The differences between an annuity derivation and a perpetuity derivation of the time value of money is due to differences in time periods. Since the life of an annuity differs from the life of a perpetuity, an annuity uses a compounding interest rate to calculate its present value or future value, while a perpetuity uses the stated interest rate or discount rate only.
An annuity is an equal and annual series of cash flows over a predetermined time period. Therefore, the value of an annuity is derived as follows:
Present value of an annuity = (annual cash flow) x {{(1+ interest rate) ^ number of time periods – 1)} / (interest rate)}
When deriving the value of an annuity, it’s required to compound the stated interest rate. Each year, the annuity’s owner receives a cash flow plus an interest rate, which compounds each year as the annual cash flow and annual interest is earned.
A perpetuity, on the other hand, is an infinite series of periodic payments of equal face value. Therefore, a perpetuity’s owner will receive constant payments forever. While the actual face value of a perpetuity is indeterminable due to its time period, its present value can be derived. The present value is equal to the sum of the discounted value of each periodic payment. The value of a perpetuity is derived as followed:
Present value of a perpetuity = (fixed periodic payment) / (interest rate)
By using the actual interest rate and not one plus the interest rate compounded, a perpetuity can be derived as an infinite stream of payments.