ANN 
tvcgians renounced all title to the fucceflion to the illes of 
Scotland. It was paid till the year 1468, when -t-he annu¬ 
el, with all its arrears, was renounced in the contratSt of 
marriage between king James III. and Margaret daughter 
of Chriftian I. king ot Norway, Denmark, and Sweden. 
ANNUITANT,yi He that pofleflcs or receives an an¬ 
nuity. 
ANNU'lTY,y. \_aimiitc, Fr.] A yearly allowance. A 
yearly rent to be paid for term ot life or years. 'I he dif¬ 
ferences between a rent and an annuity are, that every rent 
is going out of land; but an annuity changes onlyyhe 
granter, or his heirs, that have alfets by defeent. T he 
fecond difference is, that, for the recovery ot an annuity, 
no action lies, but only the writ of annuity againft the 
granter, his heirs, or fucceftbrs; but, of a rent, the fame 
actions lie as do of land. The third difference is, that an 
annuity is never taken for affets, becaitfe it is no freehold 
in law, nor (hall be put in execution upon a fratutc-mer- 
chant, flat ute-Jlaple, or elegil, as a rent may. Cornell.^ 
If a man by deed grant to another the fum ot 2oh a- 
year, without exprefling out of what lands it Ihall ilfue; 
no land at all fliall be charged with it, but it is a mere 
perfonal annuity. 2 Blackjl. 40. 
ANNUITIES, a term for any periodical income, ari- 
fing from money lent, or from houfes, lands, falaries, 
penfions, &c. payable from time to time; either annually, 
or at other intervals of time. Annuities may be divided 
into fitch as are certain , and fuch as depend on lome con¬ 
tingency, as the continuance of a life, Sec. Annuities are 
alfo divided into annuities in pojfifficn, and annuities in re- 
vyrfion ; the former meaning fuch as have commenced ; and 
and the latter fuch as will not commence till fome parti¬ 
cular event has happened, or till fome given period of 
time has elapfed. Annuities may be farther confidered as 
payable either yearly, or half-yearly , or quarterly, see. The 
The prefent value of an annuity, is that fum, which, being 
improved at interefl, will be fufficient to pay the annuity. 
The prefent value of an annuity certain, payable yearly, 
is calculated in the following manner: I.et the annuity be 
1, and let r denote the amount of il. for a year, or il. 
Jncreafed by its intereft for one year. Then, 1 being the 
prefent value pf the fum r, and having to find the prefent 
Value of the fum 1, it will be, by proportion thus, 1 :r 
:: i : - the prefent value of il. due a year hence. In 
like manner will be the prefent value of il. due 2 
years hence; for r: 1 :: - : —. In like manner—, — > 
r r r 
Sec. will be the prefent value of il. due at the end of 
3, 4, 5, &c. years refpe&ively; and, in general, ~ will be 
the value of xl. to be received after the expiration of n 
years. Confequently the fum of all thefe, or - + -j- 
Sec. continued to n terms, will be the prefent 
r 3 r* 1 
value of all the n years annuities. And the value of the 
perpetuity, is the lum of the feries continued, ad infini¬ 
tum. But this feries, it is evident, is a geometrical pro- 
greffion, whofe firft term and common ratio are each 
r r 
and the number of its terms n\ and therefore the fum i of 
all the terms, or the prefent value of all the annual pay¬ 
ments, will be s~ —---— x -• 
r —1 r—1 r n 
When the annuity is a perpetuity, it is plain that the 
laft term — vaniflies, and therefore —— x — alfo va- 
r n r —1 r n 
ftifbes; and confequently the e.xpreffion becomes barely 
Vol. I. No. 47. 
a n 737 
s — -; that is, any annuity divided by its intereft for 
r — 1 
one year, is the value of the perpetuity. So, if the rate 
of intereft be 5 per cent, then = 2o is the value of the 
perpetuity at 5 per cent. Alfo-= 25 is the value of 
the perpetuity at 4 per cent. And ~ —33I. is the value 
of the perpetuity at 3 percent, intereft: and fo on. If 
the annuity is not to be entered on immediately, but after 
a certain number of years, as m years; then the prefent 
value of tlie reverfion is equal to the difference between 
two prefent values, the one for the tnfi term of rn years, 
and the other for the end of the laft term n\ that is, equal 
to the difference between-——x~N and— - 
r —1 r —1 r n r —1 
or 2= 
1 1 
1 
Annuities certain differ in value, as they are made pay¬ 
able yearly, half-yearly, or quarterly. And, by proceed¬ 
ing as above, tiling the intereft or amount of a half-year, 
or a quarter, as thole for the whole year were ufed, the 
following fet of theorems will arife; where r denotes as 
before, the amount of 11. and its intereft for a year, and 
n the number of years during which any annuity is to be 
paid; alfo P denotes the perpetuity-, Y denotes—— 
r —1 r —1 
—• - X—the value of the annuity fuppofed payable 
yearly, H the value of the fame when it is payable half- 
yearly, and Q^the value when payable quarterly; or uni- 
verfally, M the value when it is payable every m part of 
a year. 
Theor. 1. Y = P — P x (-) • 
r 
Theor. 2. 
Theor. 3. 
Theor. 4. 
H = P — P X (—7—) • 
r+I 4 , 
Q-== p — P x (-7—) . 
r +3 
jjt th7{ 
M = P — P X (—;— -) • 
r-\-m —1 
Example 1. 
I.et the rate of intereft be 4 per cent, and the term 5 
years; and confequently 1-04, n— 5, P—25; alfo let 
m—12, or the intereft payable monthly in theorem 4: then 
the prefent value of fuch annuity of il. a-year, for 5 
years, according as it is fuppofed payable il. yearly, or. 
£l. every half year, or £l. every quarter, or T y. every- 
month or -J^-th part of a year, will be as follows: 
Y — 25 — 25 X -821928 — 4 ‘ 45 iS 
H — 25 — 25 X -820348 — 4-4913 
Q-J= 25 — 25 X -819543 — 4-5114 
M — 25 — 25 X "818996 rz: 4-5251 
Example 2. Suppofing the annuity to continue 25 years, 
the rate of intereft and every thing el'fe being as before; 
then the values of the annuities for 25 years will be 
Y — 25 — 25 X "375 11 8 = 15-6221 
H — 25 — 25 X -371527 = 15-7118 
Q-j= 25 — 25 x -369709 = 15-7573 
15 — 25 x -368477 = 15-7881 
Example 3. And, if the term be 50 years, the values 
will be 
Y = 25 — 25 x -140713 = 21-4822 
II = 25 —-25 x -138032 = 21-5493 
Q_=i 25 — 25 x -136685 = 21-5829 
M — 25 — 25 x -135775 = 21-6056 
Example 4. Alfo, if the term be 100 years^ the values 
will be 
9 B 
