The Objective of Funding

1.1. As every actuarial student is taught:

‘Pay-as-you-go is acceptable for a State pension scheme because the State is, for practical purposes, assured of a continuing existence.’

However:

‘The position is quite different in the case of an occupational scheme, since an employer's business may cease to exist.’

1.2. It seems to us to follow, therefore, that the prime purpose of funding an occupational pension scheme must be to secure the accrued benefits, whatever they might be, in the event of the employer being unable or unwilling to continue to pay at some time in the future. To that end, the contributions would have to be sufficient both to pay the benefits as they fell due for as long as the scheme continued, and also to establish and maintain a fund which would be sufficient to secure the accrued benefits in the event of contributions ceasing and the scheme being discontinued, whenever that might occur.

After reviewing some general issues concerning solvency and the problems associated with establishing the financial strength of a general insurance company using the traditional balance sheet concept, the authors put forward an emerging costs approach for examining the strength of a company. This enables the true nature of the assets and liabilities to be taken into account, including their essential variability. Simulation is suggested as a powerful tool for use in examining the financial strength of a company and in exploring the impact of alternative scenarios. A particular example of such a simulation model is then presented and used to explore the resilience of a company's financial position to variations in a wide variety of parameters. The model enables the user to quantify the probability that the assets will prove adequate to meet the liabilities with or without an assumption of continuing business. This in turn permits an appropriate asset margin to be assessed individually for any particular company in the light of the strategy that the company intends to follow. Some of the implications of this approach for the management and supervision of general insurance companies are explored. The suggestion is made that the effectiveness of statutory supervision based on the balance sheet and a crude solvency margin requirement is limited, since it cannot have proper regard to the risk profile of individual companies. More responsibility should be placed on an actuary or other suitably qualified professional individual to report on the overall financial strength of the company, both to management and to the supervisory authorities.

The pattern of mortality at the highest ages has been considered by many authors, including Redington (1969), Humphrey (1970) and Benjamin (1964, 1982). The questions raised have included the following:

(a) Is there a definite upper limit to the span of human life, so that qx reaches unity at a finite age? Or does qx tend gradually to unity as age tends to infinity, as happens under the Gompertz and Makeham laws? Or does qx tend to a constant less than unity, as under the Perks formula or the formula which was used to graduate the English Life Tables No. 11 and 12?

(b) Has the fall in mortality rates at lower ages been accompanied by a similar fall at the very highest ages? Has the upper tail of the curve of death (μxlx) shifted?

(c) Does the lower mortality of females compared with males extend to the very highest ages, or do the rates eventually tend to converge?

It may be useful to begin with some introductory comments on the nature of graduation and its uses with particular reference to the distinction between graduation by parametric methods, as described in traditional actuarial textbooks, and graduation by non-parametric methods.

Some of the comments have appeared in the early sections of Copas and Haberman's paper which first described the kernel method of non-parametric graduation.

Graduation may be regarded as the principles by which a set of data is adjusted in order to provide a basis suitable for inferences to be drawn and further practical calculations to be made. In actuarial terminology, graduation usually refers to a set of decremental rates, and one of the principal actuarial examples of its use is the construction of a life table from a set of age-specific, observed death rates.

The paper considers systems of payments which are not fully funded, i.e partially funded or fully unfunded. Generally, the objective is to be able to establish a premium formula which is consistent with long term planning as to e.g. a target rate of funding, limited variation in premiums from year to year, etc. The premium formulas considered are those which relate premiums to prior years' premiums, claims experience and accumulated funds. These questions are reviewed in Section 1 which suggests the use of control theory. Section 2 formulates and provides a formal solution to the problem.

Section 3 provides a couple of results which relate premiums to targeting of accumulated funds.

Subsequent sections consider the construction of premium formulas. It is emphasized that the intention is not to supply a definitive set of instructions as to how the premium formula might be constructed, but rather to illustrate some of the principles relevant to it.

In particular, two numerical examples are provided in § 6.2. Premium formulas are constructed which appear to respond reasonably satisfactorily to simulated claims experience.

It is found that accurate prediction of future claims escalation is crucial to the operation of formulas of the linear control theory type (§ 6.2.1). Brief comment on possible further research aimed at dealing with this aspect of the question is given in Section 7.

An approach to loan and valuation problems through linear algebra is used to cast commonly applied rules into a general form suitable for routine use under complex conditions. An illustrative example is included.