 # Present Value of an Ordinary Annuity

Many times, the first payment in an annuity o course at the end or each period. The PRESENT VALUE OF AN ORDINARY ANNUITY TABLE provides the necessary factor to determine that \$5,000 to be received at the end of each year, for a five-year period, is worth only \$18,954, assuming a 10% interest rate (\$5,000 X 3.79079 = \$18,954). The following graphic confirms this conclusion: Be aware that most electronic spreadsheets also include functions for calculating present and future value amounts by simply completing a set of predetermined queries. You can see a screen shot of the present value function applied to the preceding illustration by clicking on the link at this location

Many scenarios represent a combination of lump sum and annuity cash flow amounts. There are a variety of approaches to calculating the future or present value for such scenarios. Perhaps the safest approach is to diagram the anticipated cash flows and apply logical manipulations. To illustrate, assume that Markum Real Estate is considering buying an office building. The building will be vacant for two years while it is being renovated. Then, it will produce annual rents of \$ 100,000 at the beginning of each of the next three years. The building will be sold in five years for \$700,000. Markum desires to know the present value of the anticipated cash inflows, assuming 5% annual interest rate.

As you can see below, the rental stream has a present value of \$285,941 as of the beginning of Year 3. That value is discounted back to the beginning of Year 1 value (\$259,357) by treating it as a lump sum. The sales price is separately discounted to its present value of \$548,471. The present value of the rents and sales price are combined to produce the total present value for all cash inflows (\$807,828). This type of cash flow manipulation is quite common in calculating present values for many investment decisions. For the more inspired mind, you will at least find it interesting to note that an alternative way to value the rental stream would be to subtract the value for a two year annuity from the value for a five year annuity (4.54595 - 1.95238 = 2.59357; \$100,000 X 2.59357 = \$259,357). This result occurs because it assumes a five-year annuity and backs out the amount relating to the first two years, leaving only the last three years in the resulting present value factor. Like all things mathematical, the more you study them, the more power you find buried within! In the above spreadsheet, formulas were used to determine present value factors. For example, the "balloon" shows the specific formula for cell H17 - (1/(1 +i) ) - where "i" is drawn from cell C17 which is set at 8%. Similar formulas are used for other present value factor cells. This simple approach allows rapid recalculation of net present value by simply changing the value in the interest rate cell.

To illustrate NPV, let's return to our illustration for Markum Real Estate. Assume that the firm's cost of capital is 5%. You already know the present value of the cash inflows is \$807,828. Let's additionally assume that the up-front purchase price for the building is \$575,000. \$60,000 per year will be spent on the remodel effort at the end of Year 1 and Year 2. Maintenance, insurance, and taxes on the building will amount to \$10,000 per year, payable at the end of each of the five years. The present value of the cash outflows is \$729,859: This project has a positive net present value of \$77,969 (\$807,828 - \$729,859). This suggests the project's returns exceed the 5% cost of capital threshold. Had the up-front investment been \$675,000 (instead of \$575,000), the project would have a negative net present value of \$22,031 (\$807,828 - \$829,859).