It is impossible to compare the value or potential purchasing power of the future dollar to today's dollar; they exist in different times and have different values. Present value (PV) considers the future value of an investment expressed in today's value. This allows a company to see if the investment's initial cost is more or less than the future return. For example, a bank might consider the present value of giving a customer a loan before extending funds to ensure that the risk and the interest earned are worth the initial outlay of cash.
Similar to the Future Value tables, the columns show interest rates (i) and the rows show periods (n) in the Present Value tables. Periods represent how often interest is compounded (paid); that is, periods could represent days, weeks, months, quarters, years, or any interest time period. For our examples and assessments, the period (n) will almost always be in years. The intersection of the expected payout years (n) and the interest rate (i) is a number called a present value factor. The present value factor is multiplied by the initial investment cost to produce the present value of the expected cash flows (or investment return).
Present Value = Present Value Factor × Initial Investment Cost |
When referring to present value, the lump sum return occurs at the end of a period. A business must determine if this delayed repayment, with interest, is worth the same as, more than, or less than the initial investment cost. If the deferred payment is more than the initial investment, the company would consider an investment.
To calculate the present value of a lump sum, we should use the Present Value of $1 table. For example, you are interested in saving money for college and want to calculate how much you would need put in the bank today to return a sum of $40,000 in 10 years. The bank returns an interest rate of 3% per year during these 10 years. Looking at the PV table, n = 10 years and i = 3% returns a present value factor of 0.744. Multiplying this factor by the return amount of $40,000 produces $29,760. This means you would need to put in the bank now approximately $29,760 to have $40,000 in 10 years.
Present Value of $1 Table Factor =  |
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| Period (n) | Rate(i) | |||||||
| 1% | 2% | 3% | 5% | |||||
| 1 | 0.990 | 0.980 | 0.971 | 0.952 | ||||
| 2 | 0.980 | 0.961 | 0.943 | 0.907 | ||||
| 3 | 0.971 | 0.942 | 0.915 | 0.864 | ||||
| 4 | 0.961 | 0.924 | 0.888 | 0.823 | ||||
| 5 | 0.952 | 0.906 | 0.863 | 0.784 | ||||
| 6 | 0.942 | 0.888 | 0.837 | 0.746 | ||||
| 7 | 0.933 | 0.871 | 0.813 | 0.711 | ||||
| 8 | 0.914 | 0.853 | 0.789 | 0.677 | ||||
| 9 | 0.914 | 0.837 | 0.766 | 0.645 | ||||
| 10 | 0.905 | 0.820 | 0.744 | 0.614 | ||||
| 11 | 0.896 | 0.804 | 0.722 | 0.585 | ||||
As discussed previously, annuities are a series of equal payments made over time, and ordinary annuities pay the equal installment at the end of each payment period within the series. This can help a business understand how their periodic returns translate into today's value.
For example, assume that Sam needs to borrow money for college and anticipates that she will be able to repay the loan in $1,200 annual payments for each of 5 years. If the lender charges 5% per year for similar loans, how much cash would the bank be willing to lend Sam today? In this case, she would use the Present Value of an Ordinary Annuity table in Appendix B, where n = 5 and i = 5%. This yields a present value factor of 4.329. The current value of the cash flow each period is calculated as 4.329 × $1,200 = $5,194.80. Therefore, Sam could borrow $5,194.80 now given the repayment parameters.
Present Value of an Ordinary Annuity Table Factor = ![\frac{[1-1/(1+i)^n]}{i}](https://learn.saylor.org/filter/tex/pix.php/8694ef29898101c7ae2328cac4150637.svg "\frac{[1-1/(1+i)^n]}{i}") |
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| Period (n) | Rate(i) | |||||||
| 1% | 2% | 3% | 5% | |||||
| 1 | 0.990 | 0.980 | 0.971 | 0.952 | ||||
| 2 | 1.970 | 1.942 | 1.913 | 1.859 | ||||
| 3 | 2.941 | 2.884 | 2.829 | 2.723 | ||||
| 4 | 3.902 | 3.808 | 3.717 | 3.546 | ||||
| 5 | 4.853 | 4.713 | 4.580 | 4.329 | ||||
Determine the present value for each of the following situations. Use the present value tables provided in Appendix B when needed, and round answers to the nearest cent where required.
You are saving for college and you want to return a sum of $100,000 in 12 years. The bank returns an interest rate of 5% after these 12 years.
You need to borrow money for college and can afford a yearly payment to the lending institution of $1,000 per year for the next 8 years. The interest rate charged by the lending institution is 3% per year.
Solution
a. Use PV of $1 table. Present value factor where n = 12 and i = 5 is 0.557. 0.557 × $100,000 = $55,700. b. Use PV of an ordinary annuity table. Present value factor where n = 8 and i = 3 is 7.020. 7.020 × $1,000 = $7,020.