Present Value i.e. Principal or Original amount (PV) can be ascertained if the future value, interest rate and the number of time periods are known.
FVn
PV = ————
[1 + (r) (n)]Translating a value to the present is referred to as discounting.
PV and FV of Uneven Cash Flows
The FV and PV annuity formulas assume level and sequential cash flows, but if a problem breaks this assumption, the annuity formulas no longer apply. To solve problems with uneven cash flows, each cash flow must be discounted back to the present (for PV problems) or compounded to a future date (for FV problems); then the sum of the present (or future) values of all cash flows is taken. In practice, particularly if there are many cash flows, this exercise is usually completed by using a spreadsheet.
It helps to set up this problem as if it were on a spreadsheet, to keep track of the cash flows and to make sure that the proper inputs are used to either discount or compound each cash flow. For example, assume that X is to receive a sequence of uneven cash flows from an annuity and it is asked for the present value of the annuity at a discount rate of 8%. X will draw a table similar to the one below, with periods in the first column, cash flows in the second, and formulas in the third column and computations in the fourth.
| Time Period | Cash Flow | Present Value Formula | Result of Computation |
| 1 | Rs. 1,000 | (Rs. 1,000)/(1.08)1 | Rs. 925.93 |
| 2 | Rs. 1,500 | (Rs. 1,500)/(1.08)2 | Rs. 1,286.01 |
| 3 | Rs. 2,000 | (Rs. 2,000)/(1.08)3 | Rs. 1,587.66 |
| 4 | Rs. 500 | (Rs. 500)/(1.08)4 | Rs. 367.51 |
| 5 | Rs. 3,000 | (Rs. 3,000)/(1.08)5 | Rs. 2,041.75 |
Taking the sum of the results in column 4, there is a PV
= Rs.6, 208.86
Suppose it is required to find the future value of this same sequence of cash flows after period 5. Here is the same approach using a table with future value formulas rather than present value, as in the table above:
| Time Period | Cash Flow | Future Value Formula | Result of computation |
| 1 | Rs. 1,000 | (Rs. 1,000)*(1.08)4 | Rs. 1,360.49 |
| 2 | Rs. 1,500 | (Rs. 1,500)*(1.08)3 | Rs. 1,889.57 |
| 3 | Rs. 2,000 | (Rs. 2,000)*(1.08)2 | Rs. 2,332.80 |
| 4 | Rs. 500 | (Rs. 500)*(1.08)1 | Rs. 540.00 |
| 5 | Rs. 3,000 | (Rs. 3,000)*(1.08)0 | Rs. 3,000.00 |
Taking the sum of the results in column 4, FV (period 5) = Rs. 9,122.86.
Check the present value of Rs. 9,122.86, discounted at the 8% rate for five years:
PV = (Rs. 9,122.86)/ (1.08)5
= Rs. 6,208.86
The principle of equivalence applies even in examples where the cash flows are unequal.
