Skc. 1] VALUE OF CAPITAL 203 
of one year, the sum of $104, the present value of this 
right, if the rate of interest is 4 per cent, will be $100. 
If the property is the right to $1 one year hence, its present 
value is evidently 5; or $0.962, and if the sum to which the 
property entitles the owner is any other amount than $1, 
its present value is simply that amount divided by 1.04 
or multiplied by .962. Thus the present value of $432 
due in one year is ju, or 432 x .962, which is $416." 
If the future sum is due in two years, and the rate of 
interest is still 4 per cent, it is evident that $1 to-day is 
the present value of $1.04 next year, which in turn (by 
compounding) will then be the present value of $1.04 x 1.04 
(i. e. [1.04] %, or $1.082) at the end of the second year. The 
$1.082 is called the “amount” of $1 at the end of two 
years, and $1.04 is the “amount” of $1 in one year. 
Similarly, in three years (1.04)° is the “amount” or sum 
worth $1 in present value; and so on for any number of 
years. These results show what $1 to-day is worth at 
the end of any number of years. And conversely, from 
them it is easy to see what $1 due at the end of any number 
of years is worth to-day. We have already seen that the 
present worth of $1 due in one year is fg or $0.962. 
Similarly, the present value of $1 due at the end of two, 
three, etc., years is respectively wig, ops etc.’ Knowing 
the present value of $1, we may evidently find that of any 
other sum by simple proportion. 
To illustrate these results geometrically, let us represent 
time by horizontal lines, and the value of the capital by 
vertical lines: then the curve 4 A” A” A’, as shown in 
Figure 1, will exhibit the relative values at any two in- 
stants, exchangeable on the basis of a given rate of interest, 
compounded annually. : , : 
The point B represents the present instant; B’, the in- 
! For the general mathematical treatment, see Appendix to Chap, 
XIII, § 1. 
2 For a mathematical formulation, see Appendix to Chap. XIII, § 2.