Disability Insurance is a subject which has at various times found a place in the proceedings of the Faculty and Institute of Actuaries, but only on rare occasions at our own proceedings. The present paper does not present anything new on the subject but has been written with the object of providing an opportunity of considering and discussing a form of insurance which, in some form or other, is transacted by most Life Offices in this country. Most Life Offices will only issue a somewhat mild form of the insurance, that is, the Waiver of Premium benefit, but even if we are not usually concerned with more than this form of the insurance, it is felt that a discussion of the principles involved would be of value.

Considering the number of eminent actuaries who have, at various times, stated that they would like to see an extension of the writing of Disability Insurance, it is somewhat surprising that more of the business has not been written in this country. The reasons for the neglect are not far to seek. First, there are very real administrative difficulties involved including the difficulty of obtaining suitable statistics on the subject. A large part of the present paper is taken up with a discussion of these difficulties. Secondly, there is the unfortunate experience of our American colleagues who have in the past lost considerable sums of money in their Disability Accounts. This experience has led a large number of American Offices to cease to transact the more liberal forms of benefit.

The simplest form of nomogram is a graphical device for representing a functional relationship between three variables in a manner which is often more convenient for practical reference than that of plotting a series of contours for chosen values of one of the three variables on a Cartesian graph. We develop the basis of such a nomogram by means of analytical geometry in this section.

The condition that three points (ξ1, η1), (ξ2, η2) and (ξ3, η3) shall be collinear is commonly expressed by means of the determinant

If the relationship between three variables a, b and c which we wish to represent nomographically is

F (a,b,c) = o, (2)

and we can express (2) in the determinantal form [similar to (1)]

where f1f2 (a), f2 (b), f3 (c), g1 (a), g2 (b) and g3 (c) represent (generally) different functions of a, b and c, it is apparent that we can associate

any three particular values of a, b and c satisfying (2) with the points

on a Cartesian graph; and they will be collinear because (3) is of the same form as (1).

The relation between actuarial and scientific matters has frequently been investigated and there are various ways of looking at their interconnexions. We can ask, among other questions, whether our activities as actuaries entitle us to consider ourselves to be scientists in any sense of the word. This is the context in which the question has been debated in the Students' Society (cf. ‘Are we Scientists?’ by W. G. Bailey & H. W. Haycocks, J.S.S. Vol. VI, p. 93). The feeling on that occasion was that the answer must be in the negative, but even if our everyday activities are not strictly scientific it could still be claimed that there exists a set of correlated topics to be called ‘actuarial science’ which is at the actuary's disposal and which is kept alive by contributions of individual actuaries.

German scholars have asserted the existence of a ‘Science of Assurance’ which, according to them, comprises and combines into an organic whole those portions of all other sciences which have some bearing on (not necessarily Life-) Assurance. We are, in this country, less prone to create abstract notions by definition, but even if we do not see any merit in ascribing a scientific status to our profession, we can ask whether there exist branches of science which have something to offer to us in our capacity as actuaries. It will be agreed that two sciences in the wider sense of the word are in some way connected with our work, viz. Economics and Mathematics. Our profession can claim historic connexions with both and members of our community have proved themselves competent in them. Should we encourage, and even extend, this connexion?

A problem of occasional practical importance is to find the extra premium for a particular class of policy, which corresponds to an extra premium fixed by the Medical Officer for a standard class of policy, here assumed to be whole life. Except when the extra mortality can be assessed by a simple ‘rating up’ of the age of the assured, this problem is not easily solved. The direct calculation of the required extra premium from specially constructed mortality tables would be very laborious, particularly in view of the necessity to reproduce the given whole life extra.

In the following note a procedure is developed which provides in the circumstances mentioned and without undue labour, an extra premium for any class of policy once it is decided whether the extra risk is of increasing, constant or decreasing type. A few examples illustrate how the method may be applied in practice.

The problem considered by Mr E. M. Thomas on pp. 253-4 of J.S.S. Vol. VII can be approached analytically as follows:

Consider the ratio where and

. This is a function of h and may be expanded in a Taylor series

The first term on the right-hand side is unity and it is easy to show that the second term is

where v = (I + i)−1.

The organizing Committee of the Twelfth International Congress of Actuaries which was to have been held in Lucerne in 1940 did me the honour of suggesting that I should give a lecture on the development of insurance mathematics, particularly the mathematics of non-life insurance. Unfortunately the war prevented the Congress, but the individual articles were published in Zurich. Those which relate to the subject mentioned are to be found in Section I which includes papers intended to answer the following questions: ‘What hypotheses underlie actuarial science and to what extent can the applications of the theory of probability and of the theory of risk be justified? Should actuaries use these theories—particularly more recent developments—in practice? Are observations made on insurance data likely to confirm the theories of probability and of risk?’ The articles in this Section are concerned only to a slight degree with non-life insurance and in Prof. Jecklin's summary review he correctly points out that up to the present the mathematician has not generally concerned himself with non-life insurance. He writes: ‘Many problems, in particular those relating to the mathematics of non-life insurance, will be only of academic interest so long as it proves impossible to test the theory by means of appropriate statistical data. We must not forget that interesting and valuable though theory may be—permitting us as it does to study effects in a model form based on specific assumptions—yet for practical purposes (for instance when premiums and reserves are to be calculated mathematically according to the theory of probability and statistics) a mass of recent statistical data must be available.