Homework 2 (Ch 5, 6) Solutions

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1. You will receive annual payments of $800 at the end of each year for 12 years. The first payment will be received three years from today. What is the present value of these payments if the discount rate is 7 percent? rev: 03_25_2020_QC_CS-205328 $5,465.20 $6,018.52 $6,299.80 $5,549.96 $6,856.60 PV 2 = $800 × ({1 − [1 / (1 + .07) 12 ]} / .07) = $6,354.15 PV 0 = $6,354.15 / (1 + .07) 2 = $5,549.96 2. How much money does Yvette need to have in her retirement savings account today if she wishes to withdraw $36,000 a year for 30 years? She expects to earn an average rate of return of 8.25 percent. $393,388.77 $405,280.20 $339,752,80 $354,049.89 $395,904.99 3 0 8.2 5 36,000.0 0 0 N I/Y PV PMT F V −395,904.9 9

3. Chandler Tire Co. is trying to decide which one of two projects it should accept. Both projects have the same start-up costs. Project 1 will produce annual cash flows of $52,000 a year for six years. Project 2 will produce cash flows of $48,000 a year for eight years. The company requires a 15 percent rate of return. Which project should the company select and why? Project 1, because the annual cash flows are greater by $4,000 than those of Project 2 Project 1, because the present value of its cash inflows exceeds those of Project 2 by $14,211.62 Project 2, because the total cash inflows are $72,000 greater than those of Project 1 Project 2, because the present value of the cash inflows exceeds those of Project 1 by $18,598.33 It does not matter as both projects have almost identical present values. PV 1 = $52,000 × {1 − [1 / (1 + .15) 6 ]} / .15 = $196,793.10 PV 2 = $48,000 × {1 − [1 / (1 + .15) 8 ]} / .15 = $215,391.43 Difference = $215,391.43 − 196,793.10 = $18,598.33’ 4. Assume all else is equal. When comparing savings accounts, you should select the account that has the: a. lowest annual percentage rate. b. highest annual percent rate. c. highest stated rate. d. lowest effective annual rate. e. highest effective annual rate. 5. Find the EAR in each of the following cases. (Use 365 days in a year. Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) Stated rate (APR) Number of Times Compounded Effective Rate (EAR) 10.2 %Quarterly +/-1%10.60 % 18.0 %Monthly +/-1%19.56 % 13.5 %Daily +/-1%14.45 % 9.5 %Semiannually +/-1%

6. Although you may know William Shakespeare from his classic literature, what is not well-known is that he was an astute investor. In 1604, when he was 40 and writing King Lear , Shakespeare grew worried about his eventual retirement. Afraid that he would become like King Lear in his retirement and beg hospitality from his children, he purchased grain “tithes,” or shares in farm output, for 440 pounds. The tithes paid him 60 pounds per year for 31 years. Even though he died at the age of 52, his children received the remaining payments. What interest rate did the Bard of Avon receive on this investment? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) Interest rate +/-1%13.36 % 7. The written agreement that contains the specific details related to a bond issue is called the bond: indenture. debenture. document. registration statement. issue paper. 8. The Treasury yield curve plots the yields on Treasury notes and bonds relative to the ___________ of those securities. face value market price maturity coupon rate issue date

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9 Lake Industries bonds have a face value of $1,000, a coupon rate of 7.2 percent, semiannual interest payments, and mature in 15 years. What is the current price of these bonds if the yield to maturity is 6.98 percent? $988.39 $1,000.00 $1,020.26 $1,012.78 $1,010.68 3 0 3.4 9 36 1,000.0 0 N I/Y PV PM T FV - 1,020.26 10. A bond yielded a real rate of return of 3.87 percent for a time period when the inflation rate was 3.75 percent. What was the actual nominal rate of return? 87.58 percent 7.62 percent 7.77 percent 8.28 percent .36 percent R = (1 + .0387)(1 + .0375) − 1 R = .0777, or 7.77 percent

11. Locate the Treasury bond in Figure 6.3 maturing in May 2037. Assume a $1,000 par value. a. Is this a premium or discount bond? b . What is its current yield? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) c. What is its yield to maturity? (Do not round intermediate calculations and enter your answer as a percent rounded to 3 decimal places, e.g., 32.161.) d . What is the bid-ask spread in dollars? (Do not round intermediate calculations and round your answer to 3 decimal places, e.g., 32.161.) a . This is a premium bond because it sells for more than 100 percent of face value. b. The current yield is based on the asked price, so the current yield is:

Current yield = Annual coupon payment/Price Current yield = $50/$1,291.5630 Current yield = .0387, or 3.87% c. The YTM is located under the “Asked Yield” column, so the YTM is 2.973%. d. The bid-ask spread is the difference between the bid price and the ask price, so: Bid-Ask spread = ($129.1563 – 129.0938)(10) Bid-Ask spread = .0625 × 10 Bid-Ask spread = $.625 12. Bond J has a coupon rate of 4 percent. Bond K has a coupon rate of 14 percent. Both bonds have 17 years to maturity, a par value of $1,000, and a YTM of 8 percent, and both make semiannual payments. a. If interest rates suddenly rise by 2 percent, what is the percentage change in the price of these bonds? (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) b. If interest rates suddenly fall by 2 percent instead, what is the percentage change in the price of these bonds? (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) a. Initially, at a YTM of 8 percent, the prices of the two bonds are: P J = $20(PVIFA 4%,34 ) + $1,000(PVIF 4%,34 ) = $631.78 P K = $70(PVIFA 4%,34 ) + $1,000(PVIF 4%,34 ) = $1,552.34 If the YTM rises from 8 percent to 10 percent: P J = $20(PVIFA 5%,34 ) + $1,000(PVIF 5%,34 ) = $514.21 P K = $70(PVIFA 5%,34 ) + $1,000(PVIF 5%,34 ) = $1,323.86 The percentage change in price is calculated as: Percentage change in price = (New price – Original price)/Original price Δ P J % = ($514.21 – 631.78)/$631.78 = –.1861, or –18.61% Δ P K % = ($1,323.86 – 1,552.34)/$1,552.34 = –.1472, or –14.72% b.

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If the YTM declines from 8 percent to 6 percent: P J = $20(PVIFA 3%,34 ) + $1,000(PVIF 3%,34 ) = $788.68 P K = $70(PVIFA 3%,34 ) + $1,000(PVIF 3%,34 ) = $1,845.27 Δ P J % = ($788.68 – 631.78)/$631.78 = .2484, or 24.84% Δ P K % = ($1,845.27 – 1,552.34)/$1,552.34 = .1887, or 18.87% Calculator Solution: Initially, at a YTM of 8 percent, the prices of the two bonds are: P J Enter 17 × 2 8%/2 ±$40/2 ±$1,000 N I/Y PV PMT FV Solve for $631.78 P K Enter 17 × 2 8%/2 ±$140/2 ±$1,000 N I/Y PV PMT FV Solve for $1,552.34 If the YTM rises from 8 percent to 10 percent: P J Enter 17 × 2 10%/2 ±$40/2 ±$1,000 N I/Y PV PMT FV Solve for $514.21 P K Enter 17 × 2 10%/2 ±$140/2 ±$1,000 N I/Y PV PMT FV Solve for $1,323.86 Δ P J % = ($514.21 – 631.78)/$631.78 = –.1861, or –18.61% Δ P K % = ($1,323.86 – 1,552.34)/$1,552.34 = –.1472, or –14.72% If the YTM declines from 8 percent to 6 percent: P J Enter 17 × 2 6%/2 ±$40/2 ±$1,000 N I/Y PV PMT FV Solve for $788.68

P K Enter 17 × 2 6%/2 ±$140/2 ±$1,000 N I/Y PV PMT FV Solve for $1,845.27 Δ P J % = ($788.68 – 631.78)/$631.78 = .2484, or 24.84% Δ P K % = ($1,845.27 – 1,552.34)/$1,552.34 = .1887, or 18.87% All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in interest rates. 13. A bond that makes quarterly coupon payments and has an annual coupon rate of 6% (face value is 1,000) sells for $1,070, 2 months after the last coupon payment. What is the clean price? Full price = Clean Price + Acc. Int. Quarterly coupon payments = 6%/4*1,000 = 15. Acc Int = 2/3*15 = 10 Clean price = 1,070 – 15 = 1,055

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