Long-Term Financing: Bonds
Future value
The future value or worth of any investment is the amount to which a sum of money invested today grows during a stated period of time at a specified interest rate. The interest involved may be simple interest or compound interest. Simple interest is interest on principal only. For example, USD 1,000 invested today for two years at 12 percent simple interest grows to USD 1,240 since interest is USD 120 per year. The principal of USD 1,000, plus 2 X USD 120, is equal to USD 1,240. Compound interest is interest on principal and on interest of prior periods. For example, USD 1,000 invested for two years at 12 percent compounded annually grows to USD 1,254.40 as follows:
| Principal or present value | $1,000.00 |
| Interest, year 1 = $1,000 x 0.12 = | 120.00 |
| Value at end of year 1 | $1,120.00 |
| Interest, year 2 = $1,120 x 0.12 = | 134.40 |
| Value at end of year 2 (future value) | $1,254.40 |
In Exhibit 122, we graphically portray these computations of future worth and show how USD 1,000 grows to USD 1,254.40 with a 12 percent interest rate compounded annually. The effect of compounding is USD 14.40—the interest in the second year that was based on the interest computed for the first year, or USD 120 X 0.12 = USD 14.40.
Interest tables ease the task of computing the future worth to which any invested amount will grow at a given rate for a stated period. An example is Table A.1 in the Appendix at the end of this text. To use the Appendix tables, first determine the number
of compounding periods involved. A compounding period may be any length of time, such as a day, a month, a quarter, a half-year, or a year, but normally not more than a year. The number of compounding periods is equal to the number of years in the
life of the investment times the number of compoundings per year. Five years compounded annually is five periods, five years compounded quarterly is 20 periods, and so on.
Second, determine the interest rate per compounding period. Interest rates are usually quoted in annual terms; in fact, federal law requires statement of the interest rate in annual terms in some situations. Divide the annual rate by the number of compounding
periods per year to get the proper rate per period. Only with an annual compounding period will the annual rate be the rate per period. All other cases involve a lower rate. For example, if the annual rate is 12 percent and interest is compounded
monthly, the rate per period (one month) will be 1 percent.
To use the tables, find the number of periods involved in the Period column. Move across the table to the right, stopping in the column headed by the Interest Rate per Period, which yields a number called a factor. The factor shows the amount to which
an investment of USD 1 will grow for the periods and the rate involved. To compute the future worth of the investment, multiply the number of dollars in the given situation by this factor. For example, suppose your parents tell you that they will
invest USD 8,000 at 12 percent for four years and give you the amount to which this investment grows if you graduate from college in four years. How much will you receive at the end of four years if the interest rate is 12 percent compounded annually?
How much will you receive if the interest rate is 12 percent compounded quarterly?
To calculate these amounts, look at the end-of-text Appendix, Table A.1. In the intersection of the 4 period row and the 12 percent column, you find the factor 1.57352. Multiplying this factor by USD 8,000 yields USD 12,588.16, the answer to the first
question. To answer the second question, look at the intersection of the 16 period row and the 3 percent column. The factor is 1.60471, and the value of your investment is USD 12,837.68. The more frequent compounding would add USD 12,837.68 - USD
12,588.16 = USD 249.52 to the value of your investment. The reason for this difference in amounts is that 12 percent compounded quarterly is a higher rate than 12 percent compounded annually.
An annuity is a series of equal cash flows (often called rents) spaced equally in time. The semiannual interest payments received on a bond investment are a common example of an annuity. Assume that USD 100 will be received at the end
of each of the next three semiannual periods. The interest rate is 6 percent per semiannual period. Using Table A.1 in the Appendix, we find the future value of each of the USD 100 receipts as follows:
Exhibit 122: Compound interest and future value
| Future value (after three periods) of $100 | |
| received at the end of the - | |
| First period: | 1.12360 x $100=$112.36 |
| Second period: | 1.06000 x 100 = 106.00 |
| Third period: | 1.00000 x 100 = 100.00 |
| Total future value | $318.36 |
Such a procedure would become quite tedious if the annuity consisted of many receipts. Fortunately, tables are available to calculate the total future value directly. See the Appendix, Table A.2. For the annuity just described, you can identify one single factor by looking at the 3 period row and 6 percent column. The factor is 3.18360 (the sum of the three factors shown above), and when multiplied by USD 100, yields USD 318.36, which is the same answer. In Exhibit 123, we graphically present the future value of an annuity.