366 NATURE OF CAPITAL AND INCOME

interest, in the price sense, was defined as the ratio of
income to capital, when the first installment of income is
due at the end of the first time-interval. But if the first
installment is due at the beginning, the ratio of income to
capital is no longer the rate of interest, but may be called a
sort of rate of discount.
Thus, if $100 will buy a perpetual annuity of $4 a year, the
first installment being due one year hence, then $104 will buy
such a perpetual annuity, the first installment being due at
once. In the first case, the ratio of income to capital, ti, is
called the “rate of interest in the price sense.” In the second
case, the ratio of income to capital, 14, may be called “the rate
of discount in the price sense.” The rate of discount in the
price sense and the rate of discount in the premium sense are
related in a manner strictly analogous to the relation between
the rates of interest in the two respective senses.
As is well. known, bonds, just before an installment of
interest is due, are sold in either of two ways: they may be
sold without the interest payment (“ex-interest”), or with the
interest payment (“flat”). These two methods are associated
respectively with the rate of interest apd rate of discount, as
just described.

§ 7 (ro Cu. XII, § 7)

Mathematical Relations between the Rates of Discount for Different
Time Reckonings
The rates of discount for different time reckonings are
related in a manner quite analogous to that in which the corresponding
 rates of interest are related, as shown in this Appendix,
 § 2. Thus if V to-day will buy W in half a year,
V d'
as i ge
w >
and if W will then buy V' in another half year on the same
terms,

a’
ey

Ww
y="
whence, multiplying, w == (1 7)