382 NATURE OF CAPITAL AND INCOME

§ 10 (to Cm. XIII, § 7)
Formula for a Bond when Interest is reckoned oftener than yearly

The formula in the case of semi-annual income when interest
 is reckoned semi-annually is evidently,—

which applies just after an interest payment; just before, it

is evidently V+; and at intervening intervals it is this value

discounted, or for practical purposes, the simple formula,
V+at!, where V is the value taken after the last « interest”
payment, and # the time elapsing since that date. For the case
of continuous interest, if we let, as in the previous chapter,
3 represent continuous interest, we have,—
PZ
Y= 3 + a 4
which formula remains unchanged during the entire period of
the bond.
These various formule may, of course, be somewhat transformed
 and simplified for practical purposes. Moreover, they
may all be transformed in terms of the various rates of
interest. Some actuaries apparently prefer to use, as the interest
 rate, only the “effective” rate, i, which is what we call
the “rate of interest reckoned annually.” The preceding
formula, which employ the semi-annual, quarterly, and other
forms of interest rates, may be transformed by substituting
their values in terms of i, in accordance with the relations
shown in Appendix to Chap. XII, § 2.

§ 11 (to Cm. XIII § 8)

Formula for Capital-value of Any Series of Income Installments

We may express, in general formule, the capital-value of
any income stream, as follows: Let a,, a, a5 represent the
successive installments of income accruing at various times dis-