198 NATURE OF CAPITAL AND INCOME [Cmar. XII 
hence, the buyer of such an annuity at the end of one year 
may, immediately upon the receipt of his first $4, sell out 
his rights. By hypothesis they will bring $100. Conse- 
quently, he receives $104 in all for his $100 a year ago. He 
has thus virtually exchanged $100 one year for $104 the 
year after. That is, the premium rate of interest for this 
year is also 4 per cent. 
We see, then, that if the rate of interest in either of the 
two senses — price or premium — remains constant, the 
rate in the other sense will also remain constant and equal 
to the former. 
It is clear that the same reasoning applies to interest 
reckoned for any period of time, — semi-annually, quar- 
terly, continuously. 
§6 
But if the rate of interest does not remain constant, its 
two senses of price and premium are no longer interchange- 
able. Thus, suppose that the rate is 4 per cent in the 
premium sense for the first year, but 3 per cent for the 
second year and for all succeeding years. This means that 
$100 to-day will buy $104 next year, and that $100 next year 
will buy $103 the year after. Then $100 to-day will evi- 
dently not buy $4 a year forever, nor $3, but an intermediate 
amount, approximately $3.03, so that the rate of interest in 
the price sense is 3.03 per cent.’ 
Again, suppose that the rate of interest in the price sense 
is 4 per cent this year, but 3 per cent next year. This 
means that $100 to-day will buy $4 a year forever, and 
that $100 next year will then buy $3 a year forever. 
Then $100 to-day will not buy $104 next year, nor $103, 
but $1374. That is, the rate of interest in the premium sense 
is 37% per cent.? Thus a very slight change in the price rate 
of interest implies a great change in the premium rate of in- 
1 See Appendix to Chap. X11, § 3. 
2 See Appendix to Chap. XII, § 4.