Annuities & Perpetuities

Annuities & Perpetuities

An annuity is a series of equal payments made at regular intervals. A perpetuity pays forever.

Annuity Present Value

For payments of PMT per period over n periods at rate r:

$PV_{annuity} = PMT \cdot \frac{1 - (1+r)^{-n}}{r}$

This formula sums the geometric series of discounted cash flows: $PV = \sum_{t=1}^{n} \frac{PMT}{(1+r)^t}$

Perpetuity Present Value

When n → ∞, the annuity formula simplifies to:

$PV_{perpetuity} = \frac{PMT}{r}$

Examples

• Annuity: $100/year for 5 years at 5$432.95
• Perpetuity: $100/year forever at 5$2000

These are used to value bonds, loans, leases, and dividend-paying stocks.