It is not a new idea for debenture or loan stocks to be issued with an option to the lender to convert the whole or part of his holding into ordinary shares at fixed prices on certain future dates or during certain periods. There has in recent years been an increase in the number of these stocks available to the British investor and there is also a substantial number that have been issued in the U.S.A. and Canada. A typical example is the Bowater Paper Corporation Limited 5¾ % Convertible Unsecured Loan Stock 1978–82. In July 1959, holders have the right to convert each £50 of loan stock into twenty-one £1 Ordinary shares; in July 1960, into twenty Ordinary shares; and in July 1961, into nineteen Ordinary shares. There are no conversion rights after July 1961 and any stock not converted by that date will remain as a normal unsecured loan stock redeemable in the period from 1978 to 1982.

The existence of this option is, of course, attractive to the investor and this is reflected in the market price of the stock. The problem is how to determine the portion of the market price that represents the cost of the option. Of the many features that, no doubt, affect the position, the most significant seems to be whether the option is remote or not.

In a recent paper to the Students' Society (J.S.S. 15, 43), D. F. Gilley and D. Funnell pointed out that it would be theoretically possible, though normally impracticable, to make a valuation of a pension fund by projecting the fund membership year by year, in order to obtain a series of notional revenue accounts for each future year of the Fund's existence, and that the algebraic sum of the discounted values of the expected future current surpluses and/or deficits would be equal to the surplus or deficiency revealed by a valuation made in the normal way.

It is frequently possible, when an estimate of the future trend of the fund is required, to reverse this process. Instead of directly estimating the future income and outgo of the fund, and so building up the expected amount of the fund year by year, we may extend the normal valuation technique, and use a series of notional valuations to determine the future trend of the Fund. In other words, the growth or decline of a pension fund between two future dates may be expressed as the difference between the net liabilities of the fund, as revealed by notional valuations, on these two dates.

Given any probability distribution, new distributions can be derived from it by assuming its parameters to follow some specific probability distributions. A simple example of this process is provided by the Poisson distribution

P(r∣λ) = e-λλr/r! (r = o, 1, 2, …).

If the parameter λ is assumed to follow the Pearson's Type III law

then the probability of r successes is obtained as

When the question of the future constitution of the female population of this country as regards marital status was studied about eight years ago, the conclusion was reached (J.S.S.10, 24) that ‘a considerable fall in the number of marriages, to be followed by a consequential fall in the number of births, is almost inevitable during the next few years’. This conclusion was based on the fact that the number of births declined by 20% during the 1920's and this decline was likely to be reflected in the number of marriages about 20 to 25 years later.

In fact, marriages in England and Wales, which averaged about 350,000 a year during 1941–45, rose to 391,000 a year during 1947–49 (which presumably included a number of ‘demobilization’ marriages) and fell only to 356,000 on average during 1950–52, and 348,000 a year during 1953–55, whilst in 1956, the latest year for which figures are available, the marriages numbered 353,000.

The text book, The Practice of Life Assurance, by Coe and Ogborn, pp. 367–9, and the references there to other literature, give a brief summary of the subject of group life assurance; this paper attempts to bring out in more detail a little of the practice (of some U.K. offices) and some of the special problems that belong to the subject, for the benefit of students who have no practical experience of it. There is no dearth of literature on pension schemes, with which group life contracts are very often linked, so that reference to pension schemes in this paper, though necessary, will be brief.

The technique described by Miss Merriman in the preceding note can be applied, with suitable alterations, to many different types of life assurance and annuity funds as well as to pension funds. By the use of approximate methods it is possible to abbreviate the working within a comparatively small compass, whilst retaining sufficient accuracy to afford a guide for investment policy. This note describes a particular application of these methods.

The Management Committee of a large Widows' and Orphans' Fund sought the Actuary's advice on the probable future progress of the Fund. The Fund was established many years ago, but the membership has been expanding and both contributions and benefits have been increased from time to time. It was clear on general grounds that, if sufficient new entrants were admitted to maintain the present membership without limit of time, financial maturity would not be reached for many years. On these assumptions the Fund would eventually reach stability, but it would never decrease, so that from this point of view irredeemable investments could be purchased in the belief that they would never have to be sold. The adoption of this theory, however, would be tantamount to making the future surplus or deficiency of the Fund in some way contingent upon the future flow of new entrants.

Conjectural estimates of the early population of Britain are subject to wide margins of error. Professor Grahame Clark (1957) has put forward tentative figures of early prehistoric numbers, based upon the native population density in newly colonized lands. The Upper Palaeolithic population density of Britain was compared to that of Alaska in 1867 and the North-West Territories in 1911, and the Mesolithic to that of Tasmania. Since prehistory has no specific dates, being measured in periods of hundreds or even thousands of years, overall numbers must be similarly imprecise in date. Again, the colonial figures themselves are only estimates (the two Palaeolithic estimators being in the ratio 20:3), so that the estimates of prehistoric population might not even be of the correct order of magnitude.

Estimates of between a half and one and a half million have been made for the population of Roman Britain in the second century A.D., based upon different interpretations of the evidence for population density of town and country (Wheeler, 1930). Here we have agreement on the order of magnitude, but the estimates are still imprecise in date and size.

Three of the writer's earlier notes in this Journal (see References) have discussed respectively the effect, upon pension fund contribution rates, of changes in the salary scale, the rate of interest and the entry age, and approximate methods have been supplied for estimating rapidly the effects of the two last-mentioned changes. The object of the present note is to propose a method for estimating the effect on the survivorship ratio ly/lx according to a service table, of changing one of the sets of decremental rates in the same proportion at all ages, or of removing a set altogether, or of introducing a new set in addition to those already present. For example, given a service table based on certain rates of mortality and withdrawal, we shall be able to estimate, with a negligible amount of arithmetical work, the new values of such ratios as l60/l25 resulting from removing the mortality rates altogether, from doubling or halving the withdrawal rates or from introducing a new set of decremental rates.

This article attempts to develop a practical system for dealing with extra-risks whilst retaining consistency in the terms offered to proposers. The extra premiums derived vary according to the view taken of the incidence of the extra-risk, and tables are produced to enable the underwriter to deduce quickly the required special terms.

W. Perks' paper, ‘The Treatment of Sub-Standard Lives in Practice’ (J.I.A.78, 205), describes a system whereby the special terms are fixed according to uniform percentage increases in mortality throughout the term. This approach does not make a broad assessment of the incidence of the extra-risk. The present writer feels that some allowance for incidence should be made, and the process that follows attempts to adhere to the narrow theoretical path. The scope of this article does not include the interesting fields of underwriting practice and experience which were admirably discussed in W. Perks' paper.