LIFE ANNUITIES. 
117 
117. Mr. Davies’s formula is an improved modification of that of 
Barrett, which first pointed out the principle of making the preparatory 
labour directly available for finding the values of temporary and deferred 
annuities, &c. Messrs. Baily and Babbage, at the end of their respec 
tive works, treat on the application of Barrett’s formula, which is thus 
obtained : 
In the expression (1) for a m in Art. 112, writing for r its value 
(l-f z) -1 , and call cc the oldest age in the table, we have 
— ~h ^ 1 ~h 0 ~4~ tm+si 1 + 0 4~ • • • • 4~ ( 1 + 0 
VIH 7 
'm 
which, by multiplying numerator and denominator by (1 -f 
becomes 
4„+, ■ ( l-H)*-^ +LU1 +ir~° n+i) +LU i+Q- ( ”‘+ 3) +.... .+4-AHQ+4 
C(l+»)'"” 
which expresses the following rule : 
Let the number of living at each year of age be multiplied by the. 
amount of £l at the end of as many years as are equal to the difference 
between the age and the oldest in the table, then the sum of all the 
products above any given age divided by the product at the given age 
will give the value of an annuity on a life of that age. The following- 
illustration is from the Carlisle-^per cent; the number in column A 
opposite to any age being the product at that age, and the number in 
column B the sum of the numbers in column A at that age and all 
ages above : the value of £ 1 per annum at any age is therefore the 
number in column B 4 at an age one year older than the given one 
divided by the number in column A at the given age. 
/ 104 X 1.04°= 1 X 1 = 1.000000 
X 1.04‘ = 3X 1.04 = 3.12 
4.120000 
Xl.04 2 = 5X1.0816 5.408000 
9.528000 
/,„ xl.04 3 =7x 1.124864= 7.874048 
17.402048 
l m x 1.04* = 9 X 1 • 169859= 10.528731 
27.930779' 
Age 
A 
B 
104 
103 
102 
101 
100 
1. 000000 
3.120000 
5.408000 
7.874048 
10.528731 
1.000000 
4.120000 
9.528000 
17.402048 
27.930779