Solving Time Value of Money Problems

There are, basically, two algebraic formulas associated with the solving of time value of money problems. For all future value solutions, where one is looking for the compounded future value of a present amount, or a stream of cash flows of equal amounts, or the future value of a mixed stream of cash flows, the basic formula is:

FV= P(1+k)^n

For the future value of a single amount, this formula says that the future value of an amount presently held is the present value(P), multiplied by 1 plus the applied annual interest rate (interest rates are expressed annually by the symbol k), raised to the appropriate power (number of years). For example the future value of $1000 in 15 years compounded at 9% annual interest is:

$1000(1.09)^15= $3642.48

With a stream of cash flows received each year as opposed to one single amount at the beginning compounding each year, the same formula would apply, but to each period. For example in the above problem, the first year would be as shown, while the second year's cash flow would be $1000(1.09)^14=$3341.72, and so on. If all the years were totaled, the answer would be the compounded value of the cash flow stream for the 15 year period compounded at 9%. However, if each of the invested amounts are equal one factor can be computed to deliver one answer simply by using one dollar for each year at the appropriate interest rate, then adding them up. This factor could then be applied to any series of equal payments to determine the compound future value.

With a stream of cash flows of different values in each period, it is necessary to compute each value separately for each year, using the above formula, then add the results to determine the compounded future value of the cash flow stream.

For present value calculations, we are computing at an obverse process, that is , the process of discounting a single future amount or a stream of future cash to the present.

This process, used in all investment analysis, reveals the present value of a future single amount, or a series of equal or unequal cash flows. Its basic formula reflects the reverse process to compounding, called discounting, and is:


For example, the present discounted value of the right to receive $3642.48 in 15 years, if one has to receive 9% compounded annually on the present value of the investment is:


The present value of a stream of equal payments, or a mixed stream of cash flows to be received, follows the same obverse pattern using the above formula.

The algebra becomes more rigorous when it is necessary to solve for time, or interest in the above formulas. For example, if in the above example we wanted to solve for the interest rate which would make the present value of $3642.48 equal $1000 in 15 periods:


The rigor of this type of calculation in time consumed, and in the potential for processing errors, benefits from the use of calculators with time value mode line buttons, where solving for the various elements of such calculations can be more cost effective and business like.

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