RatedrillPerpetuity
A perpetuity is a level payment that continues forever; its present value is PV = C / r, the payment divided by the discount rate per period. If the payment grows at a constant rate g, the value is PV = C1 / (r - g), where C1 is the first payment.
Syntax
PV = C / r growing: PV = C1 / (r - g)| Part | What it is |
|---|---|
C | Payment per period, received at the end of each period |
r | Discount rate per period |
g | Constant growth rate of the payment (growing perpetuity) |
C1 | First payment, one period from now (= C0 x (1 + g)) |
Examples
100 per year forever at 5%
PV = 100 / 0.05Result: PV = 2,000.00
Preferred stock paying 4 per share, required return 8%
PV = 4 / 0.08Result: PV = 50.00 per share
Dividend just paid 50, growing 3% a year, r = 8%
PV = 50 x 1.03 / (0.08 - 0.03)Result: PV = 1,030.00
This is the Gordon growth model
Common mistakes
- Using the last dividend instead of the next one in the growing formula; multiply the current payment by (1 + g) first.
- Entering a growth rate equal to or above the discount rate: the formula gives a negative or infinite value and does not apply.
- Mixing periods: a monthly payment must be divided by a monthly rate, not an annual one.
Practice Perpetuity
These drills check your answer on live data, free, no account needed.
- PerpetuityA foundation wants to fund a scholarship that pays $500 at the end of every year, forever. If the fund earns 3.25% per year, how much must it set aside today?
- Growing perpetuityA rental property's net cash flow next year will be $8,000, and it is expected to grow 1% a year forever. At a required return of 9%, what is the property worth today?
Questions
How can something that pays forever have a finite value?
Because each payment is discounted more than the last, the present values shrink toward zero and the infinite sum converges to C / r.
What is a deferred perpetuity worth?
Value it at the period just before the first payment with C / r, then discount that lump sum back to today: 100 per year starting in year 4 at 5% is 2,000 / 1.05^3 = 1,727.68.
Related
- Present value of an annuityHow to find the present value of equal payments with PV = PMT x (1 - (1 + r)^-n) / r. Worked examples, Excel PV and why this is the loan formula in disguise.
- Growing annuityThe present value of a growing annuity is C1 / (r - g) x (1 - ((1 + g) / (1 + r))^n). Examples for rising rents, the special case r = g, and mistakes.
- Present valueHow to calculate present value with PV = FV / (1 + r)^n, what discounting means, examples with yearly and quarterly rates, the Excel PV function and mistakes.
- WACCWACC = E/V x Re + D/V x Rd x (1 - T): the blended after-tax cost of a company equity and debt. Step-by-step examples, where each input comes from, errors.