ANN

170

A N N

ANNUITIES,

Anv periodical income, arising from money lent, or fromhouses, lands, salaries, pensions, Ac. payable from time totime, either annually or at other intervals ot time. Annuitiesmay be divided into such as are certain, and such as de-pend on some contingency, as the continuance of a life, &c.Annuities arc also divided into annuities in possession, and an-nuities in reversion : the former meaning such as have com-menced ; and the latter, such as will not commence till someparticular event has happened, or till some given period of timehas elapsed. Annuities may be farther considered as payableeither yearly, or half-yearly, or quarterly, &c. Annuities may besupposed to be improved,'either in the way of simple or com-pound interest. The last of these hypotheses, being the mostequitable, both for seller and bu} er, is commonly assumed.

Annuities Certain.

The present value of an annuity, is that sum, which, be-ing improved at interest, will be sufficient to pay the annuity.The present value of an annuity certain, payable yearly, is cal-culated in the following manner. Let the annuity be 1, andlet r denote the amount of 1/. for a year, or 1/. increased by itsinterest for one year. Then, 1 being the present value of thesum r, and having to find the present value of the sum 1, it will

1

be, by proportion thus, r : 1:: 1: — the present value of 1/. due

r

a year hence. In like manner, ^ will be the present value of

if. due two years hence; for r: 1

1 1

In like manner,

1 i A j- c . w ill be the present value of 1 f. due at tire endr 3 , r*, r J ’ 1

1

of 3, 4, 5, &c. years respectively; and in general, — will be

the value of If. to be received after the expiration of n years.

11 1 1

Consequently the sum of all these, or — + ^ + -^ +

&c. continued to n terms, will be the present value of all the nyears annuities. And the value of the perpetuity, is the sum ofthe series continued ad infinitum.

But this series, it is evident, is a geometrical progression,

1

whose first term and common ratio are each —, and the number

r

- «f its terms n; and therefore the sum s of all the terms, or thepresent value of all the annual payments, will be s = —

1 1

When the annuity is a perpetuity, it is plain that the last

1 1 1

term — vanishes, and therefore-

r r —

— X —- also vanishes; andn r n 9

the difference between

i i,i

-; X — and -

— 1 r" r —

— x Hence also the present

consequently the expression becomes barely s = y_ ^ ; that is,

the annuity divided by the interest of if. for one year, is the va-lue of the perpetuity.

If the annuity is not to be entered on immediately, butafter a certain number of years, as ni years ; then the pre-sent value of the reversion is equal to the difference betweentwo present values, the one for the first term of m years andthe other for the end of the last term n: that is, equal to

worth of a perpetuity in reversion, of If. to commence after m1 1

years, is, —- X —''

Annuities certain differ in value, as they are made payableyearly, half-yearly, or quarterly. And by proceeding as above,using the interest or amount of a half year, or a quarter, as thosefor the whole year were used, the following set of theorems willarise; where r denotes, as before, the amount of 1/. and its in-terest for a year, and n the number of years, during whichany annuity is to be paid; also P denotes the perpetuity

-; X — the value of the an-

r — 1 r«

, Y denotes

/ i i i /- —- i y/*

nuity supposed payable yearly, II the value of the same whenit is payable half-yearly, and Q the value when payable quar-terly ; or universally, M the value when it is payable every m,part of a year.

Theor. 1. Y = P — P X (—) n .

Theor. 2 . II = P — P x (—;—)

r -(- r

Theor. 3. Q = P — P x ( —7-j)

r + 3

Theor. 4. M = P — Px (

0

r + wi — 1

EXAMPLE.

Let the rate of interest he 4 per cent, and the term 5 years;and consequently r = 1-04, n — 5, P = 25; and also let m =12, or the interest payable monthly in theorem 4: then thepresent value of such annuity of 1 1. a-year, for 5 years, accord-ing as it is supposed payable 1 1. yearly, or one-half of a poundevery half year, or one-fourth of a pound every quarter, one-twelfthof a pound, ever}’ month, or one-twelfth part of a year, will h«ras follows:

Y = 25 — 25 X ’821928 = 4-4528

11 = 25 — 25 X -820348 = 4’9413

Q = 25 — 25 x ’819543 = 4-5114

M =r 25 — 25 X -818990 = 4-5251

Annuities may be considered as in arrears, or as forbom forany number of years; in which case each payment is to be con-sidered as a sum put out to interest for the remainder of the termafter the time it becomes due. And as 1/. due at the end of 1year amounts to r at the end of another year, and r- at the endof the third year, and to r 3 at the end of tire 4th year, and so on ;therefore, by adding always the last year’s annuity, or 1, to theamounts of all the former years, the sum of all the annuities andtheir interests will be the sum of the following geometrical series,

1 -f- r + r 2 -(- r 3 -f- r‘ - --to r"" -1 , continued till the last

term be r n ~', or till the number of terms he n, the number ofyears the annuity is forborn. But the sum of this geometrical pro-

rn — 1

gression is --- which therefore is the amount of \l. annuity

forborn for n years. And this quantity being multiplied by anyother annuity a, instead of 1, will produce the amount of thatother annuity.

The amounts of annuities, or their present values, are easilyfound, by means of the follow ing tables:

TABI.£