Therefore the amount of error in computing such additional value by Jones's formula may be found from (3) for given values of i and wx. Without wishing to devote more time to this subject than its importance warrants, I will consider the following other points con1 wx nected with Jones's formula

i + @x

1st. What is really the interpretation to be put on it? This may be easily seen on reference to vol. i, Articles 244 and 245, of his work on Annuities, but will perhaps be more obvious if the formula be converted into one more familiar to the reader. This can be done by multiplying its numerator and denominator by v. Thus we have

if vwx='x, or 'a represent a reduced in the ratio 1:1+i.

1

d + ox

1,

The latter formula coincides with the well known one, except that 'x is substituted for ; or, in other words, Jones, in designing to alter the formula for a curtate annuity so as to obtain one for a complete annuity, really retained the same formula, only making a modification in one of its terms, viz., Tx.

2nd. The increase in the value of the annuity on account of its being complete, if Jones's formula be used, is

Differentiating the Napierian logarithm of the latter quantity, viz., loge d+loge wx-loge (d+wx)—loge (d+vox), to x or wr, and equating the result to 0, according to the theory of maxima and minima, we have

Differentiating again: d being the symbol of differentiation,

(d+ox) (d+vox)'

when x= Vid; that is,

the difference between the values of a complete and curtate annuity, if Jones's formula be used for the former, attains a maximum value at age at which the annual premium =√id, and thence decreases as the premium increases; whereas it will be seen that the difference, viz.,

the

annuity, increases as r increases.

Since the value of a complete annuity as above deduced is equal

1 wx

2 d + ox

it may appear at

to that of a curtate annuity increased by
first sight that there is no difference between the two annuities, or
between the methods of carrying out the transactions in respect of

1 wx

2d+x

them, except in such additional payment of
being made for
the complete annuity. In reference to this point, the following details
as to the values, &c., of the two annuities are furnished.

It will thus be seen that not only the values and costs of the two annuities differ, but also the sums assured, premiums, and interest.

In the case of a complete annuity of 1,

d

2

-, representing the

sum to be assured and the proportion of the annuity in respect
year of death, the purchaser will receive at that time

of the

the original cost of the annuity: therefore he will be repaid the latter with a year's interest on it, as should be the case. Supposing, 1 however, the assurance to be effected for as in the case of a

d + w x'

curtate annuity, since will be received as before for the proportion of the annuity in respect of the year of death, the purchaser will

d

1+ +

2

d+x

1

is therefore greater

d + ox

Your obedient servant,

THOMAS CARR.

JOURNAL

OF THE

INSTITUTE OF ACTUARIES

AND

ASSURANCE MAGAZINE.

A comparison of Reserves brought out by the Use of different Data in the Valuation of the Liabilities of a Life Office. By JAMES VALENTINE, Assistant Actuary of the Northern Assurance Company.

[Read before the Institute, 27 April 1874.]

It is not intended in this paper to do more than go briefly again over part of the ground covered by Mr. Manly, in his admirable essay, printed in the 14th volume of the Journal of this Institute, on the Values of Policies as found by means of various Tables and Methods of Valuation.

Assuming an office to have been started with sufficient rates of premium, it will be universally admitted that no part of the duties afterwards devolving upon the actuary is more important than the determination of the data upon which the Reserves are to be calculated at the periodical investigations into the state of its affairs. The importance of this subject has led to its consideration by the leading men in our profession, many of whose contributions towards its elucidation are to be found in the pages of the Journal, and I will not presume to enter into it here further than by instituting a new comparison between the results arrived at by the use of different data.

VOL. XVIII.

R

A similar task was executed by Mr. Manly, in the essay to which I have referred, in the most complete manner, as far as the materials at his command at the time would permit, but the new Experience Tables of Mortality since published by the Institute present so valuable a field for the prosecution of fresh inquiries in the same direction, that it is worth while, I think, to undertake the work again.

Those who are familiar with Mr. Manly's essay will remember that he constructed a table, which may be said to represent a model office, showing assumed amounts of policies taken out at various ages and remaining in force at the end of stated periods, and that, having done so, he formed a second table giving the reserves, in the case of such model office, at different stages in its career, according to various data. In continuation of the second table I have calculated the corresponding reserves required by the Institute Experience Table HM, reckoning interest at 3 per cent., 3 per cent., and 4 per cent. respectively; and also those by the HM(5) Table, with HM pure premiums, at the same rates of interest. Mr. Manly's second table likewise contains a comparison of the several reserves worked out by him with the Carlisle 3 per cent. as a standard. To make a new comparison with a truer standard, formed from the more modern and better data which we now have at hand, is my present intention.

What, then, should be our standard of comparison?

Firstly, as regards mortality. Surely, if we can find a table of the mortality actually experienced among a body of assured lives, numerous enough and drawn from the records of a sufficient number of offices, to give them a representative character-and still more, if it appear that such table is confirmed by another and independent table answering the same conditions-it will need no argument to show that we have there a true exponent for this part of our standard. Other tables, formed from statistics of the whole population, or of particular districts of the country, or even from the experience of individual offices, altho' they may be framed with skill and be very useful in some ways, cannot, it seems to me, the slightest pretentions, for general life office purposes, to an equal degree of value, to say nothing of superiority. Now, in the new Institute Mortality Experience, I think we find all that is above desired. It is based upon a very large number of facts, contributed by 20 offices, while, as regards confirmation by another table similarly constructed, this is supplied in a striking manner by the

have