Chapter Review Problems Unit 9. Time-value-of-money terminology For Problems 9, assume you deposit $,000 today in a savings account. You earn 5% compounded quarterly. You deposit an additional $50 each quarter, starting in 3 months. At the end of 3 years you withdraw the balance of $,803.77. Identify each value.. Compounding period? 3 months 2. Periods per year? 4 (There are four 3-month periods in a year) 3. Term, in years? 3 years 4. n? 2 (3 years 4 periods per year) 5. Present value? $,000 (This is the amount that happens at the beginning of the first period) 6. Future value? $,803.77 (This is the amount that happens at the end of the last period) 7. Periodic payment? $50 (This is the amount that happens every period) 8. Nominal rate? 5% 9. Periodic rate?.25% (5 4 =.25) For Problems 0 4, assume you borrow $400 from a friend and repay your friend $05 at the end of each month for 4 months. 0. Periods per year? 2. n? 4 2. Present value? $400 3. Future value? None 4. Periodic payment? $05 Unit 9.2 Compound interest formulas For problems in this unit, use the formulas of Illustration 9-. 5. You invest $5,800 in a savings plan, earning 8% compounded semiannually. What will your balance be at the end of 5 years? We are solving for FV and know PV, so we use Formula 2A of Illustration 9-. PV = $5,800 i = 8% 2 = 4% =.04 n = 5 2 = 0 FV = PV( + i) n = $5,800(.04) 0 = $8,585.42 6. You invest $00 each 6 months (starting in 6 months) in a savings plan, earning 8% compounded semiannually. What will your balance be at the end of 5 years? We are solving for FV and know PMT, so we use Formula 2B. PMT = $00 i = 8% 2 = 4% =.04 n = 5 2 = 0 [ n ] FV = PMT ( + i) - = $00 (.04) - = $,200.6 i.04 [ 0 7. Refer to Problems 5 and 6. Suppose you invest $5,800 today and $00 each 6 months (starting in 6 months). What will your balance be at the end of 5 years? We simply add the two previous answers. To help visualize the process, pretend you deposit $5,800 with one bank and $00 each 6 months with a second bank. You would have an ending balance of $8,585.42 with the first bank and $,200.6 with the second, for a total of $9,786.03 ($8,585.42 + $,200.6 = $9,786.03). The result would, of course, be the same if your deposits were made with one bank. ] 88 Chapter 9 Time-Value-of-Money Problems: An Introduction
8. Bob s grandfather gives all of his grandchildren $5,000 on their 30th birthdays. Bob just turned 20. What is the value of the $5,000 gift in today s dollars, assuming that Bob can earn 0% compounded annually? We are solving for PV and know FV, so we use Formula A: FV = $5,000; i = 0% =.0; n = 0. PV = FV = $5,000 = $,927.72 ( + i) n (.0) 0 9. You win 3rd place in the lottery and will receive ten $50,000 annual payments, starting in year. What is the real value of your prize, in today s dollars, assuming that you can earn 8% compounded annually? We are solving for PV and know PMT, so we use Formula B: PMT = $50,000; i = 8% =.08; n = 0. 20. Refer to Problem 9. What is the real value of your prize if payments are made at the beginning of each year? The formula used in Problem 9 assumes payments are made at the end of each period. If payments are made at the beginning of each period, we must multiply the result ($335,504.07) by ( + i). PV = $335,504.07.08 = $362,344.40 2. Joe Salazar purchased some vacant land 4 years ago for $28,500. He just sold the land for $40,000. What interest rate, compounded annually, did Joe earn on the investment? Use 5 decimal places in your final answer. We are solving for i and know FV and PV, so we use Formula 3: FV = $40,000; PV = $28,500; n = 4. 22. Jack Willis purchased some corporate stock 0 years ago for $2,000. He has received quarterly dividends of $200 (starting 3 months after he purchased the stock). Immediately after receiving the 40th quarterly dividend check, Jack sold the stock for $5,000. Can Jack use Formula 3 of Illustration 9- to determine the rate he earned? Explain. No, because a periodic payment is involved in the problem. Formula 3 is not designed to solve a problem in which there is a periodic payment. There is no practical formula that can be used to determine Jack s rate of return. Unit 9.3 Compound interest tables [ ] [ ] PV = PMT - = $50,000 - ( + i) n (.08) 0 = $335,504.07 i.08 ( ) n 4 ~ ~ ( ) i = FV - = $40,000 -.0884382 8.84382% PV $28,500 For problems in this unit, use the compound interest tables of Illustration 9-2. Notice that some of the decimal values of Illustration 9-2 have 0 digits and others have digits. If your calculator will not accommodate all of the digits of the decimal value, use as much of the value (rounded) as your calculator will allow. 23. Rework Problem 5 using compound interest tables. You invest $5,800 in a savings plan, earning 8% compounded semiannually. What will your balance be at the end of 5 years? We are solving for FV and we know PV, so we use Column A of Illustration 9-2. PV = $5,800 i = 8% 2 = 4% n = 5 2 = 0 We use the row within 4% where n = 0; the decimal value in Column A for that row is.4802442849. FV = PV Appropriate decimal value = $5,800.4802442849 = $8,585.42 24. Rework Problem 6. You invest $00 each 6 months (starting in 6 months) in a savings plan, earning 8% compounded semiannually. What will your balance be at the end of 5 years? We are solving for FV and we know PMT, so we use Column B; PMT = $00; i = 8% 2 = 4%; n = 0. We use the row within 4% where n = 0; the decimal value in Column B for that row is 2.00607230. FV = PMT Appropriate decimal value = $00 2.00607230 = $,200.6 Chapter Review Problems 89
25. Refer to Problems 23 and 24. If you invest $5,800 today and $00 each 6 months (starting in 6 months), what will your balance be at the end of 5 years? $8,585.42 (answer for Problem 23) + $,200.6 (answer for Problem 24) = $9,786.03 26. Rework Problem 8. Bob s grandfather gives all of his grandchildren $5,000 on their 30th birthdays. Bob just turned 20. What is the value of the $5,000 gift in today s dollars, assuming that Bob can earn 0% compounded annually? We are solving for PV and we know FV, so we use Column D: FV = $5,000; i = 0%; n = 0. We use the row within 0% where n = 0; the decimal value in Column D for that row is.3855432894. PV = FV Appropriate decimal value = $5,000.3855432894 = $,927.72 27. Rework Problem 9. You win 3rd place in the lottery and will receive 0 annual payments of $50,000, starting in year. What is the real value of your prize, in today s dollars, assuming you can earn 8% compounded annually? We are solving for PV and we know PMT, so we use Column E: PMT = $50,000; i = 8%; n = 0. We use the row within 8% where n = 0; the decimal value in Column E for that row is 6.70083989. PV = PMT Appropriate decimal value = $50,000 6.70083989 = $335,504.07 28. Refer to Problem 27. What is the real value of your prize if payments are made at the beginning of each year? As noted at the top of Illustration 9-2, if payments are made at the beginning of each period we must, for Column E, multiply the result by ( + i). Logically, if payments are made sooner, the PV should be greater; multiplying the result of Problem 27 by ( + i) will give a greater value. PV = $335,504.07.08 = $362,344.40 29. If compound interest tables are used, for which of the five variables must a value be estimated: PV, PMT, FV, rate, n? The tables are not designed to solve for rate and n. The values for rate and n can be estimated by finding a decimal value within the tables that is close to a target decimal value. Unit 9.4 Financial calculators 30. If you make a total of six $00 payments, you should enter $600 in the PMT register. (T or F) False. You do not make a payment of $600, so you should not enter $600 in the PMT register. Instead, you should enter $00 (actually, negative $00) in the PMT register. 3. If loan payments are made on the 5th of each month, you must set your calculator in middle-of-the-month mode. (T or F) False; there is no middle-of-the-month mode. For Problems 32 39, rework the indicated problems. Record given data. Then find the answer. Finally, record your answer. Problem 32 is completed as an example (with answer in bold). Rework Problem 32. 5 5 2 = 0 8 2 = 4-5,800 8,585.42 33. 6 0-00,200.6 34. 7-5,800 9,786.03 35. 8 0 0 -,927.72 5,000 36. 9 0 8-335,504.07 50,000 37. 20-362,344.40 50,000 Begin * 38. 2 4 8.84382-28,500 40,000 39. 22 0 4 = 40 2.07 4 = 8.30-2,000 200 5,000 *Note: Don t forget to put your calculator back in End mode after finishing Problem 37. 90 Chapter 9 Time-Value-of-Money Problems: An Introduction
Unit 9.5 Checking answers 40. In Problem 6, you deposited $00 each 6 months (starting in 6 months) in a savings plan for 5 years, earning 8% compounded semiannually. Use estimating to determine if the answer of $,200.6 is reasonable. You will deposit a total of $,000 (0 deposits $00 = $,000). In addition, you will earn some interest, so the answer of $,200.6 seems reasonable. 4. Refer to Problem 40. Use a longhand proof to determine if the answer of $,200.6 is correct. Remember, in your proof, don t round intermediate results. Balance in 6 months: $00.00 Balance in 2 months: + 4% + $00 = $204.00 Balance in 8 months: + 4% + $00 = $32.6 Balance in 24 months: + 4% + $00 = $424.65 Balance in 30 months: + 4% + $00 = $54.63 Balance in 36 months: + 4% + $00 = $663.30 Balance in 42 months: + 4% + $00 = $789.83 Balance in 48 months: + 4% + $00 = $92.42 Balance in 54 months: + 4% + $00 = $,058.28 Balance in 60 months: + 4% + $00 = $,200.6 (Answer is identical) 42. In Problem 2, Joe Salazar purchased some vacant land 4 years ago for $28,500. He just sold the land for $40,000. We determined that Joe earned an annual rate of 8.84382% on his investment. Use estimating to determine if the answer is reasonable. Annual rate: 40.35% 4 years 0.09% per year Because interest is compounded each year, a lower interest rate will result in the same ending value, so the rate of 8.84382% seems reasonable. 43. Refer to Problem 42. Use a longhand proof to determine if the answer of 8.84382% is correct. Theoretical value in year: $28,500 + 8.84382% = $3,020.49 Theoretical value in 2 years: + 8.84382% = $33,763.88 Theoretical value in 3 years: + 8.84382% = $36,749.90 Theoretical value in 4 years: + 8.84382% = $40,000.00 (The rate of 8.84382% works!) 44. Percent increase = Amount of increase = $40,000 - $28,500 = $,500.4035 40.35% Original amount $28,500 $28,500 ~ ~ Challenge problems For Problems 44 47, use the given data to solve for the unknown. Then write a word problem that matches the data. 4 2 = 48.5 2 0.96* 5,000-39.34 *Note: Don t make the mistake of entering a rounded periodic rate (0.96). By dividing.5 by 2 and transferring the result in the interest rate register, we enter the internal, more accurate, value (0.9583333). One possibility: If I borrow $5,000 at.5% to buy a new sports car and the loan requires monthly payments for 4 years, my payment will be $39.34. 45. 4 2 = 48 5.5 2 0.46-5,000 8,68.76 One possibility: If I deposit $5,000 in a savings account for 4 years and the account earns interest at 5.5% compounded monthly, I will have $8,68.76 at the end of 4 years. 46. 5 = 5 5.92-5,000 20,000 One possibility: I purchased some IBM stock 5 years ago for $5,000. If I sold the stock today for $20,000, I earned 5.92% per year on my investment. Chapter Review Problems 9
47. 74.26 6.50 2 = 3.25-500 50,000 One possibility: If I deposit $500 each at the end of each 6 months and earn 6.50% compounded semiannually, I will have $50,000 in 37.3 years (74.26 six-month periods). 48. You buy a bond for $955. You receive semiannual interest checks of $38.75 at the end of each 6 months. You receive the $,000 maturity value in 4 years. What is your rate of return? 4 2 = 28 4.5 2 ~ 8.30* -955 38.75,000 *Note: Remember i represents the interest rate per period, so we must multiply by the number of periods per year. For Problems 49 53, assume that you want to accumulate $0,000 for a down payment on a home. You start a savings plan today by depositing $2,000. We will find what additional amount you must deposit each 6 months (starting in 6 months) to have $0,000 in 4 years, assuming that you can earn 8% compounded semiannually? 49. Determine your monthly payment using compound interest formulas. We are solving for PMT, so we use Formula 4. Remember, for Formulas 4 and 5, we must treat dollar amounts with proper sign convention (as a positive or negative value). PV = - $2,000; the value is negative because you give the money to the bank FV = $0,000; the value is positive because you will receive this money from the bank in 4 years i = 8% 2 = 4% =.04 n = 4 2 = 8 According to the mathematical order of operations, we first perform multiplication and then addition, so we perform multiplication on FV and then add PV to the result. PMT = PV + FV [ = -$2,000 + $0,000 ( + i) ] [ n (.04) ] 8 = -$788.22 - - ( + i) n (.04) 8 i.04 Because the arithmetic is a bit complicated on this problem, calculator keystrokes are shown. Although there are several different ways to do the arithmetic on our calculators, the keystrokes below are done by storing values. Storage register : Storage register 2: Entire numerator Storage register 3: Entire denominator (.04) 8 HP 0BII.04 _ y x 8 =.37 _ /x 0.73 _ STO 0.73* 0,000 = 7,306.90-2,000 = 5,306.90 _ STO 2 5,306.90 RCL 0.73 - = -0.27.04 = -6.73 _ STO 3-6.73 RCL 2 5,306.90 RCL 3-6.73 = -788.22 TI BAII PLUS.04 y x 8 =.37 /x 0.73 STO 0.73* 0,000 = 7,306.90-2,000 = 5,306.90 STO 2 5,306.90 RCL 0.73 - = -0.27.04 = -6.73 STO 3-6.73 RCL 2 5,306.90 RCL 3-6.73 = -788.22 * Note: The internal, more accurate, value is stored. 92 Chapter 9 Time-Value-of-Money Problems: An Introduction
50. Determine your monthly payment using compound interest tables. In Illustration 9-2, Columns C and F are used to solve for PMT. But Column C is used when we know FV, and Column F is used when we know PV. Unfortunately, in this problem, we know both FV and PV. So, the only way we can do this problem using compound interest tables is as a two-step problem. First, we will find the payment, without the initial $2,000 deposit required to accumulate $0,000. Then, we will find the PMT that will be saved by depositing $2,000. The answer is the difference between the two steps. Step (find PMT required to accumulate $0,000): Use Column C, i = 4%, n = 8 PMT = FV Appropriate decimal value = $0,000.085278320 = $,085.28 Step 2 (find PMT resulting from a deposit of $2,000): Use Column F, i = 4%, n = 8 PMT = PV Appropriate decimal value = $2,000.485278320 = - 297.06 Difference $ 788.22 5. Determine your monthly payment using time-value-of-money registers. 4 2 = 8 8 2 = 4-2,000-788.22 0,000 52. Determine whether the answer ($788.22) seems reasonable. Initial deposit $2,000.00 Semiannual deposits: 8 $788.22 +6,305.76 Total amount deposited $8,305.76 The ending balance will actually be greater after interest is added, so deposits of $788.22 seem reasonable to end up with $0,000. 53. Determine whether the answer ($788.22) is correct. Balance in 6 months: $2,000 + 4% + $788.22 = $2,868.22 Balance in 2 months: + 4% + $788.22 = $3,77.7 Balance in 8 months: + 4% + $788.22 = $4,70.24 Balance in 24 months: + 4% + $788.22 = $5,686.86 Balance in 30 months: + 4% + $788.22 = $6,702.56 Balance in 36 months: + 4% + $788.22 = $7,758.88 Balance in 42 months: + 4% + $788.22 = $8,857.46 Balance in 48 months: + 4% + $788.22 = $9,999.98 Because the deposit amount ($788.22) was rounded to the nearest penny (instead of the more accurate 788.222656), the ending balance is a few pennies different from the desired $0,000, but the deposit amount is correct, to the nearest penny. Practice Test. You deposit $500 today in a savings account that earns 6% compounded quarterly and leave the money there for 5 years. What is the n-value? n = 5 years 4 periods per year = 20 2. You deposit $00 at the end of each month in a savings account that earns 4.5% compounded monthly. Use the compound interest formulas of Illustration 9- to find the balance at the end of 5 years. We need to know FV and we know PMT, so we use Formula 2B. PMT = $00 i = 4.5% 2 =.375% =.00375 n = 5 2 = 80 [ ] [ ] n 80 FV = PMT ( + i) - = $00 (.00375) - = $25,64.47 i.00375 Practice Test 93
3. Find the value of -$5,000 + $2,000 [ ] (.04) 6 - (.04) 6.04 ~ -$5,000 + $2,000 (.53390876) ~ -$5,000 + $6,406.8980822 ~ $,406.8980822 ~ -$20.74.53390876 - -0.46609824 -.652295608.04.04 4. Your Uncle Ben gives you $5,000 at the beginning of each year for college. You will receive a total of four payments, and you can earn 8% compounded annually. Use the compound interest tables of Illustration 9-2 to determine the real value of his gift, in today s dollars. We are solving for PV and we know PMT, so we use Column E: PMT = $5,000; i = 8%; n = 4. We use the row within 8% where n = 4; the decimal value in Column E for that row is 3.32268400. PV = PMT Appropriate decimal value = $5,000 3.322684 = $6,560.63 Because payments are at the beginning of each period, we multiply by ( + i): $6,560.63.08 = $7,885.48 5. You deposit $525 in a savings account earning 3.5% compounded semiannually. You let the money sit for 3 years. If you use the compound interest tables of Illustration 9-2 to find the ending balance, what decimal value do you multiply by $525? Note: Don t solve the problem; just find the decimal value. We are solving for FV and we know PV, so we use Column A: PV = $525; i = 3.5% 2 =.75%; n = 3 2 = 6. We use the row within 4 3 % where n = 6; the decimal value in Column A for that row is.097023542 6. You open a savings plan by depositing $00. Then you deposit $50 at the end of each quarter for 52 years. You earn 7.85%, compounded quarterly. Use time-value-of-money registers to determine your savings plan balance at the end of 52 years. 52 4 = 208 7.85 4 ~.96-00 -50 48,286.23 7. You buy a bond for $975. You receive semiannual interest checks of $40 at the end of each 6 months. You receive the $,000 maturity value in 2 years. What is your rate of return? 2 2 = 24 4.7 2 ~ 8.33* -975 40,000 *8.33% compounded semiannually 8. You deposit $0,000 and let the money sit for 2 years, earning 4.50%, compounded semiannually. Use a longhand approach to determine the ending balance. Balance in 6 months: $0,000 + 2.25% = $0,225.00 Balance in 2 months: + 2.25% = $0,455.06 Balance in 8 months: + 2.25% = $0,690.30 Balance in 24 months: + 2.25% = $0,930.83 94 Chapter 9 Time-Value-of-Money Problems: An Introduction