Time Value of Money

Present Values

The present value, P, of an amount, F, to be received in n years when the interest rate is r is

[Page 564]For example, if you will receive $10,000 in three years and the interest rate is 6% then you will be indifferent to the choice between receiving the $10,000 three years from now and receiving $10,000/(1.06)3= $8,396.19 right now. We say, “The present value of $10,000 to be received in three years is $8,396.19.” Another way of putting it is that $10,000 to be received in three years is worth $8,396.19 today if the interest rate is 6%.

Note that to use the formula, n doesn't have to be measured in years, and r doesn't have to be an annual interest rate. It is common to calculate monthly payments for mortgages and auto loans, so n might be in months and r might be a monthly interest rate. Frequently, there will be a series of equal payments, such as with a mortgage, or the interest payments on a bond. This is called an annuity. In these cases the present value of an annuity is just the sum of the present values of the individual payments. If there are more than a few such payments, then the calculations become tedious and it is more convenient to use a formula. The present value interest factor of an annuity when the interest rate is r% and there are n equal payments is denoted PVIFA (r%, n) and is the present value of $1 to be received at the end of each of n periods when the interest rate is r% per period:

Financial calculators and spreadsheet programs such as Microsoft Excel can perform this calculation automatically. To simplify the exposition we will use the expression PVIFA (r%, n) instead of the preceding formula when discussing how to perform the various calculations. Using this expression, the present value of an annuity of $100 per year for 20 years at an annual interest rate of 12% is $100 PVIFA (12%, 20) = $100 × 7.4694 = $746.94.

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