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# What is Growing Perpetuity: Formula and Calculation

A good investment will grow in value over a period of time. Anticipate this by calculating your growing perpetuity with common financial formulas.

Growing perpetuity is a formula that helps you understand the true value of a growing investment. Say you purchase a certain stock that pays dividends each year. How do you assess the value of something that you can’t see yet? Calculating growing perpetuity is one way.

Here we’ll touch upon the ins and outs of growing perpetuities, including how to calculate present and future values, calculating values for different investment time periods, and everything in between!

Here’s What We’ll Cover:

Perpetuities and Growing Perpetuities

What Investments Might You Consider Growing Perpetuity For?

Example of the Present Value of Growing Perpetuity Formula

How to Calculate Future Value of Growing Perpetuity Formula

What Are Annuities Used For?

Key Takeaways

## Perpetuities and Growing Perpetuities

Let’s start with perpetuity by itself. Perpetuity is a formula that offers a fixed, finite value to infinite cash flows. While you might propose a value for a set number of payments, you can’t do so with a perpetuity, since it applies to cases where the payments don’t have a set number — they don’t stop.

You might have heard the term consoles. These are perpetuities in bonds offered by the British government. With a console, a small amount of money is paid regularly without an end date.

In other words, perpetuities help you assess value for investments that have:

• Infinite cash flows
• Payments over time

Now, let’s take a look at calculating perpetuity with a formula.

### Perpetuity Formula

Perpetuity Value = Cash Flow/Required Rate of Return

PV=C/R

Now, let’s see how growing perpetuities differ from regular perpetuities.

### Understanding Growing Perpetuities

While perpetuity offers value as an infinite cash flow model, growing perpetuity offers a value that considers the diminishing or increasing value of those cash flows over time. For example, your \$50 bill 20 years ago was worth much more than it was today. The same goes for your \$50 bill today when compared to your \$50 bill after 20 years.

As the name suggests, this growing perpetuity considers growth within its formula.

Also known as increasing or graduating perpetuity, growing perpetuity gives you the value of infinite cash flows that grow at a constant rate.

In other words, growing perpetuity helps you assess value for investments that entail:

• Regular payments
• Payments for an infinite time frame
• Proportional rate of growth

Let’s take a look at how to calculate growing perpetuity.

### Growing Perpetuity Formula

Present Value of a Growing Perpetuity = Periodic Payment / (Required Rate of Return for the Discount rate - Growth Rate)

PV = PMT/ (R-G)

## What Investments Might You Consider Growing Perpetuity For?

You might calculate growing perpetuity for a few different kinds of investments = namely, stocks, annuities, and real estate.

• Real estate cash flows from rental payments are infinite and increase over time, so growing perpetuity offers you a way to put a value on that growth.
• Stocks with dividends offer infinite cash flows, and growing perpetuity lets you value the dividends that you’ll receive in the future.
• Annuities are similar to stock investments, but the infinite cash flow increases regularly at a consistent percentage for a set amount of time.
• College Investments and Funds require growing perpetuities to assess value since tuition fees increase over time.

Let’s take a look at the growing formula of perpetuity in practice.

## Example of the Present Value of Growing Perpetuity Formula

Say you invested in a business with the following specifications:

• Cash flow: \$5,000 per year
• Predicted growth rate for cash flow: 10% with each year that passes
• Required return for the discount rate: 15%

Let’s plug these into the formula, and remember to put your percentage rates into number values with decimals:

PV = 5,000 / (15%-10%)

PV = 5,000 / (0.15 - 0.10)

PV = 5,000 / 0.05

PV = \$100,000

So, the present value or growing perpetuity of your investment is \$100,000. This number accounts for the regular rate of growth and consistent cash flow payments expected each year.

## How to Calculate Future Value of Growing Perpetuity Formula

What if you wanted to calculate the value of your investment at a later date? Say, 5 years from now. To calculate the future value, you’ll need a future date.

Now, growing perpetuities are for investments with infinite cash flows. If you set a time limit for your investment with a future date, you’re changing the growing perpetuity into an annuity, which is like perpetuity except for its fixed time period.

So, growing perpetuities don’t really work for future values. They only consider the present value if your cash flows are infinite, with no time limit or cease date.

## What Are Annuities Used For?

As mentioned, annuities are similar to perpetuities except for their time period. While perpetuities don’t consider an end date in their calculations, annuities always have a predetermined end date.

Most commonly, insurance companies offer annuities to retired people as a steady source of income. Through the annuity, a retiree gives the insurance company a lump sum amount. In return, the insurance company pays the retiree a set amount of money every month or year for the rest of their life.

But what is the value of the retiree’s annuity? How is their future cash flow valued?

### The basic method to Calculate Annuity

Where n symbolizes the number of time periods, here’s one of the most common annuity formulas:

Present Value = Cash Flow Payment * [(1-(1+Interest Rate))^-n / Interest Rate]

Keep in mind that finding the present value of annuity involves compounding interest rates. That’s because the cash flow payments each year experience a growing interest rate as the cash flow increases over time.

So, growing perpetuities help provide a valuation for a company’s cash flow on investment, while annuities help determine the value of a consol or retirement income.

## Key Takeaways

In summary, growing perpetuities can be used to find the present value of any investment with a constant cash flow. But they aren’t always the most useful when considering the value of future cash flows with long-term growth rates. Instead, annuities can be used to find the present value for a specific time period.

If you found this information useful, feel free to check out our resource hub for more educational material.

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