The aim of this paper is to examine and document a scientific approach to establishing an estimate of the economic value or range of possible economic values of a life assurance enterprise.

This is essentially a practical paper which provides a ‘practitioner's guide’ to performing an actuarial appraisal of a life assurance company rather than providing a rigorous theoretical derivation of the methods. There is, however, a philosophical issue, that of an appropriate risk rate of return, which cannot be completely side-stepped in a paper on appraisal values. This issue is briefly covered in Section 3. It should be noted, however, that an entire paper could be devoted to this subject.

The paper finishes on a potentially controversial note, that of UK Generally Accepted Accounting Principles (UK GAAP) for life assurance companies, suggesting that the appraisal value techniques outlined in this paper could play a major part in formulating UK GAAP.

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.

It is proposed to discuss the actuarial problems associated with the management of long-term sickness and invalidity benefits. The emphasis is on benefits provided by social security schemes (with particular reference to National Insurance benefit in the UK), but the main points are relevant to private sector benefits including permanent health insurance (PHI).

The plan of the paper is as follows. We describe in Section 2 the nature of long-term invalidity benefit provided by the British social security system and then consider in Section 3 the problems associated with defining disability. A discussion of the differences between the incidence of morbidity and the making of an insurance claim leads to consideration of selection and moral hazard (Section 4). We then examine in Section 5 the methodology for analysing costs and estimating future costs with particular reference to the development within the Government Actuary's Department (GAD) of a model based on an incidence and survivorship approach to invalidity benefit and the debate aroused by CMI Report No. 7 on the same topic5. The paper then presents an analysis of recent trends in disability claim rates (Section 6) and incidence and termination rates (Sections 7 and 8) based on invalidity benefit within the British social security system, as well as PHI and the experience of other countries. Attempts are then made in Section 9 to explain these trends and the upward trend in financial costs for disability benefits experienced by many social security systems. The paper concludes with an examination of two areas of current interest, viz. sex differences in morbidity rates and claim rates (Section 10) and the relationship between claim rates and the prevailing level of unemployment (Section 11).

8.1 This paper is concerned with the selection of investment portfolios to meet specified criteria which involve the liabilities of a long term investing institution such as a pension fund or a life office. In Part 1 (5) I showed a certain relationship between, on the one hand, portfolios selected according to a criterion of pure matching to the liabilities and, on the other hand, portfolios selected according to a more general criterion of ‘efficiency’. This connection points to a particular actuarial approach to the selection of portfolios, which is now further examined in Part 2.

8.2 Writing Part 2 separately presents an opportunity to re-state the main ideas. The next few paragraphs recapitulate the basic points with a view to reducing, within the subsequent discussion, the amount of cross-reference to Part 1, and to the three preceding papers on which the study is based.

In this paper I discuss a simple application of option pricing theory to bonuses on with-profits life assurance policies. The approach is a new one and I have been able to make only an introductory exploration of its possible applications. Nevertheless it seems to me to give actuaries some sort of a handle for gripping the rather intractable problem of the relationship between reversionary bonus, terminal bonus and the proceeds of comparable unit-linked policies. I look at the problem very much from the policyholder's side, and I am not concerned here with aspects of valuation or solvency. These require further investigation.

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 purpose of this short note is to explain a simple method of constructing a life table, given an explicit function for the force of mortality, μx, for all x. A similar method can be applied to the construction of multiple decrement tables, given explicit functions for the separate forces of transition, μbx, μcx, etc, from one ‘active’ status (a) to various ‘dead’ statuses (b, c, etc) for all x.

Although numerical results for the examples discussed in this note could be calculated equally well by more direct methods than the ones described herein, results for more complicated examples involving multiple status tables, in which transition back and forth between statuses is possible, cannot in general be calculated by such direct methods, and our methods provide a practical way by means of which numerical results for these more complicated cases can be calculated.

1.1 My paper entitled ‘The Matching of Assets to Liabilities’ described a new study of the subject of matching. The paper outlined the mathematical framework, showed some calculations, and suggested various applications in the field of actuarial valuation.

1.2 Subsequently A. D. Wilkie published a note entitled ‘Portfolio Selection in the Presence of Fixed Liabilities: a Comment on “The Matching of Assets to Liabilities”’. Wilkie's note appears to represent a new extension of portfolio selection theory which enables him to encompass the conventional portfolio theory and to show where my approach to matching fits into the picture.

1.3 This paper is in the nature of a reply to that of Wilkie. My objective at the stage of the first draft was to re-examine the matching portfolio, as defined in relation to specified liabilities, in view of the wide range of alternative portfolios which Wilkie has described. The scope of this work progressed well beyond my original plans when I found a most interesting mathematical relationship between the ‘matching portfolios’ described in my earlier paper and the ‘efficient portfolios’ described in Wilkie's.

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.

1.1 A number of papers have been written in recent years that discuss the valuation of the business of a life office, examples of these are Burrows and Whitehead (1987) and Bunch (1986). These papers recognize the need to use a risk adjusted discount rate in the calculations. Burrows and Whitehead devote a section of their paper to the determination of such a rate. In their abstract they state that “…an entire paper could be devoted to this subject”. This latter point is supported by the existence already of a number of papers in the finance literature on the very topic of determining risk adjusted discount rates e.g. Robichek and Myers (1966), Fama (1977).

1.2 This paper outlines some results of the theory of determining risk adjusted discount rates and the valuation of risky cash flows as developed in the finance literature, and discusses the application of this theory to the determination of life office appraisal values. Some comments are made on Section 3 of the Burrows and Whitehead paper.

Bias in estimates of the initial decremental rate q and the central decremental rate m is considered for several survival functions. It is first shown that the central rate estimate is unbiased when the survival function is an exponential curve; is similarly unbiased when the survival function assumes the Balducci shape. The evidence presented here also indicates that the proportional bias (the bias divided by the true value) is approximately twice as great for as for when the survival function is a straight line.

Life tables in common use in the United Kingdom are, however, concave at ages arising most frequently in actuarial calculations, while none of the above curves is concave. In order to investigate the bias properties of general survival functions a quadratic is chosen, with a parameter indicating the extent of curvature. For this particular class of survival functions, one can show that bias is approximately minimized when exposure to risk is symmetric about the middle of the rate year.

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.