368 NATURE OF CAPITAL AND INCOME

Hence its equal, 1 —¢i, is a pure number. Since of this sum,
the term 1 is a pure number, the other term, ti, is also a pure

; ; ; : ial
number. Hence ¢ must be of a dimension reciprocal to ¢, i.e. 7

ort?
The rate of discount evidently has the same dimensionality
as the rate of interest.

APPENDIX TO CHAPTER XIII
§ 1 (ro Cu. XIII, § 1)
Formula for Present Value of Sum due in One Year

If the rate of interest be denoted by ¢, it is evident that 1+:

due next year is worth $1 to-day,
hence $1 due next year is worth Se to-day,
1
and any sum ¥ due next year is worth vi to-day,
?

which is the general formula for the present value of a single
sum due at the end of one year.

§ 2 (ro Cu. XIII, §1)
Formula for Present Value of Sum V due at End of Any Time ¢

In general, it is obvious that (1+47)* is the amount which,
two years hence, has a present value of $1, hence that
: 1
1 due at the end of 2 years is worth to-day ———
$1 due at the en years is w ay ari
14
A+)”

>

and that ¥ due at the end of 2 years is worth to-day

Similarly, ¥ due at the end of 3 years is worth to-day

 

and V due at the end of ¢ years is worth to-day a+r
The last formula is perfectly general. It is not even necessary
 that ¢ should be an integer. The reader who is mathematically
 inclined will have no difficulty in proving that the
formula applies if the period of time is 3% years or any other
time whatever.