximation to ruin probability is higher than the De-Vylder’s approximation at the same level of initial capital and safety loading. The implication is that the De-Vylder’s approximation to ruin probabilities is an improvement over Tijim’s ruin model and hence De-Vylder’s approximation is recommended for the insurer’s ruin assessment

On the moments of dividends and capital injections under a variant type of Parisian ruin

This paper considers a risk model driven by a spectrally negative Lévy process, where any surplus above b (0<b<∞) is deducted away as dividends and any deficit is covered by injected capitals/raised money. For such a risk model, we define a variant of Parisian ruin time as the first time that the surplus process stays continuously below a (0<a<b<∞) for a time interval with length larger than some pre-specified exponential random variable that is marked on this time interval. A recursive formula for the moments of the Net Present Value (NPV) of dividends paid until Parisian ruin is provided. The expected NPV of capitals injected until the Parisian ruin time is also characterized compactly in terms of the scale functions of the underlying process.

Irreversible reinsurance: minimization of capital injections in presence of a fixed cost

AbstractWe propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cramér-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any particular functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model’s parameters.

Ruin probability for heavy-tailed and dependent losses under reinsurance strategies

The frequency and severity of extreme events have increased in recent years in many areas. In the context of risk management for insurance companies, reinsurance provides a safe solution as it offers coverage for large claims. This paper investigates the impact of dependent extreme losses on ruin probabilities under four types of reinsurance: excess of loss, quota share, largest claims, and ecomor. To achieve this, we use the dynamic GARCH-EVT-Copula combined model to fit the specific features of claim data and provide more accurate estimates compared to classical models. We derive the surplus processes and asymptotic ruin probabilities under the Cramér–Lundberg risk process. Using a numerical example with real-life data, we illustrate the effects of dependence and the behavior of reinsurance strategies for both insurers and reinsurers. This comparison includes risk premiums, surplus processes, risk measures, and ruin probabilities. The findings show that the GARCH-EVT-Copula model mitigates the over- and under-estimation of risk associated with extremes and lowers the ruin probability for heavy-tailed distributions.

Optimal investment for asset–liability management with delay and partial information under Ornstein–Uhlenbeck process

In this paper, we investigate the optimal investment strategy of asset liability management (ALM) with bounded memory and partial information. Suppose that investors invest their assets in a financial market consisting of a risk-free bond and a risk-free stock, while also taking on liabilities, in which the value of liabilities and the price of risky assets satisfy the Ornstein–Uhlenbeck (O–U) processes whose drift terms are unobserved. By constructing a dynamic portfolio of risk-free bonds, risky stocks and liabilities, a stochastic delay differential equation is obtained to depict the surplus process of investor. The ALM problem is formulated as finding the best strategy to maximize the terminal utility of the sum of terminal surplus and some historical wealth under partial information, and the corresponding full information case is also studied as a supplement. For both cases of partial information and full information, we apply the dynamic programming method to derive HJB equations, verification theorems, and closed-form solutions of optimal strategies and value functions. Moreover the relationship between optimal strategy and value function under full information and partial information is also given. Finally, numerical examples are carried out to illustrate the influence of some important parameters on the obtained results.

On an insurance ruin model with a causal dependence structure and perturbation

The classical compound Poisson risk model and the Sparre-Andersen risk model for insurance businesses assume that the interclaim times and the claim amounts are independently distributed. To relax the independence assumption, we consider a continuous-time risk model under which the interclaim-time distribution depends on the size of the previously occurred claim. For practical purposes, the surplus process is further assumed to be perturbed by a Brownian motion to address small financial fluctuations or investment returns. To analyze the risk associated with the event when the insurer’s ruin occurs, explicit solutions for the Gerber–Shiu discounted penalty function may be derived through the defective renewal equations provided here when claim amounts follow an arbitrary distribution. Applications with Kn family claim amounts and the Laplace transform of the ruin time are discussed in detail. An illustrative numerical example is presented to assess the impact of different perturbations on the underlying dependent-structure surplus process.

Non-zero-sum investment-reinsurance game with delay and ambiguity aversion

In this paper, we investigate a investment-reinsurance game problem with delay and ambiguity aversion for two insurers under mean–variance preference. Each insurer can purchase proportional reinsurance whose surplus process is a diffusion approximation process, and invest oneself’s wealth in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a jump–diffusion process. Based on the introduction of delay, we obtain wealth dynamics for each insurer depicted by a stochastic delay differential equation. By the stochastic control theory in the framework of game theory, the robust time-consistent Nash equilibrium investment-reinsurance strategies and the corresponding robust equilibrium value functions for each insurer are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment-reinsurance strategy.

Non-zero-sum reinsurance and investment game under thinning dependence structure: mean–variance premium principle

This paper considers a non-zero-sum stochastic differential reinsurance and investment game between two competitive insurers under the expected exponential utility. It is assumed that the surplus process of each insurer is characterized by a compound Poisson model and their claim businesses are correlated through thinning dependence structure. Each insurer can purchase proportional reinsurance and is allowed to invest in a risk-free asset and a risky asset. The insurers compete with each other and are concerned with the relative performance. By using the stochastic control technique, not only the existence and uniqueness of the optimal strategies are proved, but also the specific analyses for the constraint reinsurance strategies are provided. We find that (i) the thinning dependence structure has significant impact on optimal reinsurance strategies for both insurers and can help them reduce risks; (ii) the behavior of non-zero-sum game will reduce the proportion of purchasing reinsurance for each insurer.

Optimal robust reinsurance contracts with investment strategy under variance premium principle

This paper investigates the optimal robust proportional reinsurance contracts with investment in a liquid financial market under variance premium principle in a principal-agent framework. The surplus process of the insurer (agent) is assumed to follow an approximating compound Poisson risk process. Both the insurer and reinsurer (principal) are allowed to invest in a risk-free asset and a risky asset whose price process is governed by the constant elasticity of variance model. The insurer and reinsurer aim to maximize the expected exponential utility of terminal wealth. The reinsurer is ambiguity-averse and has deterministic ambiguity aversion preferences against the diffusion risk caused by the financial market and the approximated diffusion risk which comes from the claim process. The reinsurance price is described by the reinsurer's safety loading which can be decided by the optimal strategies of the insurer and the reinsurer. By utilizing stochastic optimal control principle and HJB (or HJBI) equations, the optimal (robust) proportional reinsurance-investment strategies and the corresponding value functions are obtained explicitly. Numerical examples are provided.

Ordering Results of Aggregate Claim Amounts from Two Heterogeneous Portfolios

In this paper, we discuss the stochastic comparison of two classical surplus processes in an one-year insurance period. Under the Marshall-Olkin extended Weibull random aggregate claim amounts, we establish some sufficient conditions for the comparison of aggregate claim amounts in the sense of the usual stochastic order. Applications of our results to the Value-at-Risk and ruin probability are also given. The obtained results show that the heterogeneity of the risks in a given insurance portfolio tends to make the portfolio volatile, which in turn leads to requiring more capital. We also obtain some sufficient conditions for comparing aggregate non-random claim amounts with different occurrence frequency vectors in terms of increasing convex order.

Numerical computation of Gerber–Shiu function for insurance surplus process with additional investment

This paper studies the Gerber–Shiu function for the insurance surplus process with additional investment under the Bachelier model. The Gerber–Shiu function allows us to study the moments of the time of ruin, which is the first time that the surplus is negative. First, we use the martingale theory in deriving the integro-differential equation of the Gerber–Shiu function. Then, we give the exact solution of the ruin probability in case the amount of claims follows the exponential distribution. Under a general distribution case, we propose a numerical method of the Gerber–Shiu function using the finite differential method based on the integro-differential equation. Then, numeric illustrations are provided to study the effect of the parameters on the Gerber–Shiu function.

Quantum Computing in Insurance Capital Modelling under Reinsurance Contracts

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.

Poissonian occupation times of refracted Lévy processes with applications

Inspired by the work of Lkabous (2021), we consider Poissonian occupation times below level 0 of a refracted Lévy process where its premium rate is adaptive. In this model, occupation time is accumulated once the surplus process is observed to be negative at Poisson arrival times. Our analysis depends on various exit identities for refracted Lévy processes observed at Poisson arrival times. As an application of Poissonian occupation times, we derive an explicit expression for the probability of Parisian ruin with Erlang(2, λ) implementation delays. The resulting main quantities are in terms of scale functions.

Optimality of Threshold Strategies for Spectrally Negative Lévy Processes and a Positive Terminal Value at Creeping Ruin

This article investigates a dividend optimization problem with a positive creeping-associated terminal value at ruin for spectrally negative Lévy processes. We consider an insurance company whose surplus process evolves according to a spectrally negative Lévy process with a Gaussian part and a finite Lévy measure. Its objective function relates to dividend payments until ruin and a creeping-associated terminal value at ruin. The positive creeping-associated terminal value represents the salvage value or the creeping reward when creeping happens. Owing to formulas from fluctuation theory, the objective considered is represented explicitly. Under certain restrictions on the terminal value and the surplus process, we show that the threshold strategy should be the optimal one over an admissible class with bounded dividend rates.

Ruin in a continuous-time risk model with arbitrarily dependent insurance and financial risks triggered by systematic factors

This paper is devoted to asymptotic analysis for a continuous-time risk model with the insurance surplus process and the log-price process of the investment driven by two dependent jump-diffusion processes. We take into account arbitrary dependence between the insurance claims and their corresponding investment return jumps caused by a sequence of systematic factors, whose arrival times constitute a renewal counting process. Under the framework of regular variation, we obtain a simple and unified asymptotic formula for the finite-time ruin probability as the initial wealth becomes large. It turns out that, in the weakly dependent case, the tails of the claims determine the exact decay rate of the finite-time ruin probability while the investment return jumps only contribute to the coefficient of the asymptotic formula; however, in the strongly dependent case, they both produce essential impacts on the finite-time ruin probability which is under-estimated in the weakly dependent case.

On a time-changed Lévy risk model with capital injections and periodic observation

In this paper, an insurance company’s surplus process is modeled by a time-changed Lévy process, where the continuous-time change is realized by integrating the Cox–Ingersoll–Ross (CIR) process. Assume that the surplus level is monitored periodically with constant monitoring frequency, and the insurer makes decisions on capital injections based on the observed surplus levels. Given a critical level b (b>0), whenever the observed surplus level belongs to the interval [0,b), capitals are injected into the insurance company to bring the surplus level back to b; whenever the observed surplus level is below level 0, ruin is declared, and the surplus process is stopped. The finite-time expected present value of operating costs until ruin is studied. Utilizing the Fourier series expansion in cosine form, we derive the closed-form approximation formula for this function and analyze the approximation error. Finally, many numerical examples are given to demonstrate the effectiveness of our method, and the impacts of some parameters are also studied.

Comparison of the Minimum Initial Capital of Investment Discrete Time Surplus Process in Case Separated Claims

In this paper, we suggest the bare minimum initial capital a firm providing insurance must hold to avoid going bankrupt. A case-separated investment discrete-time surplus process in motor insurance claims serves as the study's model. The 50th, 60th, 70th, and 80th percentiles are used as the dividing line between a claim's standard claim and large claim situations. We also discover a link between an insurance company's initial capital and the likelihood of ruin. The least squares regression method is utilized to calculate the minimum initial capital, and the simulation approach would be used to determine the ruin probability. The results indicate that the least initial capital is better provided by the 70th percentile than the 50th, 60th, and 80th percentiles, respectively.

Measuring the suboptimality of dividend controls in a Brownian risk model

AbstractWe consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some constant. The utility function destroys the linearity and the time-homogeneity of the problem considered. The value function depends not only on the surplus, but also on time. Numerical considerations suggest that the optimal strategy, if it exists, is of a barrier type with a nonlinear barrier. In the related article of Grandits et al. (Scand. Actuarial J.2, 2007), it has been observed that standard numerical methods break down in certain parameter cases, and no closed-form solution has been found. For these reasons, we offer a new method allowing one to estimate the distance from an arbitrary smooth-enough function to the value function. Applying this method, we investigate the goodness of the most obvious suboptimal strategies—payout on the maximal rate, and constant barrier strategies—by measuring the distance from their performance functions to the value function.

Conditional mean risk sharing of losses at occurrence time in the compound Poisson surplus model

This paper proposes a new risk-sharing procedure, framed into the classical insurance surplus process. Compared to the standard setting where total losses are shared at the end of the period, losses are allocated among participants at their occurrence time in the proposed model. The conditional mean risk-sharing rule proposed by Denuit and Dhaene (2012) is applied to this end. The analysis adopts two different points of views: a collective one for the pool and an individual one for sharing losses and adjusting the amounts of contributions among participants. These two views are compatible under the compound Poisson risk process. Guarantees can also be added by partnering with an insurer.