Perpetuity Calculator
Calculate the present value of a perpetual periodic payment (with or without growth).
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At a glance:A perpetuity is a cash flow that continues forever, paying a fixed amount each period. Its present value is the sum of every future payment discounted back — although payments never stop, the time value of money makes distant payments approach zero, so the total converges to a finite number. A plain perpetuity has PV = D ÷ R (D is the periodic payment, R the discount rate); if the payment grows at a constant rate G, use the growing-perpetuity formula PV = D ÷ (R − G), where G must be less than R. It is used in Hong Kong to value preferred shares, perpetual bonds or tenanted property, and is the core model behind the terminal value in discounted cash flow (DCF) valuation.
Formula
Plain perpetuity: PV = D ÷ R, D = periodic payment, R = discount rate.
Growing perpetuity: PV = D ÷ (R − G), G = growth rate (requires R > G).
The condition R > G is necessary; otherwise PV diverges to infinity.
How to Use
- Enter the fixed periodic payment.
- Enter the discount rate (required return).
- Enter a growth rate if the payment grows each period (0 if none); the present value shows instantly.
FAQ
Why does a perpetual payment have a finite present value?
Although the payments never stop, the time value of money makes each distant payment worth less, approaching zero. Summing all the discounted payments converges to a finite number — the perpetuity's present value.
Why must the growth rate be below the discount rate?
If the growth rate is equal to or above the discount rate, each period's payment no longer shrinks in present-value terms and the sum diverges to infinity, so no finite present value exists. This calculator therefore requires R > G.
What are some real-world examples of a perpetuity?
Fixed dividends on preferred shares, perpetual bonds (e.g. the old UK Consols), and the rental income of a tenanted property can all be approximated as perpetuities, though in reality the payments may not literally last forever.
Why does a small change in growth rate move the value so much?
Because the denominator is (discount rate − growth rate); the smaller the gap, the larger — and non-linearly larger — the present value. With a payment of HK$10,000 and a discount rate of 6%, growth of 2% gives a value of HK$250,000, but at 3% the value jumps to HK$333,333 — one extra percentage point adds over HK$80,000. A growing-perpetuity valuation is extremely sensitive to the growth assumption, so stay conservative and run a sensitivity analysis.
How is a perpetuity related to the terminal value in DCF?
Closely. When a DCF valuation computes the 'terminal value', it uses the growing-perpetuity formula (Gordon model): the post-forecast cash flow is treated as a perpetuity growing at a constant rate, valued at D ÷ (R − G) at the end of the forecast, then discounted back to today. So the perpetuity present value computed here is essentially the mechanics behind the 'terminal value' block of a DCF. Understanding perpetuities explains why much of a DCF's value often comes from the terminal value.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.