## What is Perpetuity?

A **Perpetuity** refers to a constant stream of cash flows payments anticipated to continue indefinitely.

## How to Calculate Present Value of Perpetuity (PV)?

In a perpetuity, the series of cash flows received by the investor is expected to be received forever (i.e. a never-ending stream of cash flows).

For instance, if an investment comes with terms stating that a $1,000 payment will be paid out at the end of each year with an indefinite end, this represents an example of a zero-growth perpetuity (i.e. the annual payout remains the same through the life of the investment).

Despite the cash flows theoretically lasting “forever,” the present value (PV) – i.e. the approximate valuation of the total potential stream of cash flows as of the current date – can still be calculated.

The “time value of money,” a fundamental concept in corporate finance, states that the further away from the date of when a cash flow payment is received, the greater the reduction in its value today.

As a result, the present value (PV) of the future cash flows of a perpetuity eventually reaches a point where the cash flow payments in the far future have a present value of zero.

## Perpetuity vs. Annuity: What is the Difference?

**Perpetuity →**To reiterate, a perpetuity is a cash flow expected to continue forever with no ending date.**Annuity →**In contrast, an annuity comes with a pre-determined maturity date, which is when the final cash flow payment is received.

## Growing Perpetuity vs. Zero-Growth Perpetuity: What is the Difference?

In the prior example, the size of the cash flow (i.e. the $1,000 annual payment) is kept constant throughout the entire duration of the perpetuity.

However, for growing perpetuities, there is a perpetual (or “continuous”) growth rate attached to the series of cash flows.

If we assume equal initial payment amounts, a growing perpetuity will thus be valued higher than one with zero-growth, all else being equal.

For example, if the investment stated that $1,000 would be issued in the following year but at a 2% growth rate, then the annual cash flows would increase 2% year-over-year (YoY).

Since the cash flows increase each year, the growth rate helps offset the discount rate used to calculate the present value (PV).

## Perpetuity Formula

To calculate the present value (PV) of a perpetuity with zero growth, the cash flow amount is divided by the discount rate.

**Present Value of Zero-Growth Perpetuity (PV) =**Cash Flow

**÷**Discount Rate

The discount rate is a function of the opportunity cost of capital – i.e. the rate of return that could be obtained from other investments with a similar risk profile.

For a growing perpetuity, the formula consists of dividing the cash flow amount expected to be received in the next year by the discount rate minus the constant growth rate.

**Present Value of Growing Perpetuity (PV) =**Year 1 Cash Flow

**÷**(Discount Rate

**–**Growth Rate)

## Present Value of Perpetuity Calculator (PV)

We’ll now move to a modeling exercise, which you can access by filling out the form below.

## 1. Present Value of Perpetuity Calculation Example

In our illustrative scenario, we will compare two perpetuities, sharing the following assumptions:

- Cash Flow Amount (Year 0) = $100
- Discount Rate (r) = 10.0%

The difference between the two perpetuities is their respective growth rate assumptions:

- Zero Growth = 0.0% Growth Rate
- Growing = 2.0% Growth Rate

For the first zero growth perpetuity, the $100 annual payment amount remains fixed, while the payment for the second perpetuity grows at 2.0% per year perpetually.

For the zero-growth perpetuity, we can calculate the present value (PV) by simply dividing the cash flow amount by the discount rate, resulting in a present value of $1,000.

- Present Value (PV), Zero-Growth = $100 Cash Flow ÷ 10.0% Discount Rate = $1,000

If someone came to us and offered to sell the investment to us, we’d only proceed with investing if the purchase price is equal to or less than $1,000.

Otherwise, the investment would not make much sense economically.

## 2. Present Value of Growing Perpetuity Calculation Example

Next, for the growing perpetuity, the first step is to grow the Year 0 cash flow amount by 2% once to arrive at the Year 1 cash flow amount.

- Year 1 Cash Flow = $100 Year 0 Cash Flow × (1 + 2.0% Growth Rate)
- Year 1 Cash Flow = $102

The denominator is equal to the discount rate subtracted by the growth rate.

- Present Value of Growing Perpetuity (PV) = $102 ÷ (10.0% – 2.0%) = $1,275

In conclusion, we can see the positive impact that growth has on the value of a perpetuity, as the present value (PV) of the growing perpetuity is $275 more than the zero-growth perpetuity.