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 title of this lecture was chosen over a year ago before I had formed a clear picture of its scope. It is, however, adequately descriptive of my purpose which is to develop consistent general principles, as opposed to detailed methods, which can serve as a guide to the individual computer—primarily one working alone with an ordinary calculating machine. I shall endeavour to illustrate the principles by actual examples, but it will be outside my present compass to concentrate upon those methods which rely upon brilliance of technique rather than on soundness of principle.

It is clearly desirable to say precisely what ‘computation’ embraces, and I therefore put forward the following definitions of ‘calculation’ and ‘computation’—in each case it being understood that the word is prefixed by ‘numerical’:

Calculation is the application of the four rules of arithmetic to the numerical evaluation of an algebraic expression.

The records of papers read before the Students' Society or published in its Journal contain few references to the system whereby surplus is allotted to policyholders primarily in the form of reduction of premium. It is thought that this system would form a suitable subject for discussion and would be of interest to members as well as of value to those students who are concerned in their studies with Valuation and Surplus.

The description of the system contained in Mr R. K. Lochhead's book issued in the Consolidation of Reading Series is necessarily brief, and it may be convenient to enlarge on this by tracing the development of the system in one of the offices which has transacted this class of business for upwards of one hundred years dating from the early 1800's.

Collins in his Reversions and Life Interests (1925) sets out on p. 22 the five points to be considered by a purchaser in assessing the probable value of the fund which will be available when the reversion falls into possession on the death of the life tenant. These points are the risks of deferment, diminution, dispossession, depreciation of investments and default by the Trustees. The first three have been fully discussed in Collins's lectures and elsewhere, and the last is an inevitable risk which is best allowed for in the valuation rate of interest, being one of the reasons why a Life Office purchasing a reversion looks for a return on this investment a good deal greater than it expects from a first-class Stock Exchange security. It is not proposed to consider these four risks in detail here.

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).

A Few years ago two estate duty valuations were made in somewhat unusual circumstances.

A bequeathed an annuity of £75 equally amongst his five daughters, expressed to be payable for their joint lives, the whole of the annuity being subject to remainder to the survivors or survivor for lives or life. The annuity was charged on a leasehold property which was bequeathed absolutely to B, the son of A, subject to the joint-life annuity.

One of the daughters died, and for the purposes of estate duty it was necessary to know the present value of the proportion of annuity which passed on her death, i.e. the value of the annual sum of £15 payable until the last survivor of four lives aged respectively 81, 78, 77 and 71.

If a variable assumes the discrete values xj (j = 1, 2, 3, …) with specified probabilities f(xj), where f(xj) it is said to be a discrete random variable. If a discrete random variable is also a function of a continuous (non-random) variable, for convenience usually assumed to be ‘time’, it is called a discrete random process.

A class of discrete random processes of particular interest to the pure mathematician and to the mathematical statistician has been called stochastically definite by Kolmogoroff (1931). Such a random process is distinguished by the fact that the probability that the random variable concerned assumes a given value n at time t depends only on the value m assumed by the variable at time s (s < t) and not on the values assumed at any intermediate or earlier points of time. This circumstance is allowed for by writing the probability of the value n at time t in the form Pmn (s, t).

Although the first essay on the theory of risk (41) was published in 1786, this theory has been for the most part developed during this century. Moreover, the original presentation was rather difficult and the formulae inconvenient for the practical computer. Only recently has the classical theory of risk attained a satisfactory and seemingly final form. Few text-books, none of them in the English language, discuss these developments; it may therefore be useful to give an exposition of these results. As a rule no proofs will be given, instead reference will be made to the relevant literature.

This survey will be restricted to the classical theory of risk, which deals essentially with the individual insurance. It is possible to approach the problem of risk from an entirely different angle, considering not the individual insurance but all the policies in force. This leads to the collective theory of risk (27, 29, 15, 40), which is based upon the theory of homogeneous random processes. Only the classical theory will be considered here, the inclusion of the collective theory of risk would considerably extend the scope of this paper, and in any case many of the relevant papers are in English.

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?

The object of medical examination in life assurance is to guard the insurer against financial loss which might occur if the life assured suffered a heavier mortality than that underlying the premium calculation. Such a loss would occur under a whole-life assurance; on the other hand, if the sum assured became payable on the death of A provided B survived him, then the higher the mortality of B the greater the profit of the insurer.

There are, however, more complex types of assurances for which it is less easy to say whether a higher mortality in a particular life will produce a profit or loss to the insurer. Examples will be given below of assurances under which an increase in mortality will produce a profit or loss according to the age of the life in question. It is therefore necessary to give a rule to settle this point, and with it the question whether medical examination is required or not.

These notes were written after a meeting of the Students' Society at which Mr A. W. Joseph submitted some problems in reversions concerning the valuation of an annuity. It has been my experience that where a case is submitted to an actuary ‘for valuation of an annuity’ it is imperative to obtain precise instructions on the circumstances and purpose of the valuation and the exact nature of the interests to be valued. It frequently turns out that the valuation to be made does not entail the valuation of an annuity. The following paragraphs may help to indicate what may be required in a number of cases.

The orthogonal transformation used in a previous paper(1) to derive the distribution of x2 has its counterpart in a double orthogonal transformation which can be applied to a variable y dependent on another variable x in order to deduce the known tests of significance in the simple regression problem. The general theorem is due to R. A. Fisher (2), and the particular transformation now used is a special case of the problem considered by Vajda (3) in the multivariate case. Attention is also directed to papers by Bartlett (4) and Elfving (5). In the present problem the use of two successive transformations is of interest in showing what happens, and in providing a relatively elementary proof of the known results.

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 present note is contributed, not because the numerical results obtained will be of use to actuaries, but in the belief that the methods adopted may be of interest to students as illustrating the application of the concepts of theoretical statistics to a problem which arose in practice.

The theory and technique of statistical Quality Control have been described by H. C. H. Carpenter in J.S.S. Vol. v, 1946, p. 207, and it is shown there how the control lines inserted on the charts are based on the assumption of a normal distribution.

In this article we are concerned with the problem of interpreting numerically the relative variations which the policy value based on a standard mortality table undergoes when this table is replaced by one which allows for 100T% extra mortality.

This question is of particular interest from two points of view. First, it is of consequence for the technique of insurance of abnormal risks to obtain some idea of the numerical effect of percentage additions to the mortality on such important lifecontingency functions as surrender values. Further, in the treatment of this question as a typical approximation problem of lifecontingency mathematics, it appears that the solution most suited to the essence of the inquiry and for practical applications is obtained by a nomographic evaluation of the quantities under consideration. As regards the importance of nomography in the practical application of insurance mathematics the author has already made a thorough exposition and given examples in a series of earlier publications; in future, in life-contingency problems, the necessary consideration will always be given to the development of the appropriate nomographic methods, particularly in connexion with the introduction of typical approximate methods.

The statistical fate of a group of individuals who are subject to a process of ‘wastage’ forms the theme of an important class of actuarial problems. Usually the rates of the several individual sources of wastage are relatively complicated functions of such variables as age, duration, or rank, and the full apparatus of the service table must be employed. In other cases, however—including some of practical importance—these functions degenerate, in fact or in supposition, into constants, and a simpler approach accordingly becomes possible. By considering the operation of wastage—which includes such events as death, breakdown of health, resignation, and discharge—as equivalent to the financial operation of discounting, a formal identity can be created between certain ‘manpower’ problems and corresponding problems involving compound interest; and numerical work on the former is much simplified by the use of tables of common compound interest functions.

The simplicity of L. Solomon's solution to the manpower problem discussed in the preceding article is due partly to the use of compound interest functions but mainly to the fact that transfers to semi-skilled status and thence to the skilled classification each occur at a single point, after two and five years respectively, from entry. When dealing with the corresponding problems of, say, office staffs who are recruited upon leaving school or university and may serve until age 60 or 65 the conditions are very different. Transfers from one category to another, in this case promotions from grade to grade, generally occur at varying rates over a range of ages. Further, the work cannot be simplified by the use of compound interest functions, the full service-table technique must be employed.

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 variance of a continuous frequency distribution is often calculated by concentrating the frequencies in intervals of width h at the mid-points (vide Kendall, Advanced Theory of Statistics, Vol. 1, §3.16). Under certain conditions this approximation can be corrected by subtracting from the calculated variance. This note gives an interpretation in elementary terms of the ‘reason’ for such a correction.

We consider three distributions, all having the same frequency in any one of the intervals of width h (Fig. 1):

C the original continuous distribution;

H where the frequency density is uniform within any one interval;

P where the frequency in an interval is concentrated at the mid-point.