Classical Risk Theory Research Articles

Stable dividends under linear-quadratic optimisation

The optimisation criterion for dividends from a risky business is most often formalised in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for the stability of dividends. In particular, within actuarial risk theory, the maximisation of future dividends has been studied as the so-called de Finetti problem. However, there the optimal strategies typically become so-called barrier strategies. These are far from stable, and suboptimal affine dividend strategies have recently received attention. In contrast, in the class of linear-quadratic problems, the demand for stability is explicitly stressed. These have often been studied in diffusion models different from the actuarial risk models. We bridge the gap between these thinking patterns by deriving optimal affine dividend strategies under a linear-quadratic criterion for an additive process. We characterise the value function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal controls to the classical objective of maximising the expected present value of future dividends. Thereby we provide a framework within which stability of dividends from a risky business, e.g. in classical risk theory, is explicitly demanded and obtained.

Risk Assessment of Crowd-Gathering in Urban Open Public Spaces Supported by Spatio-Temporal Big Data

The urban open public spaces are the areas where people tend to gather together, which may lead to great crowd-gathering risk. This paper proposes a new method to assess the rank and spatial distribution of crowd-gathering risk in open public spaces in a large urban area. Firstly, a crowd density estimation method based on Tencent user density (TUD) data is built for different times in open public spaces. Then, a reasonable crowd density threshold is delimited to detect critical crowd situations in open public spaces and find out the key open public spaces that need to have intensive crowd-gathering prevention. For estimating the crowd-gathering risk in key open public spaces, the quantified risk assessment approach is conducted based on the classical risk theory that simultaneously considers the probability of an accident occurring, the severity of the accident consequence, and the risk aversion factor. A case study of the area within the Outer-ring Road of Shanghai was conducted to determine the feasibility of the new method. The thematic maps that describe the ranks and spatial distribution of crowd-gathering risk were generated. According to the risk maps, the government can determine the crowd control measures in different areas to reduce the crowd-gathering risk and prevent dangerous events.

РИЗИК: СУТНІСТЬ, ПРИЧИНИ ВИНИКНЕННЯ ТА ОСНОВНІ ВИДИ

The activities of any business are always associated with various risky events. Today, risk management is an integral part of the enterprise management system. Identification, analysis, evaluation and accounting of risk factors in accordance with changes in the external environment are necessary for making sound management decisions. The formation of the basis of modern managerial thinking, able to meet the requirements of the new economy, is possible on the basis of a study of the conceptual apparatus in the field of risk management theory. Based on the analysis of different views on the nature of risk of representatives of different risk theories, we have identified three such conditional research groups: supporters of classical or neoclassical risk theory; a group of scientists who determine the nature of risk based on the subject of research; representatives who support an integrated approach to defining the essence of the category “risk”, based on the essence of the phenomenon of risk and its complex nature. In the course of the research the logical-semantic field of the category “risk” is constructed, on the basis of which associative terms are determined, which allow to determine the possible semantic space of multifaceted interpretation. Based on the generalization of theoretical aspects, it is argued that there are two main approaches to the interpretation of the category “risk” in modern science, such as process (risk as the activity of enterprises in uncertainty) and object (result resulting from risk events). The author’s definition of the essence of the category “risk” is proposed, which is based on an integrated approach, according to which risk is an objective-subjective category that characterizes uncertainty in the modern economic space due to business environment factors that have both direct and indirect impact on the activities of the enterprise. The practical value of the article is to identify the key areas of activity of the trading company that are most sensitive to risks, and to identify the main types of risks, such as information and DG risks; resource; organizational; commodity; price; sales; logistical, distributive; risks of marketing communications; HR risks.

Stochastic Finance of a Company Associated with Ruin, Restricted Reserve, Dividends and Dividend Strategy

In this paper a dual model of classical risk theory is considered to study the expected value of discounted dividends using imbedding method. The optimal dividend strategy is obtained in different forms. By the same technique the classical model is also analyzed. Further in the dual model expenses of a company are taken as proportional to the surplus to obtain optimal barrier strategy. We also consider a model with two components of incomes to the company in analyzing the optimal dividend strategy. In the dual model jump-diffusion is also introduced to determine the two components of dividends due to jump-gains and jump-diffusion.

Some Optimization and Decision Problems in Proportional Reinsurance

Reinsurance is one of the tools that an insurer can use to mitigate the underwriting risk and then to control its solvency. In this paper, we focus on the proportional reinsurance arrangements and we examine several optimization and decision problems of the insurer with respect to the reinsurance strategy. To this end, we use as decision tools not only the probability of ruin but also the random variable deficit at ruin if ruin occurs. The discounted penalty function (Gerber & Shiu, 1998) is employed to calculate as particular cases the probability of ruin and the moments and the distribution function of the deficit at ruin if ruin occurs.We consider the classical risk theory model assuming a Poisson process and an individual claim amount phase-type distributed, modified with a proportional reinsurance with a retention level that is not constant and depends on the level of the surplus. Depending on whether the initial surplus is below or above a threshold level, the discounted penalty function behaves differently. General expressions for this discounted penalty function are obtained, as well as interesting theoretical results and explicit expressions for phase-type 2 distribution. These results are applied in numerical examples of decision problems based on the probability of ruin and on different risk measures of the deficit at ruin if ruin occurs (the expectation, the Value at Risk and the Tail Value at Risk).

A heavy traffic approach to modeling large life insurance portfolios

We explore a new framework to approximate life insurance risk processes in the scenario of plentiful policyholders, via a bottom-up approach. Given the insurance contract structure, we aggregate the balance of individual policy accounts, and derive an approximating Gaussian process with computable correlation structure. The methodology is borrowed from heavy traffic theory in the literature of many-server queues, and involves the so-called fluid and diffusion approximations. Our framework is different from the individual risk model in that it takes into account the time dimension and the specific policy structure including the premium payments. It is also different from classical risk theory in that it builds the risk process from micro-level contracts and parameters instead of assuming aggregated claim and premium processes outright. As a result, our approximating process behaves differently depending on the issued contract structure. We also illustrate the flexibility of our approach by formulating a finite-horizon ruin problem that incorporates actuarial reserve in the consideration.

FROM RUIN TO BANKRUPTCY FOR COMPOUND POISSON SURPLUS PROCESSES

AbstractIn classical risk theory, the infinite-time ruin probability of a surplus process Ct is calculated as the probability of the process becoming negative at some point in time. In this paper, we consider a relaxation of the ruin concept to the concept of bankruptcy, according to which one has a positive surplus-dependent probability to continue despite temporary negative surplus. We study the resulting bankruptcy probability for the compound Poisson risk model with exponential claim sizes for different bankruptcy rate functions, deriving analytical results, upper and lower bounds as well as an efficient simulation method. Numerical examples are given and the results are compared with the classical ruin probabilities. Finally, it is illustrated how the analysis can be extended to study the discounted penalty function under this relaxed ruin criterion.

On the Optimal Dividend Strategy in a Regime-Switching Diffusion Model

In this paper we consider the optimal dividend strategy under the diffusion model with regime switching. In contrast to the classical risk theory, the dividends can only be paid at the arrival times of a Poisson process. By solving an auxiliary optimal problem we show that the optimal strategy is the modulated barrier strategy. The value function can be obtained by iteration or by solving the system of differential equations. We also provide a numerical example to illustrate the effects of the restriction on the timing of the payment of dividends.

On the Optimal Dividend Strategy in a Regime-Switching Diffusion Model

In this paper we consider the optimal dividend strategy under the diffusion model with regime switching. In contrast to the classical risk theory, the dividends can only be paid at the arrival times of a Poisson process. By solving an auxiliary optimal problem we show that the optimal strategy is the modulated barrier strategy. The value function can be obtained by iteration or by solving the system of differential equations. We also provide a numerical example to illustrate the effects of the restriction on the timing of the payment of dividends.

Some Results behind Dividend Problems

We consider the basic dividend problem of the compound Poisson model with constant barrier strategy. Some results concealed behind the dividend problem are made explicit in the present work. Different methods and some of which are firstly given in this paper. All these results presented certain direct relationship between some important actuary variables in classical risk theory is also revealed.

Log-concavity of the extremes from Gumbel bivariate exponential distributions

In the classical risk theory, it is often used that different type dimensions can be aggregated into a single-dimensional statistic, as well as the assumption of properties on log-concavity of this aggregation. The extreme-order statistics, minimum and maximum, might be used as aggregate statistics. In this paper, we discuss the log-concavity of the survival function of the minimum and maximum from Gumbel bivariate exponential models, through the log-concavity of generalized mixtures of four or fewer exponential distributions, extending the papers of Baggs and Nagaraja [Baggs, G.E. and Nagaraja, H.N., 1996, Reliability properties of order statistics from bivariate exponential distributions. Communications in Statistics—Stochastic Models, 12, 611–631] and Franco and Vivo [Franco, M. and Vivo, J.M., 2002, Reliability properties of series and parallel systems from bivariate exponential models. Communications in Statistics—Theory and Methods, 31, 2349–2360] devote to the log-concavity for generalized mixtures of three or fewer exponential distributions.

A credit risk model for large dimensional portfolios with application to economic capital

In this paper we develop a multi-period and multi-state portfolio credit risk model which is applicable to large dimensional portfolios like for example retail and mortgage portfolios. The model includes a methodology for estimation and simulation of systematic transition risk through a model for stochastic migration, a methodology for the modelling of recoveries in the case of stochastic collaterals as well as an approach to dimension reduction of the portfolio. One important application of our model is economic capital (EC) and a concept of EC based on the analogy with classical risk theory is introduced and the questions of allocation as well as risk-adjusted pricing based on the allocation of EC are structured and described. The model is illustrated by an extensive numerical example giving a concretization of the model as well as of several of the concepts introduced.

On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier

In the framework of classical risk theory we investigate a model that allows for dividend payments according to a time-dependent linear barrier strategy. Partial integro-differential equations for Gerber and Shiu's discounted penalty function and for the moment generating function of the discounted sum of dividend payments are derived, which generalizes several recent results. Explicit expressions for the nth moment of the discounted sum of dividend payments and for the joint Laplace transform of the time to ruin and the surplus prior to ruin are derived for exponentially distributed claim amounts.

Risk theory with a nonlinear dividend barrier

In the framework of classical risk theory we investigate a surplus process in the presence of a nonlinear dividend barrier and derive equations for two characteristics of such a process, the probability of survival and the expected sum of discounted dividend payments. Number-theoretic solution techniques are developed for approximating these quantities and numerical illustrations are given for exponential claim sizes and a parabolic dividend barrier.

Risk and Complex Fractals in Finance∶ Application to a Black-Scholes Equation of Order n

The short term risks in finance cannot be suitably analyzed by using the classical risk theory, and a way to circumvent this pitfall is to use fractional processes. The higher the risk is the higher the order of the fractal is. Recently, a model of complex-valued fractional Brownian motion of order n has been suggested in the literature, and the purpose herein is to examine which kind of results it can provide in mathematics of finance. After a short background to this model, one comments on fractals in finance, and then one suggests a general model of complex fractals for stock market. Various state variables can be identified with the parameters so involved in this model, and mainly, one can introduce a new concept of stock market time different from the standard absolute physical time. As an application, a Black-Scholes equation of order n is derived.

An optimal stopping problem in risk theory

Abstract In classical risk theory often stationary premium and claim processes are considered. In some cases it is more convenient to model non-stationary processes which describe a movement from environmental conditions, for which the premiums were calculated, to less favorable circumstances. This is done by a Markov-modulated Poisson claim process. Moreover the insurance company is allowed to stop the process at some random time, if the situation seems unfavorable, in order to calculate new premiums. This leads to an optimal stopping problem which is solved explicitly to some extent.

On the Time Value of Ruin

This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ultimate ruin. Explicit results are obtained when dividends are paid out to the stockholders according to a constant barrier strategy.

An optimal stopping problem in risk theory

In classical risk theory often stationary premium and claim processes are considered. In some cases it is more convenient to model non-stationary processes which describe a movement from environmental conditions, for which the premiums were calculated, to less favorable circumstances. This is done by a Markov-modulated Poisson claim process. Moreover the insurance company is allowed to stop the process at some random time, if the situation seems unfavorable, in order to calculate new premiums. This leads to an optimal stopping problem which is solved explicitly to some extent.

Estimating the Probability of Ruin for Variable Premiums by Simulation

AbstractThere is a duality between the surplus process of classical risk theory and the single-server queue. It follows that the probability of ruin can be retrieved from a single sample path of the waiting time process of the single-server queue. In this paper, premiums are allowed to vary. It has been shown that the stationary distribution of a corresponding storage process is equal to the survival probability (with variable premiums). Thus by simulation of the corresponding storage process, the probability of ruin can be obtained. The special cases where the surplus earns interest and the premiums are charged by layers are considered and illustrated numerically.

Risk theory with the gamma process

The aggregate claims process is modelled by a process with independent, stationary and nonnegative increments. Such a process is either compound Poisson or else a process with an infinite number of claims in each time interval, for example a gamma process. It is shown how classical risk theory, and in particular ruin theory, can be adapted to this model. A detailed analysis is given for the gamma process, for which tabulated values of the probability of ruin are provided.