Perpetuities Lec A perpetuity is a type of annuity that provides an infinite series of payments. Unlike regular annuities, perpetuities have no end date, meaning the payments continue forever. The concept is widely used in finance to value perpetual bonds, endowments, and other cash flows that do not have a finite life. — Key Characteristics of Perpetuities • Payments are equal and occur at regular intervals (e.g., annually). • The series of payments continues indefinitely. • The present value of a perpetuity depends on the payment amount and the interest rate. — Formula for Perpetuities The present value (P V ) of a perpetuity is given by: PV = P r where: • P V : Present value of the perpetuity. • P : Payment amount per period. • r: Interest rate per period (expressed as a decimal). 1 Derivation: A perpetuity can be thought of as an infinite series: PV = P + P P P + + + ··· 2 (1 + r) (1 + r) (1 + r)3 This series converges to: PV = P r — Applications of Perpetuities • Perpetual Bonds: Bonds that pay interest forever, often used by governments or corporations. • Endowments: Funds that generate perpetual payments for scholarships or charities. • Preferred Stocks: Stocks that provide a fixed dividend indefinitely. • Real Estate: Properties that yield perpetual rent payments. — Types of Perpetuities 1. Simple Perpetuity: Fixed periodic payments. 2. Growing Perpetuity: Payments that grow at a constant rate over time. For a growing perpetuity, the formula is: PV = P r−g where: • g: Growth rate of payments (as a decimal). • r > g: The interest rate must exceed the growth rate for the formula to hold. 2 — Example Problems Example 1: Simple Perpetuity You invest in a perpetual bond that pays $500 annually. The interest rate is 5%. What is the present value of the bond? Solution: Using the perpetuity formula: PV = 500 P = = 10, 000 r 0.05 Answer: The present value of the bond is $10,000. — Example 2: Growing Perpetuity An endowment fund generates $2,000 in the first year, and the payments grow at a rate of 2% annually. If the discount rate is 6%, what is the present value of the endowment? Solution: Using the growing perpetuity formula: PV = P 2, 000 2, 000 = = = 50, 000 r−g 0.06 − 0.02 0.04 Answer: The present value of the endowment is $50,000. — Problems 4.3 textbook 1. A perpetual preferred stock pays a dividend of $8 per year. If the discount rate is 4%, what is the value of the stock? 2. A real estate property generates $10,000 annually in rent indefinitely. Assuming a discount rate of 7%, calculate the present value of the property. 3. A scholarship fund pays $1,000 annually and grows by 3% per year. The discount rate is 5%. What is the present value of the fund? 3