We now show the manner in which a simple result of calculation answers its end. To simplify the case, suppose an office starts with 5642 individual subscribers, each aged 30 years, the mortality among them being that of the Carlisle Table. [Monratarv.] The bargain is for a temporary assurance, as it is called, of 20 years, and of 10001.: that is to'say, the executors of each one:who dies within 20 years are to receive 10001. at the end of the year of death. Money makes three per cent. once a year. According to the table, then, there are 57, 57, 56, &c. deaths in the successive years, and the following is the result, the pro-1 per premium being calculated at Ill. 12s. 34d. each person, or more exactly 11,614/. 16.t. for 1000 persons. It is supposed that there are no expenses of management. By P is meant that premiums are paid, and the number paid precedes the letter : by y, that a year's interest is received, and by c, that claims, in number as stated, are paid; small letters denote a transaction at the end of a year, and the large letter one at the beginning; the age of the parties paying premiums is in parentheses at the beginning. Fractions of pounds are neglected, I/. being written for everything above 10e.
At the outset the office receives 65,530/. from the 5642 persons assured ; this is Immediately invested at 3 per ceut., and yields 1966/. by the end of the year, making 67,496/. But at the end of the year the claims of the executors of 57 persons who have died during the year are to be satisfied, which requires a disbursement of 57,0001., reducing the society's accumulation to 10,496/. The contributors who are left, 5585 iu number, uow pay their second premiums, 64,8691., so that, these being immediately invested, the company has 75,365/. at interest daring the second year. This yields 22611., so that by the end of the year 77,626/. is accumulated. Theu comes the demand of 57,0001. on behalf of 57 contributors deceased during the year, which reduces the accumulation to 20,626/. This is more than it was at the same time last year, which is denoted by +. lu this way the company goes on, accumulating to an amouut which would lead a person unacquainted with the subject to conclude that the premium must be too large : in fact ten years give an accumulation of 91,809/. But now the state of affairs begins to change; the contributors have been diminishing, while the claims have been increasing, until the yearly incomings no longer equal the outgoings. The accumulations then come in to make good the difference in such manner that by the time the renirtieiug contributors come to be 50 years of age, and the claims of 61 who died in their fiftieth year have been satisfied, there only remains 8/. of the 91,809/. ; and this 8/. is merely the error arising from omitting shillings, &c., in the calculation. Something of the same kind must take place iu every office which dies a uatural and a solvent death : the only difference being that, when new business ceases, instead of a number of contributors all of the same age, and under similar contracts, both ages and contracts vary cou siderably.
There are certain tables which are variously named (sometimes after Mr. Barrett, the inventor; sometimes after Mr. Griffith Davies, the improver; sometimes after n and N, letters of reference used in them), but which we call commutation tables. They are de. scribed ur the' Treatise on Annuities,' in the 'Library of Uecful Know ledge,' and a copious collection is given : also in an article in the ' Com panion to the Almanac' for 1540. They very much exceed in utility those which preceded them ; awl wo shall here give part of one of them, namely, that for the Carlisle Table, at 3 per cent., which con tains the materials for judging of the demands made by an insu rance company in cases involving one life only. Opposite to each age of life are three rows of figures in columns marked n, N, and ss : and by si (r) we mean the number in column sr opposite to the 113C X.
To find the value of an annuity of 1/. on a life of any age, divide the x of that age by its D. Thus at the age of 35 the value of an i annuity of 1/. is x(35) 4. D(35), or 35126'57 or 18.4331., or 18/. 8s. Sd. Thus, tho following formulte will be readily under stood : Value of an annuity which is to commence N(;t) immediately; that is, which is to make the first payment in a year (ago .r) . . D(x) Value of an annuity which is to commence in n years; that is, to make the first x(x +a) payment in n +1 years, if the party be D(x) then alive (present age x) . .
Premium for such an annuity, payable 1 x(x+a)now and n times in all . . . . f N(X —I) The same premium, payable a+1 times . x(x+a) x(s-1)—x(x +a) Value of a life annuity for n years; or 1 payable n times at most . . . f D(x) Present value of an assurance of 1/ at 1 11(,r) death . f n(.r) 1 at(x) Premium for the same . . .
Present value of an assurance of 1/. at1 at(x +a)death if after years . . . . f D(r) Premium for the same, payable (n + 1)} at(x+n) times . ..... f s(.r —1) —s(x +T) Present value of an assurance of 1/. at1 +n)death if within a years . . . . f D(x) 31(x)—xt(x+n) Premium for the same, payable n times . , , NvZ — — 74-1) As an instance, let us take the case of the last formula, which was proposed at the beginning of this article : the age is 30, and the term of insurance 20 years; we have then to divide the excess of at (30) over at (50) by the excess of N (29) over N (49) : at (30) 932'6867 N (29) 48783'19 DI (50) 555'9583 x (49) 15347.98 div. by 32435'21 gives '0116148: this is for 1/., giving or 11/. 12s. 3id. for 1000/., and to be the total premium for 1000 persons.