Application of Improper Integrals

Application of Improper Integrals#

Perpetuity#

Definition and Notation

A perpetuity is an annuity (i.e., a sequence of payments made at regular intervals) in which the periodic payments begin at a fixed date and continue indefinitely.

$P =$ the size of each payment
$r$ = annual interest rate (compounded continuously)
$m =$ the number of payments per year

Annually	Semiannually	Quarterly	Monthly	Weekly	Daily
$m = 1$	$m = 2$	$m = 4$	$m = 12$	$m = 52$	$m = 365$

Present Value of a Perpetuity

By taking the present value formula for an income stream, $\int_{0}^{T} R (t) e^{- r t} d t$ , with $R (t) = m P$ and letting the term $T$ go to infinity (i.e., evaluating the improper integral $\int_{0}^{\infty} m P e^{- r t} d t$ ), we arrive at the following formula for the present value of a perpetuity.

P V = \frac{m P}{r}

Example 1#

Funding a scholarship indefinitely

A group wishes to provide a semiannual math scholarship in the amount of $6,000 beginning in six months. If the fund will earn 4% interest per year compounded continuously, find the amount of the endowment the group is required to make now.

Application of Improper Integrals

Contents

Application of Improper Integrals#

Perpetuity#

Example 1#