Endowment Assurances Research Articles

Der Zusammenhang zwischen Deckungskapital und Sterblichkeit im Jecklinschen Reservemodell

The author determines the decrement tables leading, for endowment assurances, to the hyperbolic reserve function ofJecklin’s reserve model. He shows in particular that, for a given maturity age (s), a necessary and sufficient condition for the reserve to follow a rectangular hyperbola with asymptotes parallel to the co-ordinate axes is given by the following mortality laws a $$\begin{gathered} x) = D_x = v^x l(x) = e^{ - \delta x} l(x): \hfill \bar f(x) = k\left( {l - \frac{x}{c}} \right)\left( {l - \frac{x}{s}} \right)^{\lambda (c - s) - 1} e^{ - \lambda x} \hfill \end{gathered} $$ and b $$f(x) = K(c - x)\frac{{\Gamma (s - x)}}{{\Gamma (\sigma - x)}}(1 - \lambda )^x ,$$ according to whether the reserve is defined in (a) a continuous or (b) a discontinuous way. If the above rectangular hyperbola is replaced by a non-rectangular hyperbola with one asymptote parallel to the ordinate then the functions (a) and (b) are replaced by a $$f^ * (x) = k\left( {l - \frac{x}{c}} \right)\left( {l - \frac{x}{s}} \right)^{\lambda A - 1} \left( {l - \frac{x}{{s + a}}} \right)^{\lambda B - 1} $$ and b $$\begin{gathered} f^ * (x) = K(c - x)\frac{{\Gamma (s - x \Gamma (s + a - x)}}{{\Gamma (s - \alpha + 1 - x) \cdot \Gamma (s - \beta + l - x)}} \hfill = K(c - x)(s - 1 - x)^{(\lambda A - 1)} (s + a - 1 - x)^{(\lambda B - 1)} \hfill \end{gathered} $$ These functions include de Moivre’s law as a special case, leading to a hyperbolic reserve curve for every maturity age. The reserves defined by these functions also include the special cases of rectangular hyperbolic and linear reserves.

Extras and Debts for Endowment Assurance.

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Sterbegesetze mit vorgegebenem Reserveverlauf

The author determines the decrement tables corresponding to a given curve of the mathematical reserve for endowment assurances with a given maturity age and thus gives a solution of the inversion problem of reserve theory. The paper starts from a criterion in the form of a differential relation which functions of the three variables t (expired duration), x (age at entry), and n (insurance term) must satisfy in order to represent the mathematical reserve defined in a continuous way. The mortality laws corresponding to linear, hyperbolic, exponential and parabolic reserve curves are determined. In this connection, special investigations are made with respect to Jecklin’s model of hyperbolic reserves (F-method).

Ein Theorem über die Änderung des Deckungskapitals bei Sterblichkeitsvariation

Following Lidstone’s well-known article in the Journal of the Institute of Actuaries for 1905, the author establishes, by analytical methods, sufficient conditions for variations in the mortality ratesnot giving rise to variations in the sign of the premium reserves of an endowment assurance at any time during the term of the policy. If lx is replaced, in the varied life table, by $$\tilde l_x = l_x \cdot f(x)$$ (where f(x) is a positive function which is differentiable as many times as required) there is no variation in the sign of the reserve if 0≤f(r+1)(x)·f(x) ≤ f(r)(x)·f′(x); r=0, 1, 2, 3, ... For the practical determination of such functions f(x) a method is proposed for the reduction of the number of conditions to a given finite number.

Consistent Extra Premiums and Equivalent Decreasing Debts for Endowment Assurances.

This article attempts to develop a practical system for dealing with extra-risks whilst retaining consistency in the terms offered to proposers. The extra premiums derived vary according to the view taken of the incidence of the extra-risk, and tables are produced to enable the underwriter to deduce quickly the required special terms.W. Perks' paper, ‘The Treatment of Sub-Standard Lives in Practice’ (J.I.A.78, 205), describes a system whereby the special terms are fixed according to uniform percentage increases in mortality throughout the term. This approach does not make a broad assessment of the incidence of the extra-risk. The present writer feels that some allowance for incidence should be made, and the process that follows attempts to adhere to the narrow theoretical path. The scope of this article does not include the interesting fields of underwriting practice and experience which were admirably discussed in W. Perks' paper.

Some premium reserve methods related to Lidstone's

Abstract Lidstone's group method, which applies to endowment assurances, is based on the approximation, where a n−t and bn−t depend only on n−t . For a valuatIOn group, consisting of policies with the same unexpired terms, n−t , the mean maturity age y + n is given by the equation, W denoting S or P. Inserting here values by (1) om both sides we arrive at the well-known Z-method: where the quantities on the right hand side, or proportional values, are used as individual policy constants.

Trends in Industrial Assurance

SynopsisSince the paper “Industrial Assurance Today” by the same author was submitted to the Faculty in January 1953, publication of the annual reports of the Industrial Assurance Commissioner has been resumed after a gap of about 15 years due to the war. These annual reports have contained more detailed figures than those shown in the previous statistical summaries, and it is possible to follow the trends in the business over a period of years.The following trends are quite clearly visible in these statistics and are illustrated by tables made up of figures extracted from the reports :—(i) A fairly rapid rise in sums assured and premiums per new policy, and also a more gradual rise in sums assured and premiums for paying policies in force. The premium income has increased by 33 per cent. between 1950 and 1957.(ii) A steady tendency to more endowment assurances and, because such policies are usually by lunar monthly premiums, towards monthly premiums.(iii) Some rise in the expense ratios over the whole business, although societies are able to show decreases due to progress in the elimination of book interest and to an increase in the proportion of endowment assurance business which carries lower commission rates.(iv) A gradual fall in the lapse rate for the whole business from 24.0 per cent. experienced in 1949 to 16.8 per cent. in 1957. Some offices have distinctly lower rates.(v) Rising interest rates which, in conjunction with falling mortality rates, have resulted in increased surpluses and improvements in bonus schemes. The share of surplus allocated to policyholders in the proprietary offices is now about the same as in proprietary ordinary life offices.Other features referred to in the paper are the contraction over a period in the number of agents resulting in a considerable increase in the average collections per agent per week. The earnings of agents have considerably increased. The gradual disappearance of small collecting societies is noted.

Note on the Calculation of With-Profit Endowment Assurance Premiums By a Programme Controlled Computer

Note on the Calculation of With-Profit Endowment Assurance Premiums By a Programme Controlled Computer

On educational annuity, non-dependent and in combination with endowment assurance

Abstract Extract Let us denote by n the length of the insurance period and by the continuous premium to be paid during this whole period for an educational annuity of 1 per annum, continuously paid. Then we have—V as usual standing for the net reserve—

Some Experiments With Net Premium Valuations

SynopsisThe paper deals with the valuation of whole life and endowment assurance business in offices whose solvency is not in question and which are writing an expanding business.A survey is made of the valuation methods and bases of major British life offices. A large majority of these offices use the net premium method for their published valuations and make these valuations on mortality and interest bases similar to those used in their current premium scales. A model office is used to study the provision for expenses made by net premium valuations. It is shown that the renewal expenses may often be less than the annual provision for expenses in the valuation, with the result that the valuation makes some provision for the initial expenses of new business.Examples are given showing the rate at which surplus is released by net premium valuations in an expanding business and the rates of reversionary bonus provided are compared with the uniform rates supported by the premiums.Further examples are given showing the rates of reversionary bonus provided by net premium valuations after changes in the experienced rates of interest, mortality and expense, and these reversionary bonuses are compared with the uniform reversionary bonuses supported by the reserves and the premiums under the new conditions.

Elementary generalizations upon Lidstone's approximation for two joint lives

Lidstone's formula for approximating to the premium for a joint-life endowment assurance isIt has been found empirically that the greater the disparity in age the less accurate is the approximation. There is no such progressive loss of accuracy as the period of assurance lengthens.

Notes on Constant and Increasing Extra Mortality

SynopsisThe paper deals with the types of under-average mortality which may be expressed either as a constant addition or as a fixed percentage addition to A1924-29 ultimate qx.For various constant and percentage additions, tables are given which show:–(a)The extra premiums required, for different classes of policy, in addition to A1924-29 2½ per cent. ult. net premiums.(b)The level debts equivalent to certain of the extra premiums.(c)The smaller extra premiums required if Double Endowment Assurances are issued in lieu of Endowment Assurances.

Beitrag zum Zinsfußproblem der Prämie

A method of approximation is given to the value of the premium for an endowment assurance at rates of interest differing from those tabulated.

Calculation of policy-values for different rates of interest

Abstract If we calculate the Policy-Values of a Whole Life Assurance for a given Age at Entry and a given Duration, but for several equidistant rates of interest, we find that the proportion of two successive values is very nearly constant. This fact is illustrated by Tables 1 a and 1 b, giving certain Policy Values on the basis HM and A 24–29 respectively, for the rates of 3, 4, 5 and 6 % of interest. Tables 2 a and 2 b show, that the same applies, if to a smaller degree, to Endowment Assurances.

Whole of Life and Endowment Assurance Mortality

SynopsisThe object of the paper is to determine the extent to which the differences between the ultimate rates of mortality for the whole life and endowment assurance classes are attributable to class selection.The effect on the ultimate rates of mortality of the different distributions of the exposed to risk in the two classes is examined on the basis of common “select” rates of mortality derived from the combined experience.While the effect is found to be appreciable, there remains a substantial proportion of the differences between the actual ultimate rates of mortality unaccounted for, and the conclusion is reached that this unaccounted balance constitutes a measure of the effect of class selection.Graduations of the separate class experiences with reference to the official combined rates are given in the paper.

A New Table for obtaining the Mean Maturity Age for use with the “Z” Method of Valuation of Endowment Assurances in conjunction with the A1924–29 (Ultimate) Table

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The Valuation of Double Endowment Assurances by the a 1924–29 Mortality Table

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Approximations to the Values of Certain Actuarial Functions leading to Suggested Check Methods of Valuation of Endowment Assurances

It is demonstrated, with particular reference to the OM, OM(5) and A1924–29 Tables, that over the range age 20 to age 65 of a normal mortality table, μx can, with a close degree of approximation to the tabular values, be expressed in the formleading to the following analogous expressions:—Generally, all functions involving mortality within the stipulated range can be expressed in a similar form as purely mathematical and easily evaluated functions of the initial age and interest in combination.It is claimed that the evaluation of such temporary functions at any rate of interest is thereby greatly facilitated.The retrospective and prospective values of an endowment assurance policy in terms of the formula are investigated, and check methods of valuation, continuous in process and independent of the working principles and practice of the principal method, are put forward.Numerical results and examples of the procedure are given in the paper.

Family Income Policies

The so-called Family Income Benefit introduced, or reintroduced, into this country some four years ago has received an astonishing welcome from the public. Its rapid popularity as an addition to the ordinary whole life or endowment assurance policies has a parallel only in the disablement income benefit granted by the American and Canadian Offices in North America. Although there is no reason to anticipate that the history of this supplementary benefit will follow the unfortunate course of the disablement benefit in America, an early examination of the problems connected with it is most desirable, for the policies are not without their pitfalls and limitations.

The Application of the Z-method of Valuing Endowment Assurances to Valuations Based on the New a 1924–29 Mortality Table

The Application of the Z-method of Valuing Endowment Assurances to Valuations Based on the New a 1924–29 Mortality Table