Annuity Derivation vs. Perpetuity Derivation of the Time Value of Money

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. Several different kinds of annuities do exist, however, and some seek to replicate features of a perpetuity.

An annuity is an equal and annual series of cash flows over a predetermined time period. Annuities can be used for a variety of purposes, but the most common purpose they are used for is ensuring income for retirees. In the case of retirees, a lump sum of money or assets is exchanged for a series of smaller payments in the future. This payment is often guaranteed for the life of the beneficiary, meaning that, for a fee, the seller of an annuity assumed the longevity risk, or the risk that the beneficiary will outlive the savings. Annuities are generally sold by insurance companies. The lump sum up gained by an insurance company upfront, followed by small payments made years later, can be a good complement for other insurance products, which generally take in small annual payments in the form of premiums, followed by large, unpredictable, payouts.

The value of an annuity is derived as follows:

$\begin{aligned} &\text{PV} = \text{Periodic Cash Flow} \times \frac{ 1 - (1 + r) ^{-n}}{ r } \\ &\textbf{where:}\\ &\text{PV} = \text{Present value} \\ &r = \text{Interest rate per time period} \\ &n = \text{Number of time periods} \\ \end{aligned}$​PV=Periodic Cash Flow×r1−(1+r)−n​where:PV=Present valuer=Interest rate per time periodn=Number of time periods​

When deriving the value of an annuity, it is 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. A perpetuity can be thought of as a kind of annuity that never ceases, though in the case of a perpetuity, interest is not used to calculate the value. The concept of a perpetuity is used in numerous financial models. The British government issues a perpetuity, a bond called a consol, which upon purchase pays a small coupon forever.

A perpetuity calculation in finance is used in valuation methodologies to find the present value of a company's cash flows. This is done by discounting back at a certain rate.

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:

$\begin{aligned} &\text{PV} = \frac{ \text{Periodic Payment} }{ r } \\ &\textbf{where:}\\ &\text{PV} = \text{Present value of a perpetuity} \\ &\text{Periodic Payment} = \text{Payment per time period} \\ &r = \text{Interest rate per time period} \\ \end{aligned}$​PV=rPeriodic Payment​where:PV=Present value of a perpetuityPeriodic Payment=Payment per time periodr=Interest rate per time period​

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.