A perpetuity is a theoretical stream of cash flows that continues indefinitely.
Unlike an ordinary annuity, which contains a finite number of payments, a perpetuity has no specified final payment date.
For example, if a financial asset is modeled as paying $5,000 every year forever and the required discount rate is 6%, the present value under the basic perpetuity formula is approximately $83,333.33.
The mathematics is simple, but the assumptions are demanding. A true perpetuity requires payments to continue indefinitely, while a growing perpetuity additionally requires a constant growth rate that remains below the discount rate.
What Is a Perpetuity?
A perpetuity is an infinite series of periodic cash flows.
The basic version assumes:
- equal payments;
- regular intervals;
- payments continue forever;
- the discount rate remains constant.
Conceptually:
C, C, C, C, C, … forever
where C represents the recurring payment.
Because very distant cash flows are heavily discounted, the present value can remain finite even though the nominal payments continue indefinitely.
Perpetuity Formula
For a level perpetuity:
Present Value of Perpetuity = Cash Flow per Period ÷ Discount Rate
Using symbols:
PV = C ÷ r
Where:
- PV = present value;
- C = recurring cash payment;
- r = discount rate per period as a decimal.
The first cash flow is generally assumed to occur one period from today.
Perpetuity Example
Suppose an asset is expected to pay:
$5,000 per year forever
Required return:
6% per year
Convert 6% to decimal form:
6% = 0.06
Then:
PV = $5,000 ÷ 0.06
PV = $83,333.33
The theoretical present value is approximately $83,333.33.
Verify the Economic Logic
At a value of $83,333.33, an annual $5,000 payment represents:
$5,000 ÷ $83,333.33
≈ 6%
The recurring payment therefore matches the assumed required rate.
The formula can also be rearranged:
Cash Flow = Present Value × Discount Rate
$83,333.33 × 6% ≈ $5,000
Why a Perpetuity Can Have a Finite Value
At first, receiving money forever may seem as though it should create infinite present value.
The reason it does not is discounting.
A payment received:
- one year from now has substantial present value;
- 50 years from now has much less;
- 200 years from now has extremely little at a positive discount rate.
Mathematically, the infinite discounted series converges to:
C ÷ r
provided the required conditions hold.
Perpetuity as an Infinite Discounted Series
The long-form expression is:
PV = C ÷ (1 + r) + C ÷ (1 + r)² + C ÷ (1 + r)³ + …
Because this is an infinite geometric series, it simplifies to:
PV = C ÷ r
This compact formula is one of the classic time-value-of-money relationships.
How the Discount Rate Changes Value
Perpetuity valuation is highly sensitive to the discount rate.
Suppose annual cash flow remains $5,000.
At 4%:
PV = $5,000 ÷ 0.04
PV = $125,000
At 5%:
PV = $100,000
At 6%:
PV ≈ $83,333.33
At 8%:
PV = $62,500
As the discount rate rises, present value falls.
Discount Rate ↑ → Perpetuity Value ↓
Why Rate Percentages Must Be Converted Correctly
The denominator is a decimal rate.
The percentages conversion is:
6% = 0.06
not:
6
Using $5,000 ÷ 6 would produce $833.33, which is not the correct present value for a 6% required return.
Solve for the Required Return
The formula can be rearranged:
Required Return = Cash Flow ÷ Present Value
Suppose an asset is priced at $100,000 and pays $4,500 indefinitely.
r = $4,500 ÷ $100,000
r = 0.045
r = 4.5%
The implied required return is 4.5% under the level-perpetuity assumptions.
Solve for Cash Flow
If present value and required return are known:
Cash Flow = Present Value × Required Return
Suppose:
- PV = $250,000;
- required return = 5%.
Then:
Cash Flow = $250,000 × 0.05
Cash Flow = $12,500 per year
Growing Perpetuity
A growing perpetuity assumes the payment increases at a constant rate forever.
The formula is:
PV = C₁ ÷ (r − g)
Where:
- C₁ = cash flow expected one period from now;
- r = discount rate;
- g = perpetual growth rate.
For the standard formula to produce a finite positive value:
r > g
Growing Perpetuity Example
Suppose:
- next year’s cash flow = $3,000;
- required return = 7%;
- perpetual growth rate = 3%.
Then:
PV = $3,000 ÷ (0.07 − 0.03)
PV = $3,000 ÷ 0.04
PV = $75,000
The growing perpetuity has a theoretical value of $75,000.
Why r Must Exceed g
Suppose:
r = 5%
and:
g = 5%
The denominator becomes:
0.05 − 0.05 = 0
The formula becomes undefined.
If the growth rate exceeds the discount rate, the ordinary growing-perpetuity expression no longer converges to a finite positive value.
That is not merely a calculator problem. It violates the mathematical conditions of the model.
C₁ vs Current Cash Flow
A common mistake in the growing-perpetuity formula is using today’s cash flow instead of the next-period cash flow.
Suppose the current payment is $2,000 and grows 3%.
Next payment:
C₁ = $2,000 × 1.03
C₁ = $2,060
If:
r = 7%
then:
PV = $2,060 ÷ (0.07 − 0.03)
PV = $51,500
Using $2,000 instead would understate the valuation.
Perpetuity vs Ordinary Annuity
A finite annuity includes a specified number of payments.
A perpetuity does not.
For example:
$5,000 annually for 20 years
is not a perpetuity.
It requires a finite annuity formula because the twentieth payment is the final one.
Using the perpetuity formula would overstate the value by implicitly assuming payments continue after year 20.
Perpetuity vs Pension Payouts
A lifetime stream of pension payouts is also not automatically a perpetuity.
Although payments might continue for the retiree’s lifetime, a human lifetime is finite and uncertain.
A perpetuity assumes payments continue indefinitely regardless of lifespan.
That distinction makes actuarial lifetime-income valuation different from the simple C ÷ r formula.
Perpetuity vs Payment Calculations
The formula used when payment amounts are calculated for a finite loan includes a specified number of periods.
A perpetuity has:
n → infinity
There is no principal amortization to zero after a fixed number of installments in the basic perpetuity model.
The two formulas therefore answer different cash-flow questions.
Perpetuity and Portfolio Risk
A theoretical perpetual income stream can still carry portfolio risk.
The formula assumes the cash flows occur as modeled.
Real investments can face:
- credit risk;
- dividend cuts;
- business deterioration;
- changing required returns;
- inflation;
- liquidity risk.
A perpetuity valuation should never be mistaken for a guarantee that payments will actually continue forever.
Perpetuity and Portfolio Rebalancing
An income-producing asset valued with a perpetuity-style model can still form part of a portfolio subject to portfolio rebalancing.
Rebalancing concerns portfolio weights.
Perpetuity mathematics concerns the theoretical present value of an infinite cash-flow stream.
A change in the asset’s market value can alter its portfolio weight even when its scheduled cash flow remains unchanged.
Perpetuity and Inflation
A level perpetuity pays the same nominal amount forever.
That means its real purchasing power can decline when inflation is positive.
Suppose the annual payment remains $5,000 while prices rise over decades.
The nominal cash flow stays the same, but it buys less.
A growing perpetuity can partly model nominal payment growth, although assuming one constant growth rate forever is a strong simplification.
Real vs Nominal Perpetuity Models
A valuation model should remain internally consistent.
If cash flows are projected in nominal dollars, the discount rate should generally be a nominal rate appropriate to those cash flows.
If cash flows are expressed in real purchasing-power terms, a real discount rate may be more appropriate.
Mixing nominal cash flows with a real discount rate can materially distort the valuation.
Perpetuity Starting Today
The standard formula assumes the first payment occurs one period in the future.
If an equal payment also occurs immediately, the value becomes:
PV = Immediate Payment + Standard Perpetuity Value
For a $5,000 immediate payment followed by $5,000 annually forever at 6%:
PV = $5,000 + $5,000 ÷ 0.06
PV = $88,333.33
Payment timing matters.
Deferred Perpetuity
A perpetuity can also begin several years in the future.
First calculate its value one period before the payments begin.
Then discount that value back to today.
Suppose:
- payments = $5,000 annually;
- required return = 6%;
- first payment occurs at the end of year 6.
Value at the end of year 5:
PV₅ = $5,000 ÷ 0.06
PV₅ = $83,333.33
Then discount five years:
PV₀ = $83,333.33 ÷ 1.06⁵
PV₀ ≈ $62,269.42
The delayed start materially reduces current value.
Limitations of Perpetuity Models
The formula is elegant because it simplifies an infinite series.
Its simplicity also creates limitations.
It assumes:
- no final payment date;
- stable cash-flow structure;
- a consistent required return;
- for growing perpetuities, a constant sustainable growth rate forever.
Real-world cash flows rarely satisfy those assumptions exactly.
Common Perpetuity Mistakes
One mistake is using a percentage as a whole number instead of a decimal.
Another is valuing a finite payment stream as a perpetuity.
For growing perpetuities, people may use current cash flow rather than next-period cash flow.
A further error is allowing the perpetual growth assumption to equal or exceed the discount rate while still treating the formula as valid.
Frequently Asked Questions
What is a perpetuity?
A perpetuity is a theoretical sequence of periodic cash flows that continues indefinitely.
What is the perpetuity formula?
PV = C ÷ r
What does C represent?
C is the equal cash payment received each period.
What does r represent?
r is the required return or discount rate per period, expressed as a decimal.
How much is $5,000 forever worth at 5%?
$5,000 ÷ 0.05 = $100,000
What is a growing perpetuity?
It is a perpetual payment stream assumed to grow at a constant rate.
What is the growing perpetuity formula?
PV = C₁ ÷ (r − g)
Why must the discount rate exceed the growth rate?
The standard growing-perpetuity series requires r > g to converge to a finite value.
Is a lifetime pension a perpetuity?
No. A lifetime pension has a finite but uncertain duration, while a perpetuity theoretically continues forever.
Does a perpetuity guarantee payments forever?
No. The formula is a valuation model. Real securities remain exposed to financial and contractual risks.
Why does a higher discount rate reduce perpetuity value?
Future cash flows are assigned lower present values when the required return increases.
Where is perpetuity math useful?
It is a fundamental time-value-of-money concept within broader Savings & Investing and valuation analysis.
