921 



INTEREST. 



INTEREST. 



923 



from one of which forma of the equation, either of the four, A, a, r, or 

 n, can be found, when the other three are known. 



From the second form it appears that the fraction of 11., which will 

 in a year amount to a pound, is 1 divided by 1 + r. Let this be called 

 : we have then 



1 1-v 



= 



Hence it is easily seen, that according as a pound is to be the amount 

 at the end of one, two, three, &c., years, the principal now necessary 

 to produce that amount is v, *, t 3 , &c., or t>* expresses " the present 

 value of U. to be received at the end of n years." Here are no less 

 than fifteen words necessary to express a fundamental result; and 

 when we speak of (1 + r)* it must be as "the amount of I/. in n 

 years." To shorten these phrases, the former might be advantageously 

 culled the nth present value, and the latter the nth amount. 



The sum which yields II. every year U called the value of a per- 

 petuity of one pound, or simply the perpetuity of It. If it be t, we 

 have 



1 1 v' p 



n- = 1,P =7> r = p p= j ;,= _ 



The reader will find an arithmetical account of ANNUITIES under 

 that word ; we now proceed to the algebraical formula; connected with 

 them. An annuity, and also a perpetuity, is always said to be created 



a year before any payment is made : thus an immediate grant of an 

 annuity payable yearly implies that the first payment is made a 

 year hence ; and similarly of a perpetuity. But hi cases where we have 

 to speak of an annuity or perpetuity, of which one payment is to be 

 made now, we propose to call them an annuity due, and a perpetuity 

 due. Again, an annuity or perpetuity deferred for, say 10 years, 

 makes its first payment in 11 years : but a perpetuity due in 10 years, 

 makes the first payment at the end of 10 years. An annuity of 20 

 years makes 20 payments ; an annuity due of 20 years makes 21 pay- 

 ments. Let all annuities mentioned be of 11., unless otherwise 

 specified. 



The present value of an annuity for n years is evidently 



for v in one year becomes If., and provides for the first payment; v- 

 for the second, and so on. The preceding is equivalent to 



1 (1 + r)" 1 



1 v 



Similarly the present value of an annuity due for n years is If. more 

 than the preceding, or 



r(l-rr) 



TABLE I. 

 Tun PRKSEXT VALUE op 1, DUE AT THE END OF ANY XTMBER op YEARS. 



TABLE II. 

 THE PEESENT VALUE op 1 PEE ANNUH FOB ANT NUMBER dp YEARS. 



