Overview
This lecture explains how to value perpetuities and how the perpetuity valuation formula can be used to derive the formula for valuing annuities.
Perpetuity Valuation
- A perpetuity pays a constant cash flow each period indefinitely.
- Present value (PV) of perpetuity: PV = C / r, where C = annual payment, r = interest rate.
- Example: $100 invested at 5% yields $5/year forever; PV = $5 / 0.05 = $100.
- The mathematical solution uses infinite geometric series and algebraic manipulation.
Generalizing the Perpetuity Formula
- For any constant cash flow C and interest rate r: PV = C / r.
- Assumes interest rate remains constant forever (not always realistic, but useful for modeling).
- Formula applies to valuing long-lived investments like stocks.
Annuity Valuation Using Perpetuity
- An annuity is a series of fixed payments for a specified period (not forever).
- Thought experiment: Subtract one perpetuity (starting at year 1) from another (starting at year t+1); result is an annuity.
- Difference in cash flows between the two perpetuities matches an annuity's cash flows (first cash flow at year 1, last at year t).
Deriving the Annuity Formula
- Present value of perpetuity A (starts in one year): PV_A = C / r.
- Present value of perpetuity B (starts in year t+1): PV_B = [C / r] / (1 + r)^t.
- Present value of annuity = PV_A − PV_B.
- Simplified annuity formula: PV = [C / r][1 − 1 / (1 + r)^t].
Applications and Importance
- Perpetuity formula is foundational for valuing stocks and other long-term investments.
- Annuity formula is widely used in real-world financial decisions (loans, retirements, etc.).
Key Terms & Definitions
- Perpetuity — A financial instrument paying constant cash flows forever.
- Annuity — A financial product paying fixed cash flows for a set period.
- Present Value (PV) — Today's value of a future cash flow, discounted at the appropriate interest rate.
- Interest Rate (r) — The rate used to discount future cash flows to the present.
Action Items / Next Steps
- Think about how perpetuity and annuity formulas can help analyze personal financial decisions.
- Review examples of annuity and perpetuity valuation in practice.