Formula and Use
The present value of a growing perpetuity formula is used to calculate the current worth of an infinite stream of payments (Pmt) that increase by a constant rate (g) each year. Additionally it is assumed that the payments are made at the end of each period, continue forever, and have a discount rate i applied.
The formula is used in finance and economics to determine the value of investments, annuities, and other financial instruments. A growing perpetuity is sometimes referred to as an increasing perpetuity or graduated perpetuity.
As can be seen the formula discounts the value of each payment back to the start of period 1 (PV).
In the formula the terms have the following meanings.
PV (Present Value) The present value (PV) of a perpetuity is the current value of all future payments. Consequently it represents the amount of money needed to be invested today to provide the stream of payments in perpetuity.
Pmt (Payment) The payment (Pmt) is the periodic payment. This could be an annual payment, a semi-annual payment, or any other frequency of payment. The payment amount grows each period at a constant rate growth rate (g).
i (Discount Rate) The discount rate (i) is the rate at which future payments are discounted to their present value. It reflects the time value of money and the risk associated with the investment. The discount rate is the interest rate on an investment or a rate representing the risk associated with the investment.
g (Growth Rate) The growth rate (g) is the constant rate at which the payments grow each year. It is important to realize that when using the formula, the discount rate (i) must be greater than the growth rate (g).
Present Value of a Growing Perpetuity Formula Example
To illustrate suppose a payment of 6,000 is received at the end of period 1. The payments continue forever growing at a rate of 3% per period. Additionally assume the discount rate is 6%. The value of the payments today is given by the present value of a growing perpetuity formula as follows:
PV = Pmt / (i - g) PV = 6,000 / (6% - 3%) PV = 200,000
In summary, the present value of a growing perpetuity formula is a useful tool for calculating the value of an investment or annuity that provides an infinite stream of payments that grow at a constant rate each year.
The present value of a growing perpetuity formula is one of many used in time value of money calculations, discover another at the links below.
About the Author
Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a degree from Loughborough University.