# Busi 530 Homework 2-1

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b. How much was each entitled to if the interest rate was 6%? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Future value \$ 4. Professor’s Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of \$82,000 at age 65, the firm will pay the retiring professor \$650 a month until death. a. If the professor’s remaining life expectancy is 20 years, what is the monthly rate on this annuity? What is the effective annual rate? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Monthly rate on annuity % Effective annual rate % b. If the monthly interest rate is 1%, what monthly annuity payment can the firm offer to the retiring professor? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Monthly annuity payment \$ 5. Investments in the stock market have increased at an average compound rate of about 5% since 1908. It is now 2012. a. If you invested \$1,000 in the stock market in 1908, how much would that investment be worth today? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Investment \$ b. If your investment in 1908 has grown to \$1 million, how much did you invest in 1908? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Present value \$ Unformatted text preview: 1. Award: 1 out of 1.50 points Show my answer Old Time Savings Bank pays 3% interest on its savings accounts. If you deposit \$2,100 in the bank and leave it there (Do not round intermediate calculations. Round your answers to 2 decimal places.): a. How much interest will you earn in the first year? First year interest \$ 63.00 ±1% b. How much interest will you earn in the second year? \$ Second year interest 64.89 ±1% c. How much interest will you earn in the tenth year? Tenth year interest \$ 82.20 ±1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. a. Future value Year 1 = Present value × (1 + r) = \$2,100 × 1.03 = \$2,163 Interest Year 1 = Future value Year 1 – Present value = \$2,163 – 2,100 = \$63 b. Future value Year 2 = Present value × (1 + r)2 = \$2,100 × 1.032 = \$2,227.89 Interest Year 2 = Future value Year 2 – Future value Year 1 = \$2,227.89 – 2,163 = \$64.89 c. Future value Year 9 = Present value × (1 + r)9 = \$2,100 × 1.039 = \$2,740.02 Future value Year 10 = Present value × (1 + r)10 = \$2,100 × 1.0310 = \$2,822.22 Interest Year 10 = Future value Year 10 – Future value Year 9 = \$2,822.22 – 2,740.02 = \$82.20 Calculator computations: a. Enter 1 3 –2,100 N I/Y PV PMT Solve for FV 2,163 b. Enter 2 3 –2,100 N I/Y PV PMT Solve for FV 2,227.89 c. Enter 9 3 –2,100 N I/Y PV PMT Solve for Enter FV 2,740.02 10 3 –2,100 N I/Y PV Solve for PMT FV 2,822.22 2. Award: 0 out of 1.50 points Show my answer Compute the future value of a \$125 cash flow for the following combinations of rates and times. (Do not round intermediate calculations. Round your answers to 2 decimal places.) a. r = 8%, t = 10 years b. r = 8%, t = 20 years c. r = 4%, t = 10 years d. r = 4%, t = 20 years Future Value \$ 269.87 ±1% \$ 582.62 ±1% \$ 185.03 ±1% \$ 273.89 ±1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. FV = PV × (1 + r)t FV = \$125 × (1.08)10 = \$269.87 FV = \$125 × (1.08)20 = \$582.62 FV = \$125 × (1.04)10 = \$185.03 FV = \$125 × (1.04)20 = \$273.89 Calculator computations: Enter 10 8 –125 N I/Y PV PMT Solve for Enter 20 8 –125 N I/Y PV PMT Solve for Enter FV 582.62 10 4 –125 N I/Y PV PMT Solve for Enter FV 269.87 FV 185.03 20 4 –125 N I/Y PV Solve for PMT FV 273.89 3. Award: 0 out of 1.50 points Show my answer If you earn 5% per year on your bank account, how long will it take an account with \$110 to double to \$220? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Number of years 14.21 ±1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. FV = PV × (1 + r)t \$220 = \$110 × 1.05t 1.05t = 2 t × ln1.05 = ln2 t = ln2 / ln1.05 t = .69315 / .04879 t = 14.21 years Calculator computations: Enter N Solve for 5 –110 I/Y PV 220 PMT FV 14.21 4. Award: 0 out of 1.50 points Show my answer In 1880 five aboriginal trackers were each promised the equivalent of 100 Australian dollars for helping to capture the notorious outlaw Ned Kelley. In 1999 the granddaughters of two of the trackers claimed that this reward had not been paid. The Victorian prime minister stated that if this was true, the government would be happy to pay the \$100. However, the granddaughters also claimed that they were entitled to compound interest. a. How much was each granddaughter entitled to if the interest rate was 3%? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Future value A\$ 3,370.00 ±.1% b. How much was each entitled to if the interest rate was 6%? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Future value A\$ 102,659.22 ±.1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. FV = PV × (1 + r)t a. FV = A\$100 × 1.03119 = A\$3,370.00 b. FV = A\$100 × 1.06119 = A\$102,659.22 Calculator computations: a. Enter 119 3 –100 N I/Y PV PMT Solve for FV 3,370.00 b. Enter 119 6 –100 N I/Y PV Solve for PMT FV 102,659.22 5. Award: 0 out of 1.50 points Show my answer You can buy property today for \$2.4 million and sell it in 5 years for \$3.4 million. (You earn no rental income on the property.) a. If the interest rate is 8%, what is the present value of the sales price? (Do not round intermediate calculations. Enter your answer in millions rounded to 3 decimal places.) Present value \$ million 2.314 ±1% b. Is the property investment attractive to you? No c-1. What is the present value of the future cash flows, if you also could earn \$140,000 per year rent on the property? The rent is paid at the end of each year. (Do not round intermediate calculations. Enter your answer in millions rounded to 3 decimal places.) Present value \$ million 2.873 ±1% c-2. Is the property investment attractive to you now? Yes Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. a. P = FV / (1 + r)t V = \$3,400,000 / 1.085 = \$2,313,982.87, or \$2.314 million b. The investment is not attractive because the present value of the sales price is less than the purchase price of the property. c-1. P = Per-year rent × ((1 / r) – {1 / [r(1 + r)t]}) + Sales price / (1 + r)t V = \$140,000 × ((1 / .08) – {1 / [.08 (1.08)5]}) + \$3,400,000 / 1.085 = \$2,872,962.28, or \$2.873 million c-2. The investment is attractive now because the present value of the future cash flows exceeds the current purchase price of the property. Calculator computations: a. Enter 5 8 N I/Y Solve for c-1. Enter –3,400,000 PV PMT FV –140,000 –3,400,000 PMT FV 2,313,982.87 5 8 N I/Y Solve for PV 2,872,962.28 6. Award: 0 out of 1.50 points Show my answer Find the interest rate implied by the following combinations of present and future values: (Leave no cells blank - be certain to enter "0" wherever required. Do not round intermediate calculations. Round your answers to 2 decimal places) Present Value Years Future Value Interest Rate % \$420 10 \$826 7.00 ±1% 193 3 257 10.02 ±1% 320 6 320 0 ±1% % % Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. FV = PV × (1 + r)t r = (FV / PV)1/t – 1 r = (\$826 / \$420)1/10 – 1 = .0700, or 7.00% r = (\$257 / \$193)1/3 – 1 = .1002, or 10.02% r = (\$320 / \$320)1/6 – 1 = .00, or 0% Calculator computations: Enter 10 –420 N I/Y Solve for Enter PV 826 PMT FV 7.00 3 –193 N I/Y Solve for PV 257 PMT FV 10.02 7. Award: 0.50 out of 1.50 points Show my answer A famous quarterback just signed a \$8.0 million contract providing \$2.7 million a year for 4 years. A less famous receiver signed a \$7.0 million 4-year contract providing \$3 million now and \$2.2 million a year for 4 years. The interest rate is 8%. a. What is the PV of the quarterback's contract? (Do not round intermediate calculations. Enter your answer in millions rounded to 2 decimal places.) Present value \$ million 8.94 ±1% b. What is the PV of the receiver's contract? (Do not round intermediate calculations. Enter your answer in millions rounded to 2 decimal places.) Present value \$ million 10.29 ±1% c. Who is better paid? Receiver Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. a. PV = C((1 / r) – {1 / [r(1 + r)t]}) = \$2,700,000 × ((1 / .08) – {1 / [.08(1.08)4]}) = \$8,942,742.47, or \$8.94 million b. PV = C0 + C((1 / r) – {1 / [r(1 + r)t]}) = \$3,000,000 + \$2,200,000 × ((1 /.08) – {1 / [.08(1.08)4]}) = \$10,286,679.05, or \$10.29 million c. Even though the receiver has the smaller contract amount, he is actually better paid because the present value of his contract exceeds the present value of the quarterback’s contract. Calculator computations: a. Enter 4 8 N I/Y Solve for b. CF0 CO1 I=8 CPT NPV = 10,286,679.05 –2,700,000 PV PMT FV 8,942,742.47 = 3,000,000 = 2,200,000 FO1 = 4 8. Award: 0.50 out of 1.50 points Show my answer A local bank advertises the following deal: Pay us \$100 at the end of each year for 10 years and then we will pay you (or your beneficiaries) \$100 at the end of each year forever. a. Calculate the present value of your payments to the bank if the interest rate available on other deposits is 8.75%. (Do not round intermediate calculations. Round your answer to 2 decimal places.) \$ Present value 648.89 ±1% b. What is the present value of a \$100 perpetuity deferred for 10 years if the interest rate available on other deposits is 8.75%. (Do not round intermediate calculations. Round your answer to 2 decimal places.) \$ Present value 493.97 ±.1% c. Is this a good deal? No Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. a. PV = C((1 / r) – {1 / [r(1 + r)t]}) = \$100 × ((1 / .0875) – {1 / [.0875(1.0875) 10]}) = \$648.89 b. PV10 = C / r = \$100 / .0875 = \$1,142.86 This is the present value of the annuity in 10 years. You now need to find the present value as of today. PV = FV / (1 + r)t = \$1,142.86 / 1.087510 = \$493.97 c. This is not a good deal if you can earn 8.75% because today’s value of the bank’s payments to you less than the value of your payments to the bank. Calculator computations: a. Enter 10 8.75 N I/Y –100 PV PMT FV Solve for 648.89 b. Enter 10 8.75 N I/Y –1,142.86 PV Solve for PMT FV 493.97 9. Award: 0 out of 1.50 points Show my answer You take out a 25-year \$200,000 mortgage loan with an APR of 9% and monthly payments. In 15 years you decide to sell your house and pay off the mortgage. What is the principal balance on the loan? (Round the monthly loan payment to 2 decimal places when computing the answer. Round your answer to 2 decimal places.) \$ Principal balance on the loan 132,494.95 ±.1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. First, you need to compute the monthly mortgage payment. Because the payments are monthly, the interest rate and the number of time periods must also be expressed in terms of months. It is assumed that annuity payments occur at the end of each time period unless a problem indicates otherwise. PV = C((1 / r) – {1 / [r(1 + r)t]}) = C × [[1 / (.09 / 12)] – (1 / {(.09 / 12)[1 + (.09 / 12)](25 × 12)})] = \$1,678.39 \$200,000 Second, compute the loan balance at the end of 15 years. Use the monthly loan payment rounded to 2 decimal places and adjust the number of months to equal the number of payments remaining. = C((1 / r) – {1 / [r(1 + r)t]}) = \$1,678.39 × [[1 / (.09 / 12)] – (1 / {(.09 / 12)[1 + (.09 / 12)][(25 – 15) × 12]})] = \$132,494.95 PV Calculator computations: Enter 25 × 12 9 / 12 –200,000 N I/Y PV PMT FV Solve for Enter 1,678.39 [(25 – 15) × 12] 9 / 12 N I/Y Solve for –1,678.39 PV PMT FV 132,494.95 10. Award: 0 out of 1.50 points Show my answer Home loans typically involve “points,” which are fees charged by the lender. Each point charged means that the borrower must pay 1% of the loan amount as a fee. For example, if the loan is for \$120,000 and 2 points are charged, the loan repayment schedule is calculated on a \$120,000 loan but the net amount the borrower receives is only \$117,600. Assume the interest rate is 1.00% per month. What is the effective annual interest rate charged on such a loan, assuming loan repayment occurs over 216 months? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places. Use a financial calculator or Excel.) % Effective annual interest rate 13.06 ±1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. First, compute the monthly payment on a \$120,000 loan using the monthly interest rate provided: PV = C((1 / r) – {1 / [r(1 + r)t]}) = C × ((1 / .0100) – {1 / \$120,000 [.0100(1.0100)216]}) = \$1,358.34 Enter 216 1.00 –120,000 N I/Y PV Solve for PMT FV 1,358.34 Second, compute the monthly interest rate based on the actual amount received and the loan payment calculated above. PV = C((1 / r) – {1 / [r(1 + r)t]}) = \$1,358.34 × ((1 / r) – {1 / [r(1 \$117,600 + r)216]}) To solve for r, it is easiest to use either a financial calculator or a computer. Here is the calculator solution based on a monthly period of time: Enter 216 N Solve for I/Y 117,600 –1,358.34 PV PMT FV 1.028 Thus, the actual monthly interest rate is 1.028%. Lastly, compute the effective annual rate as follows: = (1 + Monthly interest rate)12 – 1 = 1.0102812 – 1 = .1306, or 13.06% EAR 11. Award: 0 out of 1.50 points Show my answer A couple will retire in 50 years; they plan to spend about \$38,000 a year in retirement, which should last about 25 years. They believe that they can earn 9% interest on retirement savings. a. If they make annual payments into a savings plan, how much will they need to save each year? Assume the first payment comes in 1 year. (Do not round intermediate calculations. Round your answer to 2 decimal places.) \$ Annual payment 457.94 ±.1% b. How would the answer to part (a) change if the couple also realize that in 20 years they will need to spend \$68,000 on their child’s college education? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Annual payment \$ 1,564.82 ±.1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. a. First, determine the amount of savings required on the date of retirement: PV = C((1 / r) – {1 / [r(1 + r)t]}) = \$38,000 × ((1 / .09) – {1 / [.09(1.09)25]}) = \$373,258.02 Now compute the annual savings needed to accumulate the needed retirement savings: FV = C × {[(1 + r)t – 1] / r} \$373,258.02 = C × [(1.0950 – 1) / .09] C = \$373,258.02 / [(1.0950 – 1) / .09] C = \$457.94 b. The first step is to determine the present value of the cash needed for both college and retirement at Time 0: = \$68,000 / 1.0920 + \$373,258.02 / 1.0950 = \$17,153.08 PV Now determine the annual savings for 50 years that equates to that present value: PV = C((1 / r) – {1 / [r(1 + r)t]}) = C × ((1 / .09) – {1 / \$17,153.08 [.09(1.09)50]}) C = \$1,564.82 Calculator computations: a. Savings needed on retirement date: Enter 25 N 9 I/Y Solve for –38,000 PV PMT 373,258.02 Annual savings needed to fund retirement: Enter 50 9 N I/Y 373,258.02 PV Solve for PMT FV 457.94 b. Present value of college education: Enter 20 N 9 I/Y Solve for 68,000 PV PMT FV 12,133.30 Present value of required retirement savings: Enter 50 9 N Solve for FV I/Y 373,258.02 PV 5,019.78 Total savings needed at Time 0 = \$12,133.30 + 5,019.78 = \$17,153.08 Annual savings needed to fund both college and retirement: Enter 50 9 17,153.08 PMT FV N I/Y PV Solve for PMT FV 1,564.82 12. Award: 0 out of 1.50 points Show my answer A store will give you a 4.00% discount on the cost of your purchase if you pay cash today. Otherwise, you will be billed the full price with payment due in 1 month. What is the implicit borrowing rate being paid by customers who choose to defer payment for the month? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) % Effective annual rate 63.21 ±1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. If you assume a purchase price of \$1, then: Cash price today = Purchase price – Discount = \$1 – (.0400 × \$1) = \$.9600 You know the price in one month is \$1 so you can compute the monthly interest rate as: FV = PV × (1 + r)t \$1 = \$.9600 × (1 + r)1 r = \$1 / \$.9600 – 1 r = .0417, or 4.17% = (1 + Monthly interest rate)12 – 1 = 1.041712 – 1 = .6321, or 63.21% EAR Calculator computations: Enter 1 N Solve for –.9600 I/Y 4.17 PV 1 PMT FV 13. Award: 0 out of 1.50 points Show my answer a. How much will \$100 grow to if invested at a continuously compounded interest rate of 9% for 7 years? (Do not round intermediate calculations. Round your answer to 2 decimal places.) \$ Future value 187.76 ±1% b. What if it is invested for 9 years at 7%? (Do not round intermediate calculations. Round your answer to 2 decimal places.) \$ Future value 187.76 ±1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. a. FV = C × ert = \$100 × e(.0900 × 7) = \$187.76 = C × ert = \$100 × e(.07 × 9.00) = \$187.76 FV 14. Award: 0 out of 1.50 points Show my answer In April 2013 a pound of apples cost \$1.56, while oranges cost \$1.20. Four years earlier the price of apples was only \$1.35 a pound and that of oranges was \$1.06 a pound. a. What was the annual compound rate of growth in the price of apples? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) Compound annual growth rate % per year 3.68 ±1% b. What was the annual compound rate of growth in the price of oranges? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) % per year Compound annual growth rate 3.15 ±1% c. If the same rates of growth persist in the future, what will be the price of apples in 2030? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Price \$ 2.88 ±1% d. If the same rates of growth persist in the future, what will be the price of oranges in 2030? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Price \$ 2.03 ±1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. a. FV = PV (1 + r)t \$1.5 = \$1.35 × (1 + r)4 6 r = (\$1.56 / \$1.35)1 / 4 – 1 r = .0368, or 3.68% b. FV = PV (1 + r)t \$1.2 = \$1.06 × (1 + r)4 0 r = (\$1.20 / \$1.06)1 / 4 – 1 r = .0315, or 3.15% c. FV = PV (1 + r)t = \$1.56 × 1.0368(2030 – 2013) = \$2.88 d. FV = PV (1 + r)t = \$1.20 × 1.0315(2030 – 2013) = \$2.03 Calculator computations: a. Enter 4 N Solve for –1.35 I/Y PV 1.56 PMT FV 3.68 b. Enter 4 –1.06 1.20 N I/Y Solve for PV PMT FV PMT FV 3.15 c. Enter 17 3.68 –1.56 N I/Y PV Solve for 2.88 d. Enter 17 3.15 –1.20 N I/Y PV Solve for PMT FV 2.03 15. Award: 0 out of 2.00 points Show my answer An engineer in 1950 was earning \$6,100 a year. Today she earns \$61,000 a year. However, on average, goods today cost 8.9 times what they did in 1950. What is her real income today in terms of constant 1950 dollars? (Round your answer to 2 decimal places.) Today's real income \$ 6,853.93 ±.1% Explanation: Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. Real income = Nominal income / Inflation multiple = \$61,000 / 8.9 = \$6,853.93 Her real income only increased from \$6,100 to \$6,853.93. ...
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