# Perpetuity

Learn the Concept of Zero-Growth & Growing Perpetuities

• What is the definition of a perpetuity?
• What is the distinction between a growth perpetuity and a zero-growth perpetuity?
• What is the formula for calculating the present value (PV) of a perpetuity?
• How is a perpetuity different from an annuity?

## Perpetuity Definition

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 perpetuity 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

Perpetuities are cash flows that are expected to continue forever.

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

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 a growing perpetuity, there is a perpetual growth rate attached to the series of cash flows.

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).

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

## Perpetuity Present Value (PV) Formula

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

### Zero-Growth Perpetuity Formula

• Present Value (PV) of Zero-Growth Perpetuity = 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.

### Growing Perpetuity Formula

• Present Value of Growing Perpetuity = Year 1 Cash Flow / (Discount Rate – Growth Rate)

## Excel Template – Present Value (PV) of Perpetuity

Now, we can move on to an example present value (PV) calculation of perpetuities with varying growth rates.

To get started, fill out the form below to access the Excel file.

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## Perpetuity Present Value (PV) 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%

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

• Zero Growth Perpetuity: 0% Growth Rate
• Growing Perpetuity: 2% Growth Rate

For the first zero growth perpetuity, the \$100 annual payment amount remains fixed, whereas the payment for the second perpetuity grows at 2% 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) of Zero-Growth Perpetuity = \$100 Cash Flow / 10% 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.

Next, for the growing perpetuity, the first step is to grow the Year 0 cash flow amount by 10% once to arrive at the Year 1 cash flow amount.

• Year 1 Cash Flow = \$100 Year 0 Cash Flow * (1 + 2% Growth Rate) = \$102

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

• Present Value of Growing Perpetuity = \$102 / (10% – 2%) = \$1,275

From our example, we can see the positive impact that growth has on the value of a perpetuity, as the present value of the growing perpetuity is \$275 more than that of the zero-growth perpetuity.

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