Constant Interest Force Research Articles

The Finite-time Ruin Probability for the Jump-Diffusion Model with Constant Interest Force

In this paper, we consider the finite-time ruin probability for the jump-diffusion Poisson process. Under the assumptions that the claimsizes are subexponentially distributed and that the interest force is constant, we obtain an asymptotic formula for the finite-time ruin probability. The results we obtain extends the corresponding results of Kluppelberg and Stadtmuller[8] and Tang[14].

Flow of Dividends under a Constant Force of Interest

This study addresses the issue of maximization of dividends of an insurer whose portfolio is exposed to insurance risk. The insurance risk arises from the classical surplus process commonly known as the Cramer-Lundberg model in the insurance literature. To enhance his financial base, the insurer invests in a risk free asset whose price dynamics are governed by a constant force of interest. We derive a linear Volterra integral equation of the second kind and apply an order four Block-by-block method of Paulsen et al.[1] in conjunction with the Simpson rule to solve the Volterra integral equations for each chosen barrier thus generating corresponding dividend value functions. We have obtained the optimal barrier that maximizes the dividends. In the absence of the financial world, the analytical solution has been used to assess the accuracy of our results.

The finite-time ruin probability of the compound Poisson model with constant interest force

In this paper, we establish a simple asymptotic formula for the finite-time ruin probability of the compound Poisson model with constant interest force and subexponential claims in the case that the initial surplus is large. The formula is consistent with known results for the ultimate ruin probability and, in particular, is uniform for all time horizons when the claim size distribution is regularly varying tailed.

The finite-time ruin probability of the compound Poisson model with constant interest force

In this paper, we establish a simple asymptotic formula for the finite-time ruin probability of the compound Poisson model with constant interest force and subexponential claims in the case that the initial surplus is large. The formula is consistent with known results for the ultimate ruin probability and, in particular, is uniform for all time horizons when the claim size distribution is regularly varying tailed.

Ruin in the perturbed compound Poisson risk process under interest force

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

Ruin in the perturbed compound Poisson risk process under interest force

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

The win-first probability under interest force

In a classical risk model under constant interest force, we study the probability that the surplus of an insurance company reaches an upper barrier before a lower barrier. We define this probability as win-first probability. Borrowing ideas from life-insurance theory, hazard rates of the maximum of the surplus before ruin, regarded as a remaining future lifetime random variable, are studied, and provide an original derivation of the win-first probability. We propose an algorithm to efficiently compute this risk-return indicator and its derivatives in the general case, as well as bounds of these quantities. The efficiency of the proposed algorithm is compared with adaptations of other existing methods, and its interest is illustrated by the computation of the expected amount of dividends paid until ruin in a risk model with a dividend barrier strategy.

On a joint distribution for the risk process with constant interest force

We consider the classical risk process with constant interest force. We derive the explicit expression for the joint distribution of three actuarial diagnostics: the time of ruin, the surplus immediately before ruin and the deficit at ruin.

Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation

Klüppelberg and Stadtmüller (1998, Scand. Actuar. J., no. 1, 49–58) obtained a simple asymptotic formula for the ruin probability of the classical model with constant interest force and regularly varying tailed claims. This short note extends their result to the renewal model. The proof is based on a result of Resnick & Willekens (1991, Comm. Statist. Stochastic Models 7, no. 4, 511–525).

On the distribution of surplus immediately after ruin under interest force and subexponential claims

The compound Poisson risk process with a constant interest force is an interesting stochastic model in risk theory. It provides a basic understanding about how investments will affect the ruin probability and related ruin functions. This paper considers the compound Poisson risk model with a constant interest force for an insurance portfolio and studies the distribution of the surplus immediately after ruin under the model. By using the techniques of Kalashnikov and Konstantinides [Kalashnikov, V., Konstantinides, D., 2000. Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econ. 27, 145–149] and a formula obtained by Yang and Zhang [Yang, H.L., Zhang, L.H., 2001a. On the distribution of surplus immediately after ruin under interest force. Insurance Math. Econ. 29, 247–255], we give asymptotic formulas of the low and upper bounds for the distribution of the surplus immediately after ruin under subexponential claims. To some extent, we can view our work here as the continuation of the recent important work of Kalashnikov and Konstantinides [Kalashnikov, V., Konstantinides, D., 2000. Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econ. 27, 145–149], Yang and Zhang [Yang, H.L., Zhang, L.H., 2001a. On the distribution of surplus immediately after ruin under interest force. Insurance: Mathematics and Economics 29, 247–255] and Konstantinides et al. [Konstantinides, D., Tang, Q.H., Tsitsiashvili, G., 2002. Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Math. Econ. 31, 447–460].

Two-Sided Bounds for Ruin Probability under Constant Interest Force

0. Therefore, the compound Poisson process X(t) = PN(t) k=1 Zk represents the total claim amount accumulated up to time t 0, with X(t) = 0 when N(t) = 0. We write = b and let c be the constant gross premium rate which is not necessarily positive. We assume that there exists a constant interest force > 0 which affects the risk process. Letu > 0 be the initial surplus of the insurance company, then the total

Recursive calculation of finite time ruin probabilities under interest force

In this paper, we consider a classical insurance surplus process affected by a constant interest force. We present numerical algorithms for the calculation of finite time ruin probabilities using a discrete time Markov chain to approximate the risk process. Based on this method, upper and lower bounds are also obtained.

Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails

In this paper we investigate the ruin probability in the classical risk model under a positive constant interest force. We restrict ourselves to the case where the claim size is heavy-tailed, i.e. the equilibrium distribution function (e.d.f.) of the claim size belongs to a wide subclass of the subexponential distributions. Two-sided estimates for the ruin probability are developed by reduction from the classical model without interest force.

On a gamma series expansion for the time-dependent probability of collective ruin

In the framework of the extended classical risk model with constant force of real interest i, we investigate when it is suitable to represent the probability of collective survival U(x,t) of an insurance company with initial capital x and time horizon t as a gamma series. Moreover, we derive exact analytical solutions for exponentially distributed claim sizes and integer values of λ/i, where λ is the risk parameter. As a by-product we observe that numerical procedures for estimating U(x,t) are very accurate.

On the distribution of surplus immediately before ruin under interest force

In this paper, we consider a compound Poisson model with a constant interest force for an insurance portfolio. We investigate the distribution of surplus process immediately before ruin in particular. Equations satisfied by the distributions of surplus immediately before ruin and their Laplace transform have been obtained. Some special cases are also discussed and Lundberg-type bounds are presented.

Approximating the finite-time ruin probability under interest force

We present an algorithm to determine both a lower and an upper bound for the finite-time probability of ruin for a risk process with constant interest force. We split the time horizon into smaller intervals of equal length and consider the probability of ruin in case premium income for a time interval is received at the beginning (resp. end) of that interval, which yields a lower (resp. upper) bound. For both bounds we present a renewal equation which depends on the distribution of the present value of the aggregate claim amount in a time interval. This distribution is determined through a generalization of Panjer’s [ASTIN Bulletin 12 (1981) 22] recursive method.

The Joint Distribution of Surplus Immediately before Ruin and the Deficit at Ruin under Interest Force

In this paper we consider a compound Poisson risk model with a constant interest force. We investigate the joint distribution of the surplus immediately before and after ruin. By adapting the techniques in Sundt and Teugels (1995), we obtain integral equations satisfied by the joint distribution function and a Lundberg-type inequality. In the case of zero initial reserve and the case of exponential claim sizes, we obtain explicit expressions for the joint distribution function.

Ruin under interest force and subexponential claims: a simple treatment

A simple proof of the asymptotic formula for the ruin probability of a risk process with a positive constant interest force [derived earlier by Asmussen (Asmussen, S., 1998. The Annals of Applied Probability 8, 354–374)] is given. The proof is based on a formula obtained by Sundt and Teugels (Sundt, B., Teugels, J.L., 1995. Insurance: Mathematics and Economics 16, 7–22).

Ruin probabilities with compounding assets

We consider a classical surplus process modified by the action of a constant force of interest. We derive recursive algorithms for the calculation of the probability of ruin in finite time. We also discuss the numerical evaluation of the probability of ultimate ruin using methods proposed by De Vylder [De Vylder, F., 1996. Advanced Risk Theory. Editions de l’Université de Bruxelles.] and Sundt and Teugels [Sundt, B., Teugels, J.L., 1995. Insurance: Mathematics and Economics 16, 7–22]. Finally, we consider the problem of recovery from ruin.

The adjustment function in ruin estimates under interest force

We continue our discussion of infinite time ruin probabilities in continuous time in a compound Poisson process with a constant premium rate and a constant interest force. Under appropriate conditions the ruin probability is exponentially bounded. The usual adjustment coefficient is replaced by an adjustment function depending in an intricate way on the initial reserve, the interest force and all ingredients of the compound Poisson process. After deriving general bounds we also give expansions for the case where the interest force is small.