The present value of growing perpetuity formula is used to derive the present value of a series of cash flows that are generated by an investment in the future. These payments are expected to be made on predetermined future dates and in predetermined amounts.

For example, if the rate of growth is 10% that means that the first payment will be 10%, the second payment will be 10% more than the first one, the third payment will be 10% more than the second and so on.

The total value of the perpetuity is infinite since the payments are made endlessly, however, the present value of the perpetuity is finite because the farther in the future the payment is, the lower its present value. This happens due to time value of money, according to which, the same amount of money available at the present time is worth more than the same amount in the future.

### How to calculate present value of growing perpetuity?

For one period of time, the formula of present value of growing perpetuity is calculated by dividing the Amount of the consistent payment by the difference between the discount (or interest) rate and the growth rate.

$PV\;of\;Growing\;Perpetuity = \frac{ A_{1} }{ r - G }$

Where A1 = amount of the consistent payment, r = discount rate or interest rate, and G = the growth rate.

For this formula it’s important to notice that the discount/interest rate must be always greater than the growth rate. Otherwise, the growing perpetuity would have an infinite value.

### Examples of a present value of growing perpetuity calculation

#### Example 1

• An investor plans an investment where the cash flow payments will be $5,000 per year. • The required rate of return is 10%. • The cash flow payments are expected to grow by 3% every year and will be paid indefinitely. $PV\;of\;Growing\;Perpetuity = \frac{ \5,000 }{ 10\% - 3\% } = \71,428.57$ Which means the investment will be profitable if the investor pays$71,428.57 or less.

#### Example 2

• The amount of each consistent payment is \$100,
• The interest rate is 6%
• The value of payments will grow by 2% every year forever?
$PV\;of\;Growing\;Perpetuity = \frac{ \100 }{ 6\% - 2\% } = \2,500$

### How perpetuity calculation is used in real life

Despite the fact that it might seem that perpetuity is not used in real life, there are some real-life forms of investment, that are based on perpetuity.

• Real estate. Let’s assume an investor buys a flat and lets it. The flat is now their property and they also receive an infinite stream of rental payments.
• Common stocks. In theory, a company has an infinite life. Therefore, when an investor buys a company’s stocks, they’re entitled to receive dividends endlessly.
• Consols (Consolidated Stocks). Consols were certain UK and USA types of perpetual redeemable at the option of the government bonds with fixed interest payments. First Consols were issued in 1750s by the Bank of England and were redeemed only in 2015.

Perpetuity can also be a useful tool to calculate the loss of real value of money. By definition, the growth rate is always lower than the required rate of return, therefore the growing perpetuity assumes that we will lose a small percentage of the real value of money annually. In case of growing perpetuity, however, we will lose less money value than in case of an ordinary perpetuity, because the rate of return is constantly growing.