Chapter 9 Time Value of Money CHAPTER PREVIEW Money can increase or grow over time if we can save or invest it and we are paid a “return” on the use of our money by others. We begin the chapter with examples of simple interest being earned on a savings amount or investment. We then turn our attention to compounding. Most individuals have experienced compounding by watching a savings account grow or increase over time when interest is reinvested. Then, we follow with a discussion of discounting which can be viewed as being the opposite of compounding. When saving or investing involves equal periodic payments (e.g., quarterly) we call this an “annuity.” Both the future and present values of annuities are covered in the chapter. In some instances, it is important to be able to find or solve for the “interest rate” being earned on an investment. Likewise, you might find it important to know how long it will take to accumulate enough savings to pay for a vacation trip. These topics are also covered in the chapter. Other topics covered include how to handle more frequent than annual compounding or discounting intervals and the comparison of the annual percentage rate (APR) and the effective annual rate (EAR). “Annuity due” problems are covered in Learning Extension 9. Solutions to problems are presented through equations and tables that the student should be able to grasp easily. LEARNING OBJECTIVES Explain what is meant by the “time value of money.” Describe the concept of simple interest and the process of compounding. Describe discounting to determine present values. Find interest rates and time requirements for problems involving compounding or discounting. Describe the meaning of an ordinary annuity. Find interest rates and time requirements for problems involving annuities. Calculate annual annuity payments. Make compounding and discounting calculations using time intervals that are less than one year. Describe the difference between the annual percentage rate and the effective annual rate. Describe the meaning of an annuity due. (in the Learning Extension). Learning Extension: Annuity Due Problems CHAPTER OUTLINE I. BASIC CONCEPTS II. COMPOUNDING TO DETERMINE FUTURE VALUES A. Inflation or Purchasing Power Implications III. DISCOUNTING TO DETERMINE PRESENT VALUES IV. EQUATING PRESENT VALUES AND FUTURE VALUES V. FINDING INTEREST RATES AND TIME REQUIREMENTS A. Solving for Interest Rates B. Solving for Time Periods C. Rule of 72 VI. FUTURE VALUE OF AN ANNUITY VII. PRESENT VALUE OF AN ANNUITY VIII. INTEREST RATES AND TIME REQUIREMENTS FOR ANNUITIES A. Solving for Interest Rates B. Solving for Time Periods IX. DETERMINING PERIODIC ANNUITY PAYMENTS EXAMPLES INVOLVING ANNUAL PAYMENTS REAL ESTATE MORTGAGE LOANS WITH MONTHLY PAYMENTS MORE FREQUENT COMPOUNDING OR DISCOUNTING INTERVALS XI. COST OF CONSUMER CREDIT UNETHICAL LENDERS APR VERSUS EAR SUMMARY LEARNING EXTENSION: ANNUITY DUE PROBLEMS FUTURE VALUE OF AN ANNUITY DUE PRESENT VALUE OF AN ANNUITY DUE INTEREST RATES AND TIME REQUIREMENTS FOR ANNUITY DUE PROBLEMS LECTURE notes I. BASIC CONCEPTS It is important to understand the time value of money (i.e., the mathematics of finance whereby interest is earned over time by saving or investing money) concept before trying to understand the pricing and valuation of bonds, stocks, and real asset investments. Simple interest is interest earned only on the principal of the initial investment. (Use Discussion Questions 1 and 2 here.) II. COMPOUNDING TO DETERMINE FUTURE VALUES Compounding and discounting are often referred to as the mathematics of finance. Compounding is the process whereby interest is earned each period on the principal amount plus the interest previously earned. Compounding is, thus, related to the accumulation of future values. The time value of money concepts are best taught by actually working out problems. The instructor can usually generate student interest in the compounding process by referring to the personal financial planning box titled “So You Want to Be a Millionaire!” Getting students involved by having them work with various interest rates and/or investment amounts seems to work well. The instructor needs to decide at the beginning of this section whether students will be allowed to use financial calculators and, if so, the level of calculator sophistication. We try to offer the instructor maximum flexibility in this respect. First we describe the calculation procedures in detail to enhance understanding of the logic involved in the time value of money concepts. This is the process of working the problems the “long way.” Financial calculators or computer software programs can also be used. We encourage students to make use of all three computational approaches. (Use Figure 9.1, Table 9.1, and Discussion Questions 3 and 4 here.) III. DISCOUNTING TO DETERMINE PRESENT VALUES Most financial management decisions involve present rather than future values. Present values are associated with the process of discounting, which conceptually is the opposite of compounding. More specifically, discounting is an arithmetic process whereby a future value decreases at a compound interest rate over time to reach a present value. (Use Figure 9.2, Table 9.2, and Discussion Question 5 here.) IV. EQUATING PRESENT VALUES AND FUTURE VALUES The equations for calculating present values and future values are two ways of looking at the same process involving compound interest rates. (Use Discussion Question 6 here.) V. FINDING INTEREST RATES AND TIME REQUIREMENTS Interest rates can be found if we first have information on the future value and present value amounts and the number of periods. The time periods necessary to produce a certain compound interest rate between a present value amount and a future value amount also can be determined. The Rule of 72 provides a shortcut method for approximating the time required for an investment to double in value. (Use Discussion Questions 7, 8, and 9 here.) VI. FUTURE VALUE OF AN ANNUITY The term annuity refers to a cash flow stream that is constant or level in each time period. An ordinary annuity exists when the level cash flow amount per period begins at the end of the first period. In contrast, an annuity due exists when the level cash flow amount per period starts at the beginning of the first period. The process for determining the future value of an annuity is described in the chapter. The instructor will find it useful to cover the process of determining annual (or other periods such as monthly) payments for annuities, since this concept will be useful for understanding the amortization of bank term loans and home mortgage loans. (Use Table 9.3 and Discussion Question 10 here.) VII. PRESENT VALUE OF AN ANNUITY An annuity is a series of equal payments (receipts) that occur over a number of time periods. The process for determining the present value of an annuity is described in the chapter. (Use Table 9.4 here.) VIII. INTEREST RATES AND TIME REQUIREMENTS FOR ANNUITIES We can solve for the compound interest rate that makes the future value (or present value) equal to the stream of annuity payments if we know the future value (or present value), the annuity payment, and the time period for the annuity. The calculation process is illustrated in the chapter. We can solve for the time it will take for the future value (or present value) to equal the stream of annuity payments if we know the future value (or present value), the annuity payment, and the compound interest rate for the annuity. The calculation process is illustrated in the chapter. (Use Discussion Question 11 here.) IX. DETERMINING PERIODIC ANNUITY PAYMENTS There are many instances for which we would want to determine the periodic equal payment required for an annuity. For example, we might want to find the equal payment necessary to pay off, or amortize, a loan or real estate mortgage. An amortized loan is repaid in equal payments over a specified time period. (Use Discussion Question 12 here.) X. MORE FREQUENT COMPOUNDING OR DISCOUNTING INTERVALS An understanding of the process of compounding or discounting more frequently rather than annually, is useful because many savings investments offer intrayear compounding. (Use Discussion Question 13 here.) XI. COST OF CONSUMER CREDIT The annual percentage rate (APR) is determined by multiplying the interest rate charged per period by the number of periods in a year. Banks, finance companies, and other lenders are required by law to disclose their borrowing interest rates (i.e., APRs) to their customers. However, the APR misstates the true interest rate. The effective annual rate (EAR) is the true opportunity cost measure of the interest rate because it considers the effects of period compounding. (Use Discussion Questions 14 and 15 here.) DISCUSSION QUESTIONS AND ANSWERS 1. Briefly describe what is meant by the time value of money. The time value of money refers to the mathematics of finance. Money has a time value as long as interest can be earned by saving or investing money. An understanding of the time value of money is necessary in order to understand the pricing and valuation of bonds, stocks, and real asset investments. 2. Explain the meaning of simple interest. Simple interest is interest earned only on the principal of the initial investment. 3. Describe the process of compounding and the meaning of compound interest. Compounding is an arithmetic process whereby an initial value increases or grows at a compound interest rate over time. Compound interest occurs when interest is earned on interest as well as principal. For example, if $100 is invested for one year at 10 percent, $110 is received at the end of one year. For a second year, 10 percent interest would be earned on the $110 (original $100 principal plus $10 in interest) for an amount at the end of two years of $121 ($110 × 1.10). 4. Briefly describe how inflation or purchasing power impacts on stated or nominal interest rates. It is common to refer to stated or nominal interest rates. As long as the nominal interest rate is higher that the inflation rate, there will be an increase in purchasing power over time. However, if the nominal rate and the inflation rate are equal, no change in purchasing power will take place. Purchasing power will decrease over time if the inflation rate exceeds the nominal interest rate. 5. What is discounting? Give an illustration. Discounting is the opposite of compounding. For example, assume you are offered $110 one year from now. How much is this worth to you now? The answer, of course, depends on what rate of interest you could earn if you had money to invest now. Let’s assume an interest (discount) rate of 10 percent. Dividing $110 by 1.10 gives a current or discounted price of $100. 6. Briefly explain how present values and future values are related. The process of finding present values and future values both involve using compound interest rates. If we know the future value of an investment, we can find its present value and vice versa. 7. Describe the process for solving for the interest rate in present and future value problems. If we know the present value, future value, and the time period for an investment, we can solve for the compound interest rate that would be earned on an investment. The calculation process is shown in the chapter. 8. Describe the process for solving for the time period in present and future value problems. If we know the present value, future value, and the compound interest rate, we can solve for the time period that would be required to earn the compound interest rate on an investment. The calculation process is shown in the chapter. 9. How can the Rule of 72 be used to determine how long it will take for an investment to double in value? The Rule of 72 method is applied by dividing the interest rate into the number 72 to determine the number of years it will take for an investment to double in value. The reader should be aware that at very low or very high interest rates, the Rule of 72 does not approximate the compounding process as well and thus a larger estimation error occurs in terms of the time required for an investment to double in value. 10. What is an ordinary annuity? An annuity is a cash flow stream that is constant or level in each time period. An ordinary annuity exists when the cash flows occur at the end of each time period. 11. Briefly describe how to solve for the interest rate or the time period in annuity problems. If we know the future value (or present value), the annuity payment, and the time period for the annuity, we can solve for the compound interest rate that makes the future value (or present value) equal to the stream of annuity payments. The calculation process is illustrated in the chapter. In a similar fashion, if we know the future value (or present value), the annuity payment, and the compound interest rate for the annuity, we can solve for the time it will take for the future value (or present value) to equal the stream of annuity payments. The calculation process is illustrated in the chapter. 12. Describe the process for determining the size of a constant periodic payment that is necessary to fully amortize a loan. The constant periodic payment that is necessary to fully amortize a loan is determined by dividing the present value of an annuity by the appropriate present value interest factor for an annuity. For example, the annual payment necessary to fully amortize a $2,487 loan at 10 percent for 3 years would be $1,000 (i.e., $2,487/2.487), where the PVIFA at 10 percent for 3 years is 2.487 (see Table 9.4). A loan amortization schedule is shown in Table 9.5. 13. Describe what we mean by compounding or discounting more often than annually. Compounding (or discounting) may occur more frequently than annually. For example, to compound twice a year, the interest rate per period is reduced in half and the number of periods over which the compounding takes place is doubled. More frequent compounding causes the future value to increase more than if annual compounding had taken place. The calculation process is illustrated in the chapter. 14. What is usury, and how does it relate to the cost of consumer credit? Usury is the act of lending money at an excessively high interest rate. Lenders deserve to earn a fair rate of return to compensate them for their time and the risk that the borrower will not repay the interest and/or principal on time or in full. However, because of the existence of unethical lenders, various laws have made usury illegal. 15. Explain the difference between the annual percentage rate and the effective annual rate. The annual percentage rate (APR) is determined by multiplying the interest rate charged per period by the number of periods in a year. The effective annual rate (EAR) measures the true interest rate when compounding occurs more frequently than once a year. The process for converting from the APR to the EAR (and vice versa) is illustrated in the chapter. EXERCISES AND ANSWERS Go to the Federal Reserve Web site, http://www.federalreserve.gov. Go to “Economic Research and Data,” and access “Recent Statistical Releases” and then “Consumer Credit.” Find average interest rates charged by commercial banks on new automobile loans, personal loans, and credit card plans. Compare the average level of interest rates among the three types of loans. The instructor will need to access current interest rate data for consumer loans from the Federal Reserve Web site. Interest rate data for 2006 are provided for comparative purposes. 2006 48-month new car loan 7.72% 24-month personal loan 12.41% Credit card plans (interest) 13.21% Access “Historical Data” and then “Consumer Credit,” and compare trends in the cost of consumer credit provided by commercial banks over the past three years. The instructor will need to access current interest rate data for consumer loans from the Federal Reserve Web site. Interest rate data for 2001, 2003, and 2006 are provided for comparative purposes. 2001 2003 2006 48-month new car loan 8.50% 6.93% 7.72% 24-month personal loan 13.22% 11.95% 12.41% Credit card plans (interest) 14.46% 12.73% 13.21% Go to the Federal Reserve Web site, http://www.federalreserve.gov. Go to “Economic Research and Data,” and access “Recent Statistical Releases” and then “Consumer Credit.” Determine current interest rates charged by auto finance companies on new automobile loans. Also compare the trend in the cost of loans from auto finance companies over the past three years. The instructor will need to access current interest rate data for new car loans from the Federal Reserve Web site. Interest rate data for 2001, 2003, and 2006 are provided for comparative purposes. 2001 2003 2006 New car loans 5.65% 3.40% 4.96% Assume that your partner and you are in the consumer lending business. A customer, talking with your partner, is discussing the possibility of obtaining a $10,000 loan for three months. The potential borrower seems distressed and says he needs the loan by tomorrow or several of his relatively new appliances will be repossessed by the manufacturers. You overhear your partner saying that that in order to process the loan within one day there will be a $1,000 processing fee so that $11,000 in principal will have to be repaid in order to have $10,000 to spend now. Furthermore, because the money is needed now and is for only three months the interest charge will be 6 percent per month. What would you do? USURY IS THE ACT OF LENDING MONEY AT AN EXCESSIVELY HIGH INTEREST RATE. GOOD ETHICAL BEHAVIOR IS CONSISTENT WITH TREATING BORROWERS HONESTLY AND FAIRLY. AS A LENDER, YOU ARE ENTITLED TO EARN A FAIR RATE OF RETURN ON THE MONEY YOU LEND. THAT IS, YOU DESERVE TO EARN A RATE OF RETURN THAT WILL COMPENSATE YOU FOR YOUR TIME AND THE RISK THAT THE BORROWER MAY NOT REPAY THE INTEREST AND/OR PRINCIPAL ON TIME OR IN FULL. HOWEVER, A $1,000 PROCESSING FEE ON A $10,000 LOAN AND AN INTEREST CHARGE OF 6 PERCENT PER MONTH SEEMS EXORBITANT. PROBLEMS AND ANSWERS 1. Find the future value one year from now of a $7,000 investment at a 3 percent annual compound interest rate. Also calculate the future value if the investment is made for two years. FV one year = $7,000(1.03) = $7,210 FV two years = $7,210(1.03) = $7,426.20 Or, = $7,000[(1.03)(1.03)] = $7,000(1.0609) = $7,426.30 2. Find the future value of $10,000 invested now after five years if the annual interest rate is 8 percent. a. What would be the future value if the interest rate is a simple interest rate? $10,000(.08) = $800 interest per year $800 x 5 = $4,000 total interest FV = $10,000 + $4,000 = $14,000 b. What would be the future value if the interest rate is a compound interest rate? FV = $10,000(1.08)5 FV = $10,000(1.469) = $14,690 3. Determine the future values if $5,000 is invested in each of the following situations: a. 5 percent for ten years $5,000(1.629) = $8,145 b. 7 percent for seven years $5,000(1.606) = $8,030 c. 9 percent for four years $5,000(1.412) = $7,060 4. You are planning to invest $2,500 today for three years at a nominal interest rate of 9 percent with annual compounding. a. What would be the future value of your investment? $2,500(1.295) = $3,237.50 b. Now assume that inflation is expected to be 3 percent per year over the same three-year period. What would be the investment’s future value in terms of purchasing power? 9% - 3% = 6% purchasing power rate $2,500(1.191) = $2,977.50 c. What would be the investment’s future value in terms of purchasing power if inflation occurs at a 9 percent annual rate? 9% - 9% = 0% purchasing power rate $2,500(1.000) = $2,500 5. Find the present value of $7,000 to be received one year from now assuming a 3 percent annual discount interest rate. Also calculate the present value if the $7,000 is received after two years. PV received one year from now = $7,000/(1.03) = $6,796.12 PV received two years from now = $6,796.12/(1.03) = $6,598.17 Or, $7,000/[(1.03)(1.03)] = $7,000/1.0609 = $6,598.17 6. Determine the present values if $5,000 is received in the future (i.e., at the end of each indicated time period) in each of the following situations: a. 5 percent for ten years $5,000(.614) = $3,070 b. 7 percent for seven years $5,000(.623) = $3,115 c. 9 percent for four years $5,000(.708) = $3,540 7. Determine the present value if $15,000 is to be received at the end of eight years and the discount rate is 9 percent. How would your answer change if you had to wait six years to receive the $15,000? $15,000 received at end of 8 years: $15,000(.502) = $7,530 $15,000 received at end of 6 years: $15,000(.596) = $8,940 8. Determine the future value at the end of two years of an investment of $3,000 made now and an additional $3,000 made one year from now if the compound annual interest rate is 4 percent. FV = $3,000[(1.04)(1.04)] + $3,000(1.04) = $3,000(1.0816) + $3,000(1.04) = $3,244.80 + $3,120 = $6,364.80 9. Assume you are planning to invest $5,000 each year for six years and will earn 10 percent per year. Determine the future value of this annuity if your first $5,000 is invested at the end of the first year. First investment at end of 1 year (ordinary annuity problem): $5,000(7.716) = $38,580 10. Determine the present value now of an investment of $3,000 made one year from now and an additional $3,000 made two years from now if the annual discount rate is 4 percent. PV = $3,000/(1.04) + $3,000/[(1.04)(1.04)] = $3,000/1.04 + $3,000/1.0816 = $2,884.62 + $2,773.67 = $5,658.29 11. What is the present value of a loan that calls for the payment of $500 per year for six years if the discount rate is 10 percent and the first payment will be made one year from now? How would your answer change if the $500 per year occurred for ten years? 10% for 6 years: $500(4.355) = $2,177.50 10% for 10 years: $500(6.145) = $3,072.50 12. Determine the annual payment on a $500,000, 12 percent, business loan from a commercial bank that is to be amortized over a five-year period. The PVIFA factor at 12 percent for five years is 3.605 [See Appendix, Table 4]. $500,000/3.605 = $138,696 (rounded to whole dollars) 13. Determine the annual payment on a $15,000 loan that is to be amortized over a four-year period and carries a 10 percent interest rate. Also prepare a loan amortization schedule for this loan. $15,000/3.170 = $4,732 (rounded to whole dollars) Annual Interest Principal Loan Year Payment Payment Repayment Balance 0 — — — $15,000 1 $4,732 $1,500 $3,232 11,768 2 4,732 1,177 3,555 8,213 3 4,732 821 3,911 4,302 4 4,732 430 4,302 –0– 14. You are considering borrowing $150,000 to purchase a new home. a. Calculate the monthly payment needed to amortize an 8 percent fixed-rate 30-year mortgage loan. Total payments = 30 x 12 = 360 Monthly interest rate = .08/12 = .06667 or .6667 percent per month Using a financial calculator: enter .6667 = %i, 360 = N, -150000 = PV. Then compute (CPT) payment (PMT) = $1,100.69 [Or, $1,100.65 if the monthly rate is not rounded.] Note: make sure FV = 0 before calculating the payment amount. b. Calculate the monthly amortization payment if the loan in (a) was for 15 years. Total payments = 15 x 12 = 180 Monthly interest rate = .08/12 = .0667 or .6667 percent per month Using a financial calculator: enter .6667 = %i, 180 = N, -150000 = PV. Then compute (CPT) payment (PMT) = $1,433.51 [Or, $1,433.48 if the monthly rate is not rounded.] Note: make sure FV = 0 before calculating the payment amount. 15. Assume a bank loan requires an interest payment of $85 per year and a principal payment of $1,000 at the end of the loan’s eight-year life. a. How much could this loan be sold for to another bank if loans of similar quality carried an 8.5 percent interest rate? That is, what would be the present value of this loan? 8.5% for 8 years: Note: This loan would have a present value of $1,000 (the same as the $1,000 current principal) since the interest rate on the loan of 8.5% ($85/$1,000) is the same as the 8.5% interest rate required on similar quality loans. This could be proven using a financial calculator that can handle fractional interest rates (our tables cannot handle interest rates expressed to one half of a percent). Using a financial calculator, we would enter 1000 and press FV, enter 85 and press PMT, enter 8.5 and press %i, and enter 8 and press N. Then, press CPT and PV to find a present value of 1000. b. Now, if interest rates on other similar quality loans were 10 percent, what would be the present value of this loan? 10% for 8 years: $85 × 5.335 = $453 $1,000 × .467 = 467 $920 c. Finally, what would be the present value of the loan if the interest rate is 8 percent on similar quality loans? 8% for 8 years: $85 × 5.747 = $488 $1,000 × .540 = 540 $1,028 16. Use a financial calculator or computer software program to answer the following questions: a. What would be the future value of $15,555 invested now if it earns interest at 14.5 percent for seven years? Using a financial calculator, enter 15555 and press PV, enter 14.5 and press %i, and enter 7 and press N. Then, press CPT and FV which gives an answer of 40133.63 or $40,133.63. b. What would be the future value of $19,378 invested now if the money remains deposited for eight years and the annual interest rate is 18 percent? Using a financial calculator, enter 19378 and press PV, enter 18 and press %i, and enter 8 and press N. Then, press CPT and FV which gives an answer of 72839.17 or $72,839.17. 17. Use a financial calculator or computer software program to answer the following questions: a. What is the present value of $359,000 that is to be received at the end of 23 years if the discount rate is 11 percent? Using a financial calculator, enter 359000 and press FV, enter 11 and press %i, and enter 23 and press N. Then, press CPT and PV which gives an answer of 32558.62 or $32,558.62. How would your answer change in Part (a) if the $359,000 is to be received at the end of 20 years? Follow the procedure in (a), but substitute the number 20 before pressing N. The answer is 44528.17 or $44,528.17. 18. Use a financial calculator or computer software program to answer the following questions: a. What would be the future value of $7,455 invested annually for nine years beginning one year from now if the annual interest rate is 19 percent? Using a financial calculator, enter 7455 and press PMT, enter 19 and press %i, and enter 9 and press N. Then, press CPT and FV which gives an answer of 148529.05 or $148,529.05. b. What would be the present value of a $9,532 annuity for which the first payment will be made beginning one year from now, payments will last for 27 years, and the annual interest rate is 13 percent? Using a financial calculator, enter 9532 and press PMT, enter 13 and press %i, and enter 27 and press N. Then, press CPT and PV which gives an answer of 70618.35 or $70,618.35. 19. Use a financial calculator or computer software program to answer the following questions. a. What would be the future value of $19,378 invested now if the money remains deposited for eight years, the annual interest rate is 18 percent, and interest on the investment is compounded semiannually? Using a financial calculator, enter 19378 and press PV, enter 9.00 (18/2) and press %i, and enter 16 (8 × 2) and press N. Then, press CPT and FV which gives an answer of 76936.59 or $76,936.59. b. How would your answer for (a) change if quarterly compounding were used? Using a financial calculator, enter 19378 and press PV, enter 4.50 (18/4) and press %i, and enter 32 (8 × 4) and press N. Then, press CPT and FV which gives an answer of 79255.65 or $79,255.65. 20. Use a financial calculator or computer software program to answer the following questions. a. What is the present value of $359,000 that is to be received at the end of 23 years, the discount rate is 11 percent, and semiannual discounting occurs? Using a financial calculator, enter 359000 and press FV, enter 5.50 (11/2) and press %i, and enter 46 (23 x 2) and press N. Then, press CPT and PV which gives an answer of 30583.09 or $30,583.09. b. How would your answer for (a) change if monthly discounting were used? Using a financial calculator, enter 359000 and press FV, enter .9167 (11/12) and press %i, and enter 276 (23 x 12) and press N. Then, press CPT and PV which gives an answer of 28926.46 or $28,926.46. 21. What would be the present value of a $9,532 annuity for which the first payment will be made beginning one year from now, payments will last for 27 years, the annual interest rate is 13 percent, quarterly discounting occurs, and $2,383 is invested at the end of each quarter? Using a financial calculator, enter 2383 (9532/4) and press PMT, enter 3.25 (13/4) and press %i, and enter 108 (27 × 4) and press N. Then, press CPT and PV which gives an answer of 71005.07 or $71,005.07. 22. Answer the following questions. a. What is the annual percentage rate (APR) on a loan that charges interest of .75 percent per month? APR = .75% x 12 = 9.00% b. What is the effective annual rate (EAR) on the loan described in (a)? EAR = (1 + .0075)12 – 1 = 1.0938 – 1 = 0.0938 or 9.38% 23. You have recently seen a credit card advertisement that states that the annual percentage rate is 12 percent. If the credit card requires monthly payments, what is the effective annual rate of interest on the loan? Periodic interest rate: 12%/12 months = 1% EAR = (1 + .01)12 – 1 = 1.1268 –1 = 0.1268 or 12.68% 24. A credit card advertisement states that the annual percentage rate is 21 percent. If the credit card requires quarterly payments, what is the effective annual rate of interest on the loan? Periodic interest rate: 21%/4 = 5.25% EAR = (1 + .0525)4 – 1 = 1.2271 –1 = 0.2271 or 22.71% 25. Challenge Problem [Note: a computer spreadsheet software program or a financial calculator that can handle uneven cash flow streams will be needed to solve the following problems.] The following cash flow streams are expected to result from three investment opportunities. Investment Investment Investment Year Stable Declining Growing 1 $20,000 $35,000 $10,000 2 20,000 30,000 15,000 3 20,000 20,000 20,000 4 20,000 5,000 30,000 5 20,000 0 50,000 a. Find the present values at the end of time period zero for each of these three investments if the discount rate is 15 percent. Also find the present values for each investment using 10 percent and 20 percent discount rates. Stable Declining Growing PV: 10% $75,815.74 $75,052.93 $88,050.37 PV: 15% $67,043.10 $69,128.18 $75,199.57 PV: 20% $59,812.24 $63,985.34 $64,885.55 b. Find the future values of these three investments at the end of year five if the compound interest rate is 12.5 percent. Also find the future values for each investment using 2.5 percent and 22.5 percent compound rates. Stable Declining Growing FV: 2.5% $105,126.57 $97,077.67 $128,953.99 FV: 12.5% $128,325.20 $129,715.58 $146,437.99 FV: 22.5% $156,315.32 $170,101.11 $166,855.24 Find the present values of the three investments using a 15 percent annual discount rate but with quarterly discounting. Also find the present values for both semi-annual and monthly discounting for a 15 percent stated annual rate. Note: we are only presenting results for the “stable” investment. To calculate present values using more frequent than annual discounting when uneven cash flows are involved is quite time consuming. If assigned, we suggest that only semi-annual discounting be emphasized. Solution for Stable Investment: Semiannual discounting ($10,000 cash flow; 10 periods; 7.50% rate): $68,640.81 Quarterly discounting ($5,000 cash flow; 20 periods; 3.75% rate): $69,481.02 Monthly discounting ($1,667 cash flow: 60 periods; 1.25% rate): $70,071.66 Find the future values of the three investments using a 12.5 percent annual compound rate but with quarterly compounding. Also find the future values for both semi-annual and monthly compounding for a 12.5 percent stated annual rate. Note: we are only presenting results for the “stable” investment. To calculate future values using more frequent than annual compounding when uneven cash flows are involved is quite time consuming. If assigned, we suggest that only semi-annual compounding be emphasized. Solution for Stable Investment: Semiannual compounding ($10,000 cash flow; 10 periods; 6.25% rate): $133,365.72 Quarterly discounting ($5,000 cash flow; 20 periods; 3.125% rate): $136,073.28 Monthly discounting ($1,667 cash flow: 60 periods; 1.0417% rate): $137,983.65 e. Assume that the present value for each of the three investments is $75,000. What is the annual interest rate (%i) for each investment? The cash flows would be: Year Stable Declining Growing 0 $-75,000 $-75,000 $-75,000 1 $20,000 $35,000 $10,000 2 20,000 30,000 15,000 3 20,000 20,000 20,000 4 20,000 5,000 30,000 5 20,000 0 50,000 A trial and error process is used to find the discount rate that makes the present values of the cash flows equal to $75,000 at the end of year 0. These rates are: Stable: 10.42% Declining: 10.04% Growing: 15.09% f. Show how your answers would change in (e) if quarterly discounting takes place. Because of more frequent than annual discounting and the complexities associated with uneven cash flows, we show results only for the “Stable” investment. The investment is $75,000, the period cash flow is $5000, and the number of periods is 20. This results in a discount rate of 2.9115% quarterly, or 11.65% (2.9115% x 4) annually. g. Assume that the future value for each of the three investments is $150,000. What is the annual interest rate (%i) for each investment? [Note: (e) and (g) are independent of each other.] The cash flows would be: Year Stable Declining Growing 1 $-20,000 $-35,000 $-10,000 2 -20,000 -30,000 -15,000 3 -20,000 -20,000 -20,000 4 -20,000 -5,000 -30,000 5 130,000 150,000 100,000 A trial and error process is used to find the compound rate that makes the future values of the cash flows equal to $150,000 at the end of year 5. These rates are: Stable: 20.40% Declining: 17.78% Growing: 14.36% The cash flows would be: Year Stable Declining Growing 1 $-20,000 $-35,000 $-10,000 2 -20,000 -30,000 -15,000 3 -20,000 -20,000 -20,000 4 -20,000 -5,000 -30,000 5 130,000 150,000 100,000 h. Show how your answers would change in (g) if quarterly compounding takes place. Because of more frequent than annual compounding and the complexities associated with uneven cash flows, we show results only for the “Stable” investment. The future value of the investment is $150,000, the period cash flow is $5000, and the number of periods is 20. This results in a compound rate of 4.0715% quarterly, or 16.29% (4.0715% x 4) annually. SUGGESTED QUIZ 1. Define or discuss briefly: a. Simple interest e. Ordinary annuity b. Compounding f. Annual percentage rate c. Discounting g. Effective annual rate d. Rule of 72 2. Briefly explain the difference between an ordinary annuity and an annuity due. 3. Briefly explain the difference between an annual percentage rate (APR) and the effective annual rate (EAR). 4. A $9,000 loan will require interest payments of $4,000 per year for 3 years. Calculate the compound interest rate on this loan. Solution: $9,000/$4,000 = 2.250 PVIFA for 3 years This PVIFA falls approximately at a compound interest rate of 16% (2.246) as shown in Table 4 in the Appendix. Financial calculator solution: Enter 9000 and press PV, enter 4000 and press PMT, and enter 3 and press N. Then, press the PT and %i keys which gives an answer of 15.89 or 15.89%. When the cash flows occur at the beginning of each time period, this is referred to as an annuity due. Learning Extension 5 Annuity Due Problems I. FUTURE VALUE OF AN ANNUITY DUE II. PRESENT VALUE OF AN ANNUITY DUE III. INTEREST RATES AND TIME REQUIREMENTS FOR ANNUITY DUE PROBLEMS LECTURE NOTES I. FUTURE VALUE OF AN ANNUITY DUE An annuity is a cash flow steam that is constant or level in each time period. In contrast with an ordinary annuity, an annuity due exists when the equal periodic payments occur at the beginning of each period. The process of working future value of annuity due problems is described in the Learning Extension addition to Chapter 9. II. PRESENT VALUE OF AN ANNUITY DUE Occasionally there are present value annuity due problems. For example, leasing arrangements often require the person leasing equipment to make the first payment at the time the equipment is delivered. The process of working present value of annuity due problems is described in the Learning Extension addition to Chapter 9. III. INTEREST RATES AND TIME REQUIREMENTS FOR ANNUITY DUE PROBLEMS Tables containing FVIFA and PVIFA factors are not readily available for annuity due problems. Thus, we should use either a computer software program or a financial calculator when solving for either the interest rate or the time periods involved in annuity due problems. The process of finding interest rates (or time periods) for annuity due problems is described in the Learning Extension to Chapter 9. PROBLEMS AND ANSWERS Assume you are planning to invest $100 each year for four years and will earn 10 percent per year. Determine the future value of this annuity due problem if your first $100 is invested now. Note: Using Future Value of an Annuity Table FVIFr,n¬ ¬= 4.641 [from Table 3 in the Appendix] $100[(4.641)(1.10)] = $100(5.1051) = $510.51 Note: Using a Financial Calculator Enter 100 and press the PMT key. Next enter 10 and press the %i key, and 4 and press the N key. Then press the DUE key followed by the FV key. The solution will be 510.51 or $510.51. 2. Assume you are planning to invest $5,000 each year for six years and will earn 10 percent per year. Determine the future value of this annuity due problem if your first $5,000 is invested now. $5,000[(7.716)(1.10)] = $5,000(8.4876) = $42,438, or Using a financial calculator: 5000 = PMT, 10 = %i, and 6 = N. Then, pressing the DUE key and FV key produces 42435.86 or $42,435.86. The answers differ slightly due to rounding of figures in the tables. 3. What is the present value of a five-year lease arrangement with an interest rate of 9 percent that requires annual payments of $10,000 per year with the first payment being due now? Note: Using Present Value of an Annuity Table PVIFr,n¬ ¬= 3.890 [from Table 4 in the Appendix] $10,000[(3.890)(1.09)] = $10,000(4.2401) = $42,401.00 Note: Using a Financial Calculator Enter 10000 and press the PMT key. Next enter 9 and press the %i key, and 5 and press the N key. Then press the DUE key followed by the PV key. The solution will be 42397.20 or $42,397.20. The answers differ slightly due to rounding of figures in the tables. Use a financial calculator to solve for the interest rate involved in the following future value of an annuity due problem. The future value is $57,000, the annual payment is $7,500, and the time period is 6 years. Enter 57000 and press the FV key. Next enter -7500 and press the PMT key and enter 6 and press the N key. Then, press the DUE key followed by the %i key. The solution will be 6.795 or 6.795 percent. 5. Challenge Problem (Note: This problem requires access to a spreadsheet software package or a financial calculator that can handle uneven cash flows.) Following are the cash flows for three investments (originally presented in Problem 25 [not 24 as indicated in text] at the end of the chapter) that actually occur at the beginning of each year rather than at the end of each year. Investment Investment Investment Year Stable Declining Growing 1 $20,000 $35,000 $10,000 2 20,000 30,000 15,000 3 20,000 20,000 20,000 4 20,000 5,000 30,000 5 20,000 0 50,000 Find the present values at the end of time period zero for each of these three investments if the discount rate is 15 percent. Note: The PV of an annuity due can be solved directly, or by multiplying the PV of an ordinary annuity by one plus the discount rate. Stable: $77,099.57 [alternatively: $67,043.10 x 1.15] Declining: $79,497.41 [alternatively: $69,128.18 x 1.15] Growing: $86,479.51 [alternatively: $75,199.57 x 1.15] Find the future values of these three investments at the end of year five if the compound interest rate is 12.5 percent. Note: The FV of an annuity due can be solved directly, or by multiplying the FV of an ordinary annuity by one plus the discount rate. Stable: $144,365.85 [alternatively: $128,325.20 x 1.125] Declining: $145,930.03 [alternatively: $129,715.58 x 1.125] Growing: $164,742.74 [alternatively: $146,437.99 x 1.125] Assume that the present value for each of the three investments is $75,000. What is the annual interest rate (%i) for each investment? The cash flows would be: Year Stable Declining Growing 0 $-55,000 $-40,000 $-65,000 1 $20,000 $30,000 $15,000 2 20,000 20,000 20,000 3 20,000 5,000 30,000 4 20,000 0 50,000 Stable: 16.88% Declining: 23.62% Growing: 21.88% Assume that the future value for each of the three investments is $150,000. What is the annual interest rate (%i) for each investment? [Note: (c) and (d) are independent of each other.] The cash flows would be: Year Stable Declining Growing 0 $-20,000 $-35,000 $-10,000 1 -20,000 -30,000 -15,000 2 -20,000 -20,000 -20,000 3 -20,000 -5,000 -30,000 4 -20,000 0 -50,000 5 150,000 150,000 150,000 Stable: 13.83% Declining: 13.25% Growing: 8.22% SUGGESTED QUIZ Define the term annuity due. Solution Manual for Introduction to Finance: Markets, Investments, and Financial Management Ronald W. Melicher, Edgar A. Norton 9780470561072, 9781119560579, 9781119398288
Download
Close
Close
