On a Generalization of certain inequalities by Tchebychef and Jensen

Abstract

The inequalities dealt with in this paper were published by me in Danish in ?Matematisk Tidsskrift? B. 1925 p. 49¹. I was not at the time aware, that an inequality of similar form had already been given by JENSEN; but as the conditions under which his theorem is valid differ from mine and are, in fact, included in them, it appears that my investigation is not superfluous. I am reverting to the subject partly in order to clear up the relationship between JENSEN'S and my own results, partly to show, by means of a few examples, how these inequalities may with advantage be applied to actuarial problems.     

Fair valuation of insurance liabilities via mean-variance hedging in a multi-period setting

A general class of fair valuations which are both market-consistent (mark-to-market for any hedgeable part of a claim) and actuarial (mark-to-model for any claim that is independent of financial market evolutions) was introduced in Dhaene et al. [Insurance: Mathematics & Economics, 76, 14–27 (2017)] in a single period framework. In particular, the authors considered mean-variance hedge-based (MVHB) valuations where fair valuations of insurance liabilities are expressed in terms of mean-variance hedges and actuarial valuations. In this paper, we generalize this MVHB approach to a multi-period dynamic investment setting. We show that the classes of fair valuations and MVHB valuations are equivalent in this generalized setting. We derive tractable formulas for the fair valuation of equity-linked contracts and show how the actuarial part of their MVHB valuation decomposes into a diversifiable and a non-diversifiable component.     

Stochastic Models for Continuing Care Retirement Communities

Abstract

Continuing care retirement communities (CCRCs) offer housing and a variety of services, including long-term care. Typically, the cost of this long-term care is wholly or partially covered by entry and/or periodic fees. Thus, CCRCs provide a long-term-care insurance benefit. For this and other reasons, actuarial involvement in the financial management of CCRCs is desirable. To carry out actuarial analyses of CCRCs, appropriate models are required to describe the status of individual residents and the CCRC population.

This paper presents models that assume that, at any time, a resident is in a given “state,” which is determined by the individual’s care requirements. The resident may make “transitions” between states at various times, and randomness is associated with both the transition times and the states entered. Actuarial calculations using such a model are discussed, and numerical illustrations are provided. A simple model is examined first; then generalizations are considered. The model for an individual resident can be embedded in a model for a CCRC population. This is explored with particular attention given to the “high-demand” situation in which potential residents are always waiting to enter the community. With this model, the goal is to analyze the future care requirements of the CCRC population.     

Measuring Terminable Postretirement Obligations

Abstract

New approaches are needed to value benefit plans subject to unilateral changes or termination. The paper focuses on postretirement health benefits, but the thesis may be relevant to any flow not guaranteed by law or accumulating funds.

Retiree health benefits have usually been extended to participating active employees only in concert with a reserved right by the plan sponsor to control the design and, by implication, the cash flow. Over the course of the last fifteen years, this reserved extension of benefits has almost invariably led to reductions in benefits, when compared to the plan of benefits at an earlier period. In most cases, such reductions were anticipated under the circumstances that came to prevail (high health care cost increases), but were not taken into account by most of the projection and discounting methods of the time.

Current actuarial and accounting methods generate present values for terminable retiree health plans that have little credibility as measures of the beneficiary’s asset or the sponsor’s liability. Improvements are needed that will expand the actuarial toolbox and provide solutions to economic and accounting problems.

This paper provides a basis for discussion of assumptions that are appropriate when the plan sponsor can unilaterally and dramatically change future cash flows. The paper discusses how actuaries might best approach measurement situations where further plan reductions, or outright terminations, are to be anticipated. It introduces refinements and briefly discusses how each would fit with the usual actuarial model and how differences might affect behavior. These ideas are related to financial economics and the Bader-Gold paper “Reinventing Pension Actuarial Science.” Before concluding, the wider topic of discount rate selection is briefly addressed.     

Further remarks on Laplace' and Gauss' summation formulas

1. The practical form of the remainder terms of Laplace' and Gauss' summation formulas which I have employed in earlier papers and in my book on Interpolation suffers from the inconvenience that it is inapplicable in the case of infinite limits of summation and integration. The purpose of the present paper is to show, how this inconvenience may be avoided.     

Commercialising the service class and economic restructuring-a response to my critics

The Commercialisation of Accountancy is reviewed in this issue of AOS in two extended essays—one by Hugh Willmott and Prem Sikka (1997) and the other by Yves Dezalay (1997). Whilst agreeing with many of the insightful criticisms of my work, I would like to take this opportunity to defend elements of the text which were criticised. In particular, I would like to expand upon the criticism made by Willmott and Sikka that I should have examined how the economic restructuring of the past two decades has presented some accountants (and indeed other expert labour groups) with opportunities, whilst restricting the life chances of many others. I accept this criticism; however, I would argue that the inter- and intra-group struggles for position are implicit in the work throughout much of the text and that they are explicitly mentioned in chapter six and the conclusion (the chapters which deal with class). Dezalay also refers to this inadequacy and suggests that the work suffers from the binary simplicity of Perkin's thesis which emphasises the public sector-private sector divide (see Perkin, 1989). What follows will attempt to clarify my views in this area and to raise questions and point to future areas of research.     

Financial Economics and Actuarial Practice

Abstract

Starting in the United Kingdom and continuing through the U.S. and Canadian actuarial professions, proponents of financial economics have been forcefully promoting a review of traditional actuarial practices and training. In particular, the financial theories first proposed by Modigliani and Miller and subsequently developed by others have been used to highlight serious weaknesses in typical actuarial thinking. In summary, it is claimed that much actuarial advice wrongly specifies value, that guidelines and standards need radical revision, and that traditional actuarial intuition suffers in comparison to newer modes of thought adopted by other professions.

This paper examines concepts from both financial economics and actuarial science as applied to defined benefit schemes using a simple discounted cash-flow framework as a reference point. The general finding is that many standard modes of actuarial thought are, in fact, indefensible when examined with the tools and techniques of financial economics. The call for revision of actuarial training and practices is credible and necessary.

However, the paper also touches upon areas where a heavy-handed application of finance theory could be misguided due to limitations in the simple financial economic models presented. It concludes that financial economics should be carefully integrated into actuarial thought rather than appended to existing actuarial theory or inserted as a wholesale replacement.     

Contracting out of a state pension scheme

Abstract

The title of this paper is not common in actuarial literature, but the subject is one which has become familiar to every Briton in the last two years and which is essentially actuarial. The phrase "contracting out" has become English shorthand for an arrangement by which a private pension scheme may decide that it will not join a State pension scheme, usually because it already provides equal or better benefits. If the private scheme "contracts out" it will be excused the payment of contributions and will, of course, forgo the benefits.     

Remarks on risk theory

§ 1. The Scheme U(n).

Following remarks are the result of some deliberations which I have made in order to find a simple starting point for rough estimates of the risk reserves necessary under given conditions. They do not appear with any pretensions. The idea, upon which my thoughts were based, is far from original: I have hoped to be able to simplify the theory by simplifying in a high degree the hypotheses and by confining myself to small numbers and arithmetical methods.     

Ladislaus von Bortkiewicz

Summary

In the theory of Poisson processes and compound Poisson processes with time-dependent change variables some results are obtained by use of a transformed change variable, independent of time. The theorem presented below shows that this method can be used in a wide class of problems, many of which are of actuarial interest.     

Demand functions and the integrability condition

Abstract

This is not written to prolong the discussion of integrability, a topic on which at long last there seems to be universal agreement on all essential points. I shall only give a brief answer to a question that has recently been raised by Dr. P. A. Samuelson 1 about a theorem of mine published in this journal². Stating that he is not quite clear about my theorem, Dr. Samuelson says that this might be due to an ambiguity in translation from the Swedish. My answer is that I cannot seennything wrong in the theorem. Some c'omments are added to clarify my position.     

Metaphors,semiotics and futures studies

Metaphor belongs to key concepts of semiotics. I have made my career in the field of semiotics and I appreciate the possibility to tell to the scientific community of futurists how a semiotician sees the various functions of metaphors and their connections to the future. The edited volume CLA 2.0 (Inayatullah & Milojevic, 2015) shows that in addition to metaphors, many futures researchers have found the general language-based approach of semiotics. The paper deals with three issues: first the theory of metaphors as such, much discussed in the semiotic literature; then what semiotics says about the future; and finally, what kind of semiotics we are considering here. I would propose to scrutinise the problem of metaphors and future in light of my own new theory which I call ‘existential semiotics’.     

Actuarial Issues in the Novels of Jane Austen

Abstract

The novels of Jane Austen have enjoyed a resurgence of popularity recently, and many new readers have come to appreciate the relevance of her stories to modern times. This relevance should be particularly evident to actuaries, however, because the novels deal quite explicitly with the issues of wealth, inheritance, mortality, and life expectancy that confronted the nonworking classes of the early nineteenth century.

This paper examines the six novels of Jane Austen from an actuarial perspective. It provides historical background on inheritances, clerical livings, and mortality, and it analyzes the way in which these issues are central to Austen’s novels. It uses a contemporary mortality table to assess the accuracy with which Austen’s characters estimate life expectancies and annuity calculations. It presents a close study of Sense and Sensibility, a novel in which a number of actuarial issues are central to the plot and are presented in great detail. Finally, it suggests that Austen’s own background and family life meant that actuarial issues were important in her life and therefore reflected in her novels.

This paper offers a new argument for the relevance of great literature, and it offers actuaries a new perspective from which to explore and understand the history of their profession.     

Remarks on the foundations of actuarial science

Abstract

In a paper entjtled ?Critical Thoughts on Actuarial Science? read before the Swedish Actuarial Society and published in this part of the Journal K. ENGLUND has expressed some views which gave rise to objections also from my part in the verbal discussion at the meeting of the Society. In the following lines I have tried to develop my views about the most important questions touched upon in his paper.     

Overview

Abstract

From the inception of Social Security in 1935 to the present time, a variety of measures for evaluating the actuarial status of the program have been employed. In addition, how the results of these measures have been displayed and interpreted has also evolved. These results have had great influence on policymakers’ and the general public’s perceptions of the financial condition of the program. This paper is intended to be an update of my earlier paper on the same subject, that is, a historical review of the development of these methods and their interpretation. The paper closes with suggestions as to future changes in the measurement methods, in the light of possible future changes in the program itself.     

Statistical Independence and Fractional Age Assumptions

Abstract

This paper considers in some detail the issue of statistical independence of the curtate future lifetime and the fractional part of the future lifetime of a general status.

Statistical independence is often employed in actuarial contexts, primarily because it leads to simple relationships between quantities of interest and statistical information that is of a discrete nature, such as a life table. The uniform distribution of deaths (UDD) assumption is the most commonly used because of its simplicity and intuitive appeal, but it can be somewhat restrictive. For example, all deaths or withdrawals may be assumed to be at a particular point in the year such as the middle; assumptions of this type are often made in a multiple decrement context. This paper attempts to unify these assumptions and extend their applicability in an actuarial context.

The conditions for independence need to be stated carefully, and the last-survivor status is cited as an example in which failure to do so can lead to erroneous conclusions.

The fractional independence (Fl) assumption is defined, and it is demonstrated that many of the formulas for life table functions that hold under the more restrictive UDD assumption are extended easily to the general Fl case. The simple relationship under UDD between insurances payable on other than an annual mode and those payable at the end of the year of death is extended to the Fl case as well. These results are then used to obtain results for annuities and reserves, again generalizing UDD relationships. It is then demonstrated that many contingent probabilities in the multiple life context are exactly the same under the Fl assumption as under the more restrictive UDD assumption. Finally, a very general result that holds in the multiple decrement context is shown to hold under the Fl assumption.     

Basic concepts in life assurance mathematics

Abstract

1. Introduction.

In this paper the basic concepts of the life insurance mathematics will be discussed. Due to the fact that the importance of the probability calculus as a hasis for the actuarial science has repeatedly been disclaimed in recent literature (See e.g. Ernst ZWinggi (1]), the present author feels that there is a justification for reconsidering the fundamental ideas of the actuarial science.     

Modeling with Weibull-Pareto Models

Abstract

In this paper we develop several composite Weibull-Pareto models and suggest their use to model loss payments and other forms of actuarial data. These models all comprise a Weibull distribution up to a threshold point, and some form of Pareto distribution thereafter. They are similar in spirit to some composite lognormal-Pareto models that have previously been considered in the literature. All of these models are applied, and their performance compared, in the context of a real-world fire insurance data set.     

Sickness experience of Framtiden

Abstract

Introduction.

Livförsäkringsbolaget Framtiden, ömsesidigt (the Life Insurance Co. “The Future”, Mutual), to the policyholders of which I have the honour of devoting my actuarial work, is a relatively new enterprise. It was founded in 1911. Its main work is industrial life insurance with monthly premiums. All industrial policies include the waiver of premiums on disablement by sickness or accident for at least four weeks. Disablement lasting more than four weeks involves the waiver of premiums from the beginning of the disablement. For industrial policies issued in late years, the payment of premiums is limited to 65 years of age. Besides ordinary endowment (or whole life) policies, many childrens' insurances are written, including some or other insurance of the bread-winner, at all events the waiver of premiums till the 20th birthday of the child, if the supporter should die before that time, and for the periods during this time when the supporter is disabled by sickness or accident for at least four weeks. It has been tried to propagate more protective forms of children's policies, but it has partly failed, owing to lack of interest from agents and public. Nevertheless, it has been possible to exclude pure childrens' tariffs, and so in almost every policy issued, there is a moment of sickness insurance. In late years, this holds true also for the ordinary branch. The mass of experience collected is thus by no means unimportant, despite the smallness of the Company, as compared with foreign Companies. The experience is, moreover, collected during a relatively short period of time, giving a rather homogeneous material.     

Credibility for Severity Revisited

Abstract

It is basic actuarial knowledge that the pure premium of an insurance contract can be written as the product of the expected claim number and the expected claim amount. Actuaries use credibility theory to incorporate the contract’s individual experience into this calculation in a statistically optimal way. For many years, however, the use of credibility was limited to the frequency component. Starting with the paper by Hewitt (1971), there have been various suggestions as to how credibility theory also can be applied to the severity component of the pure premium. The latest such suggestion, Frees (2003), revived the interest in the problem.

In this paper, we review four different formulas incorporating frequency and severity into credibility calculations. We then compare by simulation which one is most accurate at predicting a contract’s next-year outcome. It is found that the classical formula of Bühlmann (1967) is as good as the other ones in many cases. Alternatives, however, may offer easier analysis of the separate effects of frequency and severity on the premium.

We also show that all the formulas reviewed in this paper stem from the same minimization problem, and we present a general, integrated, solution. At the same time, we complete Gerber (1972) by providing a proof to the main result of this paper and by stating required additional assumptions.