gen-define-annuities-and-perpetuities-and-give-one-short-worked-9817

Define annuities and perpetuities and give one short worked example.

# Annuities and Perpetuities ## **Annuity** An annuity is a series of **equal payments made at regular intervals for a fixed period of time**. **Present Value Formula:** $$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$ Where: PMT = payment, r = interest rate per period, n = number of periods ## **Perpetuity** A perpetuity is a series of **equal payments that continue forever** (infinite time period). **Present Value Formula:** $$PV = \frac{PMT}{r}$$ --- ## **Worked Example** **Problem:** You can invest in a bond that pays $500 at the end of each year. If the interest rate is 8%, what is the present value if: - a) The bond pays for 10 years (annuity) - b) The bond pays forever (perpetuity) **Solution:** **a) Annuity (10 years):** $$PV = 500 \times \frac{1 - (1.08)^{-10}}{0.08}$$ $$PV = 500 \times \frac{1 - 0.4632}{0.08}$$ $$PV = 500 \times 6.710 = \$3,355$$ **b) Perpetuity (forever):** $$PV = \frac{500}{0.08} = \$6,250$$ The perpetuity is worth more because payments continue indefinitely.