In the following remarks:—

P means Office Premiums.

π means Net Premiums.

D means Claims, Surrenders and similar outgo.

E means Expenses and Commission.

i means Earned rate of interest.

i means Valuation rate of interest.

ϕ means Loading, i.e., P –π either for a particular policy or for the whole business.

1. Surplus is the balance of estimated assets over estimated liabilities, and is therefore dependent upon the valuation assumptions. Some valuation gains or losses are not real, e.g.:—

(a) Strain of new business.

(b) Mortality profit (this is usually quite fictitious).

(c) Interim Bonuses.

1. The exact value of qx is

On the assumption of an uniform distribution of deaths over the year of age

If deaths are not uniformly distributed all these quantities may differ. This may be shown by expressing the first and last in the terms of the second, thus by Taylor's theorem:—

1. In considering the question of the calculation of Survivorship Assurances (excluding Compound Survivorship Assurances), it is only necessary to know how to obtain the values of Survivorship Assurances which determine on the first death. All Survivorship Assurances in which only one order of survivorship is involved may be expressed in terms of assurances determining on the first death. This is most easily accomplished by a process of general reasoning, but to those who prefer strictly analytical methods, an algebraical solution will be more acceptable. To take quite a simple example, suppose we are required to find the value of an assurance payable on the death of (x), provided (y) predecease him and (z) survive him, in symbols .

It is the purpose of these notes to set out briefly the most prominent features of Pension Funds. No attempt will be made to go into the many intricate questions involved when formulae have to be deduced.

The primary purpose of a Pension Fund is to make provision for the declining years of life, and therefore some form of benefit accruing at superannuation is always included. It is found, however, that allowance has usually to be made for other causes of cessation of activity, and so benefits upon retirement through ill-health, upon voluntary withdrawal, and upon death before the attainment of the age which normally qualifies a member for a pension, are introduced. Thus, in a full scheme, after certain initial periods have passed, practically the only case in which no tangible benefit is derived is that in which a man is dismissed for misconduct.

It will be recognised at once that with four main types of benefit as a basis and an illimitable fund of human ingenuity upon which to draw, the number of different allowances found in practice is very great. Only a few can be indicated here.

The usual form of contract provides for the payment of the sum assured if the child survive the selected period with a death benefit of—

(a) A return of all the premiums paid, or

(b) A return of all the premiums paid accumulated at a low rate of interest, say, 2 %.

In view of the nature of the death benefit and the fact that the commission (usually 2½% of the premiums) and expenses are small, it is usual to ignore mortality and charge a premium based on interest only, the net 3 % rate being fairly common. Expenses and profit are derived from surplus interest. Sometimes a procuration fee of 10s. is payable, with renewal commission of 2½ %, and in such a case the question of adjusting the rates must be considered, particularly if the death benefit includes interest.

(i) The characteristic feature of Double Endowment Assurances is that for a given term of years the annual premium is almost independent of the age, provided the age at maturity does not exceed a certain limit. If we assume it to be precisely true that the annual premium is independent of the age, it follows that the premium must be identical with that for a Capital Redemption Policy for the same term of years and assuring an amount equal to the Double Endowment.

The subject of Bimetallism is not a burning question now as it was some few years ago, but it is worth while to consider the arguments used for and against, as it is possible thereby to obtain an idea of the principles underlying currency problems.

1. Alleged evils—

(a) A great fall in gold prices, causing

(1) A disturbance of the standard of value,

(2) Increased burden on debtors,

(3) Discouragement of industry.

The following notes were put together in order to collect into a convenient form for reference a few ideas as to what may have been the procedure adopted in the Graduation of the Annuitants' Experience.

The Student is greatly indebted to Mr. Hardy for the information given both in the volume, “Account of the Principles and Methods, &c,” and for affording subsequently, in his Lectures on Graduation, the opportunity to follow his work, and is again fortunate in having, for purposes of reference, the stores of information to be found in Mr. Ackland's paper to the Faculty, and Mr. Todhunter's Reviews in Vols. 38 and 43 of the Journal.

Valuation Age: The valuation age is usually taken as either the nearest age or the age last birthday at the date of valuation. The former would appear to be the more reasonable method as annuity rates are frequently quoted for half ages or even quarter ages.

Grouping (Single Lives): For valuation purposes the annuities are grouped according to the office dates of birth, calculated to give the valuation age on the assumption chosen. Sub-grouping may be necessary to facilitate selection being taken into account if considered desirable.

The most convenient way of explaining the special features of Collecting Friendly Societies is, I think, by contrasting them with (I.) Other Friendly Societies, and (II.) Industrial Assurance Companies.

Under Section 8 of the Friendly Societies Act, 1896, a comprehensive list of the objects of Friendly Societies is given, but the Collecting Friendly Societies have practically limited their objects to Death Benefits and Endowments. This limitation is important, as it accounts to a large extent for the characteristic features of Collecting Societies.

In the Friendly Societies Act there is a provision that all expenses of administration are to be met entirely out of a separate Management Fund, but there is no restriction as to the amount of that fund nor how it is to be built up. All Friendly Societies are allowed to make their own arrangements in their rules.