Fixed Income: Practice Problems with Solutions

Fixed Income: Practice Problems with Solutions Directions: Unless otherwise stated, assume semi-annual payment on bonds.. A 6.0 percent bond matures in exactly 8 years and has a par value of 000 dollars. The bond sells for 338. dollars. What is the semiannual coupon payment? a. 0. 0 b. 80. 88 c. 60.0 d. 40. 44 e. 66. 906 a. semiannual coupon: C 0.06 000 60.0. A 9.0 percent bond matures in exactly 5 years and has a par value of 7000 dollars. The bond sells for 673. dollars. What is the current yield? a. 9.0 b. 9. 5 c. 9. 3706 d. 4. 75 e. 4. 5 a. current yield: CY 00 0.09 7000 673. 9. 3706 3. A 9. 5 percent bond matures in exactly 6 years and has a par value of 0000 dollars. The bond sells for 40.0 dollars. What is the yield to maturity? a. 3. 5 b. 4.0 c. 5. 5 d. 4. 5 e. 5.0 a. calculator inputs: N 3;PV 40.0;PMT 0.095 0000 475.0;FV 0000 b. cpt I/Y. 75 c. so y 5. 5

4. A 4. 5 percent bond with a par value of 000 dollars matures in 7 years, 70 days. The next coupon is paid in 70 days. What is the accrued interest? Assume 83 days between coupon payment dates. a. 0. 90 b. 3. 967 c.. 5984 d.. 049 e. 4. 098 a. days since last coupon 83 70 3 b. half-year since last coupon 83 70 3 83 83 83 70 c. so accrued interest 0.05 000. 5984 83 5. A 7. 5 percent bond with a par value of 000 dollars matures in 5 years, 0 days. The next coupon is paid in 0 days. The yield is 4. 5 percent. What is the dirty price? Assume 83 days between coupon payment dates. a. 36. b. 365. 5 c. 358. 9 d. 360. 5 e. 36. 6 a. solve for value of bond V w at first coupon payment date:w 0 days 0 0 83 83 half years b. calculator inputs: N 30;I/Y 4. 5. 5;PMT 0.075 000 37. 5;FV 000 c. cpt PV 34. 7 d. so V 34. 7 37. 5. 0 5 0 83 358. 9 6. A 9.0 percent bond with a par value of 000 dollars matures in 9 years, 9 days. The next coupon is paid in 9 days. The bond sells for 378. dollars. What is the clean price? Assume 83 days between coupon payment dates. a. 97. 5 b. 335. 5 c. 337. 8 d. 35. 3 e. 34. 4 a. days since last coupon 83 9 64

b. half-year since last coupon 83 9 64 83 83 83 9 c. so accrued interest 0.045 000 40. 38 83 d. then clean price is given by: CP 378. 40. 38 337. 8 7. A zero-coupon bond with a par value of 000 dollars matures in 0 years, 59 days. If the bond yields 4. 5 percent, what is the clean price? Assume 83 days in a half-year. a. 640. 8 b. 639. 37 c. 636. 4 d. 643. 93 e. 645. 43 a. time to maturity 0 59 379 83 83 000 b. clean (or dirty) price 636. 4. 05 379 83 8. A zero-coupon bond with a par value of 000 dollars matures in 0 years, 56 days. If the bond sells for 464. 0, what is yield to maturity? Assume 83 days in a half-year. a. 7.0 b. 8.0 c. 7. 5 d. 8. 5 e. 9.0 a. time to maturity:t 0 56 7 83 6 b. yieldtomaturity 000 464. 0 7 6 00 7. 5 9. The yields on a six month and one year zero-coupon bonds are 4.0 and 9.0 percent, respectively. A dealer holds in inventory a 5.0 percent treasury note with a par value of 000 dollars and maturity of one year. What is the minimum price the dealer should ask for the bond? a. 00. 5 b. 940. 4 c. 963. 3 d. 93. 84 e. 889. 76 a. minimum price is the cost of constructing cash flow pattern using zero-coupon bonds: 5.0d 05.0d

b. so: P 5.0. 0 05.0. 045 963. 3 0. A 7. 5 percent par treasury bond matures in exactly 3 years. A 5.0 percent par municipal bond matures in exactly 3 years. Suppose both bonds have the same default risk. At what marginal tax rate would the two bonds have the same after-tax yield? a. 34. 633 b. 36. 33 c. 33. 333 d. 37. 833 e. 39. 733 a. since both bonds are selling at par: y treasury 7. 5 and y muni 5.0 b. therefore: 7. 5 5.0 c. which implies: 5.0 7. 5 0.66667 d. so: 0.66667 33. 333. A 0.0 percent par municipal bond matures in exactly 3 years. For an investor at the 9.0 percent marginal tax rate, what is the taxable-equivalent yield? a. 7. b. 7. 043 c. 4. 085 d. 5. 85 e. 8. 885 a. since the muni is selling at par: y muni 0.0 b. therefore: TEY y muni 0.0 4. 085 0.9. A 3.0 percent TIP bond matures in exactly 4 years. Six months ago the par value was 0800 dollars. The annualized CPI (inflation rate) over the last six months equals 9. 5. Assuming a coupon is paid today, what is par value of the bond? a. 86. b. 096. c. 33. d. 4. e. 0800 a. inflation adjusted principal: M 0800 0.095 33. 3. A 4.0 percent TIP bond matures in exactly years. Six months ago the par value was 0500 dollars. The annualized CPI (inflation rate) over the last six months equals. 5.

Assume the coupon is paid today. What is the dollar value of the coupon paid today? a. 45. 5 b. 5. 5 c.. 63 d. 430. 5 e. 0.0 a. inflation adjusted principal: M 0500 0.05 063. b. apply coupon rate to inflation adjusted principal: c 063. 0.04. 63 4. A 3. 5 percent TIP bond matures in exactly 4 years. Six months ago the par value was 0600 dollars. The annualized CPI (inflation rate) compounded semiannually over the last six months equals 4. 5 percent. Assume the coupon is paid today. Six months ago the bond was selling at par and today the bond is selling at 5.0 percent premium over par. What is the annual rate of return compounded semiannually over the last six months? a. 7. 048 b. 7. 98 c. 8. 048 d. 9. 98 e. 9. 788 a. inflation adjusted principal: M 0600 0.045 0839. b. apply coupon rate to inflation adjusted principal: c 0839. 0.035 76. 3 c. one plus the return over a half year equals the ratio of the begining to end of half-year value d. value at the end of the first half year equals the semi-annual coupon: 76. 3 plus the price: P. 05 0839. 380.0 e. so: R 76. 3 380.0. 090 0 600 f. finally: R. 090 00 8. 048 percent 5. A 8.0 percent bond matures in exactly 0 years and has a par value of 0000 dollars. The bond sells for 090.0 dollars. For a 50 basis increase in the yield, determine the percentage change in the bond s price? a. 3. 948 b.. 948 c. 3. 448 d.. 048 e. 0.4755 a. first step, find yield to maturity y

b. calculator inputs: N 0;PV 090.0;PMT 0.08 0000 400.0;FV 0000 c. cpt I/Y 3. 5 d. so y 6. 5 e. second step, increase yield by 50 bps f. new yield 6. 5.5 7.0 g. so I/Y 3. 5 h. third step, compute price at new yield i. calculator inputs: N 0;I/Y 3. 5;PMT 0.08 0000 400.0;FV 0000 j. cpt PV 090.0 k. Δ%P 07. 090.0 00 3. 448 090.0 6. A 8. 5 percent bond matures in exactly 3 years and has a par value of 7000 dollars. The bond sells for 93. 4 dollars. What is the approximate (effective) duration for a 0 basis point shock (either up or down)? a. 5. 5888 b. 6. 5888 c. 8. 5888 d. 7. 5888 e. 6. 5888 a. first step, find yield to maturity y b. calculator inputs: N 6;PV 93. 4;PMT c. cpt I/Y. 5 d. so y 5.0 e. second step, increase yield by 0 bps f. new yield 5.0. 5. g. so I/Y. 6 h. third step, compute price at new yield y i. calculator inputs: N 6;I/Y. 6;PMT j. so: P 963. k. fourth step, decrease yield by 0 bps l. new yield 5.0. 4. 8 m. so I/Y. 4 n. fourth step, compute price at new yield y o. calculator inputs: N 6;I/Y. 4;PMT p. so P 9483. 4 q. fifth step, determine effective duration 0.085 7000 97. 5;FV 7000 0.085 7000 97. 5;FV 7000 0.085 7000 97. 5;FV 7000

r. definition: ED P 0 slope s. formula: ED P P P 0 Δy 93. 4 963. 9483. 4.00 8. 5888 7. A T-bill matures in exactly 4 days and has a par value of 0000 dollars. The bond sells for 978 dollars. What is the discount yield? a. 0. 0 b. 3. 368 c. 3. 74 d. 3. 3446 e. 3. 39 a. definition: annualized discount based upon 360 day year b. so DY 360 0 000 978 00 3. 74 4 0 000 8. A T-bill matures in exactly 36 days and has a par value of 0000 dollars. The discount yield equals 7. 5. What is the price? a. 9330. b. 950. 0 c. 930. 8 d. 97. 8 e. 965. 0 a. definition: annualized discount based upon 360 day year b. true discount as percent 7. 5 36 360 c. so price: P 0000 0000 0.075 36 360 930. 8 9. A 7.0 percent bond matures in exactly 3 years and has a par value of 000 dollars. The bond sells for 4. 4 dollars. The bond is callable in 7 years for 990 dollars. What is the yield to call? a. 4.0 b. 5.0 c. 4. 5 d. 5. 5 e. 6.0 a. calculate the yield to maturity assuming the bond is called at the first call date b. calculator inputs: N 4;PV 4. 4;PMT 0.07 000 35.0;FV 990 c. cpt I/Y. 5 d. so y 4. 5

0. A 4.0 percent bond with a par value of 000 dollars matures in 3 years. The bond sells for 680. 34 dollars. Assume coupons are reinvested at 7. 5 percent per year compounded semiannually. What is the total return (over holding period of T years) compounded semiannually on the bond? a. 8. 040 b. 7. 9549 c. 7. 869 d. 8. 3 e. 5. 806 a. first step, compute future value of coupons to maturity date b. calculator inputs: N 6;PV 0;I/Y 7. 5 3. 75;PMT 0.04 000 0.0 c. cpt FV 855. 63 d. second step add in maturity value: FV 855. 63 000 855. 6 e. third step, find return compounded semiannually that converts price 680. 34 into 855. 6 f. total return: TR 855. 6 680. 34 3 00 7. 869. A floating rate bond has a quoted margin of 0.5 percent, a par value of 0000 dollars, and maturity of.0 years. The bond sells for 0073. dollars. The initial reference rate is 7. 5 percent per year compounded semiannually. The coupon rate is reset every six months. What is the discount margin in basis points? a. 5 b. 0 c. 0 d. 5 e. 0 a. first step, project cash flows under the assumption that future reference rate equals the current reference rate b. coupon rate: CR 7. 5 0.5 8.0 c. coupon: C 0.08 0000 400.0 d. second step, compute yield to maturity e. calculator inputs: N 4.0;PV 0073.;PMT 400.0;FV 0000 f. cpt I/Y 3. 8 g. so y 7. 6 h. third step, discount margin is difference between computed yield and reference rate i. discount margin: 7. 6 7. 5 0

. The yields on a six month, one year, and one and a half year zero-coupon bonds are 9. 5, 5. 5, and 6. 5 percent, respectively. What is the forward price of a contract to accept delivery of a six month T-bill with a par value of 0000 dollars in one year? a. 0. b. 9599. 0 c. 959. 7 d. 0079. e. 076. a. method: cost of carry model b. forward price should equal the cost of buying the spot asset and holding it to the delivery date of one year c. first step, value spot asset d. spot asset is zero a coupon bond that has same maturity date (not time to maturity) as bond underlying forward contract e. value of spot asset: P 0 000. 035 3 9085. f. second step, carry spot asset forward at spot rate to delivery date g. forward price: F 9085. 0.075 959. 7 3. The yields on a six month, one year, and one and a half year zero-coupon bonds are 5. 5, 9. 5, and 8.0 percent, respectively. What is the forward price of a contract to accept delivery of a one year T-bill with a par value of 0000 dollars in six months? a. 9499. 8 b. 954. 7 c. 934. 4 d. 043. e. 05. a. method: cost of carry model b. forward price should equal the cost of buying the spot asset and holding it to the delivery date of six months c. first step, value spot asset d. spot asset is zero a coupon bond that has same maturity date (not time to maturity) as bond underlying forward contract 0 000 e. value of spot asset: P 8890. 0. 04 3 f. second step, carry spot asset forward at spot rate to delivery date g. forward price: F 8890. 0 0.075 934. 4 4. The yields on a six month, one year, and one and a half year zero-coupon bonds are 6.0,

6. 5, and 9.0 percent, respectively. What is the forward rate on a contract to accept delivery of a one year T-bill in six months? a. 0. 793 b. 4. 56 c. 0. 56 d. 0. 68 e. 0. 53 a. method: () construct forward contract by borrowing short term (to delivery date) and investing long term (to maturity date) and () compute yield on the constructed forward contract V b. formula: f a,b b / b a V a c. forward rate (each half year): f,3 d. so over year the forward rate is 0. 56.4.03 / 5. 58 0 5. The yields on a six month, one year, and one and a half year zero-coupon bonds are 5. 5, 4.0, and 4. 5 percent, respectively. What is the forward rate on a contract to accept delivery of a six month T-bill in one year? a.. 758 b. 7. 547 c. 5. 5037 d. 8. 007 e. 4. 0036 a. method: () construct forward contract by borrowing short term (to delivery date) and investing long term (to maturity date) and () compute yield on the constructed forward contract V b. formula: f a,b b / b a V a c. forward rate (each half year): f,3.069. 758 0.0404 d. so over year the forward rate is 5. 5037 6. The price of a six month zero-coupon bond is 96. 54. The price of a one-year 4. 5 percent coupon bond is 98. 544. Both bonds has a par value of 00 dollars. What are the spot rates? a. 7. 95, 6. 05 b. 8. 05, 5. 9 c. 8.0, 6.0 d. 8., 5. 95 e. 8. 5, 6.

a. used boot-strap method to find yield on a one-year zero coupon b. price of dollar in six months: d 96. 54/00 0.9654 c. price of coupon bond: 98. 544. 5d 0. 5d d. substitute for d : 98. 544. 5 0.9654 0. 5d e. solve for d : d f. convert d and d into spot rates g. z d 0.96 54 4.0 98. 544. 5 0.9654 0. 5 0.9460 h. z d 0.94 60 3.0 7. A 6.0 percent par treasury bond with a par value of 00 dollars matures in exactly one and a half years. The bond sells for 98. 599. What is the Macaully duration? a.. 564 b.. 564 c.. 4564 d.. 3564 e.. 564 a. first step, compute yield to maturity b. calculator inputs: N 3;PV 98. 599;PMT 0.06 00 3.0;FV 00 c. cpt I/Y 3. 5 d. so y 7.0 e. second step, compute the duration f. formula: D N C t t, where cash flow are distributed semiannually. P t y/ g. so: D 98. 599 3.0 3.0 3.0 00 3 0.035 0.035 0.035 3. 4564 8. A barbell promises 87 dollars in 3. 5 years and 00 dollars in 0.0. The term structure is a 4.0 percent for all maturities. What is the Macaully duration of the barbell? a. 6. 468 b. 6. 75 c. 6. 448 d. 6. 468 e.. 6397 0 a. first step, compute price of first cash flow b. PV 87. 0 7.0 6. 79 c. second step, compute price of second cash flow d. PV 00. 0 0.0 34. 59

e. third step, determine price of barbell f. P 6. 79 34. 59 97. 39 g. fourth step, compute duration h. D 6. 79 3. 5 34. 59 0.0 6. 448 97. 39 9. A 9. 5 percent treasury bond has a yield to maturity of 4.0 and a duration of. 5 years. If the yield changes by 93 basis points, what is your best estimate of the percentage change in the bond s price. a. 0. 84 b. 0. 485 c. 0. 485 d. 0. 695 e. 0. 695 a. formula: Δ%P D Δy y/ b. so: Δ%P. 5 93 0. 485. 0 00 30. A 8. 5 percent treasury bond has a yield to maturity of 5.0, a duration of 5.0 years, and a convexity of 56. 5. If the yield changes by 37 basis points, what is your best estimate of the percentage change in the bond s price. a. 9. 649 b. 5. 496 c. 5. 453 d. 5. 446 e. 5. 5885 a. formula: Δ%P D Δy y/ CX Δy 5. 453 b. so: Δ%P 5.0. 05 37 0 000 56. 5 37 0000