Overview
This lecture summarizes four essential valuation formulas—perpetuity, growing perpetuity, annuity, and growing annuity—frequently used in economics, finance, and business.
Perpetuity
- A perpetuity is a constant cash flow (C) received indefinitely.
- The present value (PV) of a perpetuity is calculated as C divided by the interest rate (R): PV = C / R.
- Example: $100 forever at a 10% rate equals a present value of $1,000.
Growing Perpetuity
- A growing perpetuity starts with cash flow C, increasing each period at a constant growth rate (G).
- The present value formula is PV = C / (R - G), where R > G.
- Example: $100 cash flow growing at 2% per year with a 10% interest rate gives PV = $1,250.
- The growth rate (G) must be less than the interest rate (R) for a valid result.
Annuity
- An annuity provides a fixed cash flow (C) for a set number of periods (T).
- The present value depends on C, R, and T.
- Example: $50,000 yearly for 20 years at 10% is worth much less today than the total $1,000,000 received.
Growing Annuity
- A growing annuity is a payment C that increases by growth rate G each period, paid for T periods.
- The present value requires C, G, R, and T as inputs.
- Example: $50,000 in year one, growing at 2% for 20 years, results in final year's payment of $72,800 and PV ≈ $486,500.
- Growing annuities are worth more than fixed annuities with the same initial payment and duration.
Key Terms & Definitions
- Perpetuity — a fixed cash flow received indefinitely.
- Growing Perpetuity — a perpetuity where the cash flow grows at a constant rate each period.
- Annuity — a fixed cash flow received for a specified number of periods.
- Growing Annuity — an annuity where the cash flow increases by a constant rate each period.
Action Items / Next Steps
- Practice calculating present values using the four formulas with various interest and growth rates.