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 in- 
terest 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 trans- 
formed 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 in- 
terest 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-