The achievement of smoothness is one of the main purposes of graduation. Smoothness, however, has never been defined except in terms of concepts which themselves defy definition, and there is no accepted way of measuring it. This paper presents an attempt to supply a definition by constructing a quantitative measure of smoothness, and suggests that such a measure may enable us in the future to graduate without prejudicing the process by an arbitrary choice of the form of the relationship between the variables.

In adventuring upon a Second Presidential Address, the energy of hope and freshness which stimulated me in the former effort have vanished into trepidation of feeling. For I am unhappily conscious that I then exhausted any meagre resources I possessed of novel presentation of thought and research, and was reduced to virtual bankruptcy of serviceable suggestiveness which the interval of time has failed to annul.

Critical illness (CI) insurance involves cover that pays out on the diagnosis of an illness which is deemed to be critical. Estimation and graduation of CI insurance claim rates has been challenging, partly because of the diagnosis of the insured event often being unclear or not recorded. This introduces additional uncertainties in the evaluation of claim rates. In this work, which was funded by an Institute and Faculty of Actuaries research grant, we have addressed the issue of model and parameter uncertainty in claim rate estimation, when the date of diagnosis is missing, aiming at obtaining graduated rates that can be applied to estimate the future cash flow of CI policies and determine insurers’ liability more accurately. Better understanding of uncertainty in rate graduation and pricing is important for insurers, not least because of future changes in the interpretation of the “definition” of an illness or advances in medical practice leading to more efficient diagnosis and treatment.

Estimation of claim rates was addressed in this project using a parametric Poisson model for the number of claims, which accounts for claims that have not been settled by the end of the observation. This was achieved by considering probabilities of the distribution of the delay period between diagnosis of insured illness and settlement of the corresponding claim. We have developed appropriate generalised-linear-type models (including lognormal, Burr, generalised gamma and generalised beta) to investigate this delay, using data supplied by the Continuous Mortality Investigation (1999–2005). The analysis includes various claim risk factors (e.g. gender, benefit amount, policy duration and so on) and was performed under a Bayesian framework, using Markov chain Monte Carlo estimation techniques.

Based on this experience, our analysis suggests that estimates of the claim delay distribution are model sensitive, but claim rates and pricing are not. Under the best claim delay distribution fitting, the following risk factors are considered important for predictive purposes: policy duration, selling office, benefit type, benefit amount, policy type and cause of claim.

A review of Opinion No. 8, which is a statement by the American Institute of Certified Public Accountants (A.I.C.P.A.). Opinion No. 8 is of interest to actuaries for the following reasons:

(1) Its history provides a study of the painstaking methods followed in the U.S.A.

(2) Any actuary who becomes concerned with the pension affairs of a subsidiary of a U.S. company will need to know the scope of the Opinion, its requirements and the difficulties to which it may give rise.

(3) The relationship between members of different professions has to be considered from time to time. Opinion No. 8 concerns itself to some extent with actuarial matters, and it is important to be clear about what it purports to do and the precise roles cast for actuaries and accountants in dealing with it.

(4) It is another factor stimulating discussion regarding pension funding methods.

Lidstone's formula for approximating to the premium for a joint-life endowment assurance is

It 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.

In a paper read a short time ago before the Institute, I endeavoured to show the inexpediency of using, for the purposes of valuation, tables constructed on the principle which characterizes the Northampton table. I did not intend to pursue this particular branch of the subject further; but, finding that more difference of opinion exists in reference to it than I was aware of, and considering that the practice of making valuations by the Northampton table, and by tables framed upon similar principles, is still carried on to a great extent, I am led to offer a few more observations upon the matter previously to entering upon the consideration of some other questions, the correct solution of which materially depends upon a right understanding of this one.

The presentation in November 1905, of three papers on “methods of valuation with allowance for selection” (J.I.A., vol. xl, pp. 1, 15, 42) might well be considered to have left room for no further word on the subject. Any attempt to produce a new method merely to achieve an equal degree of accuracy would undoubtedly be presumptuous, for Mr. King's method affords absolute accuracy, Mr. Diver follows with a microscopic error of £9 on a reserve of more than £3,000,000, and Mr. Ackland closes the series with a larger but still relatively insignificant error of £1,183 on a total reserve of upwards of £13,500,000. It is also shown that in both Messrs. Ackland's and Diver's methods, the error may be expected to be always less than ·01 per-cent.

The practical application of all three methods involves, however, a departure from the familiar system of classifying whole-life assurances according to attained ages, with the loss of all the advantages attaching to that system. This will be viewed by many with regret, and was especially referred to by Mr. Lidstone and Mr. R. P. Hardy in the subsequent discussion. Their remarks, together with Mr. Ackland's cordial expression of goodwill to “other workers in this field” led me to consider the subject from the standpoint of a grouping according to attained ages without any sub-grouping according to duration.

The rapid growth of Accident Assurance, the passing of the Employers' Liability Act, and the periodically recurring discussions respecting schemes for a National Accident Fund, have created a demand for statistics on the subject of accidents, which cannot be said to have been satisfied.

The Tables given in this paper are believed to be unique as regards this country. The object of their compilation was to show the Rates of Fatal Accidents in various occupations. They are based upon a comparison of the male population in each occupation in England and Wales, as enumerated at the Census of 1871, with the average of the number of the violent deaths occurring to male persons following that occupation in England and Wales during the three years, 1870, 1871 and 1872. For the extraction of the particulars relating to these deaths from the records of the General Register Office, special permission was obtained by Dr. Farr; and it is with his cordial approval that the facts are now laid before the members of the Institute.

The Census of 1871 was taken on the 3rd April of that year. The population enumerated was therefore slightly in excess of the mean population to which the average of the deaths for the years 1870, 1871 and 1872 would have reference. The difference would, however, be slight; and for the present purpose it was thought that the population as enumerated might be considered suitable.