Annuities and loans
Growing annuities and perpetuities
Value payments that rise at a constant rate — rents, salaries, dividends — and prove the formula with a schedule.
18 minintermediateMembers
Learning objectives
- Value a growing perpetuity and a growing annuity
- Identify the first payment correctly before applying growth
- Verify a closed-form value against a year-by-year schedule
Many payment streams are not level: rents escalate by a fixed percentage, salaries and pensions rise with inflation, dividends grow with earnings. Valuing them as level annuities understates them, sometimes badly.
When payments grow at a constant rate g, there are closed-form formulas — the same ones behind the Gordon growth model and the terminal value in every DCF.
PV = C₁ ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))ⁿ] Perpetuity: PV = C₁ ÷ (r − g)
- C₁
- the first payment, at the end of period 1
- g
- constant growth rate per period
- r
- discount rate per period (the perpetuity needs r > g)
- n
- number of payments
Included with a Formula Quest membership
The rest of this lesson is for members. Members get the full lesson, including its 3 graded exercises, and every other Finance Quest lesson.