390 NATURE OF CAPITAL AND INCOME
all his buying and selling in two equal amounts and at semi
annual intervals, the ratio of his annual income to his capital,
reckoned at these times, is ¢'. In the same manner it may be
shown that if his buying and selling take place at quarterly
intervals, the ratio of his income to his capital reckoned at these
times will be i'!, i.e. the rate of interest per annum reckoned
quarterly ; and so on indefinitely until we reach the limiting
case, approximately true in practice, in which the merchant buys
and sells daily, when we find that the annual net income,
divided by the value of the capital at any instant, is equal
to the rate of interest reckoned continuously. It will be observed
that to make this proposition hold good it is necessary
that the valuation of the merchant’s capital shall be, not
its wholesale price nor its retail price, but something inter-
mediate, which shall take account of the fact that the stock
cannot all be sold immediately, and that, on the other hand, it
will not be necessary to wait a year before it is all sold. Similar
reasoning may evidently be applied in case the time of turn-
over of the stock is more or less than a year.
§ 15 (ro Cn. XIII, § 11)
Influence of Variability of Rate of Interest
Thus far we have treated the rate of interest in successive
years as invariable. As a matter of fact, the rate of interest
is constantly fluctuating. We shall suppose at first that these
fluctuations are foreknown, and for convenience we shall confine
ourselves to a year as the standard interval of time. Let us
assume that the rate of interest for the first year is 7,, for the
second year, i, for the third year, i, and so on indefinitely, all
of these being supposed to be known in advance. By means of
these rates of interest we can calculate the present value of
any item or series of items of income. Thus, if $1000 is due
in two years and the rate of interest for the first year (4) is
5%, while the rate for the second year (iy) is 3%, we can
obtain the present value of the $1000 by discounting it at 3%
for one year, thus obtaining wo or $970.87, as its value a
