# Stockholders' Equity: Classes of Capital Stock

## Appendix: Future value and present value

### Present value

Present value is the current worth of a future cash receipt and is the reciprocal of future value. In future value, we calculate the future value of a sum of money possessed now. In present value, we calculate the current worth of rights to future cash receipts possessed now. We discount future receipts by bringing them back to their present values.

Assume that you have the right to receive USD 1,000 in one year. If the appropriate interest rate is 12 percent compounded annually, what is the present value of this USD 1,000 future cash receipt? You know that the present value is less than USD 1,000 because USD 1,000 due in one year is not worth USD 1,000 today You also know that the USD 1,000 due in one year is equal to some amount, P, plus interest on P at 12 percent for one year. Thus, P + 0.12P = USD 1,000, or 1.12P = USD 1,000. Dividing USD 1,000 by 1.12, you get USD 892.86; this amount is the present value of your future USD 1,000. If the USD 1,000 was due in two years, you would find its present value by dividing USD 892.86 by 1.12, which equals USD 797.20. Portrayed graphically, present value looks similar to future value, except for the direction of the arrows (Exhibit 48).

Table A.3 (end-of-text Appendix) contains present value factors for combinations of a number of periods and interest rates. We use Table A.3 in the same manner as Table A.1. For example, the present value of USD 1,000 due in four years at 16 percent compounded annually is USD 552.29, computed as USD 1,000 X 0.55229. The 0.55229 is the present value factor found in the intersection of the 4 period row and the 16 percent column.

Exhibit 47: Future value of an annuity

Exhibit 48: Compound interest and present value

As another example, suppose that you wish to have USD 4,000 in three years to pay for a vacation in Europe. If your investment increases at a 20 percent rate compounded quarterly, how much should you invest now? To find the amount, you would use the present value factor found in Table A.3, 12 period row, 5 percent column. This factor is 0.55684, which means that an investment of about 55 cents today would grow to USD 1 in 12 periods at 5 percent per period. To have USD 4,000 at the end of three years, you must invest 4,000 times this factor (0.55684), or USD 2,227.36.

The semiannual interest payments on a bond are a common example of an annuity. As an example of calculating the present value of an annuity, assume that USD 100 is received at the end of each of the next three semiannual periods. The interest rate is 6 percent per semiannual period. By using Table A.3 (Appendix), you can find the present value of each of the three USD 100 payments as follows:

 Present value of $100 due in: 1 period: 094340 x$100 = $94.34 2 period: 0.89000 x 100 = 89.00 3 period: 0.83962 x 100 = 83.96 Total present value$267.30

Exhibit 49: Present value of an annuity

Such a procedure could become quite tedious if the annuity consisted of a large number of payments. Fortunately, tables are also available showing the present values of an annuity of USD 1 per period for varying interest rates and periods. See the end-of-text Appendix, Table A.4. For the annuity just described, you can obtain a single factor from the table to represent the present value of an annuity of USD 1 per period for three (semiannual) periods at 6 percent per (semiannual) period. This factor is 2.67301; it is equal to the sum of the present value factors for USD 1 due in one period, USD 1 in two periods, and USD 1 in three periods found in the Appendix, Table A.3. When this factor is multiplied by USD 100, the number of dollars in each payment, it yields the present value of the annuity, USD 267.30. In Exhibit 49, we graphically present the present value of this annuity and show how to find the present value of the three USD 100 cash flows by multiplying the USD 100 by a present value of an annuity factor, 2.67301.

Suppose you won a lottery that awarded you a choice of receiving USD 10,000 at the end of each of the next five years or USD 35,000 cash today. You believe you can earn interest on invested cash at 15 percent per year. Which option should you choose? To answer the question, compute the present value of an annuity of USD 10,000 per period for five years at 15 percent. The present value is USD 33,521.60, or USD 10,000 X 3.35216. You should accept the immediate payment of USD 35,000 since it has the larger present value.