Future Value of an Ordinary Annuity
Sometimes an annuity will be based on "end of period" payments. These annuities are called ordinary annuities (also known as annuities in arrears). The next graphic portrays a 5-year, 10%, ordinary annuity involving level payments of $5,000 each. Notice the similarity to the preceding graphic - except that each year's payment is shifted to the end of the year. This means each payment will accumulate interest for one less year, and the final payment will accumulate no interest! Be sure to note the striking difference between the accumulated total under an annuity due versus and ordinary annuity ($33,578 vs. $30,526). The moral is to save early and save often (and live long!) to take advantage of the power of compound interest.
As you might have guessed, there are also tables that reflect the FUTURE VALUE OF AN ORDINARY ANNUITY. Review the table found in the appendix to satisfy yourself about the $30,526 amount ($5,000 x 6.10510).
Future value calculations provide useful tools for financial planning. But, many decisions and accounting measurements will be based on a reciprocal concept known as present value. Present value (also known as discounting) determines the current worth of cash to be received in the future. For instance, how much would you be willing to take today, in lieu of $1 in one year. If the interest rate is 10%, presumably you would accept the sum that would grow to $1 in one year if it were invested at 10%. This happens to be $0.90909. In other words, invest 90.9$ for a year at 10%, and it will grow to $1 ($0.90909 x 1.1 = $1). Thus, present value calculations are simply the reciprocal of future value calculations:
Where "i" is the interest rate per period and "n" is the number of periods
The PRESENT VALUE OF $1 TABLE (found in the appendix) reveals predetermined values for calculating the present value of $1, based on alternative assumptions about interest rates and time periods. To illustrate, a $25,000 lump sum amount to be received at the end of 10 years, at 8% annual interest, with semiannual compounding, would have a present value of $11,410 (recall the earlier discussion, and use the 4% column/20-period row - $25,000 x 0.45639).
Present Value of an Annuity Due
Present value calculations are also applicable to annuities. Perhaps you are considering buying an investment that returns $5,000 per year for five years, with the first payment to be received immediately. What should you pay for this investment in you have a target rate of return of 10%?
The graphic shows that the annuity has a present value of $20,849. Of course, there is a PRESENT VALUE OF AN ANNUITY DUE TABLE (see the appendix) to ease the burden of this calculation ($5,000 x 4.16897 = $20,849).
Present Value of an Ordinary Annuity
Many times, the first payment in an annuity occurs at the end of 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:
Electronic Spreadsheet Functions
Be aware that most electronic spreadsheets also include functions for calculating present and future value amounts by simply completing a set of predetermined queries.
Challenge Your Thinking
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!