On the probability function in the collective theory of risk

Abstract

Upon reading Dr. LUNDBERG'S paper ?Über die Wahrscheinlichkeitsfunktion einer Risikenmaase?¹ and trying to penetrate it along my own lines of thought, I found another way of deducing some of his formulas, giving the results in a form that directly invites a fairly simple approximation of the probability function. Though time has not permitted my going deeper into the problem, I propose here to give a brief account of the method.     

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.     

On certain inequalities by Steffensen

Abstract

Extract

In this note we are concerned with the inequalities published by Steffensen in this Journal under the title: ?On a generalization of certain inequalities by Tchebycheff and Jensen?. I will show how these inequalities are represented in one form by the integrals of Stieltjes. These integrals have the advantage of containing both sums and the integrals of Riemann. The proof of this generalized theorem is also more simple, which shows that this theorem appears only in this generalization in its most natural form.     

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.     

Correlation once more

Abstract

§ 1. Correlation Generally.

In my thesis for Doctorship of 1919, ¹ This Journal, 1919, p. 1. I have made some critical remarks on the theory of correlation, trying especially to exhibit the rather unaccomplished state of this theory, and the danger of releasing its formulae for general use. In a supplementary note, ² Ibidem, p. 204. I have pushed my criticism a little further. Since that time, more than ten years have passed away, but I find my point of view still sustainable. One reproach, however, I have never been able to reject, viz, of having been exclusively negative. In fact, I find this position rather natural, from a philosophical standpoint. The information value of correlation calculations is indeed, as a rule, very small. And this seems the more regrettable, as the importance of the Στχαστι?? Τ?χυη, even for the most difficult questions of knowledge, ought to be very great. In a paper of 1924, “Quelques questions concernant les principes de la théorie des probabilités” ³ Ibidem, 1924, p. 107. I have tried to explain, how I imagine the development of the theory of correlation in order to be more apt to set about such philosophical questions.     

On the relations between differences and differential coefficients

Abstract

In a recent paper¹ I have expressed a doubt as to the method followed by Markoff,² in deriving the remainder-terms for the developments of ?⁽ⁿ⁾ (x) in terms of Δ ⁿ?(x) , Δ ⁿ⁺¹ ?(x), ... and of Δ ⁿ?(x) in terms of ?⁽ⁿ⁾ (x), ?⁽ⁿ⁺¹⁾ (x), ‥ .I have since then realized that I have done Markoff something less tha,n justice, and that his line of argument, after filling up a slight gap in one of his proofs, is not only sound but easily accessible to certain generalizations which I proceed to give.     

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.     

On Charlier's generalized frequency-function

Abstract

It is well known, that Charlier has suggested to develop a frequency-function f(x) in a series of the form where ?(x) stands for a particular, given frequency-function, while the symbol ? denotes the ascending difference, that is ??(x)=?(x)-?(x-1). In the form proposed by Charlier this method is open to objections of which he is partly himself aware; the chief objecti.on being, that no account is taken of questions of convergence. It seems, therefore, of interest to examine what becomes of the method, if it is not carried beyond legitimate bounds. In doing so, I shall try to simplify the determination of the constants, a problem which has been attacked by Charlier ¹ C. V. L. Charlier: Über die Darstellung willkürlicher Funktionen (Arkiv för matematik, astronomi och fysik, Bd 2, N:r 20). himself, and by N. R. Jørgensen ² N. R. Jørgensen: Note sur la fonction de répartition de type B de M. Charlier (ibid., Bd 10, N:r 15); Undersøgelser over Frequensflader og Korrelation (Copenhagen, 1916), S. 5–15. in a special case. For this purpose I avail myself of a class of symmetrical functions of the observations for which I have proposed the name of “factorial moments” and the systematical use of which I recommended in my paper “Factorial Moments and Discontinuous Frequency-Functions”. ³ This Journal, 1923, p. 73. See also the author's book “Matematisk Iagttagelseslære” (Copenhagen, 1923), I, § 2. I shall assume, that the reader is familiar with the notation employed in that paper which differs in some respects from the usual notation of moments.     

River stories: Genealogies of a threatened inland river system

Most people now living in Australia's “bread basket”, the much-degraded Murray Darling Basin, are like my family, descendants of convicts or free settlers who came to the inland in the 19th or early 20th centuries. Our legacy includes the dispossession of indigenous peoples, species extinction and the ongoing degradation of the ecological communities which now sustain us. My own family's river stories which “begin” with a pair of impoverished Gaels who migrated with their offspring from the Scottish Highlands, can be considered paradigmatic. I re-narrate it in this essay in response to philosopher Alasdair MacIntyre's challenge—I can only answer the question ‘What am I to do?’ if I can answer the prior question ‘Of what story or stories do I find myself a part?’Some of these family stories I find myself part of, especially those that have been enacted within the catchment of the now-threatened Lachlan River, are very discomforting, but where do they “truly” begin? In seeking to understand my relationship with the river and its catchment, and with the indigenous peoples “my mob” displaced, I explore several possible “beginnings” and ask a further question: what stories do I want to be part of as co-author, co-narrator and protagonist. I then offer my own yet-to-be enacted “truth and reconciliation” stories about the future of the inland plains I love.     

Notes on some present day Questions regarding Mortality

Abstract

In the absence of any definite development of actuarial science in Great Britain which would be a suitable subject for this paper I considered that the most useful course would be to discuss some of the problems which are at present exercising my own mind, and no doubt the minds of many of my colleagues. Probably, many of these questions are akin to others which arise in other countries, so that their discussion at a meeting such as this may prove of mutual interest, while even if they are peculiar to ourselves, a disclosure of our doubts and difficulties will indicate the directions in which future developments may be expected. I must, however, emphasize that any views expressed are purely personal, and do not in any way pretend to represent a consensus of British actuarial opinion. Throughout this paper only Ordinary (as opposed to Industrial) business in considered.     

An Italian perspective on the development of financial mathematics from 1992 to 2008

This paper is intended to be a survey of the development of financial mathematics as seen through the events that I organised, and partly co-organised, between 1992 and 2008. These events all took place in Italy between 1992 and 2003, while in 2008 I was involved in the organisation of an entire special semester in Linz (Austria); this semester is included here because it marks quite well the state-of-the-art of the period just before the so-called big financial crisis that lasted from, roughly, 2008 to 2012. Even if the survey may be affected by my personal views, it can still be seen as reflecting the actual global development since what I am going to describe here concerns major occurrences. For completeness, I also mention, although only briefly, some events that took place in Italy during the given period, but where I was not personally involved.

     

Integration of Financial Services

Abstract

Integration financial services is a term with many meanings. Definitions might include:

• ownership of different segments of financial services,

• the distribution of products of one segment by another, and

• a consumer-centric strategy driven to satisfy the needs of the customer regardless of organizational structures.

For the purposes of my paper, I will define it as the erasing of boundaries leading to consumers receiving value propositions that are not restricted by traditional industry segment barriers.     

Optimal Annuitization Policies

Abstract

At, or about, the age of retirement, most individuals must decide what additional fraction of their marketable wealth, if any, should be annuitized. Annuitization means purchasing a nonrefundable life annuity from an insurance company, which then guarantees a lifelong consumption stream that cannot be outlived. The decision of whether or not to annuitize additional liquid assets is a difficult one, since it is clearly irreversible and can prove costly in hindsight. Obviously, for a large group of people, the bulk of financial wealth is forcefully annuitized, for example, company pensions and social security. For others, especially as it pertains to personal pension plans, such as 401(k), 403(b), and IRA plans as well as variable annuity contracts, there is much discretion in the matter.

The purpose of this paper is to focus on the question of when and if to annuitize. Specifically, my objective is to provide practical advice aimed at individual retirees and their advisors. My main conclusions are as follows:

? Annuitization of assets provides unique and valuable longevity insurance and should be actively encouraged at higher ages. Standard microeconomic utility-based arguments indicate that consumers would be willing to pay a substantial “loading” in order to gain access to a life annuity.

? The large adverse selection costs associated with life annuities, which range from 10% to 20%, might serve as a strong deterrent to full annuitization.

? Retirees with a (strong) bequest motive might be inclined to self-annuitize during the early stages of retirement. Indeed, it appears that most individuals—faced with expensive annuity products—can effectively “beat” the rate of return from a fixed immediate annuity until age 75?80. I call this strategy consume term and invest the difference.

? Variable immediate annuities (VIAs) combine equity market participation together with longevity insurance. This financial product is currently underutilized (and not available in certain jurisdictions) and can only grow in popularity.     

Some results on the compound Markov binomial model

Abstract

Some years ago, in the course of an analysis of upper and lower limits for incomplete moments of statistical distributions I established an elementary summation formula¹ which proved rather useful for the purpose I had in view. Subsequently the formula was generalized by professor Steffensen, who showed² that the formula in question could be looked upon as giving the first term of an expansion in a certain type of series. Professor Steffensen established recurrence formulae for the coefficients of the series and computed the second, third and fourth term and the corresponding remainders¹, but did not arrive at a general, explicite expression for the coefficient of the n-th term and the corresponding remainder. A year later I found these expressions accidentally while I was working on some other problem. I also discovered the real nature of the procedure in question which proved to be a certain kind of least square fitted polynomial approximation. I did not, however, at the time publish the result. Taking the question up again later I found that the whole problem could be considerably generalized. The type of generalization in question is analogous to the generalization from polynomials to arbitrary functions.     

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.     

MARX AND ACCOUNTING

Does Marx's political economy provide a theoretical foundation for understanding accounting in modern capitalism? The commentators dispute my Marxism, and contest whether my accounting is more objective than the FASB's. I show they have misunderstood Marx and the purpose of my accounting. Underlying their views is a failure to take seriously Marx's social relations of production and traditional accountability. I conclude the commentators have not challenged my claim that Marx's analysis of the circuits of industrial capital could provide a general theory of accounting.     

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.     

Measuring, explaining, and controlling tax evasion: lessons from theory, experiments, and field studies

In this paper, I assess what we have learned about tax evasion since Michael Allingham and Agnar Sandmo launched the modern analysis of tax evasion in 1972. I focus on three specific questions and the answers to these questions that have emerged over the years. First, how do we measure the extent of evasion? Second, how can we explain these patterns of behavior? Third, how can we use these insights to control evasion? In the process, I illustrate my own answers to these questions by highlighting various specific examples of research. My main conclusion is that we have learned many things but that we also still have many gaps in our understanding of how to measure, explain, and control tax evasion. I also give some suggestions—and some predictions—about where promising avenues of future research may lie.     

Critical thoughts on actuarial science

Abstract

In making some remarks before the International Actuarial Congress of 1930 in Stockholm on the subject of bonus distribution I was speaking of a tendency within Actuarial Science, which I took the liberty of calling ?critical?. I am of the opinion, that a tendency entitled to this name really exists; and in Northern countries this tendency seems to me to be chiefly represented by FILIP LUNDBERG, H. CRAM:h and F. NEVANLINNA. I do not, however, urge that this tendency should be considered as entirely new; it can very well be argued that it is twenty or thirty years old. Impulses to this critical tendency may be traced still further back. And I do not intend to claim the priority to this trend for Northern Countries.     

Note on the collective theory of risk

Abstract

What follows has grown out of a discussion with Carl Philipson following a lecture [1] on the collective theory of risk. Although I give here nothing else but a refined interpretation of Paul Lévy's form (see, e.g., [2], p. 322) of identically distributed random variables the result still seems of interest for all those working in the field of collective risk theory. I thank Carl Philipson for stimulating my interest in this matter.