Fixed Income: Practice Problems with Solutions

Fixed Income: Practice Problems with Solutions

Directions: Unless otherwise stated, assume semi-annual payment on bonds.

1. A 6. 0 percent bond matures in exactly 18 years and has a par value of 2000 dollars. The bond sells for 1338. 1 dollars. What is the semiannual coupon payment?

a. 120. 0 b. 80. 288

c. 60. 0 d. 40. 144

e. 66. 906 Solution:

a. semiannual coupon: ^C₂  ¹₂  0. 06  2000  60. 0

2. A 9. 0 percent bond matures in exactly 15 years and has a par value of 7000 dollars. The bond sells for 6723. 1 dollars. What is the current yield?

a. 9. 0 b. 9. 5

c. 9. 370 6 d. 4. 75

e. 4. 5 Solution:

a. current yield: CY  ^{1000.097000}_{6723. 1}  9. 370 6

3. A 9. 5 percent bond matures in exactly 16 years and has a par value of 10 000 dollars. The bond sells for 14220. 0 dollars. What is the yield to maturity?

a. 3. 5 b. 4. 0 c. 5. 5 d. 4. 5 e. 5. 0 Solution:

a. calculator inputs:

N  32; PV  14220. 0; PMT  ¹₂  0. 095  10 000  475. 0; FV  10 000 b. cpt I/Y  2. 75

c. so y  5. 5

4. A 4. 5 percent bond with a par value of 1000 dollars matures in 17 years, 170 days. The next coupon is paid in 170 days. What is the accrued interest? Assume 183 days between coupon payment dates.

a. 20. 902 b. 3. 196 7

c. 1. 598 4 d. 12. 049

e. 24. 098 Solution:

a. days since last coupon 183 − 170  13 b. half-year since last coupon ^183−170₁₈₃  ₁₈₃¹³

c. so accrued interest ^183−170₁₈₃  0. 022 5  1000  1. 598 4

5. A 7. 5 percent bond with a par value of 1000 dollars matures in 15 years, 20 days. The next coupon is paid in 20 days. The yield is 4. 5 percent. What is the dirty price? Assume 183 days between coupon payment dates.

a. 1362. 2 b. 1365. 5 c. 1358. 9 d. 1360. 5 e. 1362. 6 Solution:

a. solve for value of bond Vw at first coupon payment date:w  20 days  ₁₈₃²⁰  ₁₈₃²⁰ half years

b. calculator inputs:

N  30; I/Y  4. 5  ¹₂  2. 25; PMT  ¹₂  0. 075  1000  37. 5; FV  1000 c. cpt PV  1324. 7

d. so V  ^{1324. 737. 5}

1. 022 5¹⁸³²⁰  1358. 9

6. A 9. 0 percent bond with a par value of 1000 dollars matures in 19 years, 19 days. The next coupon is paid in 19 days. The bond sells for 1378. 2 dollars. What is the clean price?

Assume 183 days between coupon payment dates.

a. 1297. 5 b. 1335. 5 c. 1337. 8 d. 1351. 3 e. 1324. 4 Solution:

a. days since last coupon 183 − 19  164

b. half-year since last coupon ¹⁸³⁻¹⁹₁₈₃  ¹⁶⁴₁₈₃

c. so accrued interest ¹⁸³⁻¹⁹₁₈₃  0. 045  1000  40. 328 d. then clean price is given by: CP  1378. 2 − 40. 328  1337. 8

7. A zero-coupon bond with a par value of 1000 dollars matures in 10 years, 59 days. If the bond yields 4. 5 percent, what is the clean price? Assume 183 days in a half-year.

a. 640. 82 b. 639. 37 c. 636. 24 d. 643. 93 e. 645. 43 Solution:

a. time to maturity 20  ₁₈₃⁵⁹  ³⁷¹⁹₁₈₃ b. clean (or dirty) price ¹⁰⁰⁰

1. 022 5^3719183  636. 24

8. A zero-coupon bond with a par value of 1000 dollars matures in 10 years, 156 days. If the bond sells for 464. 10, what is yield to maturity? Assume 183 days in a half-year.

a. 7. 0 b. 8. 0 c. 7. 5 d. 8. 5 e. 9. 0 Solution:

a. time to maturity:T 20  ¹⁵⁶₁₈₃  ¹²⁷²₆₁ b. yield to maturity 2 _{464. 10}¹⁰⁰⁰

1 1272

61 − 1  100  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 1000 dollars and maturity of one year. What is the minimum price the dealer should ask for the bond?

a. 1010. 5 b. 940. 24 c. 963. 13 d. 931. 84 e. 889. 76 Solution:

a. minimum price is the cost of constructing cash flow pattern using zero-coupon bonds:

25. 0d1 1025. 0d2

b. so: P _{1. 02}^25.0  _{1. 045}^1025.02  963. 13

10. A 7. 5 percent par treasury bond matures in exactly 13 years. A 5. 0 percent par municipal bond matures in exactly 13 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. 133 c. 33. 333 d. 37. 833 e. 39. 733 Solution:

a. since both bonds are selling at par: ytreasury  7. 5 and ymuni  5. 0 b. therefore:1 − 7. 5  5. 0

c. which implies:1−   ^5.0_{7. 5}  0. 666 67 d. so:  1 − 0. 666 67  33. 333

11. A 10. 0 percent par municipal bond matures in exactly 13 years. For an investor at the 29. 0 percent marginal tax rate, what is the taxable-equivalent yield?

a. 7. 1 b. 7. 042 3

c. 14. 085 d. 15. 285 e. 18. 885 Solution:

a. since the muni is selling at par: ymuni  10. 0 b. therefore: TEY  ^y_1−^muni  _1−0.29^10.0  14. 085

12. A 3. 0 percent TIP bond matures in exactly 14 years. Six months ago the par value was 10 800 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. 11826.

b. 10962.

c. 11313.

d. 11124.

e. 10 800 Solution:

a. inflation adjusted principal: M1  10 8001  ^0.095₂   11313.

13. A 4. 0 percent TIP bond matures in exactly 12 years. Six months ago the par value was 10 500 dollars. The annualized CPI (inflation rate) over the last six months equals 2. 5.

Assume the coupon is paid today. What is the dollar value of the coupon paid today?

a. 425. 25 b. 215. 25 c. 212. 63 d. 430. 5

e. 210. 0 Solution:

a. inflation adjusted principal: M1  10 5001  ^0.025₂   10631.

b. apply coupon rate to inflation adjusted principal: ^c₂¹  10631.0.04

2  212. 63

14. A 3. 25 percent TIP bond matures in exactly 14 years. Six months ago the par value was 10 600 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. 17. 048 b. 17. 298 c. 18. 048 d. 19. 298 e. 19. 788 Solution:

a. inflation adjusted principal: M1  10 6001  ^0.045₂   10839.

b. apply coupon rate to inflation adjusted principal: ^c₂¹  10839.0.032 5

2  176. 13

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: 176. 13 plus the price: P1  1. 05  10839.  11380. 0

e. so: 1 ^R₂  176. 1311380.0

10 600  1. 090 2

f. finally: R  2  1. 090 2 − 1  100  18. 048 percent

15. A 8. 0 percent bond matures in exactly 10 years and has a par value of 10 000 dollars. The bond sells for 11090. 0 dollars. For a 50 basis increase in the yield, determine the

percentage change in the bond’s price?

a. −3. 924 8 b. −1. 924 8 c. −3. 424 8 d. −2. 024 8 e. 0. 475 15 Solution:

a. first step, find yield to maturity y

b. calculator inputs:

N  20; PV  11090. 0; PMT  ¹₂  0. 08  10 000  400. 0; FV  10 000 c. cpt I/Y  3. 25

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  20; I/Y  3. 5; PMT  ¹₂  0. 08  10 000  400. 0; FV  10 000 j. cpt PV  11090. 0

k. Δ%P  10711.−11090.0

11090.0  100  −3. 424 8

16. A 8. 5 percent bond matures in exactly 13 years and has a par value of 7000 dollars. The bond sells for 9321. 4 dollars. What is the approximate (effective) duration for a 20 basis point shock (either up or down)?

a. 5. 588 8 b. 6. 588 8 c. 8. 588 8 d. 7. 588 8 e. 6. 588 8 Solution:

a. first step, find yield to maturity y b. calculator inputs:

N  26; PV  9321. 4; PMT  ¹₂  0. 085  7000  297. 5; FV  7000 c. cpt I/Y  2. 5

d. so y  5. 0

e. second step, increase yield by 20 bps f. new yield 5. 0 . 2  5. 2

g. so I/Y  2. 6

h. third step, compute price at new yield y

i. calculator inputs: N  26; I/Y  2. 6; PMT  ¹₂  0. 085  7000  297. 5; FV  7000 j. so: P_  9163. 1

k. fourth step, decrease yield by 20 bps l. new yield 5. 0 −. 2  4. 8

m. so I/Y  2. 4

n. fourth step, compute price at new yield y₋

o. calculator inputs: N  26; I/Y  2. 4; PMT  ¹₂  0. 085  7000  297. 5; FV  7000 p. so P−  9483. 4

q. fifth step, determine effective duration

r. definition: ED  _P¹₀|slope|

s. formula: ED  _P¹

0

P−P−

2Δy  _{9321. 4}¹ 9163. 1−9483. 4

2.002  8. 588 8

17. A T-bill matures in exactly 241 days and has a par value of 10 000 dollars. The bond sells for 9781 dollars. What is the discount yield?

a. 10. 0 b. 3. 316 8

c. 3. 271 4 d. 3. 344 6 e. 3. 391 1 Solution:

a. definition: annualized discount based upon 360 day year b. so DY  ³⁶⁰₂₄₁  ^{10 000}_{10 000}⁻⁹⁷⁸¹  100  3. 271 4

18. A T-bill matures in exactly 326 days and has a par value of 10 000 dollars. The discount yield equals 7. 5. What is the price?

a. 9330. 1 b. 9250. 0 c. 9320. 8 d. 9171. 8 e. 9625. 0 Solution:

a. definition: annualized discount based upon 360 day year b. true discount as percent 7. 5 ³²⁶₃₆₀

c. so price: P  10 000 − 10 000  0. 075 ³²⁶₃₆₀  9320. 8

19. A 7. 0 percent bond matures in exactly 13 years and has a par value of 1000 dollars. The bond sells for 1141. 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 Solution:

a. calculate the yield to maturity assuming the bond is called at the first call date

b. calculator inputs: N  14; PV  1141. 4; PMT  ¹₂  0. 07  1000  35. 0; FV  990 c. cpt I/Y  2. 25

d. so y  4. 5

20. A 4. 0 percent bond with a par value of 1000 dollars matures in 13 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. 024 0 b. 7. 954 9 c. 7. 869 2 d. 8. 113 1 e. 5. 280 6 Solution:

a. first step, compute future value of coupons to maturity date b. calculator inputs:

N  26; PV  0; I/Y  ^{7. 5}₂  3. 75; PMT  ¹₂  0. 04  1000  20. 0 c. cpt FV  855. 63

d. second step add in maturity value: FV  855. 63  1000  1855. 6

e. third step, find return compounded semiannually that converts price 680. 34 into 1855. 6

f. total return: TR  2 ^{1855. 6}_{680. 34} ^2131 − 1  100  7. 869 2

21. A floating rate bond has a quoted margin of 0. 5 percent, a par value of 10 000 dollars, and maturity of 2. 0 years. The bond sells for 10073. 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. 10 d. 5

e. 0 Solution:

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: ^C₂  ^0.08₂  10 000  400. 0 d. second step, compute yield to maturity

e. calculator inputs: N  4. 0; PV  10073. ; PMT  400. 0; FV  10 000 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  10

22. 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 10 000 dollars in one year?

a. 10112.

b. 9599. 0 c. 9591. 7 d. 10079.

e. 10176.

Solution:

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  _{1. 032 5}^{10 000}3  9085. 1

f. second step, carry spot asset forward at spot rate to delivery date g. forward price: F  9085. 1  1  0. 027 5²  9591. 7

23. 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 10 000 dollars in six months?

a. 9499. 8 b. 9154. 7 c. 9134. 4 d. 10431.

e. 10252.

Solution:

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

e. value of spot asset: P  _{1. 04}^{10 000}3  8890. 0

f. second step, carry spot asset forward at spot rate to delivery date g. forward price: F  8890. 0  1  0. 027 5  9134. 4

24. 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. 10. 793 b. 4. 516 2

c. 10. 516 d. 10. 268 e. 10. 532 Solution:

a. method: (1) construct forward contract by borrowing short term (to delivery date) and investing long term (to maturity date) and (2) compute yield on the constructed forward contract

b. formula: fa, b  ^V_V^b_a ^{1/b−a} − 1

c. forward rate (each half year): f1, 3  ^{1. 141 2}_{1. 03} ^1/2− 1  5. 258 2  10⁻² d. so over year the forward rate is 10. 516

25. 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. 2. 751 8 b. 7. 514 7 c. 5. 503 7 d. 8. 007 1 e. 4. 003 6 Solution:

a. method: (1) construct forward contract by borrowing short term (to delivery date) and investing long term (to maturity date) and (2) compute yield on the constructed forward contract

b. formula: fa, b  ^V_V^b_a ^{1/b−a} − 1

c. forward rate (each half year): f2, 3  _{1. 040 4}^{1. 069} − 1  2. 751 8  10⁻² d. so over year the forward rate is 5. 503 7

26. The price of a six month zero-coupon bond is 96. 154. The price of a one-year 4. 5 percent coupon bond is 98. 544. Both bonds has a par value of 100 dollars. What are the spot rates?

a. 7. 95, 6. 05 b. 8. 05, 5. 9

c. 8. 0, 6. 0 d. 8. 1, 5. 95

e. 8. 15, 6. 1 Solution:

a. used boot-strap method to find yield on a one-year zero coupon b. price of 1 dollar in six months: d1  96. 154/100  0. 961 54

c. price of coupon bond: 98. 544  2. 25d1  102. 25d2

d. substitute for d1: 98. 544  2. 25  0. 961 54  102. 25d2

e. solve for d2: d2  98. 544−2. 250.961 54

102. 25  0. 942 60

f. convert d1 and d2 into spot rates g. z1  _d¹

1 − 1  _{0.961 54}¹ − 1  4. 0 h. z2  _d¹

2 − 1  _{0.942 60}¹ − 1  3. 0

27. A 6. 0 percent par treasury bond with a par value of 100 dollars matures in exactly one and a half years. The bond sells for 98. 599. What is the Macaully duration?

a. 1. 156 4 b. 1. 256 4 c. 1. 456 4 d. 1. 356 4 e. 1. 256 4 Solution:

a. first step, compute yield to maturity

b. calculator inputs: N  3; PV  98. 599; PMT  ¹₂  0. 06  100  3. 0; FV  100 c. cpt I/Y  3. 5

d. so y  7. 0

e. second step, compute the duration

f. formula: D  _P¹

∑

g. so: D  _{98. 599}¹ ₁^3.0_0.035¹²  ^3.01

10.035 ²  ^{3.010032}

10.035 ³  1. 456 4

28. A barbell promises 187 dollars in 3. 5 years and 200 dollars in 10. 0. The term structure is a 4. 0 percent for all maturities. What is the Macaully duration of the barbell?

a. 6. 461 8 b. 6. 75

c. 6. 441 8 d. 6. 461 8

e. 1. 639 7 10⁻² Solution:

a. first step, compute price of first cash flow b. PV  _{1. 02}¹⁸⁷7.0  162. 79

c. second step, compute price of second cash flow d. PV  _{1. 02}²⁰⁰20.0  134. 59

e. third step, determine price of barbell f. P  162. 79  134. 59  297. 39 g. fourth step, compute duration

h. D  _{297. 39}¹ 162. 79  3. 5  134. 59  10. 0  6. 441 8

29. A 9. 5 percent treasury bond has a yield to maturity of 4. 0 and a duration of 11. 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. 10. 284 b. −10. 485

c. 10. 485 d. 10. 695 e. −10. 695 Solution:

a. formula:Δ%P ≈ −_1y/2^D Δy

b. so:Δ%P ≈ −^{11. 5}_{1. 02}  −₁₀₀⁹³  10. 485

30. A 8. 5 percent treasury bond has a yield to maturity of 5. 0, a duration of 15. 0 years, and a convexity of 56. 25. If the yield changes by−37 basis points, what is your best estimate of the percentage change in the bond’s price.

a. 9. 264 9 b. 5. 491 6 c. 5. 453 1 d. 5. 414 6 e. 5. 588 5 Solution:

a. formula:Δ%P ≈ −_1y/2^D Δy  ¹₂CXΔy²

b. so:Δ%P ≈ −_{1. 025}^15.0  −_{10 000}³⁷  ¹₂  56. 25  − _{10 000}^{37 2}  5. 453 1