22. Construct a bond amortization table for a $1000 two-year bond with 7% coupons paid semi-annually bought to yield 8% semi-annually.

Transcription

1 Chapter 6 Exercises

2 22. Construct a bond amortization table for a $1000 two-year bond with 7% coupons paid semi-annually bought to yield 8% semi-annually.

3 23. Construct a bond amortization table for a $1000 two-year bond with 7% coupons paid semi-annually bought to yield 6% semi-annually.

4 27. Read Text Exercise #6-11. (a) Use the premium/discount formula to write an expression for the price of the each bond in terms of the yield rate and a n i. (b) Express the price of the second bond as a function of p, as requested in the exercise.

5 30. Read Text Exercise #6-14. (a) Indicate whether the bond is bought at a premium or at a discount. (b) Find the total interest requested in the exercise, and note that Table 6.1 can be helpful. 1000(ng p) = 1000[ng ( ) ] a

6 31. Read Text Exercise #6-16. (a) Do part (a). (b) Do part (b).

7 (c) Do part (c) by completing the following: For discount bonds the straight line values are greater than the true book values. For premium bonds the straight line values are less than the true book values,

8 32. Read Text Exercise #6-17, and recall the result in Chapter 1 Exercise #68(b), which will be helpful in doing this exercise. (a) Do part (a) by arranging the three formulas in the Flat price column of Table 6.4 (page 216) in increasing order of magnitude. From Chapter 1 Exercise #68(b), 1 + ik > (1 + i) k for 0 < k < 1. It follows that B t (1 + i) k = B t (1 + i) k < B t (1 + ik) (Theoretical) (Semi-theoretical) (Practical) (b) Arrange the three formulas in the Accrued coupon column of Table 6.4 (page 216) in increasing order of magnitude, by first using algebra to prove a useful identity. (1 + i) For 0 < k < 1, (1 + i) k < 1 + ik k 1 < k. i

9 It follows that (1 + i) Fr 1 i < kfr = kfr (Theoretical) (Semi-theoretical) (Practical) (c) Do part (b) in the textbook exercise by using the previous results to arrange the three formulas in the Market price column of Table 6.4 (page 216) in increasing order of magnitude. (Semi-theoretical) < (Practical) (Semi-theoretical) < (Theoretical) The direction of the difference between (Practical) and (Theoretical) is indeterminate.

10 33. Read Text Exercise #6-18. (a) Use the theoretical method to find the flat price, accrued interest, and market price (book value). (b) Use the practical method to find the flat price, accrued interest, and market price (book value).

11 (c) Use the semi-theoretical method to find the flat price, accrued interest, and market price (book value).

12 34. Read Text Exercise #6-19. (a) Find the most recent coupon payment date prior to the purchase date of the bond; then use Appendix A to find the number of the day in a year for the coupon payment date, the bond purchase date, and the bond maturity date. (b) Find the price of the bond on the most recent coupon payment date prior to the purchase date of the bond. C + (Fr Ci) a n (c) Calculate the price of the bond requested in the exercise using the simple interest assumption as stated.

13 44. Do Text Exercise #6-29. Hint: Use Makeham s formula.

14 45. Read Text Exercise #6-30. Hint: Use Makeham s formula.

15 46. Read Text Exercise #6-35. (a) Find the price of the stock. (b) Find the level annual dividend requested in the exercise.

16 47. Do Text Exercise #6-36. Hint: Use formula (6.28) with appropriate modification. 48. Read Text Exercise #6-37, and let E be the current earnings. (a) Write an expression for the current purchase price of the stock and an expression for the price the stock is sold at in 6 years. (b) Write and solve an equation for finding the yield rate requested in the exercise.

17 Chapter 7 Exercises

18 26. Read Text Exercise #7-10. (a) Adapt formula (7.6) to do part (a). (b) Do part (b).

19 28. Read Text Exercise #7-12. (a) Find the total accumulated value the lender receives from the $1000 payments over the 20-year period. (b) Find the effective annual rate of interest the lender earned on the $10000 loan, as requested in the exercise. (c) Write an equation of value to find the rate of interest the borrower paid on the $10,000 loan, and then use the Solver in Excel to find this rate = a 20 j j =

20 29. Read Text Exercise #7-13. (a) Find the total accumulated value in five years. (b) Find the purchase price requested in the exercise. 30. Do Text Exercise #7-14.

21 31. Do Text Exercise #7-15. (a) Use formula (7.10) to write an equation of value which can be used to solve for i. (b) Use Excel or a TI calculator to find i.

22 32. Read Text Exercise #7-16. (a) Write an equation of value which can be used to solve for i. (b) Use Excel or a TI calculator to find i.

23 33. Read Text Exercise #7-17. (a) Determine the amount of money Lender #1 can reinvest at 6% for the three years following the complete payment of the loan. (b) Determine Lender #1 s overall yield rate as requested in the exercise.

24 34. Read Text Exercise #7-18. (a) Find the accumulated value of the $50,000 payments at the end of the 4-year period. (b) Calculate P(0.1) as requested in the exercise.

25 41. Read Text Exercise #7-26. (a) Write an appropriate equation of value to find the dollar-weighted rate of return using the simple interest approximation indicated in the exercise; then find this dollar-weighted rate. (b) Let X be the amount in the account at time t = 1/2. Write an appropriate equation to solve for X, and find X as requested in the exercise.

26 42. Read Text Exercise #7-27. (a) Do part (a) by writing an appropriate equation of value to find the dollar-weighted rate of return; then use the quadratic formula to find this dollar-weighted rate. (b) Do part (b).

27 43. Read Text Exercise #7-28. (a) Find the 6-month time-weighted yield and the equivalent annual yield. (b) Write an expression for the 1-year time-weighted yield and use this to find X as requested in the exercise.

28 44. Read Text Exercise #7-29. (a) Using the time-weighted method, find X. (b) Using the dollar-weighted method with the simple interest approximation, find Y, as requested in the exercise.

29 45. Read Text Exercise #7-30. (a) Do part (a) by applying each of the dollar-weighted method and the time-weighted method, and showing the result is the same. (b) Do part (b), with the simple interest approximation for the dollar-weighted method.

30 (c) Do part (c), with the simple interest approximation for the dollar-weighted method.

31 45. - continued (d) Do part (d) by completing the following: Dollar-weighted calculations do not involve fund balances after the beginning of, and before the end of, the investment period; the calculations involve only deposits and withdrawals and the dates these occur. (e) Do part (e) by contradiction; that is, assume the opposite of what you want to prove, and show that this leads to a something that cannot possibly be true.

32

33 46. Read Text Exercise #7-31. (a) Find the balances at the beginning of the 3rd investment year and the end of the 6th investment year. (b) Find the dollar amount of interest requested in the exercise.

34 47. Do Text Exercise #7-32.

35 48. Do Text Exercise #7-33 by first calculating each of P, Q, and R.

36 49. Read Text Exercise #7-34. (a) Let i =.01j, and write an expression involving i for the amount of interest credited by the fund during the year z + 3, from the deposit made in year z, from the deposit made in year z + 1, and from the deposit made in year z + 2. (b) Find j, as requested in the exercise.

37 50. Read Text Exercise #7-35. (a) Find the accumulated value of the investment at the end of the three-year period. (b) Find the equivalent level effective rate of interest requested in the exercise.

38 51. An investment plan consists of depositing $1000 at the beginning of each year for three years, starting at the beginning of year z + 2 with interest rates given by textbook Table 7.2. (a) Find the value of the investment at the end of the three-year period. 1000(1.085)(1.087)(1.0875) = (1.09)(1.09) = (1.09) = = $ (b) Suppose the investor decides to withdraw $500 in the middle of year z + 3. Explain the difficulty with calculating the value of the investment at the end of the three-year period. It is not clear whether the $500 withdrawn should come from the $1000 earning 9% interest or the $1000 earning 8.7% interest.

39 Chapter 9 Exercises

40 4. The nominal rate of interest for level deposits at the beginning of each year for 10 years is 8%, and the rate of inflation is 5%. Let A = value of the investment at the end of 10 years with no inflation B = value of the investment at the end of 10 years in terms of dollars at time 0 C = value of the investment at the end of 10 years computed at the real rate of interest Find the ratios A/B, A/C, and B/C..... s A = X s , B = X (1.05) 10, C = X s /1.05 A/B = A/C = B/C = (1.05) 10 = s s / = / = 1.336

41 5. Do Text Exercise Do Text Exercise 9-2.

42 12. Read Text Exercise 9-7. (a) Find amount of each of the two coupons and the maturity value as requested in part (a), and remember that a TIPS bond adjusts for the inflation rate. (b) As requested in part (b), find the nominal yield rate, which can be done by solving an equation of value in terms of actual dollars, and find the real yield rate, which can be done by solving the previous equation of value after replacing the actual dollar values with real dollar values.

43 13. An amount of money is to be deposited today into an account earning 5% effective so that a withdrawal can be made at the end of each year for 30 years beginning one year from today, and the account is completely exhausted after the last withdrawal. (a) Find the exact amount of money needed so that each withdrawal is $ a = $30,744.90

44 (b) Suppose the rate of inflation is expected to be 2% for the next 30 years. Find the amounts which must be withdrawn in order that each withdrawal have a value of $2000 in real dollars (i.e., each withdrawal has been adjusted for inflation. 2000(1.02) t should be withdrawn at the end of year t (c) Again supposing that the rate of inflation is expected to be 2% for the next 30 years, find the exact amount of money needed so that the withdrawals in part (b) can be made real interest rate = = a = $39,500.64

45 (d) Suppose that the rate of inflation is expected to be (4t/30)% in year t for the next 30 years. Design an Excel worksheet to find (i) the amounts which must be withdrawn in order that each withdrawal have a value of $2000 in real dollars (i.e., each withdrawal has been adjusted for inflation), and (ii) the exact amount of money needed so that these withdrawals in part can be made. Below is displayed part of an Excel worksheet (with some answers) which you can use as a guideline in designing your own worksheet. Submit a printed copy of your worksheet with this exercise.

46 ANSWER TO (i): ANSWER TO (ii):

47 14. Bond A and Bond B both have $5000 par value with 8% semiannual coupons maturing at par in 2 years. Bond A is a regular bond and Bond B is a TIPS bond. (a) Find the price of Bond A if the yield rate is to be 7%. 200 a = $ (b) Find the price of Bond B if the yield rate is to be the same as for Bond A and the inflation rate is 1.5% for each six month period = $

48 15. Read Text Exercise 9-8. (a) Using the information given about Bond A, find the yield rate corresponding to the given price; you can use a financial calculator or Excel to do this. (b) Find the price of Bond B if the yield rate is to be the same as for Bond A, as requested in the exercise.

49 16. If $15,000 is invested in a project immediately, it is expected that there will be a return of $8000 in one year and a return of $12,000 in two years. The nominal yield rate does not take inflation into account, and the real yield rate does. (a) Find the nominal yield rate. The equation of value is (1 + i) (1 + i) 2 = 0 15(1 + i) 2 8(1 + i) 12 = 0 (1 + i) = 1.2 i = 0.2 or 20% (b) Find the real yield rate if inflation is expected to be 3% for the next two years. The equation of value is (1.03) 1 (1 + i / ) (1.03) 2 (1 + i / ) 2 = 0 15(1 + i / ) 2 8(1.03) 1 (1 + i / ) 12(1.03) 2 = 0 (1 + i / ) = i / = or 16.5%

50 (c) Find the real yield rate if inflation is expected to be 3% for the next year and 4% for the year after that. The equation of value is (1.03) 1 (1 + i / ) (1.03) 1 (1.04) 1 (1 + i / ) 2 = 0 15(1 + i / ) 2 8(1.03) 1 (1 + i / ) 12(1.03) 1 (1.04) 1 = 0 (1 + i / ) = i / = or 16.1%

51 Chapter 10 Exercises

52 9. Do Text Exercise (a) Do part (a) by comparing the coupon rate with at-par yield rate calculated in the previous exercise. Since 6% < 8.88%, it is a discount bond. (b) Do part (b).

53 10. Do Text Exercise by writing the price for a $1 bond two different ways: one way using the effective yield rate and another way using the spot rates.

54 11. Do Text Exercise