# Growing Perpetuity Examples and Calculations

Growing perpetuity examples are most commonly found in real estate investments. This is because most rental properties charge rent at a constant rate which may be increased annually. Doing this ensures a continuous stream of income that has the potential to grow indefinitely.

Stocks and bonds could also be considered as growing perpetuity examples especially when they pay their holders dividends that increase at a specified rate with each payment.

Before we discuss how to calculate the present value of a growing perpetuity using the formula, let us define a growing perpetuity.

## What is a growing perpetuity?

A growing perpetuity refers to an investment vehicle that pays a consistent amount to investors at a predetermined growth rate indefinitely. This implies that each payment increases at a specific rate forever.

For instance, if Mr. Wright has invested in a growing perpetuity that grows at a 10% rate. It implies that every subsequent payment he receives would be 10% higher than the previous payment.

Growing perpetuities are often used by investors as a means of protecting their investments from the debilitating effects of inflation. It additionally provides a constantly increasing guaranteed cash flow for an infinite period.

The practical examples of growing perpetuity consist of different investments and funds that are predicated upon the framework of a growing perpetuity, this includes college endowment funds, dividend stocks, and real estate.

Growing perpetuities are also referred to as graduating or increasing perpetuities.

## How do you calculate growing perpetuity?

Calculating the present value of a growing perpetuity is an important step in determining the current value of the continuous payments a person will receive from their investment. This can be achieved in the following simple steps:

### Determine the first payment received

This is the first cash flow that will be received by the investor and is usually clearly stated in the purchase agreement. It is normally a dollar amount

### Determine the discount rate and the growth rate

The discount rate refers to the interest rate used to calculate the present value of future cash flows from the growing perpetuity while the growth rate is the rate at which future cash flows will increase. These rates are usually given in percentage

### Convert the rates to decimal

Both the discount rate and the discount rate are converted to decimal before being applied in the calculation of the present value of the growing perpetuity.

### Subtract the growth rate from the discount rate

The discount rate of the growing perpetuity is usually larger than its growth rate. Thus, the growth rate is subtracted from the discount rate.

### Divide the cash flow by the result

To get the present value of the growing perpetuity, simply divide the first cash flow received by the result obtained when the growth rate was subtracted from the discount rate.

## Present value of growing perpetuity

The present value of growing perpetuity presents an easy way to get the current value of a perpetual series of cash flows that grow at a steady rate.

Although the payments are structured to continue to infinity which makes the total value of the payments ongoing and endless, the present value of the growing perpetuity is finite and can be determined using the present value of the growing perpetuity formula.

### Present value of growing perpetuity formula

The present value of the growing perpetuity formula is mathematically expressed as PV = C/(r-g) where

PV = Present value of the growing perpetuity

C = Cash flow at the end of the first period

r = Discount rate

g = Growth rate

## Present value of growing perpetuity examples

### Example 1

Mr. John wants to endow \$1,000,000 worth of scholarships annually for orphaned students. Assuming this endowment can earn an interest rate of 5% annually and education expenses have been projected to increase by 0.5% annually for the foreseeable future. How much can Mr. John invest today to ensure that the scholarship fund can continue forever?

To determine how much Mr. John needs to contribute today for the scholarship to continue in perpetuity, we will use the present value of a growing perpetuity formula PV = C/(r-g) where

C = worth of annual scholarship = \$1,000,000

r = Interest rate = 5% = 0.05

g = Growth rate = 0.5% = 0.005

PV = \$1,000,000 / (0.05 – 0.005)

PV = \$1,000,000 / 0.045

PV = \$22,222,222.222

This implies that Mr. John needs to contribute approximately \$22,222,222 today to keep the scholarship fund going in perpetuity.

### Example 2

If Prof. Smith invested \$19,800 to get an annual payment of at least \$800 indefinitely. Assuming the cash flow is expected to grow at 4% per year and the required return used for the discount rate is 8%. Calculate the current value of his investment.

The current value of Prof. Smith’s investment can be calculated using PV = C/(r-g) where

C = Cash flow = \$800

r = Discount rate = 8% = 0.08

g = Growth rate = 4% = 0.04

PV = \$800 /(0.08 – 0.04)

PV = \$800 / 0.04

PV = \$20,000

Hence, Prof Smith’s investment of \$19,800 is a good one since the present value of the perpetuity is higher than the amount he paid to purchase it.

### Example 3

Assuming Angela has a rental property that can pay her \$15,000 annually in perpetuity. If the discount rate for real estate is 10% annually and rental payments are predicted to increase at 3% per year, calculate the present value of her rental property.

Using the present value of a growing perpetuity formula PV = C/(r-g) we can find the present value of the rental property.

C = Cash flow from rent = \$15,000

r = Discount rate = 10% = 0.1

g = Growth rate = 3% = 0.03

PV = \$15,000 /(0.1 – 0.03)

PV = \$15,000 / 0.07

PV = \$214,285.714

The present value of Angela’s rental property is \$214,285.714

## Future value of a growing perpetuity

The future value of growing perpetuity refers to the value of the ever-increasing and infinite cash flows at a specific period in the future. The future value calculation considers the compounding effect as a result of time as well as the constant growth rate of the cash flows over the period under review.

Thus it compounds the cash flows coming from the growing perpetuity and projects its worth at a specified period in the future.

The concept of the future value of a growing perpetuity is uncommon. This is because the cash flows from perpetuities are expected to continue forever without ever stopping.

However, some investors may wish to know the value of their investment at a specified time in the future. Therefore, they use the future value of a growing perpetuity formula to do this.

### Future value of a growing perpetuity formula

The formula used in calculating the future value of a growing perpetuity is mathematically expressed as FV = PV x (1 + r)^n where

FV: Future value of the growing perpetuity

PV: Present value of the growing perpetuity

r: Discount rate

n: Number of periods (usually years) into the future

## Future value of a growing perpetuity examples

### Example 1

If Mr. Simon receives an annual payment of \$5,000 which grows at a rate of 10% with each year that passes. If the discount rate associated with the cash flow is 20%, calculate the future value of Mr. Simon’s cash flows in 5 years.

FV = PV x (1 + r)^n

PV = Present value of the growing perpetuity = C/(r-g) = \$5,000 / (0.2 – 0.1) = \$5,000 / 0.1 = \$50,000

r = Discount rate = 20% = 0.2

n = Number of periods into the future = 5

FV = \$50,000 x (1 + 0.2)^5

FV = \$50,000 x (1.02)^5

FV = \$50,000 x 1.1040808

FV = \$55,204.0402

This indicates that the value of the cash flow from the rental property will be \$55,204.0402 in 5 years

### Example 2

Abigail purchased some stocks in JotScroll Media Company in 2020, at the end of the year, she received a payment of \$600, which grows at a rate of 2% annually and continues forever. If the discount rate is 10%, calculate the future value of her stocks in 20 years.

FV = PV x (1 + r)^n

PV = Present value of the growing perpetuity = C/(r-g) = \$600 / (0.1 – 0.02) = \$600/0.08 = \$7,500

r = Discount rate = 10% = 0.1

n: Number of years into the future = 20 years

FV = \$7,500 x (1 + 0.1)^20

FV = \$7,500 x (1.1)^20

FV = \$7,500 x 6.72749995

FV = \$50,456.2496

Hence, the future value of Abigail’s cash flow from the stocks in 20 years will be \$50,456.2496

## Conclusion

Growing perpetuity examples include stocks, bonds, real estate, and college funds. Depending on the framework, these usually provide a cash flow that is constantly increasing at a definite rate and continues in perpetuity.

The present value of a growing perpetuity formula is used as a means of determining the current value of such perpetual cash flows. It also serves as a pointer to knowing if the amount spent on investing in the growing perpetuity is worth it.

In some instances, an individual may choose to ascertain the value of their growing perpetuity at a specific time in the future, for this, the future value of a growing perpetuity formula is used.

It should be noted that calculated future value figures are a theoretic projection that may not be 100% exact as reality. This is because it assumes that no unforeseen circumstances will arise, hence, both the discount and growth rate of the cash flows will be unaffected throughout the period.

Blessing's experience lies in business, finance, literature, and marketing. She enjoys writing or editing in these fields, reflecting her experiences and expertise in all the content that she writes.