Pract-Midterm

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McGill University *

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341

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Finance

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Apr 3, 2024

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48

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Page 1 of 48 PRACTICE QUESTIONS FOR THE MIDTERM Q1 A given rate is quoted as 9% APR, but has an EAR of 9.3807%. What is the rate of compounding during the year? a. Biannually b. Annually c. Semiannually d. Monthly e. Continuously 093807 . 0 1 12 09 . 0 1 12 Q2 Mary has just obtained a 20-year $50,000 loan from her bank at 10%. The loan requires equal annual payments over the next 19 years and a final payment of 5,000 in year 20. Calculate the amount that Mary must pay during each of the next 19 years. a. $ 5,879.61 b. $ 5,888.49 c. $ 5,379.61 d. $ 5,285.68 e. $ 5,872.98 49 . 5888 $ 1 . 1 000 , 5 1 . 1 1 1 1 . 0 000 , 50 20 19 C C
Page 2 of 48 Q3 You have just got married and can't wait to buy a new house. In order to be able to buy the house, you take a $277,000 loan from the bank. As per the loan covenants, you are expected to make constant monthly payments to the bank for the next 40 years. The APR is 18% compounded monthly. If you want to liquidate the loan after 25 years, what will be the balloon payment? a. $ 4,158 b. $ 277,000 c. $ 258,210 d. $ 250,460 e. $ 254,205 $4,158 12 18 . 0 12 18 . 0 1 1 1 000 , 277 480 C 210 , 258 $ 12 18 . 0 1 1 1 ) 12 / 18 . 0 ( 4158 180 Balloon Q4 You are 40 years away from retirement and expect to live for 25 years after retirement. Currently, a nursery home costs $20,000 per year, payable at the beginning of the year. You expect annual costs to continually increase at an average of 5% per year. Assuming that the interest rate is 5% and that you plan to move to a nursery home immediately after retirement, how much do you need to have in the bank when you retire (just before the first year payment) to cover the 25 years that you expect to live in the nursery home? a. $ 1,984,424 b. $ 140,800 c. $ 500,000 d. $ 281,879 e. $ 3,519,994 994 , 519 , 3 25 05 . 1 000 , 20 40 Amount
Page 3 of 48 Comments \ The annual cost of the nursery home in 40 years will be 20,000 x (1.05) 40 . Since the annual cost growth rate is equal to the interest rate (5%) you would need to have 25 times the annual cost in the bank, 25 x 20,000 x (1.05) 40 , when you retire to pay for the 25 years that you expect to live. Q5 You have just won the lotto and as a prize you will be receiving 20 equal payments of $2 million each. The first payment will be received in 2 years and thereafter, the other 19 payments will be received every three years (that is, in year 5, 8, 11, etc.). The current annual interest rate of 10% will remain stable for next 3 years and then it will increase to 20%. How much should you willing to pay for the winning ticket? a. $ 7.718 million b. $ 9.739 million c. $ 2.747 million d. $ 4.130 million e. $ 3.022 million % 8 . 72 1 ) 2 . 0 1 ( Rate Effective Year Three 3 M PV 130 . 4 $ 728 . 1 1 1 728 . 0 2 2 . 1 1 . 1 1 2 . 1 1 . 1 2 1 . 1 2 18 2 3 2 3 2 Q6 A cheque-cashing store is in the business of making personal loans to walk-in customers. The store makes only one-week loans at 5 percent interest per week. What annual percentage rate (APR) must the store report to its customers? What is the effective annual rate (EAR) that the customers are actually paying? a. APR=6.84% ; EAR=55.98% b. APR=7.67% ; EAR=92.90% c. APR=8.29% ; EAR=45.98% d. APR=125.00% ; EAR=45.98% e. APR=260.00% ; EAR=1,164.28% % 28 . 164 , 1 1 ) 05 . 1 ( % 260 5 52 52 EAR APR
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Page 4 of 48 Q7 Your father promises to pay you a lump sum of money today or an annual amount at the end of each of the next ten years. At today's interest rates the present values of the two options are identical. You should: I. choose the lump sum if you expect interest rates to rise in the near future. (TRUE) II. choose the annuity if you expect interest rates to rise in the near future. (FALSE) III. choose the lump sum if you expect interest rates to fall in the near future. (FALSE) IV. choose the annuity if you expect interest rates to fall in the near future. (TRUE) Q8 Your father deposited $224 in a bank account 25 years ago. It turns out that today she has $469 in the account. What annual interest has she earned? a. 2% b. 3% c. 4% d. 5% e. 6% 224 ) 03 . 0 1 ( 469 25 Q9 The bonds of AGI carry a 9% coupon rate, have a $1,000 face value, pay one coupon per year, and mature in five years. If the bond YTM is 5%, what is the market value of AGI's bonds? a. $842 b. $1,000 c. $1,142 d. $1,173 e. $1,193 173 , 1 $ 05 . 1 000 , 1 05 . 1 1 1 05 . 09 . 000 , 1 5 5 0 P
Page 5 of 48 Q10 The bonds of IGA carry a 12% coupon rate, have a $1,000 face value, pay two coupons per year, and mature in 10 years. If currently IGA’s bonds trade at $601.83 what is the bond YTM (quoted as an APR with semi-annual compounding)? a. 11% b. 12% c. 22% d. 24% e. 26% 83 . 601 $ 2 22 . 0 1 000 , 1 2 22 . 0 1 1 1 2 22 . 0 ) 12 . 0 000 , 1 ( 2 1 20 20 0 P Q11 Suppose you are trying to price a bond. Which ONE of the following is TRUE ? a. The lower the discount rate, the less valuable the coupon payments are today. b. Bonds with high coupon payments are generally (all else the same) more sensitive to changes in interest rates than bonds with lower coupon payments. c. When market interest rates rise, bond prices will increase, all else the same. d. Bonds with long maturities are generally (all else the same) more sensitive to changes in interest rates than bonds with shorter maturities. e. All else the same, bonds with larger coupon payments will have a lower price today.
Page 6 of 48 Q12 A given rate is quoted as 24% APR, but has an EAR of 25.44%. What is the frequency of compounding? a. Annually b. Semiannually c. Quarterly d. Monthly e. Continuously 2544 . 0 1 2 24 . 0 1 2 Q13 Peter has just obtained a 10-year $100,000 loan from his bank. The loan requires equal annual payments of $18,962 over the next 9 years and a final payment of $6,555 in year 10. Calculate the interest rate on the loan. a. 9.20% b. 10.50% c. 11.60% d. 12.80% e. 15.00% 10 9 128 . 1 555 , 6 128 . 1 1 1 128 . 0 962 , 18 000 , 100
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Page 7 of 48 Q14 A corporate bond is currently trading at $2,184.72. The bond, which pays two coupons per year, matures in 9 years, has a coupon rate of 10%, and YTM of 12% (APR with semi-annual compounding). Calculate the face value of the bond. a. $1,825 b. $2,030 c. $2,108 d. $2,450 e. $2,600 450 , 2 06 . 1 06 . 1 1 1 06 . 0 2 / 1 . 0 72 . 184 , 2 18 18 FV FV FV Q15 In order to buy a car, you take a $66,000 loan from the bank. As per the loan covenants, you are expected to make constant monthly payments to the bank for the next 5 years. The APR is 6% compounded monthly. If you want to liquidate the loan after 3 years, what will be the balloon payment? a. $ 66,000 b. $ 1,276 c. $ 28,789 d. $ 41,942 e. $ 1,468 $1,276 12 06 . 0 12 06 . 0 1 1 1 000 , 66 60 C 789 , 28 $ 12 06 . 0 1 1 1 12 06 . 0 1276 24 Balloon
Page 8 of 48 Q16 In order to buy a house, you take a $250,000 loan from the bank. As per the loan covenants, you are expected to make constant annual payments to the bank for the next 25 years. If the EAR is 12%, how much principal is amortized in the 21 st payment? a. $ 13,788 b. $ 18,087 c. $ 22,815 d. $ 41,942 e. $ 31,875 $31,874 12 . 0 1 1 1 12 . 0 000 , 250 Payment Annual 25 902 , 114 $ 12 . 1 1 1 12 . 0 875 , 31 20 Year in g Outstandin Principal 5 788 , 13 $ 12 . 0 902 , 114 21 Year in Payment Interest 087 , 18 $ 788 , 13 874 , 31 21 Year in Payment Principal Q17 In order to buy a Ferrari, you take a $150,000 loan from the bank. As per the loan covenants you are expected to amortize a constant fraction of the principal in each of the 15 annual payments of the loan. If the EAR is 8%, how much should you pay in the 7 th year? a. $ 8,320 b. $ 7,200 c. $ 16,400 d. $ 17,200 e. $ 17,524 000 , 90 $ 15 000 , 150 6 000 , 150 6 Year in g Outstandin Principal 200 , 7 $ 08 . 0 000 , 90 7 Year in Payment Interest 200 , 17 $ 000 , 10 200 , 7 7 Year in Payment
Page 9 of 48 Q18 Mary is 20 years away from retirement and by the time she retires Mary would like to have $1 million in the bank. Assuming that the interest rate is 15% per year (EAR) how much does Mary need to deposit each month in her bank account? a. $ 50,000 b. $ 4,167 c. $ 2,325 d. $ 762 e. $ 668 % 1715 . 1 1 ) 15 . 0 1 ( EMR 12 / 1 762 $ 1 ) 011715 . 0 1 ( / 011715 . 0 million 1 Payment Monthly 12 20 Q19 You have just received an inheritance and as a result you will be getting 15 equal payments of $1 million each. The first payment will be received in 1 year and thereafter, the other 14 payments will be received every three years (that is, in year 4, 7, 10, etc.). The current annual interest rate of 10% will remain stable for the next 4 years and then it will increase to 15%. Calculate the present value of your inheritance. a. $ 3.218 million b. $ 3.116 million c. $ 2.898 million d. $ 2.395 million e. $ 2.265 million % 09 . 52 1 ) 15 . 0 1 ( Rate Effective Year Three 3 M PV 898 . 2 $ 5209 . 1 1 1 5209 . 0 1 1 . 1 1 1 . 1 1 1 . 1 1 13 4 4
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Page 10 of 48 Q20 Consider the following statements: I. All else equal, bonds with longer maturities have more interest rate risk. (TRUE) II. All else equal, bonds with shorter maturities have more interest rate risk. (FALSE) III. All else equal, bonds with higher coupon payments have more interest rate risk. (FALSE) IV. All else equal, bonds with lower coupon payments have more interest rate risk. (TRUE) V. Bond ratings measure a bond’s exposure to interest rate risk. (FALSE) Q21 Consider the following statements: I. The law of one price will cause investors to demand discount bonds until these bonds trade at par value. (FALSE) II. A consol is a risk-free zero-coupon government bond. (FALSE) III. All else equal, bonds with a higher probability of default have a higher yield to maturity. (TRUE) IV. Treasury bills are long-term government bonds with a maturity of up to 10 years. (FALSE) V. The prices of bonds with lower coupon rates are less sensitive to changes in interest rates. (FALSE) Q22 A loan has a 20% APR and an EAR of 22.140%. What is the frequency of compounding of the APR rate? a. Annually b. Semiannually c. Quarterly d. Continuously e. Biannually % 140 . 22 1 2 . 0 e EAR
Page 11 of 48 Q23 Justin has just obtained a 20-year $200,000 loan to open a business. The loan requires Justin to make annual payments of $17,564.53 over the next 20 years. Calculate the interest rate on the loan. a. 12% Effective Annual Rate b. 3% Effective Annual Rate c. 6% APR with semiannual compounding d. 4% APR with semiannual compounding e. 3% APR with semiannual compounding A 6% APR with semiannual compounding corresponds to the following EAR: % 09 . 6 1 2 06 . 0 1 EAR 2 If we use 6.09% to discount the 20-year annuity we get $200,000: 000 , 200 0609 . 1 1 1 0609 . 0 53 . 17564 20 Q24 Mary deposited $1,000 in a bank account 20 years ago. She withdrew $300 from the bank account 10 years ago. If the annual interest is 12%, how much does she have in the bank account today? a. $ 2,174 b. $ 6,752 c. $ 9,646 d. $ 8,715 e. $ 10,578 715 , 8 $ 300 ) 12 . 0 1 ( 1000 ) 12 . 0 1 ( 10 20
Page 12 of 48 Q25 Consider a bond with a 9% annual coupon rate, a $1,000 face value, and that pays monthly coupons. The bond has just paid a coupon, matures in 20 years, and currently trades at $514.03. What is the bond YTM (quoted as an APR with monthly compounding)? a. 9% b. 18% c. 24% d. 12% e. 6% 03 . 514 $ 12 18 . 0 1 000 , 1 12 18 . 0 1 1 1 12 18 . 0 ) 09 . 0 000 , 1 ( 12 1 240 240 0 P Q26 Consider the following statements: I. A risk-free zero-coupon bond with a face value of $1,000 that matures in one year sells today for $972. If the risk-free interest rate is 4% per year, you can make money by borrowing at the risk-free rate and buying the zero-coupon bond. ( FALSE) II. A 20-year, risk-free coupon bond with a face value of $1,000 pays one coupon of $45 each year, has just paid a coupon, and has a YTM of 3.2% (APR with annual compounding). Currently, the yield curve is upward slopping, and hence, bonds will be expected to yield a higher interest rate in the future. Therefore, the 20- year, risk-free coupon bond must be currently trading at a discount. ( FALSE) Q27 Consider the following statements: I. A premium bond has its price increasing overtime to provide a premium return that makes up for the low coupon rate. ( FALSE) II. High yield bonds are bonds in the top four categories of creditworthiness with a low risk of default. ( FALSE) III. A T-bill is a zero-coupon bond with a maturity longer than ten years and a consol is a zero coupon bond with a maturity shorter than one year. ( FALSE)
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Page 13 of 48 Q28 Marisa has just obtained a 30-year $1,000,000 loan to buy a condo. The loan requires Marisa to make annual payments of $210,691.97 over the next 30 years. Calculate the interest rate on the loan. a. 20% APR with semiannual compounding b. 22% APR with semiannual compounding c. 10% APR with semiannual compounding d. 11% APR with semiannual compounding e. 24 % APR with semiannual compounding Solution A 20% APR with semiannual compounding corresponds to the following EAR: % 21 1 2 20 . 0 1 EAR 2 If we use 21% to discount the 30-year annuity we get $1,000,000: 000 , 000 , 1 21 . 1 1 1 21 . 0 97 . 691 , 210 30 Q29 Pierre deposited $5,000 in a bank account 20 years ago at an interest rate of 8% per year. Pierre withdrew some amount from the bank account 12 years ago. If Pierre has $20,912.52 in the bank account today, how much did Peter withdraw 12 years ago? a. $192 b. $2,025 c. $825 d. $950 e. $1,352 Solution 52 . 912 , 20 $ 950 ) 08 . 0 1 ( 5000 ) 08 . 0 1 ( 12 20
Page 14 of 48 Q30 You have just graduated and want to buy a new boat. In order to buy the boat, you take a $150,000 loan from the bank. As per the loan covenants, you are expected to make constant monthly payments to the bank for the next 10 years. The APR is 6% compounded monthly. You have just made a payment and still owe the bank $37,574. (That is, if you were to liquidate the loan today, you would need to pay the bank $37,574.) How much time is there left on the loan? a. 18 months b. 48 months c. 24 months d. 28 months e. 12 months Solution A 6% APR with monthly compounding corresponds to the following effective monthly rate: % 5 . 0 12 6 EMR The monthly payment on the loan is: 31 . 665 , 1 $ 005 . 1 1 1 005 . 0 1 000 , 150 yment Monthly Pa 120 The amount owed to the bank when there are n months left in the loan is: n 005 . 1 1 1 005 . 0 31 . 665 , 1 Plugging in n = 24 months we get that the amount still owed to the bank: 574 , 37 005 . 1 1 1 005 . 0 31 . 665 , 1 24
Page 15 of 48 Q31 Juan took a $500,000 loan from CIBC. As per the loan covenants, he is expected to make constant annual payments to CIBC for the next 32 years. If the EAR is 4%, how much interest is paid in the 22 nd payment? a. $8,202 b. $12,625 c. $27,974 d. $9,803 e. $28,887 Solution $27,974.29 04 . 0 1 1 1 04 . 0 000 , 500 Payment Annual 32 12 . 068 , 245 $ 04 . 0 1 1 1 04 . 0 29 . 974 , 27 21 Year in g Outstandin Principal 11 803 , 9 $ 04 . 0 12 . 068 , 245 22 Year in aid Interest P Q32 Consider a bond with a YTM of 18% (quoted as an APR with monthly compounding), a $1,000 face value, and that pays monthly coupons. The bond has just paid a coupon, matures in 10 years, and currently trades at $815.01. What is the bond’s coupon rate? a. 18% b. 16% c. 14% d. 12% e. 10% Solution Plugging the 14% coupon rate ( i.e. , 0.14) into the bond price formula: 01 . 815 $ 12 18 . 0 1 000 , 1 12 18 . 0 1 1 1 12 18 . 0 14 . 0 000 , 1 12 1 P 120 120 0
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Page 16 of 48 Q33 An economy has an annual risk-free rate of 5% that is certain to remain constant. Consider a risk-free bond with a 10% coupon rate, a $1,000 face value, and that pays one coupon per year. The bond has just paid a coupon, and currently trades at $1,386.09. Calculate the price of the bond two years from now (just after paying the coupon for that year). a. $1,428.16 b. $1,290.84 c. $1,323.16 d. $1,372.19 e. $1,528.16 Solution Because the bond is risk free and the risk-free rate is constant at 5% per year: 16 . 323 , 1 P 0.05) (1 P 1 . 0 1,000 0.05 1 1 . 0 1,000 09 . 386 , 1 P 2 2 2 0 Q34 Taylor Swift has just signed a new contract with Emi Records. According to the contract, Taylor Swift will be receiving 15 equal payments of $5 million each. The first payment will be received in year 1, and thereafter, the other 14 payments will be received every two years (that is, in years 3, 5, 7, 9 etc.). The current annual interest rate of 4% will remain stable for the next 4 years and then it will increase to 7%. How much is the contract worth today? a. $42.12 M b. $44.97 M c. $39.19 M d. $35.38 M e. $31.42 M Solution % 49 . 14 1 ) 07 . 0 1 ( Rate ctive Year Effe Two 2 M PV 38 . 35 $ 1449 . 1 1 1 1449 . 0 5 07 . 1 04 . 1 1 07 . 1 04 . 1 5 04 . 1 5 04 . 1 5 12 4 4 3
Page 17 of 48 Q35 A silver mine will produce for the next 12 years (i.e., from year 1 to year 12), but its production of silver will decline by 7% per year. Silver prices, however, will grow by 4% per year. Next year, the mine is expected to produce 1,000 kilos of silver and the price of silver is expected to be $2,000 per kilo. Assuming that the appropriate discount rate is 8%, calculate today’s value of the mine’s future revenues, that is, the present value of the mine’s revenues from year 1 to 12. a. $15,250,409 b. $13,011,700 c. $13,320,020 d. $15,072,156 e. $17,352,429 Solution % 28 . 3 1 ) 04 . 0 1 ( ) 07 . 0 1 ( Revenues in Growth 700 , 011 , 13 $ 08 . 0 1 0328 . 0 1 1 ) 0328 . 0 ( 08 . 0 000 , 2 000 , 1 12 PV
Page 18 of 48 Q36 A risk-free zero-coupon bond with a face value of $1,000 maturing in one year trades today at a price of $890. A risk-free bond with a coupon rate of 10%, that pays one coupon per year, has a face value of $1,000, matures in 2 years, and that has just paid a coupon trades today at $995. Calculate the current price of a risk- free zero-coupon bond with a face value of $1,000 that matures in 2 years. a. $823.64 b. $704.93 c. $105.00 d. $792.09 e. $942.50 Solution The coupon bond pays $100 in one year (i.e., 1000 x 0.1) and $1,100 in two years (i.e., coupon + FV =1000 x 0.1 + 1000). Therefore, the coupon bond is equivalent to 0.1 one-year zero-coupon bond plus 1.1 two-year zero-coupon bonds. Hence the law of one price (i.e., securities or portfolios with the same cash- flows must have the same price) implies that: Price of coupon bond = = 0.1 x Price of one-year zero-coupon + 1.1 x Price of two-year zero-coupon Plugging in the prices: 995 = 0.1 x 890 + 1.1 x Price of 2-year zero-coupon Solving: Price of two-year zero-coupon = $823.64
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Page 19 of 48 Q37 John is celebrating his 40 th birthday today and wants to start saving for his retirement at age 65. John plans to make equal annual deposits in a savings account starting on his 41 st birthday and with the last deposit being on his 65 th birthday. In addition, he expects an inheritance on his 55 th birthday of $40,000, which he will also deposit in the savings account. John wants to withdraw $25,000 from his savings account on each birthday for 20 years following retirement (the first withdrawal will be on his 66 th birthday). If the savings account offers a 10% interest per year, what is the minimum amount that John must deposit annually to be able to make the desired withdrawals during retirement? Solution Step 1: Money needed in the bank when John retires in 25 years (when he turns 65): 09 . 839 , 212 $ 1 . 1 1 1 1 . 0 000 , 25 20 65 birthday PV Step 2: Money needed in the bank when John retires in 25 years (when he turns 65) after subtracting the future value of the $40,000 inheritance deposited on Jonh’s 55 th birthday 39 . 089 , 109 $ 70 . 749 , 103 09 . 839 , 212 ) 1 . 0 1 ( 000 , 40 09 . 839 , 212 10 65 birthday V Step 3: Present value (John is now 40) of the 39 . 089 , 109 $ : 51 . 068 , 10 $ ) 1 . 0 1 ( 39 . 089 , 109 25 40 birthday PV Step 4: Equal annual deposits in the savings account starting on the 41 st needed: 23 . 109 , 1 $ 1 . 1 1 1 1 . 0 1 51 . 068 , 10 25 C Note: Steps 3 and 4 can be combined into one using the FV of an annuity: 23 . 109 , 1 $ 1 1 . 1 1 . 0 39 . 089 , 109 25 C C
Page 20 of 48 Q38 The mortgage on your house in Montreal is 10 years old. It requires semi-annual payments of $30,000, has an original term of 25 years, and has an interest of 18% (APR with semi-annual compounding). You decided to sell your Montreal house for $375,000, liquidate your loan with part of the $375,000, and use the remaining amount towards a down payment for a beach house in Hawaii. You plan to finance the rest of the beach house with a new mortgage. This new mortgage requires semi- annual payments and has an interest rate of 12% (APR with semi-annual compounding). Suppose that you are willing to continue making semi-annual payments of $30,000 and want to pay the new mortgage in 20 years. How much can you afford to pay for the new beach house in Hawaii? Solution Step 1 : Amount needed to repay the old loan: 62 . 209 , 308 $ 2 18 . 0 1 1 1 ) 2 / 18 . 0 ( 000 , 30 30 PV Step 2: Amount remaining from selling the old house after repaying the old loan: 38 . 790 , 66 $ 62 . 209 , 308 $ 0000 , 375 $ Remaining Step 3 : Amount available to buy the new house (i.e., remaining from selling the house plus new mortgage): 28 . 179 , 518 $ 91 . 388 , 451 38 . 790 , 66 2 12 . 0 1 1 1 ) 2 / 12 . 0 ( 000 , 30 38 . 790 , 66 Available Total 40
Page 21 of 48 Consider the following term structure of risk-free Canadian interest rates for January 2004, 2008, 2009, and 2010. Using the information from the above table answer the following three questions
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Page 22 of 48 Q39 On January 2004, Mary bought a risk-free zero-coupon bond with a face value of $1,000 maturing in 15 years. If Mary sold this bond on January 2010, calculate the annual return on her six-year investment. Solution Step 1 07 . 465 $ ) 052363 . 0 1 ( 000 , 1 04 Jan Price Purchase 15 Step 2 15 . 724 $ ) 036513 . 0 1 ( 000 , 1 0 1 Jan Price Sale 9 Step 3 % 7079 . 55 07 . 465 07 . 465 15 . 724 return year - 6 % 6594 . 7 1 ) 557079 . 0 1 ( return Per year 6 / 1 Q40 On January 2010, a zero-coupon bond with a face value of $1,000 maturing in one year had a YTM of 10%. On January 2010, it was also already known that the bond was certain to default and that would only pay $X at maturity (instead of its $1,000 face value). Calculate $X. Solution Step 1 09 . 909 $ 1 . 0 1 000 , 1 0 1 Jan Price Step 2 80 . 915 $ 007381 . 0 1 09 . 909 0 1 Jan Price X X
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Page 23 of 48 Q41 Calculate the YTM on January 2004 of a risk-free coupon bond that pays one coupon per year, has a coupon rate of 30% and a face value of $1,000, and that matures in three years (the next coupon is exactly one year from now January 2004). IN THIS PROBLEM, YOU DO NOT ACTUALLY NEED TO SOLVE FOR THE YTM, YOU JUST NEED TO SET UP THE EQUATION THAT ONE WOULD NEED TO SOLVE TO FIND THE YTM . Solution Step 1 30 . 748 , 1 $ ) 034756 . 0 1 ( 000 , 1 30 000 , 1 ) 030484 . 0 1 ( 30 000 , 1 025875 . 0 1 30 000 , 1 04 Jan Price 3 2 Step 2 The YTM would solve: ) 1 ( 000 , 1 ) 1 ( 1 1 3 . 0 000 , 1 30 . 748 , 1 $ ) 1 ( 000 , 1 3 . 0 000 , 1 ) 1 ( 3 . 0 000 , 1 1 3 . 0 000 , 1 30 . 748 , 1 $ 3 3 3 2 YTM YTM YTM or YTM YTM YTM Q42 Massey, a tractor manufacturer, has just announced earnings of $1 million for year 0. Massey’s earnings will grow at a rate of 20% per year for the next 3 years. After that, earnings will grow at a rate of 5% per year forever. What is the present value of all future earnings if the interest rate is 10%? Solution M PV 84 . 30 $ 05 . 0 1 . 0 05 . 1 2 . 1 1 . 1 1 1 . 1 2 . 1 1 . 1 2 . 1 1 . 1 2 . 1 3 3 3 3 2 2
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Page 24 of 48 Q43 Olivia took a 30-year loan from her bank. As per the loan covenants, she is expected to amortize a constant fraction of the original amount borrowed in each of the 30 annual payments of the loan. [In other words, in each annual payment, Olivia will amortize 1/30 of the original amount borrowed.] The EAR on the loan is 5%, and Olivia will need to make a payment of $42,625 in year 20 (i.e., the 20 th payment). How much did Olivia originally borrow from the bank? a. $825,000 b. $700,000 c. $950,000 d. $875,000 e. $975,000 Solution Borrowed Amount 30 11 Borrowed Amount 30 19 1 19 Year g Outstandin Principal Borrowed Amount 30 11 05 . 0 20 Year in Paid Interest 625 , 42 30 rowed Amount Bor rowed Amount Bor 30 11 05 . 0 20 Year in Payment 20 Year in Repaid Principal 20 Year in Paid Interest Solving for the amount borrowed: 000 , 825 $ 1 11 05 . 0 30 625 , 42 Borrowed Amount
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Page 25 of 48 Q44 You are 5 years away from retirement and expect to live for 20 years after retirement. Currently, a nursery home costs $30,000 per year, payable at the beginning of the year. You expect annual costs to continually increase at an average of 5% per year. The interest rate is 8% and you plan to move to a nursery home immediately after retirement. How much do you need to have in the bank when you retire (just before paying for the first year at the nursery home) to cover the 20 years that you expect to live in the nursery home? Solution 45 . 288 , 38 05 . 1 000 , 30 Retirement at Nursery of Cost 5 724 , 593 $ 08 . 1 05 . 1 1 05 . 0 08 . 0 05 . 1 45 . 288 , 38 45 . 288 , 38 19 PV Comment The nursery is payable at the beginning of the year, so the first payment is due right away after retirement. Q45 Consider a risk-free bond. The bond pays one coupon of $100 each year, has just made a coupon payment, and is trading for $950. The current one-year risk-free interest rate is 2%. It is also known with certainty that the one-year risk-free interest rate will be 6% the following year, and 3% the year after. (Hence there is no uncertainty about future interest rates.) Calculate the price of the bond three years from now just after the coupon payment for that year. Solution 77 . 745 $ 03 . 1 06 . 1 02 . 1 100 06 . 1 02 . 1 100 02 . 1 100 50 9 3 3 P P
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Page 26 of 48 Q46 Suppose Toy, Inc. has a zero-coupon bond that will pay $1,000 at maturity on March 2, 2012. Today is March 2, 2008, and the bond is selling for $790.09. What is the bond’s YTM (quoted as an APR with annual compounding)? a. 9.00% b. 5.02% c. 6.99%9 d. 7.82% e. 6.06% Solution % 06 . 6 1 09 . 790 000 , 1 1 YTM 1 4 1 1 n n n n P FV YTM FV P YTM is 6.06% compounded annually. Q47 Mary took a 20-year loan of $300,000 from the bank. As per the loan covenants she is expected to amortize a constant fraction of the original amount borrowed in each of the 20 annual payments of the loan. (In other words, in each annual payment, she will amortize 1/20 of the original amount borrowed.) If EAR on the loan is 10%, how much does Mary need to pay in year 5? a. $36,000 b. $37,800 c. $38,500 d. $39,000 e. $39,500 Solution 000 , 240 $ 000 , 00 3 20 4 - 20 4 Year in g Outstandin Principal 000 , 24 $ 000 , 240 % 10 5 Year in Paid Interest 000 , 39 $ 20 300,000 000 , 4 2 5 Year in Payment
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Page 27 of 48 Q48 Assume that the interest rate is 10% per year. You are planning to retire in 40 years and hope to live for 25 years in retirement. You estimate that in retirement you will need to withdraw $40,000 per year (starting one year after retirement) so that you will just exhaust your savings with the 25 th withdrawal. You plan to deposit in the bank a constant amount each year starting in one year and retire immediately after making the 40 th deposit. What amount will you need to deposit in the bank account each year? a. 1,250 b. 1,330 c. $630 d. $945 e. $820 Solution Money needed in the bank when you retire in 40 years : Step 1: 60 . 081 , 363 $ 1 . 1 1 1 1 . 0 000 , 40 25 40 PV Money need to deposit in each of the next 40 years : Step 2: 26 . 022 , 8 $ 1 . 1 60 . 081 , 363 40 0 PV Step 3: 35 . 820 $ 1 . 1 1 1 1 . 0 1 26 . 022 , 8 40 C
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Page 28 of 48 Q49 Michael Jordan has just signed a new contract with Nike Inc. In particular, M. Jordan will be receiving 10 equal payments of $2.5 million each. The first payment will be received in 2 years, and thereafter, the other 9 payments will be received every three years (that is, in years 5, 8, 11, 14 etc.). The current annual interest rate of 10% will remain stable for next 6 years and then it will decrease to 5%. How much is the contract worth today? a. $12.3 b. $6.1M c. $10.1M d. $8.7M e. $9.2M Solution % 76 . 15 1 ) 5 . 0 1 ( Rate Effective Year Three 3 M PV 1 . 10 $ 1576 . 1 1 1 1576 . 0 5 . 2 05 . 1 1 . 1 1 05 . 1 1 . 1 5 . 2 1 . 1 5 . 2 1 . 1 5 . 2 7 2 6 2 6 5 2 Q50 A bond has annual coupon payments of $200 and a face value of $1,000. If the yield to maturity of the bond is 15%, this bond should: a. Sell at a discount b. Sell at premium c. Sell for par value d. Sell for the same price as the similar bond regardless of their respective maturities. e. Sell at a premium or at discount depending on frequency of the coupon payments. Comment The coupon rate 20% (=200/1000) is larger than the YTM .
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Page 29 of 48 Q51 John took a $300,000 loan from RBC. As per the loan covenants, he is expected to make constant annual payments to RBC for the next 30 years. If the EAR is 10%, how much principal is amortized in the 22 nd payment? a. $32,500 b. $13,496 c. $19,425 d. $17,201 e. $7,321 Solution $31,823.77 1 . 0 1 1 1 1 . 0 000 , 300 Payment Annual 30 88 . 273 , 183 $ 1 . 0 1 1 1 1 . 0 77 . 823 , 31 21 Year in g Outstandin Principal 9 39 . 327 , 18 $ 1 . 0 88 . 273 , 183 22 Year in Payment Interest 496 , 13 $ 39 . 327 , 18 77 . 823 , 31 22 Year in Payment Principal Q52 Capital One is advertising a 60-month, 5.99% APR compounded monthly motorcycle loan. If you need to borrow $8,000 to purchase your dream Harley Davidson, what will your monthly payment be? Solution Timeline: 0 1 2 3 4 60 –8,000 C C C C C
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Page 30 of 48 5.99 APR monthly implies a 5.99 0.499167% 12 monthly effective rate. Using the formula for computing a loan payment we calculate the monthly payment: 60 8,000 C $154.63 1 1 1 0.00499167 1.00499167 Q53 You have credit card debt of $25.000 that has an APR (monthly compounding) of 15%. Each month you pay the minimum monthly payment only and you are required to pay only the outstanding interest. You have received an offer in the mail for an otherwise identical credit card with an APR (monthly compounding) of 12%. After considering all your alternatives, you decide to switch cards, roll over the outstanding balance on the old card into the new card and borrow additional money as well. How much can you borrow today on the new card without changing the minimum monthly payment that you will be required to pay? Solution The discount rate on the original card is 15 1.25% 12 Assuming that your current monthly payment is the interest that accrues, it equals: 0.15 $25,000 $312.50 12 Timeline: 0 1 2 312.50 312.50 This is a perpetuity so the amount you can borrow at the new interest rate is this cash flow discounted at the new discount rate. The new discount rate is 12 1% 12 and
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Page 31 of 48 hence, 312.50 PV $31, 250 0.01 In other words, switching credit cards allows you to spend an extra 31, 250 25, 000 $6, 250 Q54 You have just graduated and can't wait to buy a new car. In order to be able to buy the car, you take a loan of $17,000 from the bank. As per the loan covenants, you are expected to make constant monthly payments to the bank for the next 25 years. The APR is 10% APR (monthly compounding). a) Calculate the monthly payment. b) If you want to liquidate the loan after 5 years, what will be the balloon payment? Solution a) Notice that the loan is an annuity with 300 monthly payments (i.e., 25 years), an effective monthly rate of (0.1 / 12), and a present value of $17,000. Solving for the monthly payment: $154.48 12 1 . 0 12 1 . 0 1 1 1 000 , 17 300 C b) The balloon payment after five years is the PV in year five of the remaining payment (which is a 20-year annuity with monthly payments of $154.48): 83 . 007 , 16 $ 12 1 . 0 1 1 1 ) 12 / 1 . 0 ( 48 . 154 240 Balloon
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Page 32 of 48 Q55 Mr. Wise is collecting money for his children's education. Suppose he deposits $10,000 in his bank account every two years, with the first deposit starting in one year. If the annual rate of return is 8% and it is compounded continuously, how much money will he have accumulated after 25 years? Solution 0 1 3 5 25 10K 10K 10K 10K This stream of payments can be considered as annuity starting at year 1. Effective 2-Year Rate = 1735 . 0 1 1 2 08 . 0 2 e e r We first compute the present value of the payments at year “-1”, and then find the future value at the end of year 25. 50435.43 1735 . 1 1 1 1735 . 0 10000 13 1 PV FV 25 = 50435.43 x 1.1735 13 = $403,660.22 Alternatively, we can calculate the future value of 10,000 for 13 periods @17.35% every period: 3 $40,3660.2 1 1735 . 1 1735 . 0 000 , 10 13 25 FV
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Page 33 of 48 Q56 In three years, you will receive the first of nine annual $300 payments. The current interest rate is 14% and will stay constant for the next three years. After year 3, the interest rate will drop to 12%. What is the present value of this cash stream? Solution You can view the cash-flow from year 4 on as an 8-year annuity with a 12% discount rate. Then you can sum to the PV in year 3 of this 8-year annuity to the cash-flow in year 3, and discount everything to the present at 14%: 39 . 208 , 1 $ 300 12 . 1 1 1 12 . 0 300 14 . 1 1 8 3 PV Q57 Suppose that a young couple has just had their first baby and they wish to ensure that enough money will be available to pay for their child's college education. Currently, college tuition, books, fees, and other costs, average $12,500 per year. On average, tuition and other costs have historically increased at a rate of 4% per year. Assuming that college costs continue to increase an average of 4% per year and that all her college savings are invested in an account paying 7% interest, then what is the amount of money she will need to have available at age 18 to pay for all four years of her undergraduate education? [Note: The first tuition payment will be at age 18, when she starts college.] Solution First, determine the cost of the first year of college: 71 . 322 , 25 $ 04 . 1 500 , 12 18 FV Second, find the value for four years of college: 01 . 110 , 97 $ ) 07 . 0 1 ( 07 . 0 1 04 . 0 1 1 04 . 0 07 . 0 71 . 322 , 25 ) 1 ( 1 1 1 4 r r g g r C PV n Note : The first payment is due when she starts college in 18 years, and the annuity formula gives the PV one period before the first payment (i.e., in year 17
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Page 34 of 48 in this case). Hence to calculate the present value in year 18, you will need to multiply by (1+r). Q58 Your son is about to start kindergarten in a private school. Currently, the tuition is $12,000 per year, payable at the start of the school year. You expect annual tuition increases to average 6% per year over the next 13 years. Assuming that your son remains in this private school through high school and that your current interest rate is 6%, then what is the present value of your son's private school education? Solution Notice that this is a growing annuity but with the first payment at time 0. The PV of a growing annuity formula is undefined since r = g. But since r = g, the growth in the payments is exactly offset by the current interest rate, the answer is 12,000 × 13 = $156,000. Note : You could also individually discount each of the 13 payments and arrive at the same answer. Q59 Assume that you are 30 years old today, and that you are planning on retiring at age 65. Your salary from the next years is $45,000 and you expect your salary to increase at a rate of 5% per year as long as you work. To save for your retirement, you plan on making annual contributions to a retirement account. Your first contribution will be made on your 31st birthday and will be 8% of the year's salary, i.e., $45,000. Likewise, you expect to deposit 8% of your salary each year until you reach age 65. At age 66 you will begin withdrawing equal annual payments until your 101 st birthday. If the annual rate of return is 7%, then how much money will you have to spend in each of your golden years of retirement? Solution First Deposit = 0.08 × $45,000 = $3,600 The money available at retirement is the future value of a 35-year (that is, from age 31 to age 65) growing annuity:
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Page 35 of 48 895 , 928 $ ) 07 . 0 1 ( 07 . 0 1 05 . 0 1 1 05 . 0 07 . 0 600 , 3 ) 1 ( 1 1 1 35 35 65 n n Age r r g g r C FV The value spent each year from age 66 to age 101 is calculated as the constant payment of a 36-year annuity with discount rate 7% and present value of $928,895: 260 , 71 07 . 1 1 1 07 . 0 895 , 928 36 C C Q60 If you pay back a three-year loan of $10,000 at 12% per year with equal annual payments, how much interest is paid in the third year? Solution First we need to find the annual payment: 49 . 136 , 4 12 . 1 1 1 12 . 0 000 , 10 3 Payment Payment Year Beginning Balance Total Payment Interest Paid Principal Paid Ending Balance 1 $10,000 4,163.49 1,200 2,963.49 7,036.51 2 7,036.51 4,163.49 844.38 3,319.11 3,717.40 3 3,717.40 4,163.49 446.09 3,717.40 0.00 Interest paid each period= Beginning balance x 12%
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Page 36 of 48 Q61 Suppose you owe a principal amount of $10,000 bearing an interest rate of 10% per annum. Prepare an amortization schedule showing the annual payment, on the basis that you want to amortize 1/10 th of the $10,000 original principal every year. Solution Period Principal outstanding Principal paid Interest paid Total payment 0 10000 1 9000 1000 1000 2000 2 8000 1000 900 1900 3 7000 1000 800 1800 4 6000 1000 700 1700 5 5000 1000 600 1600 6 4000 1000 500 1500 7 3000 1000 400 1400 8 2000 1000 300 1300 9 1000 1000 200 1200 10 0 1000 100 1100 Q62 You have just won the lotto and as a prize you will be receiving 20 equal payments of $2 million each. The first payment will be received in 2 years and thereafter, the other 19 payments will be received every three years (that is, in year 5, 8, 11, etc.). The current annual interest rate of 10% will remain stable for next 3 years and then it will increase to 20%. How much should you willing to pay for the winning ticket? Solution % 8 . 72 1 ) 2 . 0 1 ( Rate Effective Year Three 3 M PV 130 . 4 $ 728 . 1 1 1 728 . 0 2 2 . 1 1 . 1 1 2 . 1 1 . 1 2 1 . 1 2 18 2 3 2 3 2
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Page 37 of 48 Q63 A bond with a coupon rate of 15% and a face value of $1,000 is priced today at $1,500. As per the terms of the prospectus, the bond makes coupon payments on a monthly basis. The bond matures 8 years from today and has just paid a coupon. What is the YTM of the bond? Solution 96 96 96 96 96 ) 1 ( 000 , 1 ) 1 ( 1 1 12 000 , 1 15 . 0 500 , 1 ) 1 ( ) 1 ( 1 1 YTM YTM YTM YTM FV YTM YTM CPN P n n n n n where 96 YTM is the yield to maturity of the bond expressed as a per effective monthly rate for holding the bond from today until maturity in 96 months. Solving with excel, a financial calculator or by trial and error we get: 96 YTM =0.572% and hence the YTM (expressed as an APR with monthly compounding) is: YTM = 0. 572 x 12 = 6.8% Q64 A risk-free zero-coupon bond with a face value of $1,000 maturing in 1 year trades today at a price of $909.09. A risk-free zero-coupon bond with a face value of $1,000 maturing in 2 years trades today at price of $790.51. Calculate the current price of a risk-free coupon bond with a coupon rate of 10% that pays one coupon per year, has a face value of $10,000, and matures in 2 years. Solution The coupon bond pays $1,000 in one year (i.e., 10,000 x 0.1) and $11,000 at maturity in two years (i.e., coupon + FV =10,000 x 0.1 + 10,000). Therefore, the coupon bond is equivalent to one 1-year zero-coupon bond plus eleven 2-year zero-coupon bonds. Hence the law of one price (i.e., securities or portfolios with the same cash- flows must have the same price) implies that: Price of coupon bond = 1 x 909.09 + 11 x 790.51 = $9,604.7
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Page 38 of 48 Q65 Consider a bond, with a $1,000 face value, a coupon rate of 5%, and that pays one coupon per year. The bond has just paid a coupon, matures in 2 years, and currently is trading at $900. The bond is certain to pay the next coupon in one year but it may default in two years. In particular, in two years, with probability 0.12, the bond will default and pay nothing and, with probability 0.88, the bond will not default and pay the promised coupon plus the face value. Calculate the bonds YTM (quoted as an APR with annual compounding). a. 1.325% b. 6.321% c. 5.000% d. 12.463% e. 10.826% Solution The YTM is given by the following equation: 2 ) 1 ( 1000 1000 05 . 0 1 1000 05 . 0 900 YTM YTM P Plugging in 10.826% the number: 900 ) 10826 . 0 1 ( 1000 1000 05 . 0 10826 . 0 1 1000 05 . 0 2 P Note : The YTM are calculated using the promised cash- flows rather than the expected cash-flows.
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Page 39 of 48 Q66 Consider a bond with a face value of $1000, that matures in three years, pays one coupon of $50 each year, and that has just made a coupon payment. The current one-year risk-free interest rate is 4%. It is also known with certainty that the one- year risk-free interest rate will be 6% the following year and thereafter. Find the (a) the bond the current price of the bond, the price of the bond in one year, and the price of the bond in two years. (b) Find the YTM of the bond in year 0, in year 1, and in year 2. (c) Find the return from holding the bond during the first year (from t=0 to t=1), from holding the bond during the second year (from t=1 to t=2), and from holding the bond during the third year (from t=2 to t=3). Solution a) 9866 . 991 $ ) 06 . 0 1 )( 04 . 0 1 ( 000 , 1 50 ) 06 . 0 1 )( 04 . 0 1 ( 50 04 . 0 1 50 2 0 P b) % 2959 . 5 ) 1 ( 000 , 1 50 ) 1 ( 50 1 50 9866 . 991 0 3 0 2 0 0 0 t t t t YTM YTM YTM YTM P % 6 ) 1 ( 000 , 1 50 ) 1 ( 50 6661 . 981 1 2 1 1 1 t t t YTM YTM YTM P % 6 ) 1 ( 000 , 1 50 5660 . 990 2 2 2 t t YTM YTM P c) % 4 9866 . 991 9866 . 991 6661 . 981 50 50 Return 0 0 0 1 1 t to 0 t P P P % 6 6661 . 981 6661 . 981 5660 . 990 50 50 Return 1 1 2 2 t to 1 t P P P % 6 5660 . 990 5660 . 990 000 , 1 50 000 , 1 50 Return 2 2 3 t to 2 t P P
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Page 40 of 48 Q67 An investor has two bonds in his portfolio. Both bonds mature in 4 years, have a face value of $1,000, and their yield to maturity equals 9.6 percent per year. One bond, Bond C, pays one coupon per year and has a coupon rate of 10 percent, the other bond, Bond Z, is a zero coupon bond. Assuming that the yield to maturity of each bond remains at 9.6 percent per year over the next 4 years, what will be the price of each of the bonds at the following time periods? Fill in the following table: Time Price of Bond C Price of Bond Z 0 1,012.79 693.04 2 1,006.98 832.49 4 1,000 1,000 Solution The Value of Bond Z increases as it nears maturity, while the value of Bond C, which is currently selling at a premium, decreases as it approaches its maturity. The values of the two bonds converge on maturity date and are equal to $1000. Q68 Evaluate the following statements: I. Firm’s debt-holders are the residual claimants to the firm’s cash flows. (FALSE) II. All else equal, a firm’s P/E ratio is decreasing in the dividend payout ratio. (FALSE) III. If a project has an IRR of 15 % and a required cost of capital of 8% then it must have a positive NPV. (FALSE) IV. Shareholders are corporation’s residual claimants. (TRUE) Q69 Consider the following statements: I. Debt-holders are corporation’s residual claimants. (FALSE) II. Debt-holders are entitled to vote for the board of directors. (FALSE) III. Shareholders are corporation’s residual claimants. (TRUE) IV. Zero coupon bonds sell at a premium. (FALSE)
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Page 41 of 48 Q70 The rate at which the stock price is expected to appreciate (or depreciate) is the: a. The risk-free rate. b. Total yield. c. Dividend yield. d. Capital gains yield. e. The coupon rate Q71 Dot.com Technologies’ dividend has been growing at a rate of 10% per year in recent years. This growth rate is expected to last for another nine years. After these nine years dividend is expected to grow at a 7% annually. If the equity cost of capital is 12%. Calculate the capital gains yield between years 14 and 15. a. 10% b. 7% c. 5% d. 12% e. 2% Solution g r Div P 15 14 and 14 15 16 15 ) 1 ( ) 1 ( P g g r Div g g r Div P . Therefore: % 7 GainsYield Capital 14 14 15 g P P P Comment By year 14 the future dividends behave like in the constant dividend growth model (i.e., in year 15 and thereafter, future dividends behave like a growing perpetuity). In the constant dividend growth model the capital gains yield is equal to the dividend growth rate.
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Page 42 of 48 Q72 The following are the FCF for Dot.com in year 1. These cash flows depend on the state of the economy (i.e. poor, normal, good, and super good). The probability of these economic states is given below. State Probability FCF ($) Bad 1/4 1,000 Normal 1/2 3,000 Good 1/8 6,000 Super Good 1/8 7,000 Dot.com. closes down at the end of period 1 and has no assets other than the FCF generated in the period. The company has the following stakeholders: senior debt- holders with a claim of $4,000, junior debt-holders with a claim of $2,000 and shareholders. Assuming a 0% discount rate for all financial assets, calculate the current value of Dot.com’s junior debt. a. $ 500 b. $ 2,000 c. $ 1,000 d. $ 125 e. $ 1,750 Solution 500 $ 000 , 2 8 1 000 , 2 8 1 0 2 1 0 4 1 ) Value( Debt Junior Comment The value of the different securities is as follows: State Probability FCF ($) Senior Debt Junior Debt Equity Bad 1/4 1,000 1,000 0 0 Normal 1/2 3,000 3,000 0 0 Good 1/8 6,000 4,000 2,000 0 Super Good 1/8 7,000 4,000 2,000 1,000 3,375 2,750 500 125
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Page 43 of 48 Q73 Dot.Com has just paid a $5 dividend and the required return on its equity is 10%. The dividend is expected to grow at 25% per year during the first year, at 9% per year during the following eight years, and at 2% per year thereafter. Calculate the current share price. a. $ 86 b. $ 106 c. $ 117 d. $ 123 e. $ 100 Solution 117 $ 02 . 0 1 . 0 02 . 1 09 . 1 25 . 1 5 1 . 1 1 1 . 1 09 . 1 1 09 . 0 1 . 0 25 . 1 5 8 9 9 P Q74 Consider a firm with a cost of equity of 8% and 5 million shares outstanding. In year one the firm will not pay any dividends but it will repurchase $10 million worth of shares. After year one, the firm will pay a total dividend of $25 million each year forever. Calculate the price per share in year zero. a. $ 60 b. $ 20 c. $ 80 d. $ 40 e. $ 90 Solution M PV 611 . 298 $ 08 . 1 0.08 25 10 s) Repurchase & Dividends Future ( Equity 0 share per 60 $ 5 611 . 298 Shares of Number Equity P 0 0 0
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Page 44 of 48 Q75 Consider a firm with a cost of equity of 10%. The firm will only start paying dividends in three years but it plans to repurchase shares during the next two years. If the current price per share is $77, calculate the price per share in year one (after the shares have been repurchased). a. $ 84.7 b. $ 68.2 c. $ 70.3 d. $ 77.0 e. $ 770.0 Solution % 10 % 0 % 10 Yield Dividend r Yield Gains Capital 0 e 0 7 . 84 $ P 77 77 P P P P Yield Gains Capital % 10 1 1 0 0 1 0 Solving Q76 Evaluate the following statements: I. The declaration of dividend is at the discretion of the board of directors. (TRUE) II. The payment of dividends by the corporation is a tax-deductible business expense. (FALSE) III. A dividend on common stock, whether declared or no by the board of directors, is not a legal liability of the firm. (FALSE) Q77 Dot.com Technologies’ dividend has been growing at a rate of 20% per year in recent years. This growth rate is expected to last for another 2 years. After these two years dividend is expected to grow at a 6% per annum. If the dividend per share in year 1 is expected to be $1.60 and the equity cost of capital is 10%. Calculate the price per share in year 0 and 1, and the dividend and the capital gains yield between year 0 and 1.
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Page 45 of 48 Solution 0 1 2 3 4 …… 1.6 1.6 x 1.2 1.6 x 1.2 x 1.06 1.6 x 1.2 x 1.06 2 ..… share per 09 . 45 $ 06 . 0 1 . 0 2 . 1 6 . 1 1 . 1 1 1 . 1 6 . 1 P 0 share per 48 $ 06 . 0 1 . 0 2 . 1 6 . 1 P 1 % 55 . 3 09 . 45 6 . 1 P Div Yield Dividend 0 1 0 % 45 . 6 09 . 45 09 . 45 48 P P P Yield Gains Capital 0 0 1 0 Notice that the dividend plus the capital gains yield equal to 10%, which is the required return on the equity. Q78 Consider two “identical” firms X and Y with a cost of equity of 10% and 2 million shares outstanding. These companies only differ in how they will distribute cash next year: Firm X will pay $10 million dividend in year 1 Firm Y will repurchase $10 million in outstanding shares in year 1 After year 1, the two firms will be identical thereafter, that is, they will pay an annual total dividend of $5 million. Calculate the price per share at the end of year one, P 1 (after the dividend has been paid or the shares have been repurchased), the price per share in year 0, P 0, and the dividend and capital gains yield between year 0 and 1 for Firm X and Firm Y. How many shares does Firm Y repurchase?
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Page 46 of 48 Solution Firm X: *Value of equity at t=1 (after the dividend at t=1): M 50 $ 0.1 5 r Dividend Total Equity 1 * Share Price at t=1 (after the dividend at t=1): share per 25 $ 2 50 Shares of Number Equity P 1 1 1 M M *Share Price at t=0: share per 27 . 27 $ 1 . 0 1 25 2 10 r) (1 P Div P 1 1 0 M M * % 33 . 18 27 . 27 2 10 P Div Yield Dividend 0 1 0 M M * % 33 . 8 27 . 27 27 . 27 25 P P P Yield Gains Capital 0 0 1 0 Firm Y: *Value of equity at t=1 (after the repurchase at t=1): M 50 $ 0.1 5 r Dividend Total Equity 1 *Value of equity at t=0: M PV 54 . 54 $ 0.1 5 1 . 1 1 1 . 1 10 s) Repurchase & Dividends Future ( Equity 0
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Page 47 of 48 * Share price at t=0: share per 27 . 27 $ 2 54 . 54 Shares of Number Equity P 0 0 0 M M Notice that the price per share P 0 in year 0 is the same for Firm X and Firm Y ( i.e. , $27.27). Hence the current share price is not affected by whether the $10M in year 1 are returned to shareholders in the form of a dividend or a share repurchase. * % 0 27 . 27 0 P Div Yield Dividend 0 1 0 * % 10 % 0 % 10 Yield Dividend r Yield Gains Capital 0 0 Notice that for Firm X’s and Firm Y’s shareholders both get a 10% return between 0 and 1 but for the shareholders of Firm Y this return comes entirely in the form of a capital gain. * Share price at t=1: 30 $ P 27 . 27 27 . 27 P P P P Yield Gains Capital % 10 1 1 0 0 1 0 Solving Notice that the price per share (after the dividend has been paid or the shares has been repurchased) P 1 in year 1 is higher for Firm Y ( i.e. , $30) than for Firm X i.e. , $25) which reflects the higher dividend per share in year 2 onwards (i.e., Firm Y will have fewer shares after the share repurchase). Notice also that the price difference $5 is just the dividend per share that shareholders of Firm X have already obtained in year 1, a $5 value that the shareholders of Firm Y will obtain in the form of higher future dividends per share. * Number of Shares at t=1 (after the repurchase at t=1): shares 667 , 666 , 1 30 50 P Equity Shares of Number 1 1 1 M * Number of Shares repurchased at t=1:
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Page 48 of 48 shares 333,333 667 , 666 , 1 000 , 000 , 2 d Repurchase Shares 1 Notice that you could also have calculated the amount of shares repurchased by dividing $10M by the price per share at t=1 : shares 333,333 30 10 P 10 d Repurchase Shares 1 1 M M
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