206 NATURE OF CAPITAL AND INCOME [Cuar. XIII
have already seen, in the last chapter, that if a person owns
the right to $1 a year payable at annual intervals forever,
its present value, reckoned at four per cent, is &, or $25.
If his annuity is $2 per year, its present value is evidently
double this, or $50; and if it is any other sum, its present
value is found by multiplying in the same way. Thus, an
annuity of $17 is worth i. In other words, the value of a
perpetual annuity 1s found by dividing the annual income
by the rate of interest," or, what amounts to the same thing,
by multiplying the income by the rate of capitalization,
also called the number of years’ purchase. This proposi-
tion, however, serves to determine only that capital-value
which an annuity possesses at its inception (i.e. one year
before the first installment) or at any other point taken
one year in advance of the first of the installments to be
included in ‘he calculation. The value of the annuity,
taken immediately before any installment of income falls
due, is evidently greater than the above, by the amount
of that installment. Thus, if the rate of interest is four
per cent, a perpetual annuity of $4 a year, of which the
first payment falls due one year hence, is worth $100 to-
day, and is also worth this same sum at any instant
immediately following the payment of an installment.
But next year, immediately before the first payment
becomes due, it will be worth $104. At any intermediate
point between the present when it is worth $100 and a
year hence when it is worth $104, it will be worth an
intermediate amount, determined by the discount curve;
for its value will always be the discounted value of the
$104, which could be realized on it at the time of the next
imterest payment. As soon as this payment has passed by,
the value will drop to $100 again, after which it will gradu-
ally ascend as before, and so on, following a series of curves
like the teeth of a saw, as shown in Figure 2. In this
diagram, the value of the annuity is represented by suec-
1 For a mathematical statement, see Appendix to Chap. XIII, § 3.
