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.

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).

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.

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.