HullFund8eCh04ProblemSolutions

HullFund8eCh04ProblemSolutions - CHAPTER 4 Interest Rates...

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CHAPTER 4 Interest Rates Practice Questions Problem 4.8. The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0. A 1.5-year bond that will pay coupons of $4 every six months currently sells for $94.84. A two-year bond that will pay coupons of $5 every six months currently sells for $97.12. Calculate the six-month, one-year, 1.5-year, and two-year zero rates. The 6-month Treasury bill provides a return of 6 94 6 383 % in six months. This is 2 6 383 12 766 %   per annum with semiannual compounding or 2ln(1 06383) 12 38 % per annum with continuous compounding. The 12-month rate is 11 89 12 360 % with annual compounding or ln(1 1236) 11 65 % with continuous compounding. For the 1 1 2 year bond we must have 0 1238 0 5 0 1165 1 1 5 4 4 104 94 84 R e e e         where R is the 1 1 2 year zero rate. It follows that 1 5 1 5 3 76 3 56 104 94 84 0 8415 0 115 R R e e R     or 11.5%. For the 2-year bond we must have 0 1238 0 5 0 1165 1 0 115 1 5 2 5 5 5 105 97 12 R e e e e           where R is the 2-year zero rate. It follows that 2 0 7977 0 113 R e R or 11.3%. Problem 4.9. What rate of interest with continuous compounding is equivalent to 15% per annum with monthly compounding? The rate of interest is R where: 12 0 15 1 12 R e i.e., 0 15 12ln 1 12 R 0 1491 The rate of interest is therefore 14.91% per annum. Problem 4.10.
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A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit? The equivalent rate of interest with quarterly compounding is R where 4 0 12 1 4 R e or 0 03 4( 1) 0 1218 R e The amount of interest paid each quarter is therefore: 0 1218 10 000 304 55 4 or $304.55. Problem 4.11. Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 4%, 4.2%, 4.4%, 4.6%, and 4.8% per annum with continuous compounding respectively. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months and pays a coupon of 4% per annum semiannually. The bond pays $2 in 6, 12, 18, and 24 months, and $102 in 30 months. The cash price is 0 04 0 5 0 042 1 0 0 044 1 5 0 046 2 0 048 2 5 2 2 2 2 102 98 04 e e e e e                   Problem 4.12. A three-year bond provides a coupon of 8% semiannually and has a cash price of 104. What is the bond’s yield? The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in 36 months. The bond yield is the value of y that solves 0 5 1 0 1 5 2 0 2 5 3 0 4 4 4 4 4 104 104 y y y y y y e e e e e e             Using the Goal Seek or Solver tool in Excel 0 06407 y or 6.407%.
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