It is impossible in these notes to attempt any comprehensive review of the system of compulsory Unemployment Insurance which has operated in this country since 1912. Moreover, the paper by Messrs Kyd & Maddex read before the Institute in January 1929 gives full information up to that date. But I want to start by outlining the original scheme, as I think the way in which it has been modified and extended may be found of interest.

The system began with the National Insurance Act, 1911, Part II. Contributions commenced in July 1912 and benefit in January 1913. Only a few industries which were considered to carry a specially heavy risk of unemployment were included, and the numbers insured at the outset were about 2¼ millions. Contributions were 2¼d. per week each from employer and worker, and 1⅔d. from the Exchequer. Benefit was 7s. per week, with a limitation of one week's benefit for every five contributions paid, and a maximum of fifteen weeks' benefit in a year.

Redemption yields on redeemable securities may conveniently be found with the aid of tables of Bond Values. These tables make no allowance for the fact that income tax is payable on the full dividend payments with no adjustment for profit or loss on redemption.

Those investors who are taxed on the full dividends require to know the net yield after allowing for tax, or, more usually, the gross equivalent of this net yield, commonly known as the “equivalent gross yield”. It is often inconvenient to deduct tax from the dividend rate and find the net yield direct, as the net dividend frequently does not coincide with the rates for which tables are available. There is, however, a very convenient and extremely accurate approximation to the required equivalent gross yield which is as follows:

To find the equivalent gross yield, reduce the unexpired term in proportion to the rate of tax payable and enter the table for the full nominal dividend rate.

The problem considered is that of Todhunter's Textbook (1936 edition), ch. VIII, para. 6, p. 176, but the method employed is suitable also for finding the yield on a loan issued at a price different from par, and redeemable through the operation of a cumulative sinking fund, applied in drawings at par or at a premium.

Let a′ be the value of an annuity of 1 payable annually for n years; it is required to find i′ the unknown rate of interest.

An examination of any published set of tables of annuity values, such as Oakes's Tables, will indicate that rate of interest i, to the nearest ⅛ %, at which is most nearly equal to a′.

Let dashed symbols refer to functions calculated at rate of interest i′.

It should not be difficult to extend the work already performed by Mr Craig to devise a formula which will be accurate to seven places of decimals even in very unfavourable circumstances (e.g. very long or short terms for very high or low rates of interest). Mr Craig's formula gives the correct value of ρ for the example given in the Textbook, but is not so accurate for more unusual conditions.

Developing as in Mr Craig's note:

We have, as far as terms involving ρ2 in numerator and denominator,

or

and ρ1= t0/t1.

An examination of the form taken by the values of i/ρ — i/ρ1 for various values of n, i and ρ suggests that it is very nearly independent of ρ for intervals up to considerably more than 1/16%.

It would thus probably be possible to find the form of this function from calculated values.

These notes deal mainly with that part of the investigation of securities which involves the study of company accounts.

At the outset, it should be stressed that the figures of various companies are not computed on a common basis. In company accounts, as in an actuarial valuation, the same general principles can be modified to give varying results, with the important qualification that even the general basis does not have to be published. There is therefore plenty of scope for variations in accounting practice, and hard and fast conclusions cannot be drawn from figures alone. In this connexion, one cannot do better than quote the remarks made by Mr Arthur Chamberlain at a company meeting last year, when he said:

I do not suppose there are many of my audience who are simple enough to believe that a company's accounts, particularly those of a holding company, set forth the actual and exact profits earned in each year.… I do not think it would be too cynical to say that the ordinary shareholder is lucky if he can read from the figures of a balance sheet more than an idea of the general tendency forward or backward, unless the directors are themselves aware of it and wish him to.… It is possible within the law to present a more or less truthful but entirely misleading position of affairs.

In the present state of company law, therefore, it is of no use trying to produce complicated formulae or elaborate indices intended to indicate true share values at a glance.