Discounting, present value

Discounting is the often-encountered process of finding an unknown present value from a known future value. Using a time line, discounting is moving backward in time from a given point of time in the future to the present time. Given the time value of money (that INTEREST RATES are always positive), present values are always smaller than future values. Lottery and sweepstakes winners may be offered their winnings not as a lump sum paid presently but as a future stream of ANNUITY payments. For example, a sweepstakes participant may have won the grand prize of $1 million to be paid in yearly installments of $25,000 for the next 40 years. While the sum of the 40 payments is $1 million the present value of such a payoff is considerably less than $1 million. Finding the present value of this future stream of annuity payments will determine the true value for the grand-prize winner. Using a discount rate of 10 percent, the present value of receiving $25,000 each year for the next 40 years is only $244,476.27. For a lump sum, the present value of some future amount is determined by the discounting formula PV = FVn ÷ [1+ir]n, where PV is the present value, FVn is the future value at some future point in time n, ir is the interest rate (expressed in decimal form) applicable to the situation in question, and the exponent n is the same point in time for which the future value is known. For instance, find the present value $133.10 to be received three years from now, given a 10 percent interest rate, compounded annually: PV = 133.10÷[1.10]3. Simplification reduces the formula to PV = 133.10÷1.331 = 100.00. It is sometimes necessary to determine the present value of an annuity. While there is a formula for this, it is much easier to use a commonly published table of interest factors. For discounting, there are tables of present value interest factors for lump sums (PVIFs) and for annuities (PVIFAs). To find the present value of a future lump sum: PV = FVn[PVIFi,n], where PVIF is the lump sum present interest factor for some interest rate i and for some time period n. To find the present value of an annuity: PVA = PMT[PVIFAi,n], where PMT is the regular annuity payment and PVIFA is the annuity present value interest factor for some interest rate i and for some time period n. While using the published tables of present value interest factors is easier than manually doing the numbercrunching, it is much more convenient to find present values for lump sums and annuities using a financial calculator. Remembering that the interest-factor tables carry the interest factors to only four digits to the right of the decimal, the results obtained from the use of a financial calculator are more accurate than using the tables. The published interest-factor tables list interest factors only for whole-number interest rates. A financial calculator can compound using any interest rate.