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# Appendix Time value of money

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### Appendix Time value of money

1. 1. Time Value of Money APPENDIXD Study Objectives After studying this appendix, you should be able to: [1] Distinguish between simple and compound interest. [2] Identify the variables fundamental to solving present value problems. [3] Solve for present value of a single amount. [4] Solve for present value of an annuity. [5] Compute the present value of notes and bonds. Would you rather receive \$1,000 today or a year from now? You should prefer to receive the \$1,000 today because you can invest the \$1,000 and earn interest on it. As a result, you will have more than \$1,000 a year from now. What this example illustrates is the concept of the time value of money. Everyone pre- fers to receive money today rather than in the future because of the interest factor. Interest is payment for the use of another person’s money. It is the difference be- tween the amount borrowed or invested (called the principal) and the amount re- paid or collected. The amount of interest to be paid or collected is usually stated as a rate over a specific period of time. The rate of interest is generally stated as an annual rate. The amount of interest involved in any financing transaction is based on three elements: 1. Principal (p): The original amount borrowed or invested. 2. Interest Rate (i): An annual percentage of the principal. 3. Time (n): The number of years that the principal is borrowed or invested. Simple Interest Simple interest is computed on the principal amount only. It is the return on the principal for one period.Simple interest is usually expressed as shown in Illustration D-1 on the next page. Nature of Interest Study Objective [1] Distinguish between simple and compound interest. D1 BMappendixd.indd Page D1 11/30/10 3:29:32 PM f-392BMappendixd.indd Page D1 11/30/10 3:29:32 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
2. 2. D2 Appendix D Time Value of Money Compound Interest Compound interest is computed on principal and on any interest earned that has not been paid or withdrawn. It is the return on the principal for two or more time periods. Compounding computes interest not only on the principal but also on the interest earned to date on that principal, assuming the interest is left on deposit. To illustrate the difference between simple and compound interest, assume that you deposit \$1,000 in Bank Two, where it will earn simple interest of 9% per year, and you deposit another \$1,000 in Citizens Bank, where it will earn com- pound interest of 9% per year compounded annually. Also assume that in both cases you will not withdraw any interest until three years from the date of deposit. Illustration D-2 shows the computation of interest you will receive and the accu- mulated year-end balances. Interest 5 Principal 3 Rate 3 Time p i n For example, if you borrowed \$5,000 for 2 years at a simple interest rate of 12% annually, you would pay \$1,200 in total interest computed as follows: Interest 5 p 3 i 3 n 5 \$5,000 3 .12 3 2 5 \$1,200 Illustration D-1 Interest computation Illustration D-2 Simple versus compound interest Simple Interest Calculation Year 1 Year 2 Year 3 \$1,000.00 × 9% \$1,000.00 × 9% \$1,000.00 × 9% \$ \$ 90.00 90.00 90.00 270.00 \$1,090.00 \$1,180.00 \$1,270.00 \$25.03 Difference Simple Interest Accumulated Year-End Balance Bank Two Compound Interest Calculation Year 1 Year 2 Year 3 \$1,000.00 × 9% \$1,090.00 × 9% \$1,188.10 × 9% \$ \$ 90.00 98.10 106.93 295.03 \$1,090.00 \$1,188.10 \$1,295.03 Compound Interest Accumulated Year-End Balance Citizens Bank Note in Illustration D-2 that simple interest uses the initial principal of \$1,000 to compute the interest in all three years. Compound interest uses the accumulated balance (principal plus interest to date) at each year-end to compute interest in the succeeding year—which explains why your compound interest account is larger. Obviously,if you had a choice between investing your money at simple interest or at compound interest, you would choose compound interest, all other things—especially risk—being equal.In the example,compounding provides \$25.03 of additional interest income. For practical purposes, compounding assumes that unpaid interest earned be- comes a part of the principal, and the accumulated balance at the end of each year becomes the new principal on which interest is earned during the next year. Illustration D-2 indicates that you should invest your money at the bank that compounds interest annually.Most business situations use compound interest.Simple interest is generally applicable only to short-term situations of one year or less. BMappendixd.indd Page D2 11/30/10 3:29:33 PM f-392BMappendixd.indd Page D2 11/30/10 3:29:33 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
3. 3. Present Value Variables The present value is the value now of a given amount to be paid or received in the future, assuming compound interest.The present value is based on three variables: (1) the dollar amount to be received (future amount), (2) the length of time until the amount is received (number of periods), and (3) the interest rate (the discount rate).The process of determining the present value is referred to as discounting the future amount. In this textbook,we use present value computations in measuring several items. For example, Chapter 15 computed the present value of the principal and interest payments to determine the market price of a bond. In addition, determining the amount to be reported for notes payable and lease liabilities involves present value computations. Present Value of a Single Amount To illustrate present value, assume that you want to invest a sum of money that will yield \$1,000 at the end of one year. What amount would you need to invest today to have \$1,000 one year from now? Illustration D-3 shows the formula for calculat- ing present value. Present Value of a Single Amount D3 Study Objective [3] Solve for present value of a single amount. Thus, if you want a 10% rate of return, you would compute the present value of \$1,000 for one year as follows: PV 5 FV 4 (1 1 i)n 5 \$1,000 4 (1 1 .10)1 5 \$1,000 4 1.10 5 \$909.09 We know the future amount (\$1,000), the discount rate (10%), and the number of periods (1). These variables are depicted in the time diagram in Illustration D-4. i = 10% n = 1 year Present Value (?) \$909.09 Future Value \$1,000 Illustration D-4 Finding present value if discounted for one period If you receive the single amount of \$1,000 in two years, discounted at 10% [PV 5 \$1,000 4 (1 1 .10)2 ], the present value of your \$1,000 is \$826.45 [(\$1,000 4 1.21), depicted as shown in Illustration D-5 on the next page. Study Objective [2] Identify the variables fundamental to solving present value problems. Present Value 5 Future Value 4 (1 1 i)n Illustration D-3 Formula for present value BMappendixd.indd Page D3 11/30/10 3:29:33 PM f-392BMappendixd.indd Page D3 11/30/10 3:29:33 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
4. 4. D4 Appendix D Time Value of Money You also could find the present value of your amount through tables that show the present value of 1 for n periods. In Table 1, below, n (represented in the table’s rows) is the number of discounting periods involved.The percentages (represented in the table’s columns) are the periodic interest rates or discount rates. The 5-digit decimal numbers in the intersections of the rows and columns are called the present value of 1 factors. When using Table 1 to determine present value, you multiply the future value by the present value factor specified at the intersection of the number of periods and the discount rate. Table 1 Present Value of 1 (n) Periods 4% 5% 6% 8% 9% 10% 11% 12% 15% 1 .96154 .95238 .94340 .92593 .91743 .90909 .90090 .89286 .86957 2 .92456 .90703 .89000 .85734 .84168 .82645 .81162 .79719 .75614 3 .88900 .86384 .83962 .79383 .77218 .75132 .73119 .71178 .65752 4 .85480 .82270 .79209 .73503 .70843 .68301 .65873 .63552 .57175 5 .82193 .78353 .74726 .68058 .64993 .62092 .59345 .56743 .49718 6 .79031 .74622 .70496 .63017 .59627 .56447 .53464 .50663 .43233 7 .75992 .71068 .66506 .58349 .54703 .51316 .48166 .45235 .37594 8 .73069 .67684 .62741 .54027 .50187 .46651 .43393 .40388 .32690 9 .70259 .64461 .59190 .50025 .46043 .42410 .39092 .36061 .28426 10 .67556 .61391 .55839 .46319 .42241 .38554 .35218 .32197 .24719 11 .64958 .58468 .52679 .42888 .38753 .35049 .31728 .28748 .21494 12 .62460 .55684 .49697 .39711 .35554 .31863 .28584 .25668 .18691 13 .60057 .53032 .46884 .36770 .32618 .28966 .25751 .22917 .16253 14 .57748 .50507 .44230 .34046 .29925 .26333 .23199 .20462 .14133 15 .55526 .48102 .41727 .31524 .27454 .23939 .20900 .18270 .12289 16 .53391 .45811 .39365 .29189 .25187 .21763 .18829 .16312 .10687 17 .51337 .43630 .37136 .27027 .23107 .19785 .16963 .14564 .09293 18 .49363 .41552 .35034 .25025 .21199 .17986 .15282 .13004 .08081 19 .47464 .39573 .33051 .23171 .19449 .16351 .13768 .11611 .07027 20 .45639 .37689 .31180 .21455 .17843 .14864 .12403 .10367 .06110 For example, the present value factor for one period at a discount rate of 10% is .90909, which equals the \$909.09 (\$1,000 3 .90909) computed in Illustration D-4. For two periods at a discount rate of 10%, the present value factor is .82645, which equals the \$826.45 (\$1,000 3 .82645) computed previously. Note that a higher discount rate produces a smaller present value. For example, using a 15% discount rate, the present value of \$1,000 due one year from now is \$869.57, versus \$909.09 at 10%. Also note that the further removed from the pres- ent the future value is, the smaller the present value. For example, using the same Illustration D-5 Finding present value if discounted for two periods i = 10% 1 Present Value (?) 0 Future Value 2 n = 2 years\$826.45 \$1,000 BMappendixd.indd Page D4 11/30/10 3:29:33 PM f-392BMappendixd.indd Page D4 11/30/10 3:29:33 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
5. 5. Present Value of an Annuity D5 discount rate of 10%, the present value of \$1,000 due in five years is \$620.92, versus the present value of \$1,000 due in one year, which is \$909.09. The following two demonstration problems (Illustrations D-6 and D-7) illustrate how to use Table 1. Present Value of an Annuity The preceding discussion involved the discounting of only a single future amount. Businesses and individuals frequently engage in transactions in which a series of equal dollar amounts are to be received or paid at evenly spaced time intervals (periodically). Examples of a series of periodic receipts or payments are loan agreements,installment sales,mortgage notes,lease (rental) contracts,and pension obligations.As discussed in Chapter 15, these periodic receipts or payments are annuities. The present value of an annuity is the value now of a series of future receipts or payments, discounted assuming compound interest. In computing the present value of an annuity, you need to know: (1) the discount rate, (2) the number of dis- count periods, and (3) the amount of the periodic receipts or payments. Illustration D-6 Demonstration problem— Using Table 1 for PV of 1 i = 8% 2 PV = ? Now \$10,000 3 years1 Suppose you have a winning lottery ticket and the state gives you the option of taking \$10,000 three years from now or taking the present value of \$10,000 now. The state uses an 8% rate in discounting. How much will you receive if you accept your winnings now? Answer: The present value factor from Table 1 is .79383 (3 periods at 8%). The present value of \$10,000 to be received in 3 years discounted at 8% is \$7,938.30 (\$10,000 × .79383). n = 3 Illustration D-7 Demonstration problem— Using Table 1 for PV of 1 i = 9% 3 PV = ? Now \$5,000 4 years1 Determine the amount you must deposit now in a bond investment, paying 9% interest, in order to accumulate \$5,000 for a down payment 4 years from now on a new Toyota Prius. Answer: The present value factor from Table 1 is .70843 (4 periods at 9%). The present value of \$5,000 to be received in 4 years discounted at 9% is \$3,542.15 (\$5,000 × .70843). 2 n = 4 Study Objective [4] Solve for present value of an annuity. BMappendixd.indd Page D5 11/30/10 3:29:33 PM f-392BMappendixd.indd Page D5 11/30/10 3:29:33 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
6. 6. D6 Appendix D Time Value of Money To illustrate how to compute the present value of an annuity, assume that you will receive \$1,000 cash annually for three years at a time when the discount rate is 10%. Illustration D-8 depicts this situation, and Illustration D-9 shows the compu- tation of its present value. Illustration D-9 Present value of a series of future amounts computation Present Value of 1 Future Amount 3 Factor at 10% 5 Present Value \$1,000 (one year away) .90909 \$ 909.09 1,000 (two years away) .82645 826.45 1,000 (three years away) .75132 751.32 2.48686 \$2,486.86 This method of calculation is required when the periodic cash flows are not uniform in each period. However, when the future receipts are the same in each period, there are two other ways to compute present value. First, you can multiply the annual cash flow by the sum of the three present value factors. In the previous example, \$1,000 3 2.48686 equals \$2,486.86. The second method is to use annuity tables.As illustrated in Table 2 below, these tables show the present value of 1 to be received periodically for a given number of periods. Table 2 Present Value of an Annuity of 1 (n) Periods 4% 5% 6% 8% 9% 10% 11% 12% 15% 1 .96154 .95238 .94340 .92593 .91743 .90909 .90090 .89286 .86957 2 1.88609 1.85941 1.83339 1.78326 1.75911 1.73554 1.71252 1.69005 1.62571 3 2.77509 2.72325 2.67301 2.57710 2.53130 2.48685 2.44371 2.40183 2.28323 4 3.62990 3.54595 3.46511 3.31213 3.23972 3.16986 3.10245 3.03735 2.85498 5 4.45182 4.32948 4.21236 3.99271 3.88965 3.79079 3.69590 3.60478 3.35216 6 5.24214 5.07569 4.91732 4.62288 4.48592 4.35526 4.23054 4.11141 3.78448 7 6.00205 5.78637 5.58238 5.20637 5.03295 4.86842 4.71220 4.56376 4.16042 8 6.73274 6.46321 6.20979 5.74664 5.53482 5.33493 5.14612 4.96764 4.48732 9 7.43533 7.10782 6.80169 6.24689 5.99525 5.75902 5.53705 5.32825 4.77158 10 8.11090 7.72173 7.36009 6.71008 6.41766 6.14457 5.88923 5.65022 5.01877 11 8.76048 8.30641 7.88687 7.13896 6.80519 6.49506 6.20652 5.93770 5.23371 12 9.38507 8.86325 8.38384 7.53608 7.16073 6.81369 6.49236 6.19437 5.42062 13 9.98565 9.39357 8.85268 7.90378 7.48690 7.10336 6.74987 6.42355 5.58315 14 10.56312 9.89864 9.29498 8.24424 7.78615 7.36669 6.98187 6.62817 5.72448 15 11.11839 10.37966 9.71225 8.55948 8.06069 7.60608 7.19087 6.81086 5.84737 16 11.65230 10.83777 10.10590 8.85137 8.31256 7.82371 7.37916 6.97399 5.95424 17 12.16567 11.27407 10.47726 9.12164 8.54363 8.02155 7.54879 7.11963 6.04716 18 12.65930 11.68959 10.82760 9.37189 8.75563 8.20141 7.70162 7.24967 6.12797 19 13.13394 12.08532 11.15812 9.60360 8.95012 8.36492 7.83929 7.36578 6.19823 20 13.59033 12.46221 11.46992 9.81815 9.12855 8.51356 7.96333 7.46944 6.25933 Illustration D-8 Time diagram for a three-year annuity i = 10% 2Now 3 years PV = ? \$1,000 \$1,000\$1,000 1 n = 3 BMappendixd.indd Page D6 11/30/10 3:29:33 PM f-392BMappendixd.indd Page D6 11/30/10 3:29:33 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
7. 7. Table 2 shows that the present value of an annuity of 1 factor for three periods at 10% is 2.48685.1 (This present value factor is the total of the three individual present value factors, as shown in Illustration D-9.) Applying this amount to the annual cash flow of \$1,000 produces a present value of \$2,486.85. The following demonstration problem (Illustration D-10) illustrates how to use Table 2. Time Periods and Discounting In the preceding calculations, the discounting was done on an annual basis using an annual interest rate. Discounting may also be done over shorter periods of time such as monthly, quarterly, or semiannually. When the time frame is less than one year, you need to convert the annual interest rate to the applicable time frame. Assume, for example, that the investor in Illustration D-8 received \$500 semiannually for three years instead of \$1,000 annually. In this case, the number of periods becomes six (3 3 2), the discount rate is 5% (10% 4 2), the present value factor from Table 2 is 5.07569, and the present value of the future cash flows is \$2,537.85 (5.07569 3 \$500). This amount is slightly higher than the \$2,486.86 computed in Illustration D-9 because interest is paid twice during the same year; therefore interest is earned on the first half year’s interest. Computing the Present Value of a Long-Term Note or Bond The present value (or market price) of a long-term note or bond is a function of three variables: (1) the payment amounts, (2) the length of time until the amounts are paid, and (3) the discount rate. Our illustration uses a five-year bond issue. i = 12% 4 PV = ? Now \$6,000 5 years1 Kildare Company has just signed a capitalizable lease contract for equip- ment that requires rental payments of \$6,000 each, to be paid at the end of each of the next 5 years. The appropriate discount rate is 12%. What is the present value of the rental payments—that is, the amount used to capitalize the leased equipment? Answer: The present value factor from Table 2 is 3.60478 (5 periods at 12%). The present value of 5 payments of \$6,000 each discounted at 12% is \$21,628.68 (\$6,000 × 3.60478). \$6,000 \$6,000 2 3 \$6,000 \$6,000 n = 5 Illustration D-10 Demonstration problem— Using Table 2 for PV of an annuity of 1 1 The difference of .00001 between 2.48686 and 2.48685 is due to rounding. Study Objective [5] Compute the present value of notes and bonds. Computing the Present Value of a Long-Term Note or Bond D7 BMappendixd.indd Page D7 11/30/10 3:29:33 PM f-392BMappendixd.indd Page D7 11/30/10 3:29:33 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
8. 8. D8 Appendix D Time Value of Money The first variable—dollars to be paid—is made up of two elements: (1) a series of interest payments (an annuity), and (2) the principal amount (a single sum). To compute the present value of the bond, we must discount both the interest pay- ments and the principal amount—two different computations. The time diagrams for a bond due in five years are shown in Illustration D-11. When the investor’s market interest rate is equal to the bond’s contractual interest rate, the present value of the bonds will equal the face value of the bonds. To illustrate, assume a bond issue of 10%, five-year bonds with a face value of \$100,000 with interest payable semiannually on January 1 and July 1. If the discount rate is the same as the contractual rate, the bonds will sell at face value. In this case, the investor will receive the following: (1) \$100,000 at matu- rity, and (2) a series of ten \$5,000 interest payments [(\$100,000 3 10%) 4 2] over the term of the bonds. The length of time is expressed in terms of interest periods—in this case—10, and the discount rate per interest period, 5%. The following time diagram (Illustration D-12) depicts the variables involved in this discounting situation. Interest Rate (i) 1 yr. Present Value (?) Now Principal Amount 5 yr. Diagram for Principal 2 yr. 3 yr. 4 yr. Interest 1 yr. Present Value (?) Now 5 yr. Diagram for Interest 2 yr. 3 yr. 4 yr. Interest Rate (i) Interest Interest Interest Interest n = 5 n = 5 Illustration D-11 Present value of a bond time diagram i = 5% 1 Present Value (?) Now Principal Amount \$100,000 10 Diagram for Principal 5 6 1 Present Value (?) Now 10 Diagram for Interest 5 6 i = 5% \$5,000 2 2 3 3 4 4 7 7 8 8 9 9 \$5,000 \$5,000 \$5,000 \$5,000\$5,000 \$5,000 \$5,000 \$5,000 n = 10 n = 10 \$5,000 Interest Payments Illustration D-12 Time diagram for present value of a 10%, five-year bond paying interest semiannually BMappendixd.indd Page D8 11/30/10 3:29:34 PM f-392BMappendixd.indd Page D8 11/30/10 3:29:34 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
9. 9. Illustration D-13 shows the computation of the present value of these bonds. Now assume that the investor’s required rate of return is 12%, not 10%. The future amounts are again \$100,000 and \$5,000, respectively, but now a discount rate of 6% (12% 4 2) must be used. The present value of the bonds is \$92,639, as computed in Illustration D-14. Conversely, if the discount rate is 8% and the contractual rate is 10%, the pres- ent value of the bonds is \$108,111, computed as shown in Illustration D-15. The above discussion relies on present value tables in solving present value problems. Many people use spreadsheets such as Excel or financial calculators (some even on websites) to compute present values,without the use of tables.Many calculators, especially “financial calculators,” have present value (PV) functions that allow you to calculate present values by merely inputting the proper amount, discount rate, and periods, and pressing the PV key. Appendix E illustrates how to use a financial calculator in various business situations. Illustration D-13 Present value of principal and interest—face value 10% Contractual Rate—10% Discount Rate Present value of principal to be received at maturity \$100,000 3 PV of 1 due in 10 periods at 5% \$100,000 3 .61391 (Table 1) \$ 61,391 Present value of interest to be received periodically over the term of the bonds \$5,000 3 PV of 1 due periodically for 10 periods at 5% \$5,000 3 7.72173 (Table 2) 38,609* Present value of bonds \$100,000 *Rounded Illustration D-14 Present value of principal and interest—discount 10% Contractual Rate—12% Discount Rate Present value of principal to be received at maturity \$100,000 3 .55839 (Table 1) \$55,839 Present value of interest to be received periodically over the term of the bonds \$5,000 3 7.36009 (Table 2) 36,800 Present value of bonds \$92,639 Illustration D-15 Present value of principal and interest—premium 10% Contractual Rate—8% Discount Rate Present value of principal to be received at maturity \$100,000 3 .67556 (Table 1) \$ 67,556 Present value of interest to be received periodically over the term of the bonds \$5,000 3 8.11090 (Table 2) 40,555 Present value of bonds \$108,111 Computing the Present Value of a Long-Term Note or Bond D9 BMappendixd.indd Page D9 11/30/10 3:29:34 PM f-392BMappendixd.indd Page D9 11/30/10 3:29:34 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05
10. 10. D10 Appendix D Time Value of Money [1] Distinguish between simple and compound interest. Simple interest is computed on the principal only, while compound interest is computed on the principal and any interest earned that has not been withdrawn. [2] Identify the variables fundamental to solving present value problems. The following three variables are fundamental to solving present value problems: (1) the future amount, (2) the number of periods, and (3) the interest rate (the discount rate). [3] Solve for present value of a single amount. Pre- pare a time diagram of the problem. Identify the future amount, the number of discounting periods, and the discount (interest) rate. Using the present value of a single amount table, multiply the future amount by the present value factor specified at the intersection of the number of periods and the discount rate. [4] Solve for present value of an annuity. Prepare a time diagram of the problem. Identify the future annuity pay- ments, the number of discounting periods, and the discount (interest) rate. Using the present value of an annuity of 1 table, multiply the amount of the annuity payments by the present value factor specified at the intersection of the number of pe- riods and the interest rate. [5] Compute the present value of notes and bonds. To determine the present value of the principal amount: Multiply the principal amount (a single future amount) by the present value factor (from the present value of 1 table) intersecting at the number of periods (number of interest payments) and the discount rate. To determine the present value of the series of interest payments: Multiply the amount of the interest payment by the present value factor (from the present value of an annuity of 1 table) intersecting at the number of periods (number of inter- est payments) and the discount rate. Add the present value of the principal amount to the present value of the interest pay- ments to arrive at the present value of the note or bond. Summary of Study Objectives Annuity A series of equal dollar amounts to be paid or re- ceived at evenly spaced time intervals (periodically). (p. D5). Compound interest The interest computed on the principal and any interest earned that has not been paid or withdrawn. (p. D2). Discounting the future amount(s) The process of determining present value. (p. D3). Interest Payment for the use of another’s money. (p. D1). Present value The value now of a given amount to be paid or received in the future assuming compound interest. (p. D3). Present value of an annuity The value now of a series of future receipts or payments, discounted assuming com- pound interest. (p. D5). Principal The amount borrowed or invested. (p. D1). Simple interest The interest computed on the principal only. (p. D1). Glossary Use present value tables. Brief Exercises Use tables to solve exercises. BED-1 For each of the following cases, indicate (a) to what interest rate columns, and (b) to what number of periods you would refer in looking up the discount rate. 1. In Table 1 (present value of 1): Number of Compounding Annual Rate Years Involved Per Year (a) 12% 6 Annually (b) 10% 15 Annually (c) 8% 12 Semiannually 2. In Table 2 (present value of an annuity of 1): Number of Number of Frequency of Annual Rate Years Involved Payments Involved Payments (a) 8% 20 20 Annually (b) 10% 5 5 Annually (c) 12% 4 8 Semiannually BED-2 (a) What is the present value of \$30,000 due 8 periods from now, discounted at 8%? (b) What is the present value of \$30,000 to be received at the end of each of 6 periods, discounted at 9%? Determine present values. BMappendixd.indd Page D10 11/30/10 3:29:34 PM f-392BMappendixd.indd Page D10 11/30/10 3:29:34 PM f-392 /Users/f-392/Desktop/Nalini 23.9/ch05/Users/f-392/Desktop/Nalini 23.9/ch05