LIFE-ANNUIT Y. 641 
glac?, for the fake of avoiding death, to re-commence a life 
equally or even lefs adyantageous in point of happinefs 
than that which they have experienced. From this num¬ 
ber our author does not except thofe pretended philofo- 
phers, who limit exiftence to the prefent ftate ; who are 
continually complaining of the miferies of life, and yet 
have not the courage to put an end to it. As to thofe 
whom reafon and religion infpire with a well-founded 
hope of a future exiftence, and of a continued progrefs to¬ 
ward perfection, though they have as lively a fenfe as 
others of the pleafures of this life, which they confider as 
a natural preparation for a future Ifate, they would never 
be defirousto re-commence their career; which, whatever 
pleafures it might afford, would only retard their advance¬ 
ment toward that perfeCt Hate for which they know they 
are defrined. 
LIFE (Tree of). See Thuya. 
LIFE-ANNUITY, an annual income, the payments of 
which depend on the continuance of any given life or 
lives. 
The value of a life-annuity is properly the fum that 
will be fufficient to enable a Jelkr (allowing for the chances 
of mortality) to pay the annuity without lofs ; and, fup- 
pofing money to bear no intereft, it is always equal to the 
expectation of the life. For example : Obfervations fliow, 
that, according to the mean probabilities of the duration 
of human life, the expectation of a life aged ten is nearly 
forty years; or, in other words, that a fet of lives at this 
age will, one with another, enjoy forty years each of ex¬ 
iftence, fome of them enjoying a duration as much longer 
as others enjoy a (hot ter. It is obvious, therefore, that, 
fuppoling money to bear no intereft, 40I. in hand for each 
life would be fufficient to enable a feller to pay any num¬ 
ber of fuch lives il. per ann. for their whole duration; 
or, in other words, that 40I. is, on this fuppofition, the 
value of a life aged ten. But, if any improvement is made 
of money by putting it out to intereft, this will be more 
than the value ; becaufe it will be more than fufficient to 
pay the annuity; and as much more than fufficient as the 
improvement or the intereft is greater. If, for inftance, 
any fum now in hand may be fo improved, by being put 
out to intereft at 4 per cent, as to double itlelf in eigh¬ 
teen years; the feller of fuch an annuity will (in confe- 
quence of putting out half the purchafe-money to intereft) 
find himfelf, at the end of eighteen years, in poffeffion 
of 42I. or of 20I. more than is fufficient to pay the re¬ 
mainder of the annuities, though he fliould make no far¬ 
ther improvement of the purchafe-money. If he puts out 
the money to higher intereft, he will be a greater gainer; 
if to lefs, he will be a lefs gainer; but at any rate of in¬ 
tereft he mull be a gainer. The truth is, that, fuppoling 
the intereft to be that juft mentioned, or 4 per cent, and 
all the improvement poffible made of the money at this 
intereft, he will find 171.10s. 6d. for each annuity (inftead 
of 40I.) to be fufficient to enable him to make all his pay¬ 
ments ; (fee the Tables at the end of this article.) but 
that, if die improves the money at 5 per cent, he will find 
15I. to be fufficient. 
It may feem to follow from hence, that we have nothing 
to do, to find the value of a life-annuity, but to find the 
expectation of the life, and then to take out of the com¬ 
mon tables the value of an annuity certain for a term of 
years equal to the expectation ; and it may appear ftrange 
that this fliould not give the true value. The truth is, 
that it will give the value greater than it is ; or that a lefs 
fum than that found in this way will be fufficient to pay 
the annuity. Suppofing the intereft 4 percent, the value 
of an annuity certain for forty years is 1 9^79277, (fee Ta¬ 
ble II. under the article Annuities, vol. i. p. 739.) or 
191. 15s. iod. but the value of a life aged ten, at this rate 
of intereft, is, as hath been juft faid, no more than 
17I. 1 os. 6d. The principal reafon of this is the differ¬ 
ence between the value of forty payments of an annuity 
to be made every year regularly one after another, till in 
forty years they are all made; and the value of the fame 
rifther o'" payments to be made.at greater diftances of 
Voi. XII. No. 859. 
time from one another, and not to be all made till the end 
of feventy or eighty years. In this laft cafe there is more 
time given for the improvement of the purchafe-money, 
and therefore a lefs fum will be fufficient to enable a fel¬ 
ler to make his payments. All that is learned from know¬ 
ing the expectation of a number of lives, is the mean num¬ 
ber of payments that will he made to each of them, and 
not the time in which they will be made. For example: 
The expectation of a life at ten being forty years, it fol¬ 
lows that to a hundred lives at this age, forty payments 
for each life, or four thoufand in all, will be made. But, 
as all the lives will not be extinCl in lefs than feventy or 
eighty years, many of the payments will not be made till 
after the expiration of forty years; and therefore a part 
of the purchafe money will be improved for a longer 
time than forty years. In general, it may be obferved 
that one-half nearly of the payments of a fet of life- 
annuities will be made after the expiration of a term of 
years equal to the expectations of the lives ; and that, 
this halt having a longer time for accumulation than the 
expectations of the lives, the value of the lives mult be 
lefs than the value of annuities to be paid regularly every 
year for a time equal to the expectations. "Thus 1980!. 
will, in confequence of being improved at 4 per cent, pay 
a hundred annuities of il. for forty years. But a lefs 
fum (or 1750I.) will pay a hundred fuch annuities to a 
fet of fives whole common expectation is forty years; be- 
caul'e one-half nearly of the payments will not be made 
till after the end of forty years, and fome not til! after the 
end of feventy or eighty years; and confequently one- 
half nearly of the purchale-money will be improved for 
more than forty years, and fome of it for more than fe¬ 
venty or eighty years. 
Thefe obfervations demonftrate, that it is a miftake to 
reckon the value of a life-annuity the lame either with 
the value of an annuity certain for a term of years equal 
to the expectation of the life; or with the value of an an¬ 
nuity fora term certain, equal to that which a fife has ail 
even chance of exifting. The true method of computing 
the values of life-annuities may be explained in the fol¬ 
lowing manner. Let us fuppofe that the duration of the 
annuity is to be only one year. That is, that il. is to be 
paid a year hence, provided a life now of a given age 
fliould be then in being. Were it certain that this life 
would not fail in the year, the value of the annuity would 
be the fame with the value of it payable a year hence, or 
with the fum which, now put out to intereft, would in- 
creafe to il. in a year; and this fum, fuppoling intereft at 
4 per cent, is by the Table juft quoted ‘96154, or 19s. 3d. 
But, the payment not being to be made Ihould the life 
happen to fail in the year, this fum ought to be dimi- 
niffied in proportion to the degree of the uncertainty of 
the fife’s continuing to exift through the year; and it is 
eafy to fee that this uncertainty , or chance, is in the propor¬ 
tion of the number of perfons at that age living at the 
end of the year, to the number living at the beginning of 
it, as fliown by obfervations. For example: If it has been 
found, in any fituation, that but half the number of per¬ 
fons of the given age living at the beginning of the year 
are living at the end of it, the uncertainty will be as half; 
and the value juft mentioned ought to be lefl'ened one- 
half. If it appears that two-thirds, or nine-tenths, or 
ninety-nine hundredths, are living at the end of the year, 
the fame value mult be diminilhed only one-third,, one- 
tenth, or one-hundredth. That is, it will be neceffary to 
multiply it by J-, or Univerfaljy then, the pre¬ 
fent value of any fum to be paid a year hence, provided a 
given life fliould be then exifting, is that fum multiplied 
by the value of xl. payable at the end of the year, and 
all'o by the fraction formed by making the number of the 
living at the age of the given life (taken out of the fol¬ 
lowing Table) the denominator, and the number of the 
living at the next fucceeding age (or at the end of the 
year) the numerator. For example : Let the value be fought 
of il. payable a year hence, if a child aged ten fliould be 
then living, reckoning intereft at 4 per cent. The value 
3 A of 
