Growing Perpetuity

How to Calculate Present Value of Growing Perpetuity?

A growing perpetuity is defined as a stream of payments anticipated to grow at a constant rate for an infinite number of periods.

Perpetuities are unique in that their cash flows continue indefinitely with no ending date, whereas annuities are a stream of cash flows with a stated maturity, i.e. there is a predefined date on which the final payment is received.

Since the periodic cash flows increase at a fixed growth rate, each payment exceeds the gross amount paid in the prior period.

The reason we can estimate the valuation of a stream of perpetual cash flows is because of the “time value of money” concept, which states that a dollar today is worth more than a dollar received in the future.

All future cash flows must thereby be discounted to the present date using an appropriate discount rate that reflects the riskiness of the cash flows (and the expected return).

The more time between the present date and the date on which a payment is expected to be received, the more pronounced the effects of discounting become, i.e. a greater discount is applied.

That said, the present value (PV) of a growing perpetuity gradually declines in value until eventually reaching a point at which the present value of the future cash flows drops to zero.

The process of calculating the present value (PV) of a growing perpetuity consists of three steps:

• Step 1. Determine the Cash Flow in the Next Period (t=1)
• Step 2. Subtract the Discount Rate (r) by the Constant Growth Rate (g)
• Step 3. Divide the Cash Flow (t=1) by (r – g)

Note that the discount rate must be greater than the growth rate assumption, or else the present value of the growing perpetuity never reaches zero (and thus, its present value would be an infinite value).

Present Value of Growing Perpetuity Formula (PV)

The formula to calculate the present value of a growing perpetuity is as follows.

Where:

• CF t=1 → Periodic Cash Flow in Year 1
• r → Discount Rate (Cost of Capital)
• g → Constant Growth Rate

Growing Perpetuities vs. Zero Growth Perpetuities

The distinction between growing perpetuities and zero growth perpetuities is the periodic cash flows do not remain constant in the case of a growing perpetuity.

If we are given two identical perpetuities—where the only difference is the growth rate—the present value of a growing perpetuity will be greater than that of a zero-growth perpetuity.

For growing perpetuities, there is a constant growth rate attached to the series of cash flows, which partially offsets the effects of the opportunity cost of capital.

Therefore, the perpetuity with growth continues to retain value for a longer period into the future compared to a perpetuity with no growth. But in either case, the present value of the cash flows in the far future eventually reaches zero.

Growing Perpetuity Calculator (PV) – Excel Template

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

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Present Value of Growing Perpetuity Calculation Example (PV)

Suppose you’re presented with the following two options to pick from:

• Option 1. $15,000 in Cash Today
• Option 2. Perpetual Interest Payments of $1,000 Continuously Growing at 3% per year

Furthermore, we’ll assume that if Option 1 is chosen, the rate of return that you could earn on the $15k in cash is 10%.

In order to determine which investment is more profitable, we’ll need to calculate the present value of the growing perpetuity.

• Year 0 Payment = $1,000
• Discount Rate = 10%
• Growth Rate = 3%

The first step is to increase the initial interest payment by the 2% growth rate assumption to arrive at the next period payment amount.

• Year 1 Payment = $1,000 * (1 + 3%) = $1,030

From there, we’ll subtract the growth rate from our cost of capital assumption.

• Present Value (PV) = $1,030 ÷ (10% – 3%) = $14,714

If the only criteria for the investment decision were picking the option that is of greater value, Option 1 would be the right choice ($15k vs. $14.7k).