Free Perpetuity Calculator

Understanding the Present Value of a Perpetuity

A perpetuity represents a financial arrangement where equal cash flows are paid at regular intervals indefinitely. The present value of a perpetuity (PV of perpetuity) captures how much that endless stream of payments is worth in today’s money. Whether you are evaluating preferred stocks, perpetual bonds, or long‑term leases, the perpetuity formula is an essential tool for converting infinite future cash flows into a single current value.

The Core Concept: Time Value of Money

The foundation of perpetuity valuation lies in the time value of money. Money available today can be invested to generate additional returns, so a dollar received in the future is worth less than a dollar in hand now. Inflation further erodes purchasing power over time. As a result, each successive perpetuity payment contributes less to the present value, with very distant payments adding almost nothing. This diminishing effect explains why an infinite series can have a finite present value.

Standard Perpetuity Formula

The basic perpetuity formula is straightforward:

$$PV = \frac{C}{r}$$

where
$PV$ = present value of the perpetuity,
$C$ = periodic cash flow (also called the dividend or payment),
$r$ = discount rate (expressed as a decimal).

If you know any two of these three variables, you can solve for the third. A dedicated perpetuity payment calculator (or present value of perpetuity calculator) makes these computations instant, allowing you to enter the known values and get the missing one.

Example: Base Case

Suppose you hold a perpetual bond that pays $10 every year. With a discount rate of 5% (0.05), the present value is:

$$PV = \frac{10}{0.05} = \$200$$

Thus, the bond should be priced at $200. Changing the discount rate to 8% (0.08) reduces the value:

$$PV = \frac{10}{0.08} = \$125$$

The following table illustrates how sensitive the present value is to different discount rates:

Discount Rate	Present Value (per $10 payment)
4%	$250.00
5%	$200.00
6%	$166.67
8%	$125.00
10%	$100.00

A higher discount rate implies faster value decay, resulting in a lower present value.

Growing Perpetuity Formula

A growing perpetuity features payments that increase at a constant rate each period. This growth partially offsets the loss of value from the time value of money. The growing perpetuity formula is:

$$PV = \frac{C}{r - g}$$

where $g$ is the constant growth rate of the payments. A critical condition is that $r > g$; otherwise, the present value becomes infinite and the calculation loses meaning.

Example: Growth Case

Take the same $10 annual payment with an 8% discount rate, but now assume the payment grows at 2% each year. The present value becomes:

$$PV = \frac{10}{0.08 - 0.02} = \frac{10}{0.06} \approx \$166.67$$

The growth adds approximately $41.67 in value compared to the non‑growing case ($125). This demonstrates how even modest growth can significantly increase the worth of a perpetuity. A growing perpetuity calculator can handle this type of input by simply accepting a growth rate alongside the payment and discount rate.

Real‑Life Perpetuity Examples

While true perpetuities are rare, several financial instruments share perpetuities’ characteristics:

• Preferred Stocks: Many preferred shares pay fixed dividends indefinitely, allowing investors to value them using the perpetuity formula.
• Consols: The United Kingdom issued consols (perpetual bonds) from the 18th century until 2015 to finance government debt. These bonds paid regular interest with no maturity date.
• Real Estate: Income‑generating properties are often appraised based on the perpetuity of net rental cash flows, though actual rents may change over time.
• Endowments: The Yale University endowment famously owns one of five surviving Dutch perpetuities from 1648, yielding occasional interest payments.

Using a Perpetuity Calculator

A perpetuity calculator can serve multiple functions: it acts as a present value of perpetuity calculator when you input the payment and discount rate, as a perpetuity payment calculator when you know the present value and discount rate, and as a discount rate finder when you have the present value and payment. For growing perpetuities, the tool also accepts a growth rate, effectively becoming a growing perpetuity calculator. The same tool can be used for reverse calculations — simply enter any two known variables, and the third is automatically computed.

Practical Considerations

• The discount rate must reflect the risk and opportunity cost of capital. In corporate finance, it is often the weighted average cost of capital (WACC) or an investor’s required return.
• For a perpetuity to be meaningful, payments must be certain and indefinite. Real‑world payments may be suspended or altered (e.g., a company suspending preferred dividends), so the perpetuity model provides a theoretical baseline.
• The formula assumes the first payment occurs one period from now. If payments begin immediately (a perpetuity due), the result multiplies by $(1 + r)$.

By mastering the perpetuity formula and its variants, you can quickly assess the fair value of infinite‑horizon cash flow streams and make informed investment decisions.