Present Value - Annuities, Perpetuities and Other Common Forms

Annuities, Perpetuities and Other Common Forms

Many financial arrangements (including bonds, other loans, leases, salaries, membership dues, annuities including annuity-immediate and annuity-due, straight-line depreciation charges) stipulate structured payment schedules, which is to say payment of the same amount at regular time intervals. The term "annuity" is often used to refer to any such arrangement when discussing calculation of present value. The expressions for the present value of such payments are summations of geometric series.

A cash flow stream with a limited number (n) of periodic payments (C), receivable at times 1 through n, is an annuity immediate. Future payments are discounted by the periodic rate of interest (i). The present value (PV) of the annuity is the value at time 0 of the stream of cash flows:

where:

= number of periods,

= amount of cash flows,

= effective periodic interest rate or rate of return.

Formula (1) applies when the level cash flows C are made at the end of equal length intervals; for example, payments of $100 at the end of each year from year one to ten. A periodic amount receivable indefinitely is called a perpetuity, although few such instruments exist. The present value of a perpetuity can be calculated by taking the limit of the above formula as n approaches infinity.

Formula (2) can also be found by subtracting from (1) the present value of a perpetuity delayed n periods, or directly by summing the present value of the payments

which form a geometric series.

These calculations must be applied carefully, as there are underlying assumptions:

• That it is not necessary to account for price inflation, or alternatively, that the cost of inflation is incorporated into the interest rate.
• That the likelihood of receiving the payments is high — or, alternatively, that the default risk is incorporated into the interest rate.

See time value of money for further discussion.