## What Is Perpetuity?

A perpetuity is a security that pays for an infinite amount of time. In finance, perpetuity is a constant stream of identical cash flows with no end. The formula to calculate the present value of a perpetuity, or security with perpetual cash flows, is as follows:

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

The concept of a perpetuity is also used in a number of financial theories, such as in the dividend discount model (DDM).

### Key Takeaways

- A perpetuity, in finance, refers to a security that pays a never-ending cash stream.
- The present value of a perpetuity is determined using a formula that divides cash flows by some discount rate.
- The British consol is an example of a perpetuity.

#### Perpetuity

## Understanding Perpetuity

An annuity is a stream of cash flows. A perpetuity is a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time. In finance, a person uses the perpetuity calculation in valuation methodologies to find the present value of a company's cash flows when discounted back at a certain rate.

An example of a financial instrument with perpetual cash flows is the British-issued bonds known as consols. By purchasing a consol from the British government, the bondholder is entitled to receive annual interest payments forever. Although it may seem a bit illogical, an infinite series of cash flows can have a finite present value. Because of the time value of money, each payment is only a fraction of the last.

Specifically, the perpetuity formula determines the amount of cash flows in the terminal year of operation. In valuation, a company is said to be a going concern, meaning that it goes on forever. For this reason, the terminal year is a perpetuity, and analysts use the perpetuity formula to find its value.

## Perpetuity Formula

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 is 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 and the growth rate.

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

## Perpetuity Example

For example, 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%, the value of the perpetuity is as follows:

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

This means that $100,000 paid into a perpetuity, assuming a 3% rate of growth with an 8% cost of capital, is worth $2.06 million in 10 years. Now, a person must find the value of that $2.06 million today. To do this, analysts use another formula referred to as the present value of a perpetuity.