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, \(\displaystyle \int_0^T R(t)e^{-rt} ~dt\), with \(R(t) = mP\) and letting the term \(T\) go to infinity (i.e., evaluating the improper integral \(\displaystyle \int_0^\infty mPe^{-rt} ~dt\)), we arrive at the following formula for the present value of a perpetuity.
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.
Step 1: Recall the formula for the present value of a perpetuity.
Step 2: Plug in the given values.
\(m=2\), \(P=6000\), and \(r=0.04\).
Therefore, a single payment of $300,000 is required to fund the scholarship indefinitely.