Sec. 5] CONCEPT OF RATE OF INTEREST 197 
(1) If the rate of interest, in the sense of a premium on 
this year’s goods in terms of next year’s goods, is the same 
year after year forever, then the rate of interest considered 
as the price of capital in terms of a perpetual annuity will be 
equal to it. 
A numerical example will make this clear. We shall sup- 
pose that the rate of interest is four per cent in the premium 
sense, i.e. that $100 at any moment during the period under 
consideration will buy $104 to be paid one year later. We 
are to show that as a consequence $100 will necessarily buy 
an annuity of $4 a year forever. Let us suppose, then, the 
premium rate being 4 per cent, that $100 is spent for $104, 
to be repaid one year later. Of this $104, when it is 
received at the end of the year, the investor reinvests $100. 
By our hypothesis of an unchanged interest rate, this $100 
will bring, at the end of the second year, another $104, 
of which in turn $100 is reinvested; and so on indefinitely. 
By continually reinvesting, he obtains for his original $100, 
$4 a year indefinitely and $100 deferred indefinitely. If 
the process is perpetual, the $100 is deferred to infinity, 
and has no present value. Hence the original $100 obtains 
simply a perpetual annuity of $4 a year, and the rate of in- 
terest in the price sense is therefore also 4 per cent, which 
was to have been proved.' 
Tt is evident that this reasoning may all be put in general 
terms, and that it applies equally to interest reckoned 
semi-annually, quarterly, ete., and continuously. : 
(2) Conversely, if a given rate of interest in the price 
sense holds good to-day, next year, two years later, and SO 
on indefinitely, then the rate of interest in the premium 
sense will be equal to it. 
This also may be readily shown by an example. If $100 
will buy $4 a year forever, the first $4 being due one year 
ts in obtaining the present value of 
d adding the results. This process 
btaining the capital-value of a 
! An alternative proof consis 
each successive item of income an 
is exemplified in the next chapter in 0 
perpetual annuity.