Present value

The present value of a future transaction is the nominal amount of money to change hands, adjusted to account for the time value of money. A given amount of money is almost always more valuable sooner than later, so present values are generally smaller than corresponding future values.

The simplest model of the time value of money is compound interest, which is in fact much simpler than simple interest. To someone who has the opportunity to invest an amount of money C for t years at a rate of interest of i compounded annually, the present value of the receipt of C, t years in the future, is C(1+i)^-t. The expression (1+i)^-t enters almost all calculations of present value. It represents the present value of 1. Many equations are expressed more concisely by making the substitution v=(1+i)^-1. Something worth 1 at time=t (years in the future) is worth v^t at time=0 (the present).

Present value is additive. The present value of a bundle of cash flows is the sum of each one's present value.

Many financial arrangements (including bondss, other loans, leases, salaries, membership dues, annuities, 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 in to refer to any such arrangement when discussing calculation of present value, whether or not the arrangement is a retirement plan. The expressions for the present value of such payments amount to summations of geometric series.

A periodic amount receivable indefinitely is called a perpetuity and is of mostly theoretical interest. A perpetuity receivable starting at the present time is called a perpetuity due. If the frequency of payments equals the frequency of interest compounding, the present value of a perpetuity due with payments of 1, is given by d^-1, where d=1-(1-i)^-1, and is called the rate of discount. In this case, i is the interest rate per period, not necessarily per year. If the first payment is 1 period in the future, the annuity is a perpetuity immediate, and the present value is i^-1.

A finite number (n) of periodic payments, receivable at times 1 through n, is an annuity immediate. Again assuming payment size of 1, its present value differs from the present value of the corresponding perpetuity immediate by an amount that is the present value of all the payments numbered n+1 and above. The latter has a value of i^-1 at time n, and vⁿi^-1 at time 0. The present value of the annuity immediate is i^-1-vⁿi^-1, or i^-1(1-vⁿ). An annuity due receivable at times 0 through n-1 has a present value of d^-1(1-vⁿ).

This entire discussion thus far makes some enormous assumptions:

That we don't have to account for price inflation.
That we don't have to account for variable interest rates.
That receipt of payments when due is certain.
That we will live long enough to receive payments receivable by us in the future.

For these and many other reasons, we consider prediction of the future to be an inexact science.