196 NATURE OF CAPITAL AND INCOME [Crap XII 
$102 is due, the debt is renewed for another six months at 
the same rate. It is evident that the $102 will then, by 
“ compounding,” amount to $102 x 1.02, or $104.04. The 
interest then, at the end of a year, instead of being $4, is 
$4.04. In other words, 4 per cent interest reckoned half- 
yearly is equivalent to 4.04 per cent reckoned yearly, as was 
also the case under the price concept of the rate of interest. 
In the same way we may consider quarterly or other inter- 
vals for compounding. At the limit, the interval for com- 
pounding may be reduced to an instant.’ 
We have then two methods of defining interest. In 
both of them the time element is prominent. Before pass- 
ing on we should here remark that the {ime element enters 
not only as referring to the times of payment but also to 
the time of contract. A rate of interest implies not only 
the two points of time between which the goods for 
exchange are available, but also the point at which the de- 
cision to exchange them is made. It would be quite pos- 
sible, for instance, to agree in the year 1900 to exchange 
$1000 in the year 1901 for a given sum or series of sums 
returnable at still later dates. In this case the rate of 
interest for this exchange appertains to the year 1900, 
although execution of the contract does not begin until a 
year later and is not concluded until later still. These 
conditions have often been overlooked in treating statis- 
tics of the rate of interest. 
§5 
We have defined the rate of interest both in the 
“price” and the “premium” sense. The question now 
arises whether these two concepts are interchangeable. 
Under certain conditions they are, and under others they 
are not. Cases in which the two are interchangeable are 
shown in the following propositions. 
g 1 For the mathematical relations involved, see Appendix to Chap. 
Il, $2.