they were about to make that experiment at his expense, he did not think it was probable that man would give a willing acquiescence to the proposition. Then, if a large amount of expense is justifiable in the case of young companies and he was not prepared to say it was not it appeared to him that a large proportion of that expenditure should be borne by some one besides the assured. It was difficult perhaps to determine what proportion could be fairly charged to the assured. They had had the opinion of an eminent actuary to the effect that 50 per-cent of the first premium might be spent. He thought that, when this sum is named for such a purpose, those who approve it are apt to forget that the class of policyholders who are brought in by this extreme pressure, which is so highly paid for, are the first to leave, and that the company really do not get the commodity which they think they are purchasing. At the same time, if they really secure a body of policyholders who would remain faithful and go on paying into the company's coffers for the term of their expectancy, as the theory at starting is that they will do, he should in that case hesitate to say that an expense equal to 50 per-cent of the first premium may not be justified. He thought that if they wanted a principle to help them they must come to this-What is there in the premiums which they have a right to spend? What do the assured contribute for this purpose? Speaking broadly, that ought to be the measure of maximum expenditure which an office should feel itself justified in maintaining. But he further thought that the best thing which could happen to the insurance world would be a well considered and very considerable reduction of premium.

Mr. BAILEY briefly replied. Among other remarks, he said it was true that the cost of a policy for £100 is greater than one for £1000. What was the conclusion from that? Why, that the man who assured for £100 ought to pay more in proportion. He believed that some day they would get back to the ancient system of charging

entrance fees.

Mr. Sprague requests us to print the following remarks on the subject of Mr. Bailey's paper:

It is quite true that every new policy effected brings a new partner to share in the profits; but the new partner's share in the profits may be so arranged that an increased rate of expenditure, incurred in inducing him to join the business, shall not prejudicially affect the interests of the previous members. This may very simply be done by postponing the time at which he shall be admitted to participate, or by providing that the first premium (or, if necessary, the first two years' premiums) paid by him, shall not be taken into account in calculating his share of the profits. Suppose that the expenses attaching to the first year's insurance are so heavy as, together with the current risk, to absorb the whole of the first year's premiums. All that would be necessary, in order to guard against the assured suffering any diminution of profit in consequence of a large increase of new business and a large increase in the total rate of expenditure of the office, is to provide that the first premium paid by each new member

shall be wholly disregarded in all calculations for the purpose of ascertaining the reserve to be made for the liabilities, the surrender value, and the division of profits; so that, for example, a policy taken out in the year 1870 at the age of 30, shall in all calculations for any of these purposes be considered as effected in the year 1871 at the age of 31. While in this way the assured would suffer no loss, they would benefit by the increase of the amount of the renewal premium income, over which the fixed establishment expenses are to be spread.

Mr. Bailey's measure of the advantage derived by the assured is a very simple but not altogether a satisfactory one. It is the proportion the profit allotted to the assured bears to the total premiums paid since the last division of profits. But this is unfair and misleading, as between mutual and proprietary companies, and does not afford the slightest assistance in judging of the relative advantages those two classes of companies offer to the assured. It is true Mr. Bailey has given a column (No. 4) showing the percentage of the annual premiums which are on participating policies, but it would have been better if he had shown how these percentages are to be used, and had himself given the corrected results they lead to. If he had compared the profits allotted to the policyholders, with the total sum assured under the participating policies, or with the annual premiums on these policies given in schedules 5 and 6, he would have obtained a measure of the benefit to the assured greatly preferable to the one he has adopted.

Probably, on further consideration, Mr. Bailey may be inclined to attribute greater importance to the basis on which the valuations are conducted; in particular, the rate of interest used in the calculations, and the deduction made from the value of the premiums payable. Take, for example, the office No. 24. This is conducted at a rate of expense, 11-4 per-cent, which must be admitted to be moderate. The rate of profit to the assured is however only 7.9 per-cent, whereas in No. 22, which was established about the same date, has nearly the same premium income, and is conducted at a slightly greater rate of expense, namely 116, the rate of profit to the assured is 21.3. No. 18, again, in which the rate of expense is 10.9, shows a rate of profit to the assured of 302. These differences in the rate of bonus to the assured are probably to be explained by the smaller reserve made by No. 24 in its former valuations, and the larger reserve made by No. 18. At the present time, No. 24 makes its valuations partly at 3 and partly at 4 per-cent interest, reserving only 14 per-cent of the value of the premiums payable; whereas No. 22, which values. partly at 3 and partly at 4 per-cent, reserves 34 per-cent of the value of the premiums. No. 18, again, makes its valuations throughout at 3 per-cent. Whatever the true explanation, these figures conclusively demonstrate that, in speculating as to the prospects of future bonuses, other things must be regarded besides the rate of expenditure. In the three cases cited above, the rates of expenditure are respectively 114, 116, 109, whereas the rates of profit to the assured are 7·9, 213, 30-2. In these cases, at all events, the effect produced by the rate of expenditure on the rate of profit to the assured, must be insignificant in comparison with that produced by some other cause or causes.

On the Value of a Reversionary Annuity payable oftener than Once a Year. A paper read before the Actuarial Society of Edinburgh. By WILLIAM EVANS, A.I.A. and A.F.A., of the Scottish Widows' Fund Life Assurance Society.

AT one of the meetings of this Society last session a paper was read on this subject by Mr. Meikle, which gave rise to considerable discussion on the question; and as no definite result was arrived at, I believe a further investigation of the problem will not be without interest. Accordingly, I propose to go over the various formulas given for the value of this annuity, and offer some remarks upon each, stating wherein they are essentially different, and what appears to me to be the exact interpretation of each.

Before proceeding to consider the annuity payable oftener than once a year, let us glance at the problem in its simplest form, as a clear understanding of the conditions of the yearly annuity will greatly assist us in comprehending the more complex forms of the problem. The expression for the general term of the ordinary reversionary annuity is

(1—nPy) (nPx.v2)=nPx.vn — nPxy.vn,

from which we get, by giving to n the values 1, 2, 3, &c., the two

....),

the first of which is a and the second ary. Now, I wish particularly to draw your attention to the date when the annuities are payable. This date, you will observe, is not left to be decided when the annuity falls in; but is fixed from the very first, and is the anniversary of the day on which the risk is undertaken. Thus the first payment of the annuity will be due at the end of the year in which y dies, reckoning from the date of the risk, and may be from 1 to 365 days after that event; while the last payment will be made at the end of the year previous to that in which x dies. It is assumed therefore that, on the average, the first payment will be made six months after y's death, which is the practical interpretation of the formula. Hence we may say that the formula ax-axy expresses the value of a reversionary annuity to a after y, the first payment to be made six months after death of y, and to be paid yearly thereafter, with no proportion to date of a's death.

Let us now proceed to consider what alterations should be

made in the foregoing formula when the annuity is payable halfyearly, quarterly, &c. The following are the formulas which from time to time have been given as expressing the value of a reversionary annuity payable half-yearly,

I. ax-axy

1

Ary, given by Mr. Holmes Ivory;

II. ax-ɑxy, alluded to by several writers;

i

3

5

III. (ax—ɑxy)(1 + i2- ¿3+..........), given by Mr.

4 16 32

Sprague; and

1

IV. ax-axy+

1+axy

ly +1 ly

+1 (1+da.+1)}, by Mr. Meikle.

As the number of reversionary annuity transactions coming under the notice of the actuary is daily increasing, it is of considerable importance that the exact interpretation of the above formulas should be known. These having been given at widely different dates and by different writers, their relation to one another has not been prominently brought forward. It is therefore my intention, in the present paper, to point out what seems to me to be the exact interpretation of each, and thus to show the causes of their differences.

1

I. In vol. iv of the Assurance Magazine there is a paper "On "the Method of Approximating to the Values of Deferred and "other Life Annuities when payable Half-yearly and Quarterly," which was read before the Institute by Mr. Holmes Ivory, as far back as April 1854. In this paper we find the value of a reversionary annuity when payable half-yearly, represented by the formula a ̧—ɑxy—A (using the modern notation); and when payable quarterly, by ɑx-αxy-§A. The reasoning by which Mr. Holmes Ivory establishes his formulas is simple enough—it is to this effect: If a be alive at the end of the year in which y dies, the value of the annuity will then be that of an annuity payable in advance (an annuity-due) upon a single life at the age which a shall then have attained. Now the deduction from an ordinary yearly annuity-due, to get the half-yearly annuity-due, is approximately, whatever the age of the life is; and it follows that in the reversionary annuity the value of this at the commencement of the transaction is A. Also the deduction from the ordinary yearly annuity-due, to get the quarterly annuity-due, being nearly, the value of the reversionary annuity payable quarterly is consequently less than the yearly by A. The point to be borne in mind

when making use of Mr. Ivory's formulas is, that the first payment of the annuity, whether payable yearly, half-yearly, quarterly, or oftener, is due at the end of the year in which y dies. The value of the half-yearly annuity is therefore necessarily less than the yearly, and the quarterly annuity less than the half-yearly; for, the first payment of each being due at the same time, if a die in the first half of a year, will have been paid for that year under the half-yearly annuity, whereas had the annuity been yearly, the full 1 would have been paid. Again, if a die in the first quarter of a year, will have been paid for that year under the quarterly annuity, while would have been paid had the annuity been half-yearly.

The above remarks apply also to annuities payable by any number of instalments, so that we have for the general formula

This formula was objected to by Mr. Thomas Carr, in a letter to the Assurance Magazine (vii, 110), because it involved "the "conclusion that a mode of payment which would be selected by "the annuitant for his convenience, and apparently to his advantage, "should be a saving of expense." The fact however is, that the annuity Mr. Ivory had in view, although made payable half-yearly or quarterly, to suit the convenience of the annuitant, is really not so valuable to him as when payable yearly, and consequently should cost him less. Mr. Carr, however, admitted that the formula is correct, although it implies "that the first payment "takes place half a year, on an average, after death." Now this is the very basis upon which Mr. Ivory constructed his formulas, as set forth at the commencement of that part of his paper which relates to reversionary annuities.

How far Mr. Ivory's views may agree with the general practice of offices is another question; which, however, is not of great consequence in the present investigation. What I wish at present to elucidate are the exact conditions of the annuities of which Mr. Ivory has given solutions.

From what I have said, it appears that Mr. Ivory's formulas are approximate solutions of the following problems—

1st. The value of a reversionary annuity of 1 to x after y, payable half-yearly; the first payment of to be made at the end of the year in which y dies (i.e., practically six months after that event), and the last at the end of the half year previous to that in which a dies.