Über die Begriffe Schiefheit und Exzess in der mathematischen Statistik

Abstract

Danish life assurance for some time has been characterized by a very high yield of interest as compared with a moderate official valuation rate. This has resulted in bonus additions up to several times the face amounts assured by the policies. In order to introduce new types of assurance coverage to cope with these conditions, it will prove useful to look at the valuation of life policies from a slightly new angle.     

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.     

Overview of Reserving Practices for Substandard Life Policies

Abstract

This paper reviews the various industry practices, both past and current, for determining statutory reserves for substandard life insurance policies This review begins with single-life policies but also considers the application of single-life practice to joint-life policies The increased popularity of joint-life policies has taken place without much discussion in the technical journals of how to handle such issues as reserves on substandard business. This paper is intended to provoke such a discussion and to provide a framework for it.     

Generalized Frasier Claim Rates Under Survivorship Life Insurance Policies

Abstract

This paper proposes two modifications to the well-known Frasier formula, often used in the pricing, design, and valuation of survivorship life insurance policies: (1) allowing lapse rates to change after the first death and (2) reflecting simultaneous exposure to the same hazards, such as infectious diseases and common accidents, and possibly higher mortality among survivors. The purpose is to improve the pricing and valuation of survivorship life insurance. The paper will be of interest to actuaries doing pricing, GAAP valuation, self-support certifications, and to illustration actuaries. The results are important to reinsurers and direct writers. The paper includes numerical examples and compares the claim rates with and without the suggested modifications. The modified survivorship claim rates are considerably higher than those developed using pure Frasier, emphasizing the importance of learning to use these or similar methods.     

Withdrawal among Norwegian Life Assured 1910–1935

Introduction.

In the series of statistical experiences of the Norwegian life assurance companies report No. IV was issued some time ago. ¹ “Frivillig avgang blant norske livsforsikrede og dödeligheten blant forsikrede med fripolise 1910–1935”. Oslo 1943. The report, which was prepared at the Statistical Bureau of the Norwegian Life Assurance Companies, comprises 2 subjects: 1. Withdrawal among Norwegian life assured

2. Mortality among assured with paid-up policies (free policies) both during the period 1910–1935.

     

Stochastic Analysis of the Interaction Between Investment and Insurance Risks

Abstract

A portfolio of different insurance policies, such as temporary, endowment, and whole life, is studied in a stochastic mortality and interest environment. The first two moments of the present value of the benefits of the portfolio are derived. The riskiness of the portfolio as measured by the variance of the present value of the benefits can be divided into an insurance risk and an investment risk in two different ways. One way leads to a more natural interpretation of the two risk components. A simple portfolio is used to illustrate the results.     

Improved approximations for the aggregate claims distribution of a life insurance portfolio

Abstract

In an earlier paper the author derived a recursion formula which permits the exact computation of the aggregrate claims distribution in the individual life model. This exact procedure requires of course more computing time than approximative methods such as Kornya's algorithm, which seemed to be the best compromise between accuracy and computational effort. In the present paper it is shown that, to save time, the exact formula can be used in an approximative way and that the corresponding error bound is smaller than the one of the Kornya-type approximations.     

Martingales in life insurance

Abstract

This paper presents a general probabilistic model, including stochastic discounting, for life insurance contracts, either a single policy or a portfolio of policies. In § 4 we define prospective reserves and annual losses in terms of our model and we show that these are generalisations of the corresponding concepts in conventional life insurance mathematics. Our main results are in § 5 where we use the martingale property of the loss process to derive upper bounds for the probability of ruin for the portfolio. These results are illustrated by two numerical examples in § 6.     

Intergenerational Transfers and Insurance Policy Design

Abstract

Group health insurance policies offering an identical benefit package to every member of the group result in lower expected health benefits for younger cohorts than older cohorts. The dispersion in insurance benefits across age groups differs among insurance policies. Simulation results presented in this paper demonstrate that a shift from comprehensive health insurance to high-deductible health insurance decreases the share of expected benefits going to younger cohorts. An estimated 81.5% of the 23-to-32-year-old cohort is expected to receive less than $500 in health benefits during a year for one prototypical high-deductible health plan. Low expected benefits for younger relatively healthy cohorts could increase the number of younger individuals who eschew health coverage. Age-rated premiums are probably the most straightforward way to stimulate demand for high-deductible health plans among younger healthier individuals.     

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.     

Living to 100 and Beyond: Implications of Longer Life Spans

Abstract

Life spans have increased remarkably in the last century. There is substantial disagreement and uncertainty among researchers today about the future course of mortality in the developed countries. Will we continue to live ever-longer lives, or is the human life span headed toward a biological upper limit? The answer to this question has important implications for the elderly, their spouses and children, businesses, and our society as a whole. Continued growth in life expectancy with good health would extend our ability to enjoy all the things we cherish.

Simultaneously, this growth would increase our need to prepare carefully for some unexpected challenges. Among these challenges is a greater need by individuals to save for retirement and to prepare for the possibility of becoming dependent upon family members and others for one’s care. Businesses will enjoy access to experienced workers and expanding markets among the elderly, while they also will try to control the post employment costs that longer life spans will generate. Finally, governments will struggle to manage competing interests as the financial needs of the elderly are weighed against other societal obligations.     

Actuarial concepts in the theory of financing life insurance activities

Abstract

In this paper is presented an attempt by the author to obtain conceptual clarity in the actuarial considerations concerned with financing life insurance schemes, whether private or public, It will, of course, be realized that the author is concerned ultimately with the financing of public or semi-public schemes. However, it is with deliberate intention that the exposition is presented in general terms. As is often the case, abstraction means clarification, thus avoiding a shifting of the meaning of the concepts as one goes from one special situation to another.     

Griffiths on community care

The Government's concern with community care found expression in the commissioning of Sir Roy Griffiths to prepare a report The result was a report which recommends more responsibility for local authorities than the Government has recently proposed in other fields. Some commentators question whether authorities will have the will or ability to deliver.     

Accounting for Carbon Emission Allowances in the European Union: In Search of Consistency

Abstract

With the commencement of Phase III of the European Union Emissions Trading System (EU ETS) in 2013, it is projected that approximately one-half of emission allowances will be acquired through auctioning and the provision of free allocations to installations will be substantially tightened. As a result, it is likely that many companies will hold purchased (as opposed to freely allocated or gratis) allowances and will have more significant liabilities under the scheme. The accounting treatment of emission allowances has therefore become more relevant and the lack of uniformity in practice that resulted after the withdrawal of IFRIC 3 is now a more pressing concern. This study uses content analysis to examine disclosed accounting policies of companies with significant emission liabilities under the EU ETS and identifies three more common approaches adopted to date. These can be generally described as the following: (i) a net liability approach, based on the classification of allowances as intangibles but only showing an emission liability when it exceeds the free allocation; (ii) an approach broadly based on IFRIC 3 (recognising the free allocation at fair value and a corresponding gross liability under the EU ETS); and (iii) an approach based on inventory classification, with free allocations given at nil value. The diversity in these treatments highlights the need for guidance from the International Accounting Standards Board.     

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.     

Financial and Demographic Risks of a Portfolio of Life Insurance Policies with Stochastic Interest Rates

Abstract

We refer to a recent paper by G. Parker (1997) in which the risk of a portfolio of life insurance policies (namely the risk related to the entire contractual life) is studied by separating the demographic component from the financial component. In our paper, after making a brief summary of Parker’s model, we propose two additional contributions: 1. We first give the problem a different formalization, thus allowing a portfolio risk analysis by management periods and a study of the risk due to the interactions among years;

2. We elaborate on a powerful and flexible algorithm for calculating the probability distribution of the sum of random variables that proves useful to solve not only the problems discussed in this paper concerning the risk analysis but also various other problems.

In the paper, we also show, for both contributions, some applications made under the same financial and demographic assumptions taken by Parker; we also compare our results with Parker’s results.     

“Social Security-Adequacy,Equity, and Progressiveness: A Review of Criteria Based on Experience in Canada and the United States,” Robert L. Brown and Jeffrey Ip,January 2000

Abstract

We consider an optimal dynamic control problem for an insurance company with opportunities of proportional reinsurance and investment. The company can purchase proportional reinsurance to reduce its risk level and invest its surplus in a financial market that has a Black-Scholes risky asset and a risk-free asset. When investing in the risk-free asset, three practical borrowing constraints are studied individually: (B1) the borrowing rate is higher than lending (saving) rate, (B2) the dollar amount borrowed is no more than K > 0, and (B3) the proportion of the borrowed amount to the surplus level is no more than k > 0. Under each of the constraints, the objective is to minimize the probability of ruin. Classical stochastic control theory is applied to solve the problem. Specifically, the minimal ruin probability functions are obtained in closed form by solving Hamilton-Jacobi-Bellman (HJB) equations, and their associated optimal reinsurance-investment policies are found by verification techniques.     

Life Insurance Mathematics with Random Life Tables

Abstract

When the insurer sells life annuities, projected life tables incorporating a forecast of future longevity must be used for pricing and reserving. To fix the ideas, the framework of Lee and Carter is adopted in this paper. The Lee-Carter model for mortality forecasting assumes that the death rate at age x in calendar year t is of the form exp(α_x + (β_xK_t), where the time-varying parameter K_t reflects the general level of mortality and follows an ARIMA model. The future lifetimes are all influenced by the same time index K_t in this framework. Because the future path of this index is unknown and modeled as a stochastic process, the policyholders' lifetimes become dependent on each other. Consequently the risk does not disappear as the size of the portfolio increases: there always remains some systematic risk that cannot be diversified, whatever the number of policies. This paper aims to investigate some aspects of actuarial mathematics in the context of random life tables. First, the type of dependence existing between the insured life lengths is carefully examined. The way positive dependence influences the need for economic capital is assessed compared to mutual independence, as well as the effect of the timing of deaths through Bayesian credibility mechanisms. Then the distribution of the present value of payments under a closed group of life annuity policies is studied. Failing to account for the positive dependence between insured lifetimes is a dangerous strategy, even if the randomness in the future survival probabilities is incorporated in the actuarial computations. Numerical illustrations are performed on the basis of Belgian mortality statistics. The impact on the distribution of the present value of the additional variability that results from the Lee-Carter model is compared with the traditional method of mortality projection. Also, the impact of ignoring the dependence hat arises from the model is quantified.     

T. B. Sprague

1. Introduction.

The object of this work is to show that the effective rate of interest of loans practically can be calculated with so great an accuracy that all practical demands regarding accuracy are more than satisfied. As will be seen from section 5, the effective rate of interest can be very quickly calculated by use of inverse linear interpolation for ordinarily occurring cases with a result which in general is correct with 3, 4 or more figures all according to practical demands. These methods can generally be used for all types of loans.     

A proposal for a new basis of life insurance calculations in Norway

Abstract

The basis used by Norwegian life insurance companies for the calculation of premiums and reserves is, as regards insurances, The Institute of Actuaries Life Tables H ^M (20 British Life Offices tables) with interest at 4 per cent. This basis has been used by Norwegian companies for more than thirty-five years and has proved entirely safe.