Compound Binomial Model Research Articles

Moments of deficit duration and its proportion in general compound binomial model

This paper discusses the deficit duration in the general compound binomial model. We derive the recursive formulas of moments of the deficit duration and its proportion in finite time with and without a dividend barrier, and use operators to design algorithms for finding their solutions. Under the condition of positive safety loading, we discuss the limit of deficit duration and show the convergence of moments of its proportion. In addition, the paper also discusses a sort of reinsurance strategy to significantly reduce the deficit duration. The proposed algorithms and reinsurance strategy are effective and are illustrated by numerical examples.

Compound binomial risk model in a Markovian environment with capital cost and the calculation algorithm

The operation of insurance institution is often affected by random environment, potential opportunity cost of capital or inflation. Considering this phenomenon, this paper establishes a new Markovian environment compound binomial risk model with capital cost, abbreviated as RMCM model. The new RMCM model is a complex recursive model which is difficult to calculate. To calculate the results in many future periods at one time, it is necessary to design an algorithm that can greatly improve calculation efficiency. One of the aims of this study is to conduct an analysis of the recursive algorithm, divide and conquer strategy and other algorithm theories to, designs a new MCPRAM-algorithm that is able to solve the calculation of this kind of recursive model with high complexity, multiple cycles and multiple periods, significantly improving the practical application value of the actuarial model. Taking aviation accident insurance as an example, the measurement methods of actuarial quantity in the model are explained statistically, and the calculation algorithm of conditional ruin probability is elaborated in detail. The conditional ruin probabilities of 60 periods under different operating conditions and different environment states were obtained and then compared and analyzed. RMCM has a wide range of applications and is suitable for describing the insurance products with low premium, small compensation probability; large claim amount and one-off compensation in a random market environment. The RMCM model proposed in this paper clarifies the risk of insurance institutions, compares the calculation of seven groups of parameter data, and obtains better parameters to control the bankruptcy risk at about 5% in the next five years. Parameter data reference and MCPRAM-algorithm can be used to design and develop new insurance products in a random environment, and provide valuable decision support for the scientific operation and management of insurance institutions.

Optimal Portfolio Selection With Transaction Costs In Compound Binomial Model

Optimal Portfolio Selection With Transaction Costs In Compound Binomial Model

Optimal portfolio selection with transaction costs in compound binomial model

Optimal portfolio selection with transaction costs in compound binomial model

Compound Binomial Model with Batch Markovian Arrival Process

A compound binomial model with batch Markovian arrival process was studied, and the specific definitions are introduced. We discussed the problem of ruin probabilities. Specially, the recursion formulas of the conditional finite-time ruin probability are obtained and the numerical algorithm of the conditional finite-time nonruin probability is proposed. We also discuss research on the compound binomial model with batch Markovian arrival process and threshold dividend. Recursion formulas of the Gerber–Shiu function and the first discounted dividend value are provided, and the expressions of the total discounted dividend value are obtained and proved. At the last part, some numerical illustrations were presented.

A Two-Parameter Description for the Injury Risk of Flash Bangs with Uncertainty

We study the random injury outcome caused by multiple flash bang submunitions on a crowd. We are particularly interested in the fluctuations in injury outcome among individual realizations. Previously, to simulate the distribution of the actual number of injured, we developed a comprehensive Monte Carlo model. While the full computational model is important for thorough theoretical investigations, in practical operations, it is desirable to characterize the phenomenological behavior of injury outcome using a concise model with only one or two parameters. Conventionally, the injury outcome is indicated by the average fraction of injured, which is called the risk of significant injury (RSI). The single metric RSI description fails to capture fluctuations in the injury outcome. The number of injured in the crowd is influenced by many random factors: the aiming error of flash bang mortar, the dispersion of submunitions after mortar burst, the amount of acoustic dose reaching individual subjects, and the biovariability of individual subjects’ reactions to a given acoustic dose. We aim to include these random factors properly and concisely. In this study, we represent the random injury outcome as a compound binomial model, in which the hidden injury probability is drawn from a two-parameter model distribution. We formulate and examine six model distributions for the injury probability. The best performer is a mixture of uniform and triangle distributions, parameterized by (RSI, dp) where dp is the standard deviation of the hidden injury probability. This mixture model predicts the behavior of injury outcome with uncertainty, based solely on the two parameters (RSI, dp) in the flash bang description. For example, we can predict the probability of the injury outcome not exceeding a prescribed tolerance. We advocate the adoption of this two-parameter characterization for flash bangs to replace the single-parameter RSI description. Whenever we need to give a high level coarse description of a flash bang situation, we state that the injury risk is represented by (RSI, dp).

Probability of ruin in discrete insurance risk model with dependent Pareto claims

Abstract We present basic properties and discuss potential insurance applications of a new class of probability distributions on positive integers with power law tails. The distributions in this class are zero-inflated discrete counterparts of the Pareto distribution. In particular, we obtain the probability of ruin in the compound binomial risk model where the claims are zero-inflated discrete Pareto distributed and correlated by mixture.

Exponential change of measure for general piecewise deterministic Markov processes

We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant~0}$ with a measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales that are associated with $X$ is given by By considering this exponential martingale to be a likelihood-ratio process, we define a new probability measure and show that the process $X$ is still a general PDMP under the new probability measure. We additionally find the new measure-valued generator and its domain. To illustrate our results, we investigate the continuous-time compound binomial model.

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable, dependent claim occurrences

AbstractThis paper investigates a discrete‐time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite‐time interval. Specifically, the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution, the computation of the minimum surplus distribution is given. Asset and risk management–oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition, comparisons are made involving the corresponding results of the classical discrete‐time compound binomial risk model, for which claim occurrences are independent and identically distributed.

An identity based on the generalised negative binomial distribution with applications in ruin theory

AbstractIn this study, we show how expressions for the probability of ultimate ruin can be obtained from the probability function of the time of ruin in a particular compound binomial risk model, and from the density of the time of ruin in a particular Sparre Andersen risk model. In each case evaluation of generalised binomial series is required, and the argument of each series has a common form. We evaluate these series by creating an identity based on the generalised negative binomial distribution. We also show how the same ideas apply to the probability function of the number of claims in a particular Sparre Andersen model.

Confidence Intervals for Weighted Composite Scores Under the Compound Binomial Error Model

AbstractReporting confidence intervals with test scores helps test users make important decisions about examinees by providing information about the precision of test scores. Although a variety of estimation procedures based on the binomial error model are available for computing intervals for test scores, these procedures assume that items are randomly drawn from a undifferentiated universe of items, and therefore might not be suitable for tests developed according to a table of specifications. To address this issue, four interval estimation procedures that use category subscores for the computation of confidence intervals are presented in this article. All four estimation procedures assume that subscores instead of test scores follow a binomial distribution (i.e., compound binomial error model). The relative performance of the four compound binomial–based interval estimation procedures is compared to each other and to the better known normal approximation and Wilson score procedures based on the binomial error model.

On the Compound Binomial Risk Model with Delayed Claims and Randomized Dividends

This paper extends the work of Yuen et al. (2013), who obtained explicit results for the discount-free Gerber–Shiu function for a compound binomial risk model in the presence of delayed claims and a randomized dividend strategy with a zero threshold level. Specifically, we establish a recursion method for computing the Gerber–Shiu expected discounted penalty function, which entails a number of important quantities in ruin theory, within the framework of the compound binomial aggregate claims with delayed by-claims and randomized dividends payable at a non-negative threshold level.

A threshold-based risk process with a waiting period to pay dividends

In this paper, a modified dividend strategy is proposed by delaying dividend payments until the insurer's surplus level remains at or above a threshold level b for a predetermined period of time h. We consider two cases depending on whether the period of time sustained at or above level b is counted either consecutively or accumulatively (referred to as standard or cumulative waiting period). In both cases, we develop a recursive computational procedure to calculate the expected total discounted dividend payments made prior to ruin for a discrete-time Sparre Andersen renewal risk process. By varying the values of b and h, a numerical study of the trade-off effects between finite-time ruin probabilities and expected total discounted dividend payments is investigated under a variety of scenarios. Finally, a generalized threshold-based strategy with a delayed dividend payment rule is studied under the compound binomial model.

On a discrete interaction risk model with delayed claims and stochastic incomes under random discount rates

ABSTRACTIn this paper, we study a discrete interaction risk model with delayed claims and stochastic incomes in the framework of the compound binomial model. A generalized Gerber-Shiu discounted penalty function is proposed to analyse this risk model in which the interest rates follow a Markov chain with finite state space. We derive an explicit expression for the generating function of this Gerber-Shiu discounted penalty function. Furthermore, we derive a recursive formula and a defective renewal equation for the original Gerber-Shiu discounted penalty function. As an application, the joint distributions of the surplus one period prior to ruin and the deficit at ruin, as well as the probabilities of ruin are obtained. Finally, some numerical illustrations from a specific example are also given.

On the first time of ruin in two-dimensional discrete time risk model with dependent claim occurrences

ABSTRACTThis article is concerned with a two-dimensional discrete time risk model based on exchangeable dependent claim occurrences. In particular, we obtain a recursive expression for the finite time non ruin probability under such a dependence among claim occurrences. For an illustration, we define a bivariate compound beta-binomial risk model and present numerical results on this model by comparing the corresponding results of the bivariate compound binomial risk model.

Ruin problems in Markov-modulated risk models

AbstractChen et al. (2014), studied a discrete semi-Markov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time two-state Markov-modulated risk model. We then study the finite time ruin probability for a discrete m-state model and show how we can approximate the density of the time of ruin in a continuous time Markov-modulated model with more than two states.

Computing finite time non-ruin probability and some joint distributions in discrete time risk model with exchangeable claim occurrences

In this paper, we study a discrete time risk model based on exchangeable dependent claim occurrences. In particular, we obtain expressions for the finite time non-ruin probability, and the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. An illustration of the results is given and some implications of the results are provided. Comparisons are made with the corresponding results for the classical compound binomial model of independent and identically distributed claim occurrences.

The Optimal Dividend Problem in the Continuous- Time Compound Binomial Model Under the Reinsurance Control

The Optimal Dividend Problem in the Continuous- Time Compound Binomial Model Under the Reinsurance Control

ON THE GERBER-SHIU PENALTY FUNCTION FOR THE COMPOUND BINOMIAL RISK MODEL WITH DELAYED CLAIMS

Abstract: In this paper we consider the Gerber-Shiu penalty function in the compound binomial risk model with time-correlated claims. It is assumed that each main claim will induce a by-claim but the occurrence of the by-claim may be delayed with a certain probability. Formulas for the probability generating function of the penalty function are obtained, together with the expression for the penalty function with zero initial surplus. Then we show that the penalty function satisfies a defective renewal equation. When the claims follow geometric distributions, explicit expression for the Gerber-Shiu function is derived. Numerical example is provided to illustrate the application of the result discussed in the paper.

Survival probabilities for compound binomial risk model with discrete phase-type claims

Due to having useful properties in approximating to the other distributions and mathematically tractable, phase type distributions are commonly used in actuarial risk theory. Claim occurrence time and individual claim size distributions are modelled by phase type distributions in literature. This paper aims to calculate the survival probabilities of an insurance company under the assumption that compound binomial risk model where the individual claim sizes are distributed as discrete Phase Type distribution