Discrete-time Insurance Risk Model Research Articles

Interplay of financial and insurance risks in dependent discrete-time risk models

Consider a discrete-time insurance risk model with two kinds of risks, i.e., insurance and financial risks. Within period i, the real-valued net insurance loss caused by traditional claims is regarded as the insurance risk, denoted by Xi, and the positive stochastic discount factor over the same time period is the financial risk, denoted by Yi. Assume that {(X,Y),(Xi,Yi),i≥1} form a sequence of independent and identically distributed random vectors. This work investigates the interplay of the insurance and financial risks, allowing a dependence structure exists between these two kinds of risks. Following the work of Li and Tang (2015), we derive some asymptotic and uniformly asymptotic formulas for the finite-time and infinite-time ruin probabilities, under the assumption that the generic random vector (X,Y) follows a bivariate Sarmanov distribution and every convex combination of the distributions of X and Y is of strongly regular variation. As an extension, we further consider a general pair (X,Y), whose dependence structure is more verifiable than the Sarmanov one to some extent. In such a setting, we study the asymptotic tail behavior of the product XY, which forms an analogue of the well-known Breiman’s theorem in a different and dependent situation.

Extensions of Breiman’s Theorem of Product of Dependent Random Variables with Applications to Ruin Theory

We consider the tail behavior of the product of two dependent random variables X and $$\Theta $$ . Motivated by Denisov and Zwart (J Appl Probab 44:1031–1046, 2007), we relax the condition of the existing $$\alpha \,+\,\epsilon $$ th moment of $$\Theta $$ in Breiman’s theorem to the existing $$\alpha $$ th moment and obtain the similar result as Breiman’s theorem of the dependent product $$X \Theta $$ , while X and $$\Theta $$ follow a copula function. As applications, we consider a discrete-time insurance risk model with dependent insurance and financial risks and derive the asymptotic tail behaviors for the (in)finite-time ruin probabilities.

A necessary and sufficient condition for the subexponentiality of the product convolution

Abstract Let X and Y be two independent and nonnegative random variables with corresponding distributions F and G. Denote by H the distribution of the product XY, called the product convolution of F and G. Cline and Samorodnitsky (1994) proposed sufficient conditions for H to be subexponential, given the subexponentiality of F. Relying on a related result of Tang (2008) on the long-tail of the product convolution, we obtain a necessary and sufficient condition for the subexponentiality of H, given that of F. We also study the reverse problem and obtain sufficient conditions for the subexponentiality of F, given that of H. Finally, we apply the obtained results to the asymptotic study of the ruin probability in a discrete-time insurance risk model with stochastic returns.

A revisit to ruin probabilities in the presence of heavy-tailed insurance and financial risks

Recently, Sun and Wei (2014) studied the finite-time ruin probability under a discrete-time insurance risk model, in which the one-period insurance and financial risks are assumed to be independent and identically distributed copies of a random pair (X,Y). For the heavy-tailed case, under a restriction on the dependence structure of (X,Y), they established an asymptotic formula for the finite-time ruin probability. In this paper we make an effort to remove this restriction as it excludes the cases with asymptotically dependent X and Y. We also extend the study to the infinite-time ruin probability. Employing a multivariate regular variation framework, we simplify the formula so that it shows in a transparent way how the ruin probabilities are affected by the tail dependence of (X,Y).

The ruin probabilities of a discrete-time risk model with dependent insurance and financial risks

Following the work of Sun and Wei (2014) [7], we investigate the ruin probabilities of a discrete-time insurance risk model with dependent insurance and financial risks. Assume that the one-period net insurance losses and discount factors form a sequence of independent and identically distributed copies of a random pair (X,θ). When the product Xθ is heavy tailed, we establish an asymptotic formula for the finite-time ruin probability without any restriction on the dependence structure of (X,θ) and extend the result to the infinite time ruin probability.

Ruin Probabilities in Risk Based on a Generalized FGM Dependence Structure

In this paper, we consider a discrete time insurance risk model, in which insurance and financial risks jointly follow a bivariate generalized FGM distribution. Assuming that every convex combination of the marginal distributions of insurance and financial risks belongs to strongly regular variation class, we derive some asymptotic equivalence formulas for these probabilities with both finite and infinite time horizons, all in the form of linear combinations of the tail probabilities of the insurance and financial risks.

Optimal investment with a constraint on ruin for a fuzzy discrete-time insurance risk model

In this paper, we consider a mean---variance portfolio optimization problem for a fuzzy discrete-time insurance risk model. The model consists of independent, identically distributed net losses considered within successive time periods, and incorporates investment incomes from a two-asset portfolio. More precisely, in the beginning of each period, the surplus is invested in both a risk-free bond with fixed interest, and a risky stock with fuzzy return rate. Our purpose is to determine the proportion invested in the stock that maximizes the insurer's expected wealth, while reducing his risks. Therefore, for this fuzzy model, we formulate mean---variance optimization problems that also include constraints on ruin, and we present a method for determining the resulting optimal proportion to be invested in the risky stock. This method is illustrated in a numerical study in which the fuzzy return rate is considered to be an adaptive fuzzy number that generalizes the well-known trapezoidal fuzzy number.

Asymptotics for a discrete-time risk model with Gamma-like insurance risks

Consider a discrete-time insurance risk model with insurance and financial risks. Within period , the net insurance loss is denoted by and the stochastic discount factor over the same time period is denoted by . Assume that form a sequence of independent and identically distributed real-valued random variables with common distribution ; are another sequence of independent and identically distributed positive random variables with common distribution ; and the two sequences are mutually independent. Under the assumptions that is Gamma-like tailed and has a finite upper endpoint, we derive some precise formulas for the tail probability of the present value of aggregate net losses and the finite-time and infinite-time ruin probabilities. As an extension, a dependent risk model is considered, where each random pair of the net loss and the discount factor follows a bivariate Sarmanov distribution.

The finite-time ruin probability with heavy-tailed and dependent insurance and financial risks

Consider a discrete-time insurance risk model in which the insurer makes both risk-free and risky investments. Assume that the one-period insurance and financial risks form a sequence of independent and identically distributed copies of a random pair (X,Y) with dependent components. When the product XY is heavy tailed, under a mild restriction on the dependence structure of (X,Y), we establish for the finite-time ruin probability an asymptotic formula, which coincides with the long-standing one in the literature. Various important special cases are presented, showing that our work generalizes and unifies some of recent ones.

Uniform asymptotics for the tail probability of weighted sums with heavy tails

This paper studies the tail probability of weighted sums of the form ∑i=1nciXi, where random variables Xi’s are either independent or pairwise quasi-asymptotically independent with heavy tails. Using the idea of uniform long-tailedness, the uniform asymptotic equivalence of the tail probabilities of ∑i=1nciXi, max1≤k≤n∑i=1kciXi and ∑i=1nciXi+ is established, where Xi’s are independent and follow the long-tailed distribution, and ci’s take value in a broad interval. Some further uniform asymptotic results for the weighted sums of Xi’s with dominated varying tails are obtained. An application to the ruin probability in a discrete-time insurance risk model is presented.

The finite-time ruin probability in the presence of Sarmanov dependent financial and insurance risks

Consider a discrete-time insurance risk model. Within period i, i ≥ 1, X i and Y i denote the net insurance loss and the stochastic discount factor of an insurer, respectively. Assume that {(X i , Y i ), i ≥ 1} form a sequence of independent and identically distributed random vectors following a common bivariate Sarmanov distribution. In the presence of heavy-tailed net insurance losses, an asymptotic formula is derived for the finite-time ruin probability.

Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks

Let us consider a discrete-time insurance risk model with insurance and financial risks, where the insurance net loss within period and the stochastic discount factor over the interval follow a certain dependence structure for each fixed . Under the assumption that the distribution of net insurance loss within one time period is consistently varying-tailed, precise estimates for finite and infinite time ruin probabilities are derived. Furthermore, these estimates are uniform on the time horizon.

A Note on the Tail Behavior of Randomly Weighted Sums with Convolution-Equivalently Distributed Random Variables

We investigate the tailed asymptotic behavior of the randomly weighted sums with increments with convolution-equivalent distributions. Our obtained result can be directly applied to a discrete-time insurance risk model with insurance and financial risks and derive the asymptotics for the finite-time probability of the above risk model.

Tail behavior of the product of two dependent random variables with applications to risk theory

Let the random vector (X,Y) follow a bivariate Sarmanov distribution, where X is real-valued and Y is nonnegative. In this paper we investigate the impact of such a dependence structure between X and Y on the tail behavior of their product Z = XY. When X has a regularly varying tail, we establish an asymptotic formula, which extends Breiman’s theorem. Based on the obtained result, we consider a discrete-time insurance risk model with dependent insurance and financial risks, and derive the asymptotic and uniformly asymptotic behavior for the (in)finite-time ruin probabilities.

The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable Xi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Yi from time i to time i − 1. Assume that (Xi, Yi), i ∈ N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functions F and G. When F is subexponential and G fulfills some constraints in order for the product convolution of F and G to be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in which F belongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.

The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

Consider a discrete-time insurance risk model. Within periodi, the net insurance loss is denoted by a real-valued random variableXi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factorYifrom timeito timei− 1. Assume that (Xi,Yi),i∈N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functionsFandG. WhenFis subexponential andGfulfills some constraints in order for the product convolution ofFandGto be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in whichFbelongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.

The impact on ruin probabilities of the association structure among financial risks

We consider a discrete-time insurance risk model, in which the financial risks constitute a stationary process with finite dimensional distributions of Farlie–Gumbel–Morgenstern type. We obtain an exact asymptotic formula for the ruin probability, reflecting the impact of this kind of association structure among the financial risks.

ASYMPTOTIC RUIN PROBABILITIES IN FINITE HORIZON WITH SUBEXPONENTIAL LOSSES AND ASSOCIATED DISCOUNT FACTORS

Consider a discrete-time insurance risk model with risky investments. Under the assumption that the loss distribution belongs to a certain subclass of the subexponential class, Tang and Tsitsiashvili (Stochastic Processes and Their Applications 108(2): 299–325 (2003)) established a precise estimate for the finite time ruin probability. This article extends the result both to the whole subexponential class and to a nonstandard case with associated discount factors.

Ruin Theory in a Discrete Time Risk Model with Interest Income

ABSTRACTIn this paper we consider a discrete time insurance risk model with interest income. Using the recursive calculation method of De Vylder & Goovaerts (1988), recursive equations for the finite time ruin probabilities and the distribution of the time of ruin are derived. Fredholm type integral equations for the ultimate ruin probability, the distribution of the severity of ruin, the joint distribution of surplus before and after ruin, and the probability of absolute ruin are obtained. Numerical results are included.

MARTINGALE METHOD FOR RUIN PROBABILITY IN AN AUTOREGRESSIVE MODEL WITH CONSTANT INTEREST RATE

In this article, we consider a discrete-time insurance risk model. An autoregressive model is used to model both the claim process and the premium process. The probability of ruin is examined in a model with a constant interest rate. Both exponential and nonexponential upper bounds are obtained for the ruin probability of an infinite time horizon.