Combination Of Exponential Distributions Research Articles

Valuation of Cliquet-Style Guarantees with Death Benefits in Jump Diffusion Models

This paper aims to value the cliquet-style equity-linked insurance product with death benefits. Whether the insured dies before the contract maturity or not, a benefit payment to the beneficiary is due. The premium is invested in a financial asset, whose dynamics are assumed to follow an exponential jump diffusion. In addition, the remaining lifetime of an insured is modelled by an independent random variable whose distribution can be approximated by a linear combination of exponential distributions. We found that the valuation problem reduced to calculating certain discounted expectations. The Laplace inverse transform and techniques from existing literature were implemented to obtain analytical valuation formulae.

Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion

Recently, the valuation of variable annuity products has become a hot topic in actuarial science. In this paper, we use the Fourier cosine series expansion (COS) method to value the guaranteed minimum death benefit (GMDB) products. We first express the value of GMDB by the discounted density function approach, then we use the COS method to approximate the valuation Equations. When the distribution of the time-until-death random variable is approximated by a combination of exponential distributions and the price of the fund is modeled by an exponential Lévy process, explicit equations for the cosine coefficients are given. Some numerical experiments are also made to illustrate the efficiency of our method.

Valuing equity-linked death benefits with a threshold expense strategy

We investigate equity-linked investment products with a threshold expense strategy, under which an insurance company will collect expenses continuously from the policyholder’s account only when the account value is lower than a pre-specified level. The logarithmic value of the policyholder’s account, before deducting any fees, is described by a jump diffusion process which is independent of the time-to-death random variable. The distribution of the time-to-death random variable is approximated by a combination of exponential distributions, which are dense in the space of density functions on [0,∞). We characterize the Laplace transform of the distribution of a general refracted jump diffusion process through some integro-differential equations. Besides, the distribution of a refracted double exponential jump diffusion process at an independent exponential random variable is derived, from which closed-form formulas to evaluate the total expenses and the fair fee rates are obtained. Finally, we illustrate our results by some numerical examples.

The time of deducting fees for variable annuities under the state-dependent fee structure

We investigate the total time of deducting fees for variable annuities with state-dependent fee. This fee charging method is studied recently by Bernard et al. (2014) and Delong (2014) in which the fees deducted from the policyholder’s account depend on the account value. However, both of them have not considered the problem of analyzing probabilistic properties of the total time of deducting fees. We approximate the maturity of a general variable annuity contract by combinations of exponential distributions which are (weakly) dense in the space that is composed of all probability distributions on the positive axis. Working under general jump diffusion process, we derive analytic formulas for the expectation of the time of deducting fees as well as its Laplace transform.

Valuing equity-linked death benefits in jump diffusion models

The paper is motivated by the valuation problem of guaranteed minimum death benefits in various equity-linked products. At the time of death, a benefit payment is due. It may depend not only on the price of a stock or stock fund at that time, but also on prior prices. The problem is to calculate the expected discounted value of the benefit payment. Because the distribution of the time of death can be approximated by a combination of exponential distributions, it suffices to solve the problem for an exponentially distributed time of death. The stock price process is assumed to be the exponential of a Brownian motion plus an independent compound Poisson process whose upward and downward jumps are modeled by combinations (or mixtures) of exponential distributions. Results for exponential stopping of a Lévy process are used to derive a series of closed-form formulas for call, put, lookback, and barrier options, dynamic fund protection, and dynamic withdrawal benefit with guarantee. We also discuss how barrier options can be used to model lapses and surrenders.

Valuing equity-linked death benefits and other contingent options: A discounted density approach

Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate the calculation of the expected discounted value of a payment at the time of death. The payment depends on the price of a stock at that time and possibly also on the history of the stock price. If the payment turns out to be the payoff of an option, we call the contract for the payment a (life) contingent option. Because each time-until-death distribution can be approximated by a combination of exponential distributions, the analysis is made for the case where the time until death is exponentially distributed, i.e., under the assumption of a constant force of mortality. The time-until-death random variable is assumed to be independent of the stock price process which is a geometric Brownian motion. Our key tool is a discounted joint density function. A substantial series of closed-form formulas is obtained, for the contingent call and put options, for lookback options, for barrier options, for dynamic fund protection, and for dynamic withdrawal benefits. In a section on several stocks, the method of Esscher transforms proves to be useful for finding among others an explicit result for valuing contingent Margrabe options or exchange options. For the case where the contracts have a finite expiry date, closed-form formulas are found for the contingent call and put options. From these, results for De Moivre’s law are obtained as limits. We also discuss equity-linked death benefit reserves and investment strategies for maintaining such reserves. The elasticity of the reserve with respect to the stock price plays an important role. Whereas in the most important applications the stopping time is the time of death, it could be different in other applications, for example, the time of the next catastrophe.

Explicit form of finite-time severity of ruin for phase-distributed claim sizes

We propose an explicit form of severity of ruin and its moments for phase-type claim amount distribution. The distribution of the reserve prior to ruin and its moments are also derived. Since the combinations of exponential distributions and their translations are a special selected phase-type distribution, our results generalize those of Dufresne and Gerber (1988). Some numerical examples are presented to illustrate the application of our method.

Component lifetime models based on weibull mixtures and competing risks

AbstractThis paper gives a review of the statistical modelling of lifetimes of electronic components. The models considered are all based on Weibull (and exponential) distributions combined as either finite mixtures, competing risks or both. A convenient notation is introduced for this general class of models, which includes the traditional monotone hazard rate models, the bathtub model, as well as the more sophisticated roller coaster models. The main advantage of this concept is that a model may be determined by relatively few parameters, and that furthermore each parameter may have a physical interpretation in terms of initial flaws or defects as well as long‐term wear out or deterioration. The properties of the distributions are investigated in examples with different scale and shape parameters, and for different types of combinations. Also, the models are compared with another class of models based on combinations of exponential distributions, known as the phase‐type models. Finally a brief discussion of model selection and estimation methods is included.

The surpluses immediately before and at ruin, and the amount of the claim causing ruin

In the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f( u; x, y) their joint density ( u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place). For an arbitrary claim amount distribution we find that f(0; x, y) = ap( x + y), where p( z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f( u; x, y) can be calculated explicitly, if the claim amount distribution is exponential or, more generally, a combination of exponential distributions. We are also interested in X + Y, the amount of the claim that causes ruin. Its density h( u; z) can be obtained from f( u; x, y). One finds, for example, that h(0; z) = azp( z).

On the Probability and Severity of Ruin

AbstractIn the usual model of the collective risk theory, we are interested in the severity of ruin, as well as its probability. As a quantitative measure, we propose G(u, y), the probability that for given initial surplus u ruin will occur and that the deficit at the time of ruin will be less than y, and the corresponding density g(u, y). First a general answer in terms of the transform is obtained. Then, assuming that the claim amount distribution is a combination of exponential distributions, we determine g; here the roots of the equation that defines the adjustment coefficient play a central role. An explicit answer is also given in the case in which all claims are of constant size.