# Perpetuity: Financial Definition, Formula, and Examples

> **Definition**
>
> A perpetuity is a financial instrument that provides continuous cash flow indefinitely without a predetermined end date.

## What Does Perpetuity Mean in Finance?

A perpetuity is an investment asset that pays a stated return for an infinite time. An annuity with no termination date is an example of a perpetuity.

Sadly, perpetuities are extremely rare in modern times. The abstract concept of perpetuity lives on in the financial world and the field of economics, however. Perpetuity is defined in finance as a continuous stream of identical cash flows with no end. The concept of perpetuity is used in several financial theories including the [dividend discount model](https://www.investopedia.com/terms/d/ddm.asp) (DDM).

### Key Takeaways

*   A perpetuity in finance is an investment asset that pays a never-ending cash stream.
*   The present value of a perpetuity is determined by dividing the amount of the regular cash flows by the discount rate.
*   A growing perpetuity increases the cash flows paid out for each period going forward.
*   Perpetuities have become uncommon financial products but the concept of perpetuity is nonetheless important in finance.

![Perpetuity](https://www.investopedia.com/thmb/ab2KO9m7FpN8OVrW75gZbjA-v-A=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Perpetuity-FINAL-9ae5648469c148a681b03572a8943a5e.jpg)

*Perpetuity translates to "never-ending.". — Investopedia / Jiaqi Zhou*

## Understanding Perpetuity

An annuity is a stream of cash flows. A retiree might invest a sum of cash in an annuity in return for a monthly stream of income for a set number of years or an indefinite lifespan. A perpetuity is a type of annuity that lasts forever. The stream of cash flows doesn't have an end date.

A perpetuity calculation is used in finance to determine the present value of a company's cash flows when discounted back at a certain rate.

### A historical example of a perpetuity

British-issued bonds known as consols were an example of a financial instrument with perpetual cash flows. The Bank of England phased these out in 2015. The bondholder was entitled to receive annual interest payments forever when they purchased a consol from the British government.

The British consol was a means of passing wealth down through the generations beginning in 1751.

### The finite value of infinite cash flow

It may seem illogical but an infinite series of cash flows can have a finite present value. Each payment is worth only a fraction of the previous payment because of the cumulative time value of money.

The perpetuity formula determines the amount of cash flows in the [terminal year](https://www.investopedia.com/terms/t/terminalvalue.asp) of operation. When companies are valued, they're valued as going concerns. They're assumed to continue forever. The terminal year is therefore a perpetuity and analysts use the perpetuity formula to find its value.  

## Perpetuity Present Value Formula

The formula to calculate the present value of a perpetuity or security with perpetual cash flows is:

$\begin{aligned} &\text{PV} = \frac { C }{ ( 1 + r ) ^ 1 } + \frac { C }{ ( 1 + r ) ^ 2 } + \frac { C }{ ( 1 + r ) ^ 3 } \cdots = \frac { C }{ r } \\ &\textbf{where:} \\ &\text{PV} = \text{present value} \\ &C = \text{cash flow} \\ &r = \text{discount rate} \\ \end{aligned}$  

The basic method used to calculate a perpetuity is to divide cash flows by some discount rate.

The formula used to calculate the terminal value in a stream of cash flows for valuation purposes is a bit more complicated. It's the estimate of cash flows in year 10 of the company, multiplied by one plus the company’s long-term growth rate and then divided by the difference between the [cost of capital](https://www.investopedia.com/terms/c/costofcapital.asp) and the growth rate.

The terminal value is some amount of cash flows divided by some discount rate which is the basic formula for a perpetuity.

## Example of Perpetuity in Valuation

The value of the perpetuity is as follows if a company is projected to make $100,000 in year 10 and the company’s cost of capital is 8% with a long-term growth rate of 3%:

$\begin{aligned} &= \frac{ \text{Cash Flow}_\text{Year 10} \times ( 1 + g ) }{ r - g } \\ &= \frac{ \$100,000 \times 1.03 }{ 0.08 - 0.03 } \\ &= \frac{ \$103,000 }{ 0.05 } \\ &= \$2.06 \text{ million} \\ \end{aligned}$

This means that $100,000 paid into a perpetuity is worth $2.06 million in 10 years assuming a 3% rate of growth with an 8% cost of capital.

Now we must find the value of that $2.06 million today. Analysts use another formula referred to as the present value of a perpetuity to find this.

## Growing Perpetuities

The net present value of a perpetuity isn't as large as it might seem because the time value of money erodes the value of dollars far into the future due to inflation. The cash flows received by a fixed perpetuity many years from now can therefore become negligible in terms of future buying power.

A growing perpetuity adjusts the amount of perpetual payments each period by a given rate, helping to maintain buying power over time. The present value of a growing perpetuity will therefore be greater than a fixed or non-growing perpetuity. The higher the growth rate of future payments per period, the greater the present value.

The formula for a growing perpetuity is nearly identical to the standard formula but it subtracts the growth rate, _g,_ from the discount rate, _r,_ in the denominator:

> PV = C/(r-g)

The growth rate remains fixed in this formula so it's also only a rough estimate if we use it to value future cash flows. Businesses don't grow at the same rate each year.  

> **Important**
>
> Note that the rate of growth in a growing perpetuity remains fixed over its infinite life. It may or may not cover inflation.

## How Does Perpetuity Work in Investing?

A perpetuity is a financial instrument that offers a stream of cash flows in perpetuity or without end. Unlike other bonds, perpetuities don't have a fixed maturity date but they continue paying interest indefinitely.

The U.K. offered a government bond called a consol until 2015. The name was a contraction for consolidated annuities. It was structured as a perpetuity. The bonds were discontinued.  

## How Is a Perpetuity Valued?

It might seem at first glance that an instrument that offers an [infinite stream of cash flows](https://www.investopedia.com/terms/p/perpetual-preferred-stock.asp) would be almost infinitely valuable but this isn't the case. The value of a perpetuity is finite and its value can be determined by discounting its future cash flows to the present using a specified discount rate.

This procedure is known as [discounted cash flow](https://www.investopedia.com/terms/d/dcf.asp) (DCF) analysis and it's also widely used to value other types of securities such as stocks, bonds, and real estate investments.

## What Is the Difference Between a Perpetuity and an Annuity?

A perpetuity and an [annuity](https://www.investopedia.com/ask/answers/12/what-is-an-annuity.asp) are similar in that both offer a fixed set of cash flows over time. The key difference is that annuities have a predetermined end date known as the maturity date. Perpetuities are intended to last forever. Both annuities and perpetuities can be valued using DCF analysis.  

## How Long Does a Perpetuity Last?

A perpetuity lasts forever...or presumably until the end of the world as we know it.

## The Bottom Line

Perpetuities are investments that make payments indefinitely with no maturity or expiration date. They're essentially never-ending annuities. Perpetuities as financial products have become quite rare but the abstract concept of a perpetuity and the calculation of its present value, dividing the cash flow amount by the discount rate, remains a key concept in finance.

> **Article Sources**
>
> 1.  Federal Reserve Bank of St. Louis. "[Consols: The Never-Ending Bonds](https://fredblog.stlouisfed.org/2016/07/consols-the-never-ending-bonds/)."