UNION ASSURANCE OFFICE.

Established 1714. Commenced Life Business 1813.

The following are extracts from the Board of Trade Returns in reference to the quinquennial valuation and distribution of profits, up to 30 June last. We have ascertained that the Company has issued no other Bonus Report on the occasion.

Balance Sheet on the 30th June, 1872.

The date up to which the Valuation was made is the 30th June, 1872. The Tables of Mortality used in the Valuation were the new Experience Table (HM), published by the authority of the Institute of Actuaries, for all ordinary Insurances, both with and without Profits, and the Carlisle Table for all special policies.

The Rate of Interest assumed in the calculation was Three per cent. throughout.

The whole of the loading has been reserved, being the excess of the Premium payable over the pure Premium, according to the Table used for the Valuation.

The principle of the Distribution of the Surplus is to make a special reserve for future contingencies, and to appropriate four-fifths of the balance to the Policy-holders, and one-fifth to the Shareholders.

The principles upon which the Valuation and Distribution were made were determined partly by the Deed of Settlement, and partly by resolution of the Board of Directors.

The Consolidated Revenue Account for Five Years, from 1st July, 1867, to the 30th June, 1872:

£

S. d.

Expenses of Management

Shareholders' proportion of Profits carried to Fire Fund
Bad debts, written off

Amount of Funds on 30th June, 1872, the end of the
the period, as per Fourth Schedule

5,023 12 3

£1,206,373 8 9

£ s. d. 390,527 19 2 14,477 5 8

267 13 1

30,476 13 8

29,222 6 7
17,175 10 6
563 10 6

723,662 9 7

£1,206,373 89

Summary and Valuation of all the Policies (see p. 377.)

Valuation Balance Sheet of the Society as at 30th June, 1872.

DR.

To net liability under Assurance and Annuity trans-
actions (as per Summary and Valuation)
Reserve for claims which had arisen, but which had
not been proved on 30th June

88,565 8 4

£742,007 9 3

CR.

By Life Assurance and Annuity Funds (as per Balance-
Sheet).

10,285 6 7

5,516 14 7

2,542 18 6

£742,007 9 3

By the Deed of Settlement the Bonus is ordered to be distributed amongst such persons as shall have participating Policies in force at the time the same shall have been declared, unless such Policies were effected in the then current year, or in the then last preceding year.

The total amount of profit made by the Company in the five years is £88,565 8s. 4d., in which is included the amount of undivided profit reserved at the last division of profits.

Of this amount £53,247 38. 8d. is divided at once amongst 4,035

Summary and Valuation of all the Policies in force as at 30th June, 1872.

Policies for the sum of £2,049,126, and £22,006 8s. 9d. retained
to form a special reserve as stated above.

The Bonus allotted to every Policy for £100, which has been in
force for five years and upwards, is £5; the equivalent amounts of
cash will be found in column A, and the equivalent reductions of
Premium for five years only in column B, of the following table :—

NUMBER OF YEARS FOR WHICH POLICY FOR £100 HAS BEEN IN FORCE.

The average rate of interest at which the Life Assurance Fund was invested at the close of each year of the Quinquennial Period was as follows:

The following is a table of minimum values allowed by the Society for the surrender of its Whole-Life Policies; it has no fixed rule for the purchase of Endowment Assurances, each case being valued upon its own merits:

The Society has no business at other than European rates, and in valuing Policies on unhealthy lives the rated age is that at which the valuation is made; if, however, the Policy is valued for surrender, the true age is taken.

CORRESPONDENCE.

ON THE EMPLOYMENT OF THE " INSTITUTE OF ACTUARIES' LIFE TABLES" IN FINDING THE VALUE OF AN ANNUITY ON THE LAST SURVIVOR OF THREE LIVES.

To the Editor of the Journal of the Institute of Actuaries. SIR,-It may be well to point out that notwithstanding the terms of "Simpson's formula" are so conveniently tabulated in the Institute Tables, the employment of these tables in the mode recently suggested to you involves the introduction of an error from which the ordinary formula is free. The formulas may be written as follows. Simpson's:

axyz=(ax+ay-axy)+(ax+az-axz)-(ax+aw-αxw).

The ordinary formula:

axyz=(ax+ay+az)-(axy +αxz +αyz) +αxw.

Both are equally tainted with the error of the assumption (expressed at the close of each), that because awayz, therefore dxw=axyz; but the difference in the notation of these two formulas will be found (upon cancelling whatever is common to both) to be that Simpson's third term has aw instead of the ayz in the second term of the common formula. These are theoretically equal, but upon converting Simpson's third term in gross into figures by adopting for aw a value tabulated in the Institute Tables, a new error is here introduced through the incorporation of a value for aw which is only a more or less close approximation to the true and known ayz instead of being exactly equal to it as contemplated in the formula itself; and to this circumstance the difference 0935 between the results of the two calculations on p. 267 is traceable. Thus, in extracting 20-8569 or a30,56 (that is, a30+a56-a30,56) the value 11-7242 of a56 replaces the known true value 11-8177 of a40,50; and their difference is the 0935 just mentioned. To guard against the introduction of this difference, which will produce sometimes an excess and sometimes a deficiency, the notation of Simpson's formula, if for use with the Institute Tables, may be altered to

axyz=αxy+axz+axw-(ax+ayz)

Working in this manner with the given example, we have