36
ON THE VALUE OF ANNUITIES.
very nearly, and a still nearer approximation may be made, if upon
trial the result is not found sufficiently exact, by proceeding in the same
manner with the value just obtained.
For common purposes, Table 5, containing the amounts of £l per
annum may be used; for if we divide the amount of the annuity by the
annuity, we obtain the amount of £l per annum, and the nearest quan
tity to this opposite the given number of years will give (by observing
the rate per cent under which this is found) an approximation to the
rate sought.
At what rate per cent will £20 per annum amount in 10 years to
£232.07?
s 232.07
— = : — = 11.6035 — amount of £l per annum in 10 years:
«20 1
referring to Table 5, we find this sum lies between the amounts of £l
per annum in 10 years at 3 and 3^ per cent.
11.7314 = amount of £l per annum at 3ij per cent . 035
11.4639 = .. .. ditto .. .. 3 per cent .03
difference
.2675 difference in the amounts . 005
11.7314—11.6035=.1279
.2675 : .005 :: .1279 : .00239;
this being added to . 03 gives . 03239 ;
call this iand make the true rate i = i' + ~ ;
of interest.
then
+ -i‘
a
(1 + i'y
2 =
U (1 + i')
i'V-i -
s
a
.000412
1 + .375837 - 1.375425
13.32276 — 11.6035
= 00024.
1.7192
i = i' + 2 = .03239 + .00024 = .03263, which result is very near
the truth, the true value being .032625, or £3 5 3 per cent: if .03263
had not upon trial proved sufficiently near, we might then have ob
tained a still nearer approximation by assuming i' = .03263, and
repeating the process.
The sum . 03239 obtained by adding to the rate per cent the propor
tional part obtained from the differences, is sufficiently near for most
purposes, it differing only 6d. per cent from the true rate.
48. When the annuity is payable m' times a year, and interest is con
vertible m times, — (if a whole number) is the number of periods
m
interest is convertible in the interval between any two payments of
the annuity; the amount of £l in the m'th part of a year is therefore
1 + 1
