Perpetuity Calculator

What Is a Perpetuity?

A perpetuity is a financial arrangement in which payments continue indefinitely, or in simple terms, continue forever. Unlike an investment that makes payments for a fixed number of years, a perpetuity does not have a defined end date.

For example, if an investment generates a fixed cash payment every year and those payments are expected to continue indefinitely, it is considered a perpetual cash flow. To determine what these future payments are worth today, investors use concepts such as present value and the discount rate.

Perpetuity Formula

The perpetuity formula is used to calculate the present value of a cash flow that is expected to continue indefinitely. It helps determine how much a series of equal periodic payments is worth today based on a specific discount rate.

PV = C ÷ r

Where:

• PV = Present Value
• C = Periodic Cash Flow
• r = Discount Rate

For example, suppose you receive an annual cash flow of $10,000 and the discount rate is 5%. First, convert the discount rate into decimal form:

5% = 0.05

Now apply the formula:

PV = $10,000 ÷ 0.05

PV = $2,00,000

So, the present value of the perpetuity is $2,00,000.

When using the perpetuity formula, always enter the discount rate in decimal form, not as a percentage. For example, 5% should be entered as 0.05, while 8% should be entered as 0.08. This ensures that the present value is calculated correctly.

How to Calculate Present Value of a Perpetuity

Calculating the present value of a perpetuity depends on the type of perpetuity you choose. With a standard perpetuity, payments remain the same over time, while a growing perpetuity assumes that payments increase at a fixed growth rate.

Follow these steps to calculate the present value:

Step 1: Enter the Periodic Cash Flow

Start by entering the periodic cash flow, which is the regular amount you expect to receive from the investment.

For example, if you receive $1,000 every year, enter $1,000 as the periodic cash flow.

Step 2: Enter the Discount Rate

Enter the discount rate, also known as the required rate of return. This represents the return you expect from an investment with a similar level of risk.

For example, if your required return is 8%, enter 8%.

Step 3: Enter the Growth Rate if Applicable

If you select Growing Perpetuity, enter the expected growth rate of the periodic payments.

For example, if payments are expected to increase by 3% each year, enter 3% as the growth rate.

For a Ordinary Perpetuity, payments do not grow, so the growth rate is 0%.

Step 4: Select the Payment Frequency

Choose how often you receive the payments:

• Annual – payments are received once a year.
• Semi-Annual – payments are received twice a year.
• Quarterly – payments are received four times a year.
• Monthly – payments are received every month.

The selected frequency helps determine the appropriate payment and rate period for the calculation.

Step 5: Select the Perpetuity Type

Choose between the two available options:

Ordinary Perpetuity: The cash flow remains constant indefinitely. Its basic formula is:

PV = C ÷ r

where C is the periodic cash flow and r is the discount rate.

Growing Perpetuity: The cash flow increases at a constant growth rate. Its formula is:

PV = C ÷ (r − g)

where C is the periodic cash flow, r is the discount rate, and g is the growth rate.

The discount rate must be greater than the growth rate for the growing perpetuity formula to produce a meaningful finite value.

Step 6: Review the Result

After entering the required information, the calculator determines the present value of the perpetuity based on your selected payment frequency and perpetuity type.

The result represents the estimated value today of the future income stream that continues indefinitely.

If you enable Show Calculation Breakdown, you can also view the steps used to arrive at the final result, making it easier to understand how the calculation works.

For a quicker and easier alternative to manual calculations, you can use a Perpetuity Calculator by entering your cash flow, discount rate, growth rate, payment frequency, and perpetuity type.

Growing Perpetuity vs. Ordinary Perpetuity

Perpetuities can be divided into two common types: ordinary perpetuity and growing perpetuity. The main difference is whether the payment remains constant or increases over time.

Ordinary Perpetuity

In an ordinary perpetuity, the payment remains the same every period. For example, if an investment pays $10,000 every year indefinitely, the annual payment does not change.

PV = C ÷ r

Where:

• PV = Present value
• C = Constant cash flow per period
• r = Discount rate

Growing Perpetuity

In a growing perpetuity, the cash flow increases at a fixed growth rate every period. For example, if an investment pays $10,000 in the first year and the payment grows by 3% each year, it represents a growing perpetuity.

The formula is:

PV = C₁ ÷ (r − g)

Where:

• C₁ = Cash flow received in the next period
• r = Discount rate
• g = Growth rate

An important condition for this formula is r > g. The discount rate must be higher than the growth rate; otherwise, the formula will not produce a meaningful finite present value.

Simple Example

Suppose an investment is expected to pay $10,000 next year, with a discount rate of 8%.

For an ordinary perpetuity:

PV = $10,000 ÷ 8% = $1,25,000

Now assume the payment grows by 3% every year. For a growing perpetuity:

PV = $10,000 ÷ (8% − 3%) = $2,00,000

The growing perpetuity has a higher present value because its cash flows increase over time. This example shows how the growth rate and discount rate can significantly affect the value of future cash flows.

Frequently Asked Questions (FAQs)