Showing papers in "Scandinavian Actuarial Journal in 2010"

Extremes on the discounted aggregate claims in a time dependent risk model

Abstract: This paper presents an extension of the classical compound Poisson risk model for which the inter-claim time and the forthcoming claim amount are no longer independent random variables (rv's). Asymptotic tail probabilities for the discounted aggregate claims are presented when the force of interest is constant and the claim amounts are heavy tail distributed rv's. Furthermore, we derive asymptotic finite time ruin probabilities, as well as asymptotic approximations for some common risk measures associated with the discounted aggregate claims. A simulation study is performed in order to validate the results obtained in the free interest risk model.

Stochastic mortality under measure changes

Abstract: We provide a self-contained analysis of a class of continuous-time stochastic mortality models that have gained popularity in the last few years. We describe some of their advantages and limitations, examining whether their features survive equivalent changes of measures. This is important when using the same model for both market-consistent valuation and risk management of life insurance liabilities. We provide a numerical example based on the calibration to the French annuity market of a risk-neutral version of the model proposed by Lee & Carter (1992).

Lapse rate modeling: a rational expectation approach

Abstract: The surrender option embedded in many life insurance products is a clause that allows policyholders to terminate the contract early. Pricing techniques based on the American Contingent Claim (ACC) theory are often used, though the actual policyholders' behavior is far from optimal. Inspired by many prepayment models for mortgage backed securities, this paper builds a Rational Expectation (RE) model describing the policyholders' behavior in lapsing the contract. A market model with stochastic interest rates is considered, and the pricing is carried out through numerical approximation of the corresponding two-space-dimensional parabolic partial differential equation. Extensive numerical experiments show the differences in terms of pricing and interest rate elasticity between the ACC and RE approaches as well as the sensitivity of the contract price with respect to changes in the policyholders' behavior.

Analysis of ruin measures for the classical compound Poisson risk model with dependence

Abstract: In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the claim amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie–Gumbel–Morgenstern copula. In this framework, we derive the integro-differential equation and the Laplace transform (LT) of the Gerber–Shiu discounted penalty function. An explicit expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for claim amounts having an exponential distribution.

Moment generating functions of compound renewal sums with discounted claims

Abstract: Leveille & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered. In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.

Higher-order expansions for compound distributions and ruin probabilities with subexponential claims

Abstract: Let X i (i=1,2, …) be a sequence of subexponential positive independent and identically distributed random variables. In this paper, we offer two alternative approaches to obtain higher-order expansions of the tail of and subsequently for ruin probabilities in renewal risk models with claim sizes X i . In particular, these emphasize the importance of the term for the accuracy of the resulting asymptotic expansion of . Furthermore, we present a more rigorous approach to the often suggested technique of using approximations with shifted arguments. The cases of a Pareto type, Weibull and Lognormal distribution for X i are discussed in more detail and numerical investigations of the increase in accuracy by including higher-order terms in the approximation of ruin probabilities for finite realistic ranges of s are given.

Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes

Abstract: In this paper, we study optimal dividend problem in the classical risk model. Transaction costs and taxes are required when dividends occur. The problem is formulated as a stochastic impulse control problem. By solving the corresponding quasi-variational inequality, we obtain the analytical solutions of the optimal return function and the optimal dividend strategy when claims are exponentially distributed. We also find a formula for the expected time between dividends. The results show that, as the dividend tax rate decreases, it is optimal for the shareholders to receive smaller but more frequent dividend payments.

Gerber–Shiu analysis with a generalized penalty function

Abstract: A generalization of the usual penalty function is proposed, and a defective renewal equation is derived for the Gerber–Shiu discounted penalty function in the classical risk model. This is used to derive the trivariate distribution of the deficit at ruin, the surplus prior to ruin, and the surplus immediately following the second last claim before ruin. The marginal distribution of the last interclaim time before ruin is derived and studied, and its joint distribution with the claim causing ruin is derived.

Strong stability in a two-dimensional classical risk model with independent claims

Abstract: In this paper, we study the strong stability of ruin probabilities in risk models. The question of stability naturally arises in risk theory since the governing parameters in these models can only be estimated with uncertainty. Moreover, in most cases there are not explicit expressions known for the ruin probabilities. Our objective is to present the applicability of the strong stability method to the bivariate classical risk model with independent claims. After clarifying the conditions to approximate the two-dimensional risk model with disturbance parameters by the two-dimensional classical risk model, we obtain the stability inequalities with an exact computation of the constants.

Modeling multiple risks in the presence of double censoring

Abstract: Self-consistent (SC) iterative algorithms will be proposed to non-parametrically estimate the cause-specific cumulative incidence functions in a multiple decrement, doubly censored context. Double censoring is defined to include both left and right censored observations, in addition to exact observations. The algorithms are a generalization of the classical univariate algorithms of Efron and Turnbull. Unlike any previous competing risk models proposed in the literature to date, the proposed algorithms will be fully non-parametric while also explicitly allowing for the possibility of masked modes of failure, whereby failure is known only to occur due to a subset from the set of all possible causes. In short, the method is useful in any actuarial application that encounters censored and/or masked risks. The paper concludes by showing how the method can be applied to employee benefits modeling.

Pensions and genetics: Can longevity genes be reliable risk factors for annuity pricing?

Abstract: We consider a number of gene variants that have been found to affect longevity. Their effects have been modelled using Cox or logistic regressions, whose fitted parameters have simple asymptotic sampling distributions. The expected present value of a life annuity allowing for these genetic risk estimates inherits a sampling distribution, which can be found by simulation. We find that possibly significant uncertainty about annuity premiums may be overlooked if the standard errors of parameters estimated in medical studies are ignored by medical underwriters. Such considerations may play an important part when the acceptability of using a risk factor in underwriting is conditional on proof of its relevance and reliability. This is the current position in respect of genetic information in many countries, most prominently in the UK.

Multiple decrement modeling in the presence of interval censoring and masking

Abstract: A self-consistent algorithm will be proposed to non-parametrically estimate the cause-specific cumulative incidence functions (CIFs) in an interval censored, multiple decrement context. More specifically, the censoring mechanism will be assumed to be a mixture of case 2 interval-censored data with the additional possibility of exact observations. The proposed algorithm is a generalization of the classical univariate algorithms of Efron and Turnbull. However, unlike any previous non-parametric models proposed in the literature to date, the algorithm will explicitly allow for the possibility of any combination of masked modes of failure, where failure is known only to occur due to a subset from the set of all possible causes. A simulation study is also conducted to demonstrate the consistency of the estimators of the CIFs produced by the proposed algorithm, as well as to explore the effect of masking. The paper concludes by applying the method to masked mortality data obtained for Pueblo County, CO, for thr...

Investing for retirement through a with-profits pension scheme: a client's perspective

Abstract: Saving for retirement by with-profits pension contracts is markedly different from traditional savings vehicles in at least two aspects: premia committed to the company are managed according to the preferences of the company – which may not coincide with the long-term interests of the client – and the return on investments is not directly transferred to client's savings but awaits a decision by the company to spend it as bonus. We show that a management's general aversion to (short term) insolvency risk results in investment strategies dynamically scaling investment risk up and down with the current funding status of the company. The resultant dynamics hugely impacts the long-term funding status of the pension company and thereby the investment outcome of with-profits contracts. We show that for a one-period optimizing company there exists a stationary regime only for moderately aggressive investment strategies, and we derive an analytical approximation to the stationary funding ratio distribution when it...

An effective method for constructing bounds for ruin probabilities for the surplus process perturbed by diffusion

Abstract: In this paper, we first study orders, valid up to a certain positive initial surplus, between a pair of ruin probabilities resulting from two individual claim size random variables for corresponding continuous time surplus processes perturbed by diffusion. The results are then applied to obtain a smooth upper (lower) bound for the underlying ruin probability; the upper (lower) bound is constructed from exponentially distributed claims, provided that the mean residual lifetime function of the underlying random variable is non-decreasing (non-increasing). Finally, numerical examples are given to illustrate the constructed upper bounds for ruin probabilities with comparisons to some existing ones.

Some results on the joint distribution prior to and at the time of ruin in the classical model

Abstract: For the classical risk model (i.e. with Poisson arrivals), we study the tail of the joint distribution of the surplus prior to and at ruin. In particular, we obtain some inequalities and monotonicity results for it. Let S be the random variable with distribution function the probability of non-ruin, 1−ψ(u), and the probability the surplus just before ruin exceeds x, given that ruin occurs. We estimate the distance between the residual lifetime of S, Pr(S>u+y∣S>u) and the product , where the tail convolution includes again the random variable S. Finally, based on this distance, we derive a lower bound for the probability of ruin, and we compare this against a bound available in the literature.

On the maximum severity of ruin in the compound Poisson model with a threshold dividend strategy

Abstract: We study the distribution and moments of the maximum severity of ruin in the compound Poisson risk process with a threshold dividend strategy. The distribution can be analyzed through the probabili...

Erratum to: ‘Statistical estimate of the proportional hazard premium of loss’

Abstract: The asymptotic variance of the risk premium estimator, proposed by Necir et al. (2007), is revised, by using the right asymptotic approximation of the uniform empirical quantile process.

Commutation functions under Gompertz–Makeham mortality

Abstract: It is known, but perhaps not well-known, that when the mortality is assumed to be of Gompertz–Makeham-type, the expected remaining life-length and the commutation functions used for calculating the expected values of various types of life insurances can be expressed with an incomplete gamma function with a negative shape parameter. This is not of much use if ones software cannot calculate these values. The aim of this note is to show that one can express the commutation functions using only the exponential function, the (ordinary) gamma function and the gamma distribution function, which are all implemented in common statistical and spreadsheet software. This eliminates the need to evaluate the commutation functions and expected remaining life-length with numerical integration.

Can stocks help mend the asset and liability mismatch

Abstract: Stocks are generally used to provide higher returns in the long run But the dramatic fall in equity prices at the beginning of this century, triggering large underfundings in pension plans, raised

Inequalities for the De Pril approximation to the distribution of the number of policies with claims

Abstract: In the present paper, we give sufficient conditions for an ordering of De Pril approximations of the distribution of the number of claims in an insurance portfolio of independent policies. Possible extensions are discussed, both for the De Pril approximation and the Kornya approximation. A numerical example is given.