A Model for Analyzing the Impact of Selective Lapsation on Mortality

Abstract

This paper presents a model for examining the effect of various relationships between mortality rates and lapse rates on the mortality experience of a cohort of insured lives. The approach is individual rather than the aggregate traditionally used in analyzing selective lapsation. The model assumes that insured lives are healthy at policy issue, but later may move to an impaired state from which the lapse rate is zero. Associated with each insured is an unobservable “risk level” random variable, which reflects the heterogeneity of the insured group. Individual mortality and lapse rates are functions of the risk level. A numerical illustration provides some interesting results obtained by using this model.     

Actuarial validity of the binomial distribution for large numbers of lives with small mortality probabilities

Abstract

In a previous paper (see reference [1]), the binomial distribution was shown to be valid under rather general conditions for the case of a large number of statistically independent lives with possibly unequal mortality probabilities. This note extends these results to some situations where the lives are not necessarily statistically independent. An analysis is presented which indicates that these situations include most actuarial applications involving a large number of lives.     

Select mortality and other durational effects modelled by partially observed Markov Chains

Abstract

In a recent paper Norberg explained select mortality tables for insured lives by a simple Markov model where the lives are classified as active/disabled and insured/not insured, and where no return is possible to previously visited states. The present paper extends the set-up and its results to more complex state spaces and patterns of transition, the key tool being the Kolmogorov backward differential equations.     

Bivariate Survival Models for Coupled Lives

This article presents alternative models for modeling the dependence of the time of deaths of coupled lives and applies these to a data set from a life annuity portfolio. The data indicates a very high positive correlation in the time of deaths between coupled lives. The analysis concludes that a mixed frailty copula model with Gompertz marginals fits the data very well. Another model that fits the data well is based on a generalized Frank copula.     

On the construction of tables for the calculation of certain survivorship benefits

Abstract

1. Benefits depending on two lives often involve the necessity for calculating tables with two arguments. There are, however, cases where this can be avoided, the chief example being the joint-life annuity when the mortality table is graduated by Makeham's formula. The question therefore arises whether there are other cases where a suitable graduation of the mortality table may render similar services.     

Post-war mortality among industrial insured lives in Norway

Abstract

The results of mortality investigations among industrial insured lives with weekly premiums in the Norwegian life insurance company Fram for the period 1931/40 and for the period 1940/46 have earlier been published in this journal.l It is now possible to render the main results of the continued investigation for the post-war years 1946/50.     

The relative maxima of the likelihood function. II

Abstract

In [1] it was shown that under mild conditions, with probability approaching unity as n increases the likelihood function has a relative maximum in a fixed open interval centered at the true value of the parameter. This paper strengthens the result by having the length of the interval approach zero as n increases. An application is given.     

The mortality within some groups of substandard risks. An Investigation of the experience of norwegian life companies

Abstract

One of the subjects discussed at the 8th Scandinavian Life Insurance Congress at Helsingfors in 1929 was the » lighter forms of substandard risks ». On the part of Norway there was on that occasion submitted an investigation of the mortality within certain groups of impaired lives, and I intend here to give an account of the results of this investigation, premising some information on the nature of the material used.     

A dynamic bivariate common shock model with cumulative effect and its actuarial application

Abstract

Standard actuarial theory of multiple life insurance traditionally postulates independence for the remaining lifetimes mainly due to computational convenience rather than realism. In this paper, we propose a general common shock model for modelling dependent coupled lives and apply it to a life insurance model. In the proposed shock model, we consider not only simultaneous deaths of the coupled members due to a single shock (e.g. a critical accident), but also cumulative effect in the mortality rate when they survive shocks. Under the model, we derive a bivariate lifetime distribution and its marginal distributions in closed forms. We study the bivariate ageing property, dependence structure and the dependence orderings of the lifetime distribution. Based on it, we investigate the influence of dependence on the pricings of insurance policies involving multiple lives which are subject to common shocks. Furthermore, we discuss relevant useful stochastic bounds.     

Numerical inversion of characteristic functions

Abstract

To anyone working with characteristic functions, or with Laplace transforms of a non-negative random variable, the three papers by Harald Bohman (1971, 1974, 1975) are invaluable. Numerical integration over an infinite interval is extraordinarily beset with pitfalls (vide Davis & Rabinowitz, 1975) and the publication of actual results achieved in actuarial fields is of great value.     

Aktuareksamen ved Universitetet i Oslo

Abstract

It is a fact that when one is making a decision concerning the probability distribution of a random variable by means of observing this random variable, one is recommended by the statisticians to consider certain functions of the operating characteristic (O.C.) of the decision function as measures of the reliability of the actual decision made. For instance, the confidence coefficient of an interval estimator will as a rule be regarded as a measure of our confidence in the interval.     

On certain inequalities between mean values,and their application to actuarial problems

Abstract

Let us assume that two integrable functions f (t) and ?(t) are defined in an interval a > t > b, that f (t) never increases, and that 0 ≥ ? (t) ≥ 1.     

On the solution of a functional equation concerning mortality formulae

Abstract

1. If (x) and (y) are lives whose remaining lifetimes are stochasticallyindependent, and if the mortality of each of the lives is given by a Makeham expression, then as a well known fact (see e.g. P. F. Hooker & L. H. Longley-Cook, Life and Other Contingencies, Cambridge 1957, vol. II, pp. 137&138) the evaluation of joint-life endowments and joint-life annuities on the lives (x) and (y) may be performed by substituting a single life (u) for (x) and (y) and altering the force of interest, provided that and with the same value of the parameter c( > 1).     

A Re-Examination of the Joint Mortality Functions

Abstract

Mortality analysis involving multiple lives is easily one of the more complicated aspects in the theory of life contingencies. In this paper, we re-investigate joint mortality functions and in particular, we examine an assertion that relates the joint-life and last-survivor random variables. This common assertion states that the sum of the lifetimes of the joint-life and the last-survivor statuses is equal to the sum of the lifetimes of the single statuses. However, we show that this assertion is not precisely correct. We therefore offer a modification to the statuses definitions so that this common assertion holds.     

Some thoughts on (re) insurance loadings under a ruin criterion

Abstract

Suppose a (re)insurer has free reserves of amount U at his disposal and a portfolio characterised by the distribution function F_x (z; µ σ²). ^X is a stochastic variable describing the accumulated loss during a certain time interval; µ, and σ²) = V are the expected value and the variance of X respectively.     

A practical solution to the problem of ultimate ruin probability

Abstract

It was the Swiss actuary Chr. Moser who, in lectures at Bern University at the turn of the century, gave the name “self-renewing aggregate” to what Vajda (1947) has called the “unstationary community” of lives, namely where deaths at any epoch are immediately replaced by an equivalent number of births. It was Moser too (1926) who coined the expression “steady state” for the stationary community in which the age distribution at any time follows the life table (King, 1887). With such a distinguished actuarial history, excellently summarized by Saxer (1958, Ch. IV), it behoves every actuary to know at least the definitions and modus operandi of today's so-called renewal (point), or recurrent event, processes.     

Markovian Approaches to Joint-Life Mortality

Abstract

Many insurance products provide benefits that are contingent on the combined survival status of two lives. To value such benefits accurately, we require a statistical model for the impact of the survivorship of one life on another. In this paper we first set up two models, one Markov and one semi-Markov, to model the dependence between the lifetimes of a husband and wife. From the models we can measure the extent of three types of dependence: (1) the instantaneous dependence due to a catastrophic event that affect both lives, (2) the short-term impact of spousal death, and (3) the long-term association between lifetimes. Then we apply the models to a set of jointlife and last-survivor annuity data from a large Canadian insurance company. Given the fitted models, we study the impact of dependence on annuity values and examine the potential inaccuracy in pricing if we assume lifetimes are independent. Finally, we compare our Markovian models with two copula models considered in previous research on modeling joint-life mortality.     

Mortality investigation among persons whose applications for insurance have heen rejected

Abstract

Extract

Life insurance companies are interested III investigations of the mortality among that group of applicants whose applications for insurance, for health reasons, have not been accepted. The compa. nies thereby obtain material enabling them to decide whether their rejection policy during a period has been too strict or not, and on which points, if any, the selection of risks can be liberalized.     

Developing Mortality Improvement Formulas

Abstract

Longevity improvements have contributed to widespread underfunding of pension plans and losses in insured annuity portfolios. Insurers might reasonably expect some upside from the effect of lower mortality on their life business. Although mortality improvement scales, such as the Society of Actuaries Scale AA, are widely employed in pension and annuity valuation, the derivation of these scales appears heuristic, leading to problems in deriving meaningful measures of uncertainty. We explore the evidence on mortality trends for the Canadian life insurance companies, data, using stochastic models. We use the more credible population data to benchmark the insured lives data. Finally, we derive a practical, model-based formula for actuaries to incorporate mortality improvement and the associated uncertainty into their calculations.     

Representation of concave distortions and applications

ABSTRACT

A family of concave distortion functions is a set of concave and increasing functions, mapping the unity interval onto itself. Distortion functions play an important role defining coherent risk measures. We prove that any family of distortion functions which fulfils a certain translation equation, can be represented by a distribution function. An application can be found in actuarial science: moment-based premium principles are easy to understand but in general are not monotone and cannot be used to compare the riskiness of different insurance contracts with each other. Our representation theorem makes it possible to compare two insurance risks with each other consistent with a moment-based premium principle by defining an appropriate coherent risk measure.