Present Value

The present value of a sum of money to be received at a future date is determined by discounting the future value at the interest rate that the money could earn over the period.

Starting with the future value equation:

FV = PV ( 1 + i ) ^t

where

FV = future value
PV = present value
i = annual interest rate

we see that the present value is given by:

PV =	FV ( 1 + i ) t

The term 1 / ( 1 + i ) ^t is known as the discount factor.

If both the future value and present value are known, one can solve for the time or the interest rate using one of the techniques discussed in future value calculations.

Present Value of Multiple Future Cash Payments

When there is more than a single cash payment at a future date, the present value is calculated by taking the present values of the individual cash payments and summing them. It is helpful to draw a time line depicting the timing of the cash payments:

Time Line

0		1		2		3
PV		C1		C2		C3

In this model, the cash payment at each date may be either an inflow or an outflow; the direction is designated by the sign. The present value of the above cash flow is:

PV = C₁ / ( 1 + i ) + C₂ / ( 1 + i )² + C₃ / ( 1 + i )³

Discount Factor Table

The discount factor 1 / ( 1 + i ) ^t may be calculated for a range of time periods and interest rates and tabulated for quick reference.

Table of Discount Factors

t \ i	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%
1	0.990	0.980	0.971	0.962	0.952	0.943	0.935	0.926	0.917	0.909
2	0.980	0.961	0.943	0.925	0.907	0.890	0.873	0.857	0.842	0.826
3	0.971	0.942	0.915	0.889	0.864	0.840	0.816	0.794	0.772	0.751
4	0.961	0.924	0.888	0.855	0.823	0.792	0.763	0.735	0.708	0.683
5	0.951	0.906	0.863	0.822	0.784	0.747	0.713	0.681	0.650	0.621
6	0.942	0.888	0.837	0.790	0.746	0.705	0.666	0.630	0.596	0.564
7	0.933	0.871	0.813	0.760	0.711	0.665	0.623	0.583	0.547	0.513
8	0.923	0.853	0.789	0.731	0.677	0.627	0.582	0.540	0.502	0.467
9	0.914	0.837	0.766	0.703	0.645	0.592	0.544	0.500	0.460	0.424
10	0.905	0.820	0.744	0.676	0.614	0.558	0.508	0.463	0.422	0.386
11	0.896	0.804	0.722	0.650	0.585	0.527	0.475	0.429	0.388	0.350
12	0.887	0.788	0.701	0.625	0.557	0.497	0.444	0.397	0.356	0.319
13	0.879	0.773	0.681	0.601	0.530	0.469	0.415	0.368	0.326	0.290
14	0.870	0.758	0.661	0.577	0.505	0.442	0.388	0.340	0.299	0.263
15	0.861	0.743	0.642	0.555	0.481	0.417	0.362	0.315	0.275	0.239