378 NATURE OF CAPITAL AND INCOME

curve KB itself represents the capital-value at all points ot
time.
The formula for the present value (V') of the annuity just
after each installment of income is the same, whatever the
time intervals between installments. This is strictly true, however,
 only on the proviso that the rate of interest ¢ is to be
understood as reckoned in accordance with the frequency of
the installments of income in each case, — semi-annually, quarterly,
 etc., —instead of annually, as has been hitherto understood.

The formula for the capital-value, V, just before an install
ment is evidently found by taking the preceding formula for
Vv and adding a. This gives,—

a 1
aaa

At intermediate points the capital-value is equal to the
amount just named discounted for the interval elapsed between
 the point of time considered and the next installment

of income.

§ 8 (ro Cu. XIII, § 7)
Formule for Value of a Bond

To be general, let us suppose a bond, the income from which
is a dollars per year, payable annually for ¢ years, at the end of
which time, in addition to the final payment a, another larger
payment P, called « principal,” is paid. We are required to
find the present value, V; of these future expected payments
for a given rate of interest, i. :
The discounted value of the terminable annuity has already
been expressed, namely, — ;

a
gto,
i (d+)

The discounted value of P deferred ¢ years has also been

explained and is evidently,—
P