ISO New Theorems for determining the Rate of Interest j 



As the operation of constructing tables of the amount and 

 present value of one pound annuity for any given term is not 

 very comniodiously performed, according to the common me- 

 thod, I shall in this place show how it may be effected in a much 

 easier manner. 



Let A.A' denote the amount of an annuity for n and n+\ 



^/ [ 



years respectively; then A equal - — ;-, and hence A' = (A x 1 + 



r+1). 



If, again, the same letters be made to denote the present 

 value of an annuity of 1/. per annum, we shall similarly have 



A' X 1 + r- 1 = A and A'= 4^- 



1 +r 



In this way may we examine and extend the tables at pleasure; 

 and the amount or present value of 1/. corresponding to the 

 like or any other term of years, will then be obtained by simple 

 subtraction only : and thus a process otherwise tedious is super- 

 seded. 



I come now to the determination of the rate of interest in 

 annuities; a subject that has engaged the attention of some of 

 the greatest mathematicians that ever lived. Various formulae 

 have been devised for this purpose, most of which give the value 

 sufficiently exact, when the number of years is not very great; 

 but in this latter case no dependence can be placed upon the 

 result, and at times the formulje wholly fail. Those which I 

 have to propose are the following ; to which is annexed an ex- 

 ample in each case, to show their degree of accuracy. 



But I must first premise that S and P signify the amount and 



present value of \l. annuity for n years ( — ) __ 1 = Z 



2 



(-p-j — I = Z' R.R' being two assumed rates, deduced from 



a comparison of S and P with the values nearest them in the 

 tables, R' being less and R greater, or both less, or greater, than 

 the true rate r. 



Then in either case we shall have 



._ 12Z+(n+I.R)R _ 12Z'-(n— l.R)R' 



' ~ !■;+ !+l.(R + R'j> * ~ i2_«— l.(R + R')* 



An annuity of one pound per annum in fifty years amounts to 

 '209,348/. — what is the rate of interest allowed? 



Here we see by the tables that the rate is exactly '05 or 5 per 

 cent. But let us suppose it either -0475 or '052: hence putting 



_2_ _ 



■49 



20<>.348 



— 1 = '06019 ; we have by the first formula 



50 



0475 



