Endowment Assurances Research Articles

Ruin probabilities and optimal capital allocation for heterogeneous life annuity portfolios

A large part of the actuarial literature is devoted to the derivation of ruin probabilities in various non-life insurance risk models. On the contrary, very few papers deal with ruin probabilities for life insurance portfolios. The difficulties arise from the dependence and non-stationarity of the annual payments made by the insurance company. This paper shows that the ruin probability in case of life annuity portfolios can be computed from algorithms derived by De Pril (1989) and Dhaene & Vandebroek (1995). Approximations for ruin probabilities are discussed. The present article complements the works of Frostig et al. (2003) who considered whole life, endowment, and temporary assurances, and of Denuit & Frostig (2008) who considered homogeneous life annuities portfolios. Here, heterogeneous portfolios (with respect to age and/or face amounts) are studied. Particular attention is paid to the capital allocation problem. The total amount of reserve is shared among the risk classes in order to minimize the ruin probability. It is then fair to charge a higher margin to the risk classes requiring more capital.

Endowment Assurance Products: Effectiveness of Risk-Minimizing Strategies under Model Risk

This paper analyzes and discusses the effects of model misspecification associated with both interest rate and mortality risk on the hedging decisions of insurance companies. We consider hedging strategies in different instruments (zero bonds) which are risk- (variance-) minimizing with respect to an assumed model. In this case, the associated expected costs and the variance of the costs are the same for all strategies. While the introduction of model risk, i.e. a deviation of assumed from true models, has the same effect on the expected costs, this is not true with respect to the variance. It turns out that the choice of hedging instruments has a crucial impact on the robustness of the strategies. In addition, the results of the paper can be used to emphasize the necessity to use a combined hedging model. In terms of robust hedging, a separate specification of interest rate model and mortality model is inconvenient, even in the case that interest rate and mortality are assumed to be independent.

Endowment Assurance Products: Effectiveness of Risk-Minimizing Strategies under Model Risk

This paper analyzes and discusses the effects of model misspecification associated with both interest rate and mortality risk on the hedging decisions of insurance companies. We consider hedging strategies in different instruments (zero bonds) which are risk- (variance-) minimizing with respect to an assumed model. In this case, the associated expected costs and the variance of the costs are the same for all strategies. While the introduction of model risk, i.e. a deviation of assumed from true models, has the same effect on the expected costs, this is not true with respect to the variance. It turns out that the choice of hedging instruments has a crucial impact on the robustness of the strategies. In addition, the results of the paper can be used to emphasize the necessity to use a combined hedging model. In terms of robust hedging, a separate specification of interest rate model and mortality model is inconvenient, even in the case that interest rate and mortality are assumed to be independent.

The repayment of mortgages by endowment assurances

Looks at the problem of endowment assurances that do not meet their targets to repay residential mortgages. By analysing the present value of payments under different inflation and interest‐rate regimes, we conclude that the perceived problems with endowment policies may simply be a manifestation of “money illusion”. Nevertheless, there could be frictional problems and other problems arising from the use of endowment assurances to repay mortgages which we identify but do not analyse in detail. The failure of such a major method of repaying mortgages to perform in line with expectations will have implications for the residential housing market.

An investigation into the financing of flexible endowment assurances

1.1. In the past ten years, traditional with-profit endowment assurances have been faced with severe competition from two new products, which offer considerable flexibility to the policyholder combined with certain guarantees not normally available in the traditional endowment policy.The first of these new products, unit-linked assurance, has been analysed in some depth in recent papers submitted to the Institute, Faculty and Tokyo Congress. In particular, papers by Benjamin, Wilkie and Scott have developed stochastic models that have enabled the premiums and valuation reserves for maturity guarantees under these policies to be quantified with a much greater degree of certainty than had previously been the case.

A comparison of results by Perks' approximate valuation methods with those by individual policy computer methods

1. In papers submitted to the Institute in 1933 and 1945 (J.I.A. 64, 264 and 72, 377), Wilfred Perks put forward formulae based on uses of the '«-ages method' to enable liability valuations to be carried out by means of valuation records that did not need to contain any figures related to the valuation basis. Thus, not only was a change in valuation basis made a great deal easier actually to adopt, but the actuary was able readily to compare at any time the broad effects, for the main classes of business, of different bases, including 'bonus reserve' results, as against the net premium basis actually in use. 2. In the closing passages of the second paper he mentioned that for ordinary branch business a large office had been successfully employing the two-ages, two-terms method set out in the earlier paper and that, under wartime pressures, shorter methods, as described in the second paper, had been developed. Both kinds of method have continued to be used in that office, although not without concern about the possible effects, which proved difficult to estimate, of the considerable changes in the 'shape' of the business on the books resulting from the effects of inflation on the size of new policies. Fortunately A. E. Lacey, who had been Perks's principal assistant and collaborator on the techniques involved, was still in active employment at the office during this difficult time and his detailed examinations enabled the continued use of the methods by means of splitting up the business into groups of years of entry. 3. Both what were colloquially known in the office as the 'long method' and the 'short method' were based on punched cards that contained constants based on x . Sx and x2 . Sx for whole life assurances (x being the entry age and Sx the sum assured) and on n . Sn and SJn for endowment assurances (where correspondingly n is the term and Sn the sum assured). This punched card record was maintained independently of the office's premium collecting system and the inter-checking between the two served as a valuable piece of internal audit work. 4. Incidentally, the formulae employed were used separately for year of entry calculations and for year of birth and maturity calculations, again affording a valuable cross-check, and enabling the most suitable choices to be made for the purposes of actual valuation results. The 'short method' was, in practice, used on its own only for the less important classes of business. For the principal classes, VSA, VRB and VOP were taken from the 'long method', while NP and VNP were adjusted 'short method' results. Apart from the year-of-entry split mentioned above, the methods were

Unit-linked assurance policies: category B

The purpose of this note is to summarize and draw attention to the references in the actuarial literature to Category B unit-linked assurance policies.2. The designation ‘Category B’ was first used by A. T. Grant and G. A. Kingsnorth. In paragraph 53 of their paper the authors define this type of unit-linked policy as one where the policy is linked to a unit trust and dividends distributed by the unit trust are retained for the benefit of the life office and corresponding adjustments are made in the premium scale and in the form of benefit. It is stated that the more usual type of contract in this category takes the form of an endowment assurance under which the sum assured is payable in the units of a unit trust, the amount deemed invested in units each year being equal to the sum assured divided by the original term of the policy. At maturity the policy-holder receives the units (or their cash value) subject to any deduction for capital gains tax.

Unit-linked assurance: observations and propositions

Perhaps this paper should be entitled A Description of Methods Being Used in Connexion With a Contract of the Grant and Kingsnorth Category ‘A’ Type, For Want of Any Better Ideas. (The reference is to Unit Trusts and Equitylinked Endowment Assurances by A. T. Grant and G. A. Kingsnorth, J.I.A. 93, 387.) It has been written because the publication of the Scott Report and the passing of the Insurance Companies Act make it desirable that the Institute should discuss unit-linked assurance again at this time, and because certain aspects, in particular maturity guarantees and reserves for tax on capital gains, have not been dealt with adequately in the past. Propositions in respect of these are made in the paper, and it is hoped that, as a result of discussion, ‘generally accepted actuarial principles‘category B’ type contracts, for example considerations affecting pricing units net of tax on capital gains, but the paper does not attempt to deal with this type of contract more generally, as fundamentally different considerations are involved. It does deal with ‘category A’ policies linked to unitized internal funds as well as those linked to units of Authorized Unit Trusts.

A Note on Varying Rates of Interest

SynopsisLet us suppose, for the purpose of calculating premium rates, that interest rates will not fall below those of the straight lines from their present value io to in in n years' time (in < io), thereafter remaining constant at in. On the basis of this assumption we derive formulae for the (minimum) effective rates of interest on single and annual premium contracts of term m years (m ≤ n, or m > n) and give examples in which gross interest rates fall from 8%, 9%, 10%, 11%, to 4½% in 40 years' time, and other examples. We conclude with a discussion of modifications to the term of the contracts which enable our results to be used for pure endowments, with-return endowments, endowment assurances, deferred annuities and other policies in life assurance and pensions business, and we mention some objections to our methods.Our thanks are due to C. M. Stewart, F.I.A., F.S.S., for suggesting some improvements to this note.

Recent Mortality Trends: Some International Comparisons

SynopsisThis paper continues the work presented by Gwilt to the Centenary gathering of the Faculty in 1956 and extended by Anderson & Whitehead at the Sixteenth International Congress of Actuaries in 1960. For twenty-two countries death rates for each sex are given for quinquennial age groups between 15 and 84. Five periods centred on equidistant points in the interval 1950 to 1970 have been employed to permit ratios to be tabulated showing the improvement or worsening in mortality in each age group for any part of this period of twenty years. At some ages mortality has clearly worsened over the period, especially for males between 60 and 75.A graphical method of examining the trend of mortality at each age over the last 150 years is illustrated; this is based on a model system of calendar year life tables.A comparison with some assured lives mortality shows a greater improvement among assured lives than for the male population as a whole. Comparison of each country with the average for all countries and comparisons of mortality sex ratios bring out some points of interest which confirm the general trend.An analysis by age, sex and cause of death covers all twenty-two countries in or around 1965, and four of them in 1955 in order to continue the figures given by Gwilt for 1936 and 1950. This analysis picks out the major reasons for the changes in over-all mortality and highlights the differences between sexes by giving ratios for each sub-group. The increases in certain countries in mortality from lung cancer, heart disease and motor vehicle accidents are noted.The paper finishes by considering the effect of worsening mortality rates on endowment assurance premium rates.

The Unit-Linked Approach to Life Insurance

Unit-Linked life insurance can be simply defined as the application of unit trust principles to the ‘savings element’ of each premium paid and the application of insurance principles to the remaining ‘insurance element’ only. It has been appearing in various forms for some time now. In North America, it has been mainly of the variable annuity variety, commencing with CREF in approximately 1952. In the U.K. it has been mainly of the endowment assurance variety, commencing in approximately 1957 but not really making a big impact until 1962–63 when the unit trust groups first came on the scene. In Australia it has so far been limited to superannuation schemes with the so-called (by one life office) ‘E.F.G.’ approach the prime example, commencing in approximately 1965. In each country the position is the same—a remarkably favourable response from the public, particularly in the last few years, causing an urgent reassessment of previously held views, sometimes, unfortunately, most widely and eloquently recorded! There have also been legislative difficulties of varying degrees, most pronounced in North America, least pronounced in the U.K., with the position in Australia not clear.

Stochastische Betrachtungen über den Verlauf des Deckungskapitals bei der Lebensversicherung von Gruppen

The paper discusses a stochastic approach to policy reserves, using as an example a number of endowment assurance policies. The description of the stochastic model for the business in force is followed by the definition of two random variables, the prospective and retrospective balances of income and outgo. The expected values, the variances and the coefficient of correlation between the two balances are then derived, and it is shown that the correlation between the two balances is the same as that between prospective and retrospective reserves, if these are treated as random variables. Three possible methods are discussed for finding the probabiltiy distributions of the balances: The method of Pearson Types, approximation to the Normal distribution, and simulation by successive computation of random number sequences. Finally a conditional distribution is defined and examined. It is the distribution of the prospective balance, if the number of policies still in force is known. Numerical examples are given for the most important results.

Assurances Linked to Unit Trusts.

1. The object of this paper is to look at the two principal types of unit-linked contracts that are now being offered with reference to the actuarial problems that they raise. In particular the calculation of premium rates, valuation, analysis of surplus and investment problems are discussed.2. In looking at the policies that are currently offered in the unit-linked field there are two broad groupings into which most fall. These are the ‘Decreasing Term Assurance’ group and the ‘Endowment Assurance’ group. There are of course variations within each group. The main difference between the groups is that there is virtually no investment risk to the office in the Decreasing Term Assurance group but there is an investment risk in the Endowment Assurance group.3. Because these two groups of contracts are very different and raise different actuarial problems, it is convenient to sub-divide this Paper and deal with the two groups separately.

Bemerkungen zur Abkürzung von Kapitalversicherungen

The article deals with the shortening of the terms of endowment assurances by way of additional payments by the policyholder, which may be returnable or non-returnable on death. Bonus arising under such policies may be applied in this way.

Unit Trusts and Equity Linked Endowment Assurances

Unit trusts originated in the United Kingdom in the second half of the nineteenth century, but an early court ruling that unit trusts were not legal resulted in all but one unit trust changing to the form of investment trusts. Although the court ruling was later reversed, little further headway was made by the unit trust movement until after the Great Crash of 1929.

The marketing design of life assurance proposals

Concentrates solely on custom building, as this is the area of major growth in the life assurance business. Demonstrates how the well‐known techniques of product concept testing can be applied in a service industry when mass marketing is contemplated. Delves into a study conducted in spring 1967 in Bradford, UK, involving 40 households, where the head was between 18 and 25 years of age. Reports on the questionnaire survey that covered a range of disbursement analyses, other than those specifically concerned with life assurance proposals. Concludes that the general conclusion is that the market segmentation implicit in straight life assurance and endowment assurance is meaningful, and that the conversion policy is the most effective way of straddling the two segments, to ensure brand loyalty once contact has been converted into first business.

Unit Trusts and Equity Linked Endowment Assurances

SynopsisThe aim of the paper is to discuss some of the problems relating to unit trusts and to the issue of annual premium endowment assurance policies with benefits linked to unit trusts or equities.SECTION I describes the background to unit trust operations including the main requirements for Board of Trade authorization.SECTION II analyses expenses of administration and patterns of expense loadings. It compares the traditional life assurance approach to expense loadings and normal unit trust practice.SECTION III describes the main broad types of assurance linked to unit trusts or equities now in force. It discusses the organizational and taxation considerations involved in issuing such contracts and the desirability of doing so.

Erstellung von Änderungsangeboten mit Hilfe elektronischer Rechenanlagen

In connection with the paper ofG. Horstmann the author develops a computer programme by which the data for three possible new endowment assurances may be calculated, so as to cover more or less any desired alterations of an existing endowment policy. Apart from the reasonableness of the desired alterations it is assumed only that the existing and the new policies are in accordance with such general conditions as are usual for ordinary life policies in Germany today (e.g. age at entry and duration of the policy are integer numbers …). In order to avoid too specialised assumptions no account is taken of participation in profits but it seems that it may not be difficult to incorporate the calculation of such reserves. The programme is written in the programming language FORTRAN in the IBM 1401-version; the basic principles of this are set out in a report concerning experiences in using FORTRAN for actuarial programming (page 405 of this issue). The data for the existing policy as well as for the alterations desired are fed in by punched card; the resulting “offers” are printed out together with such technical values as are necessary to allow the actuary to control the calculation and, if suitable, to improve the “offer”.

Zur Systematik der technischen Änderungen eines Lebensversicherungsvertrages

The author investigates systematically all possible technical alterations of endowment and whole life assurances. The main point is that any alteration can be considered as the result of the alteration of one or several of the four basic values; sum assured, premium, duration of the policy and of the period of premium payments. The age at entry is not involved as long as the premium reserve is calculated on a prospective basis. A complete system of formulae for all possible (15) alterations is given, also a method to calculate the new age at entry as far as it is needed for technical reasons, as well as some practical applications.

Controlled Funding Methods for Group Pension Schemes

The costing technique known variously as Controlled Funding, Stabilized Costing and Aggregate Costing, is a comparatively recent development in the field of Life Office group pension schemes. Before explaining and discussing the methods used, it may be helpful to outline the principal types of Life Office group pension scheme, and to examine the more traditional methods of costing employed. It should be made clear at the outset that the term ‘group pension scheme’ relates to schemes insured by means of group deferred annuity contracts and approved under section 388(1) or section 379(3) of the Income Tax Act, 1952—endowment assurance schemes and group life assurance schemes are not within the scope of this paper.