What is a perpetuity?

A perpetuity is a financial instrument that pays a fixed (or steadily growing) cash flow forever, with no end date. Real-world examples include UK government consols, preferred shares with fixed dividends, endowment fund payouts, and ground rents. Because each payment is discounted by a larger and larger factor, the infinite series converges to a finite present value provided the discount rate exceeds the growth rate.

What is the formula for a perpetuity present value?

For an ordinary flat perpetuity: PV = C divided by r, where C is the periodic payment and r is the discount rate as a decimal. For a growing perpetuity (Gordon Growth Model): PV = C divided by (r minus g), where g is the constant growth rate per period. For a perpetuity-due (first payment at t=0): multiply the ordinary result by (1 plus r).

What is a growing perpetuity and when is it used?

A growing perpetuity (also called the Gordon Growth Model or dividend discount model) assumes each payment increases by a fixed percentage g each period. It is the standard way to value dividend-paying stocks under a constant-growth assumption. The formula PV = C divided by (r minus g) only works when g is strictly less than r; if g reaches or exceeds r the present value is mathematically infinite.

What is a deferred or delayed perpetuity?

A deferred perpetuity starts n periods in the future. You first calculate the present value at the start of the perpetuity (period n), then discount that lump sum back n periods to today using PV(today) = PV(start) divided by (1 plus r) to the power of n. This is useful for pensions that begin at retirement or endowments that kick in after a specified date.

What is the difference between perpetuity-due and ordinary perpetuity?

In an ordinary perpetuity the first payment occurs at the end of period 1 (t=1). In a perpetuity-due the first payment is immediate (t=0). Because the perpetuity-due payments arrive one period earlier they are worth more; its PV equals the ordinary PV multiplied by (1 plus r). For a 5% rate the perpetuity-due is worth exactly 5% more than the ordinary version.

How long until I recover my initial investment?

The break-even period is the number of payments whose present values sum to the purchase price. For a flat 5% perpetuity priced at PV = C divided by r, the break-even is roughly 14 to 15 years regardless of the payment size. For growing perpetuities the break-even extends because each payment is partially offset by a higher discount. The calculator finds the exact integer period by cumulating the discounted cash flows numerically.

Perpetuity Calculator

A perpetuity is a promise to pay a fixed (or steadily growing) cash flow every period, forever. Despite the intimidating time horizon, the mathematics is elegant: because each payment is discounted further into the future, the infinite series converges to a remarkably clean closed-form formula. This calculator handles every standard case — flat or growing cash flows, ordinary timing or perpetuity-due, immediate or deferred start — and shows you the working so you can follow each step.

How the formulas work

Ordinary flat perpetuity (first payment at end of period 1):

PV = C / r

where C is the periodic cash flow and r is the discount rate per period (expressed as a decimal, so 5% = 0.05). Divide both sides and you get the yield: r = C / PV.

Growing perpetuity — Gordon Growth Model (payments grow at rate g per period):

PV = C / (r - g)

This is the foundation of the dividend discount model used in equity valuation. The constraint (g must be less than r) is not optional: if growth ever reaches the discount rate the denominator hits zero and present value becomes infinite — the investment can never be fairly priced.

Perpetuity-due (first payment immediate, at t=0):

PV(due) = C * (1 + r) / (r - g)

Receiving every payment one period earlier multiplies the ordinary PV by (1 + r). For a 5% rate that is a 5% premium — significant on large valuations.

Deferred perpetuity (perpetuity starts n periods from now):

PV(today) = PV(start) / (1 + r)^n

You compute the perpetuity value at its start date, then discount that lump sum back n periods. A perpetuity priced at 100,000 today that is delayed 5 years at 5% is worth only 100,000 / (1.05)^5 = 78,353 today.

Break-even period: the finite holding horizon at which the cumulative present value of received payments equals the purchase price. For a flat 5% perpetuity this is roughly 14 years; a 2% growing perpetuity at 5% discount extends that to about 18 years.

Worked example

Suppose a preferred share pays a £5,000 annual dividend, you require a 5% annual return, and dividends are expected to grow at 2% per year indefinitely.

Using the Gordon Growth Model:

PV = 5,000 / (0.05 - 0.02) = 5,000 / 0.03 = £166,667

Now suppose you can only collect the dividend from year 4 onward (the company has a 3-year lock-up). Discount back 3 years:

PV(today) = 166,667 / (1.05)^3 = 166,667 / 1.1576 = £143,979

The lock-up costs you roughly £22,688 in present-value terms — about 13.6% of the unconstrained value.

The break-even analysis then tells you: even paying the full £166,667, the first 25 annual dividends cover about 63% of your purchase price in PV terms; you need the perpetuity to run well beyond 25 years to justify the price at 5% vs 2% growth.

Scenario	Payment	Rate	Growth	Present Value
Flat, ordinary	5,000	5%	0%	100,000
Growing 2%, ordinary	5,000	5%	2%	166,667
Growing 2%, due	5,000	5%	2%	175,000
Deferred 3 years	5,000	5%	2%	143,979

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