370 NATURE OF CAPITAL AND INCOME

of the capitalized annuity if the annuity is payable annually,
semi-annually, or quarterly. If the annuity be $4 a year, the
teeth drop $4 if payable annually, $2, if semi-annually, and $1,
if quarterly. If the frequency of the installments of income be

2
Annual Payments

 

2 2 2 2
2 3 4

Semi-annual Payments

LALA ad
1 2 3
Quarterly Payments
Fic. 34.

indefinitely increased, we reach the limiting case of a continuous
income, when the teeth disappear entirely and the value of the
annuity remains at a constant level. The value of the annuity
of $4 in all these cases, just after any installment of income
 is received, will be $100, if the rate of interest
is 49, provided the rate is respectively reckoned annually,”
semi-annually, and quarterly in the various cases. In the
case of continuous income, the value of the annuity of $4
will always be $100 if the rate of interest be 49, “reckoned
continuously.” The same remarks apply, of course, to an i
annuity of any number of dollars. Its value after each installment
 is equal to the annual income divided by the rate of interest,
 or the annual income multiplied by the purchase period.
In order to obtain the formula for the value of a perpetuity
payable semi-annually, quarterly, or continuously, in terms of i,
the rate of interest reckoned annually, we need only to transform

> 2 . by means of the equations in the previous chapter. Thus