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 .

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 issues that are of chief interest to Insurance Companies as investors are the loans raised by Foreign and Colonial Governments and Municipalities, principally in the form of bonds, and Foreign and Home Corporations in the form of bonds, debenture stocks, ordinary stocks, &c. When these borrowers want money—and they have apparently much needed it during the past three years and are needing it now—they set about the attainment of their desires by calling in the help of their financial agents or advisers who may be either a financial house or a bank. The usual methods in this country are for the Issuing House to buy up the loan at an agreed figure and subsequently to issue the loan at their own price, making what profit they can, or to act merely as agents for the borrower, charging a commission for their work and the advice they give.

The particular aspect of sickness insurance to be discussed is the Waiver of Premiums under life assurance contracts on disability both temporary and permanent.

Waiver of premium on Permanent Disability was introduced in Germany in 1876, and in America in 1895, and is extremely popular. In Russia some companies offer a variation, which consists of giving the assured under a participating policy the option of relinquishing the right to participation and demanding in lieu insurance against invalidity by which he is released from all future payments of premium and immediately receives 50 per cent. to 75 per cent. of the sum assured, the remainder of the sum assured being paid at death or maturity.

The following Notes afford a mere outline, which the student must fill in by means of experience and reading. Many of the papers in J.I.A. dealing with reversions discuss, in great detail, the formulæ that should be used, and seek for an extreme accuracy which, as has been pointed out by Sir G. F. Hardy, J.I.A., XXX., p. 78, almost defeats its own end. These papers may be interesting to some people, but seem to have little practical value. Probably no two actuaries would make exactly the same deductions from a fund, use exactly the formula, tables and rates of interest, and make exactly the same deduction for costs, etc., but any two actuaries having considerable experience of reversionary transactions would reach approximately the same values. From nearly every point of view there is no “true value” of a reversion; a value only becomes “wrong” if it is so low as to make a purchase at that figure impracticable for any vendor, or so high as to make it impracticable for any purchaser.