Encyclopaedia Britannica > Volume 3, Anatomy-Astronomy

240
ANNUITIES.
Popular merit of each year during the term, but subject to failure
View. with the life or lives assured.
84. But by reasoning as in No. 74, it will be found that
an annual premium, payable at the commencement of each
year in the whole duration of the life or lives assured,
will be worth one year’s purchase more than an annuity
on them payable at the end of each year; and, conse¬
quently, that if the value in present money of an assur¬
ance on any life or lives be divided by the number of
years’ purchase an annuity on the same life or lives is
worth, increased by unit, the quotient will be the equiva¬
lent annual premium for the same assurance.
85. Ex. 2. Required the annual premium for the assur¬
ance of L.l on a life of 50 years of age.
In No. 82 the single premium for that assurance was
shown to be 0'397143, and the value of an annuity on the
life is 11-66; therefore, by the preceding number, the re-
0-397143
quired annual premium will be = f°r
the assurance of L.l; and for the assurance of L.1000
it will be L.31. 7s. 5d.
86. Ex. 3. Let both the single payment in present
money, and the equivalent annual premium, be required
for the assurance of L.l, on the joint continuance of two
lives of the respective ages of 45 and 50 years.
The value of an annuity of L.l on the joint continu¬
ance of these two lives appears by Table VII. to be
T n nnn u r 20—9'737 10-263 TA/1DQ„1/1 .
L.9-737, therefore ■ 20+1 ~ —21— ^ IS
0-488714
the single premium, and iQ.737 = L.0-0455168 the
equivalent annual one, for the assurance of L.l to be paid
at the end of the year in which that life becomes extinct
which may happen to fail the first of the two.
Therefore, if the sum assured were L.500, the total
present value of the assurance would be L.244. 7s. 2d.,
and the equivalent annual premium L.22. 15s. 2d.
87. Ex. 4. Let both the single and the equivalent an¬
nual premium be required for the assurance of L.l on the
life of the survivor of two persons now aged 40 and 50
years respectively.
The value of an annuity on the survivor of these two
lives was shown in No. 66 to be 15-066, therefore, by
20—15-066 4-934
No. 81, the single premium will be — _ ——
20 —j— 1 21
= L.0-234952; and, by No. 84, the annual one will be
L.0-234952 T ____
16-066 -L-0'0146242-
That is, for the assurance of L.l to be received at the
end of the year in which the last surviving life of the
two becomes extinct.
Therefore, for the assurance of L.5000, the single pre¬
mium will be L.l 174. 15s. 2d., the equivalent annual one
L.73. 2s. 5d.
88. Ex. b. What should the single and equivalent an¬
nual premiums be for an assurance on the last survivor
of three lives of the respective ages of 50, 55, and 60
years ?
The value of an annuity on the last survivor of them
was shown in No. 68 to be 14-001, the single premium
, 1 . , 20—14-001 5-999
should therefore be =z L.0-285666,
, , , L.0-285666 _ A _
and the annual —— = L.0-0190431, for the assur¬
ance of L.l to be received at the end of the year in
which the last surviving life of the three may fail.
For the assurance of L.2000, the single premium would
therefore be L571 6s 8d., the annual one L.38. Is. 9d.
89. Lemma. If an annuity be payable at the commence- Popular
ment of each year which some assigned life or lives may View.
enter upon in a given term, the number of years’ purchase
in its present value will exceed by unit the number of
years’ purchase in the present value of an annuity on the
same life or lives for one year less than the given term,
but payable, as annuities generally are, at the end of each
year.
Demonstration. Since the proposed life or lives can
only enter upon any year after the first by surviving the
year that precedes it, the receipt of each of the pay¬
ments after the first, that are to be made at the com¬
mencement of the year, will take place at the same time
and upon the same conditions as that of the rent for the
year then expired of the life-annuity, for a term one year
less than the term proposed : this last-mentioned annuity
will therefore be worth, in present money, just the same
number of years’ purchase as all the payments subsequent
to the first which may be made at the commencements of
the several years.
And since the first of these is to be made immediate¬
ly, the present value of the whole of them will evidently
exceed the number of years’ purchase last mentioned by
unit; which was to be demonstrated.
90. If, while the rest remains the same, the payment of
the annuity which depends upon the assigned life or lives
entering upon any year is not to be made until the end of
that year ; as the condition upon which every payment is
to be made will remain the same, but each of them will
be one year later; the only alteration in the value of the
whole will arise from this increase in the remoteness of
payment, by which it will be reduced in the ratio of L.l
to the present value of L.l receivable at the end of a
year. (2.)
91. When the value of an annuity on any proposed life
or lives for an assigned term is given, it is evident that the
value of an annuity on the same life or lives for one year
less may be found, by subtracting from the given value
the present value of the rent to be received upon the
proposed life or lives surviving the term assigned.
92. Proposition. The present value of an assurance on
any proposed life or lives for a given term is equal to the
excess of the value of an annuity to be paid at the end
of each year which the life or lives proposed may enter
upon in the term, above the value of an annuity on them
for the same term, but dependent, as usual, upon their
surviving each year.
Demonstration. If an annuity, payable at the end of
each year which the proposed life or lives may enter up¬
on during the given term, be granted to P upon condition
that he shall pay over what he receives to Q at the end
of each year which the same life or lives may survive, it
is manifest that there will only remain to P the rent for
the year in which the proposed life or lives may fail; that
is, the assurance of that sum thereon for the given term
(77) ; which was to be demonstrated.
93. From the last four numbers (89-92) we derive the
following
Rule
for determining the present value of an assurance on any
life or lives for a given term.
Add unit to the value of an annuity on the proposed life
or lives for the given term, and from the sum subtract the
present value of one pound to be received upon the same
life or lives surviving the term ; multiply the remainder by
the present value of Ju.l to be received certainly at the end of
a year, and from the product subtract the present value of an
annuity on the proposed life or lives for the term.
This last remainder will be the value in present money