ANNUITIES. 



thereto is 90/. 14*. 0|r/. for this is the sum, 

 which, put out at the same rate of in; 

 will, at the end of two years, amount to 

 100/. In like manner, if a person lias 100/. 

 due to him at the end of three years, and 

 lit- wishes to have the same advance < I im- 

 mediately, the sum which ought to be 

 given as an equivalent thereto is 861. 7s. 

 Sil. for tliis is the sum which, at the same 

 rate of interest, will at the end of three 

 years amount to 100A And if these three 

 values are added together, they will make 

 272/. 6. 6d. being the sum which ought 

 to be paid down for an annuity of 1001. for 

 three years ; as this sum improved at the 

 given rate of interest is just sufficient to 

 make the three yearly payments. 



As the amount or present worth of II. 

 tor am given term is usually adopted as 

 the foundation of calculations relating to 

 annuities, let r represent the amount of 

 II. in one year; that is, one pound in- 

 creased by ayear's interest; then /", on- 

 raised to the power whose exponent is 

 any given number of years, will be the 

 amount of 11. in those years; its increase 

 in the same time is m 1 ; now the 

 interest for a single year, or the annui- 

 ty corresponding with the increase, is > 



1 ; therefore as r 1 is to r" 1, so is u 

 (any given annuity) to a its amount : 

 hence we have 



_ 



EXAMPLE. To what sum will an an- 

 nuity of 50/. amount in 6 years, at 5 per 

 cent, per annum, compound interest? 



50 xTU5l' ? ~ 1 = 340/. 19*. 1,1. 



.05 



In this manner the amount of an annuity 

 for any number of years, at any given rate 

 of interest, may be" found. But when the 

 term of years is considerable, it will be 

 more convenient to work by logarithms, 

 h\ which the labour of all calculations re- 

 lating to compound interest is greatly 

 abridged. There is, however, little occa- 

 sion in general to calculate the amount or 

 present worth of annuities, except for par- 

 ticular rates of interest, as the following 

 tables, and others of a similar nature, for 

 different rates of interest, which are given 

 in most books on compound interest, save 

 much time and labour in common prac- 

 tice, and are therefore in general use. 



TABLE I. 



Chewing the amount of an annuity of 

 I/, in any number of years not exceed- 

 ing 100, at 5 per cent, per annum com- 

 pound int : 



EXAMPLE 1. To what sum will an an- 

 niiiu of 105/. amount in 19 years, at 5 per 

 cent, compound interest .' 



The number in the table opposite to 19 

 years is 30,5390, which multiplied by 105 

 o-iv.-s the answer 3206/. 11*. IQd. 



EXAMPLE 2. In what time will an an- 

 nuity of 25/. amount to 3575/. at 5 per 

 cent, compound interest ? 



Divide3575/. by 25/.the quotient is 143 ; 

 the number nearest to this in the table is 

 142,9933, and the number of years cor- 

 responding, or 43 years, is the answer. 



The- present worth of an anuuity, or the 

 sum to be given in one present paymrni 

 as an equivalent for an annuity for any 

 given number of yours is found on similar 

 prim-iples; for as I/, is the present value 

 of r* (its amount in n years, and as the 

 prosrnt value of any other amount, and 

 consequently 



of'* must bear the same propor- 

 tion to that amount, ,wc hvc 



