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Problems on Fixed Income

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					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: C  1  0. 06  2000  60. 0
                              2     2


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  9. 370 6
                                6723. 1


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  1  0. 095  10 000  475. 0; FV  10 000
                                         2
    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  183−170  183
                                       183
                                                 13

    c. so accrued interest  183−170  0. 022 5  1000  1. 598 4
                                 183


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 V w at first coupon payment date:w  20 days  183  183
                                                                                   20     20

       half years
    b. calculator inputs:
       N  30; I/Y  4. 5  1  2. 25; PMT  1  0. 075  1000  37. 5; FV  1000
                              2                   2
    c. cpt PV  1324. 7
    d. so V  1324. 737. 5  1358. 9
                        20
                1. 022 5 183



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  183−19  164
                                       183    183
    c. so accrued interest  183−19  0. 045  1000  40. 328
                                 183
    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  183  3719
                                 59
                                            183
    b. clean (or dirty) price      1000
                                         3719    636. 24
                               1. 022 5   183




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  156  1272
                                    183     61
                                               1
                                              1272
    b. yield to maturity  2     1000
                                464. 10
                                               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. 0d 1  1025. 0d 2
    b. so: P    25.0
                             1025.0
                                           963. 13
                 1. 02       1. 045 2



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: y treasury  7. 5 and y muni  5. 0
     b. therefore: 1 − 7. 5  5. 0
     c. which implies:1 −   7. 5  0. 666 67
                                 5.0

    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: y muni  10. 0
                           y muni
     b. therefore: TEY  1−  1−0.29  14. 085
                                     10.0



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: M 1  10 8001  0.095   11313.
                                                          2


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: M 1  10 5001  0.025   10631.
                                                        2
    b. apply coupon rate to inflation adjusted principal: c21  10631.0.04  212. 63
                                                                     2


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: M 1  10 6001  0.045   10839.
                                                            2
     b. apply coupon rate to inflation adjusted principal: c21  10839.0.032 5  176. 13
                                                                        2
     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: P 1  1. 05  10839.  11380. 0
     e. so: 1  R  176. 1311380.0  1. 090 2
                  2        10 600
      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  1  0. 08  10 000  400. 0; FV  10 000
                                        2
    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  1  0. 08  10 000  400. 0; FV  10 000
                                    2
    j. cpt PV  11090. 0
    k. Δ%P  10711.−11090.0  100  −3. 424 8
                    11090.0


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  1  0. 085  7000  297. 5; FV  7000
                                           2
     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  1  0. 085  7000  297. 5; FV  7000
                                                       2
      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  1  0. 085  7000  297. 5; FV  7000
                                                       2
     p. so P −  9483. 4
     q. fifth step, determine effective duration
     r. definition: ED       1
                              P0   |slope|
                                   P  −P −
     s. formula: ED     1
                         P0         2Δy
                                                   1
                                                  9321. 4
                                                            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   360  10 000−9781  100  3. 271 4
                    241     10 000


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. 5 326
                                          360
     c. so price: P  10 000 − 10 000  0. 075 326  9320. 8
                                                360


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  1  0. 07  1000  35. 0; FV  990
                                                             2
     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  3. 75; PMT  1  0. 04  1000  20. 0
                                  2                   2
     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
                                            1
     f. total return: TR  2    1855. 6
                                680. 34
                                          213
                                                 − 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: C  0.08  10 000  400. 0
                   2      2
     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  10 000 3  9085. 1
                                  1. 032 5
     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  10 0003  8890. 0
                                  1. 04
     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
                             Vb   1/b−a
    b. formula: fa, b     Va
                                            −1
                                                                1/2
     c. forward rate (each half year): f1, 3      1. 141 2
                                                      1. 03
                                                                      − 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
                             Vb   1/b−a
    b. formula: fa, b     Va
                                            −1
     c. forward rate (each half year): f2, 3     1. 069
                                                   1. 040 4
                                                              − 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: d 1  96. 154/100  0. 961 54
    c.   price of coupon bond: 98. 544  2. 25d 1  102. 25d 2
    d.   substitute for d 1 : 98. 544  2. 25  0. 961 54  102. 25d 2
    e.   solve for d 2 : d 2  98. 544−2. 250.961 54  0. 942 60
                                       102. 25
     f. convert d 1 and d 2 into spot rates
     g. z 1  d11 − 1  0.961 54 − 1  4. 0
                            1


    h. z 2       1
                  d2
                         −1               1
                                        0.942 60
                                                        − 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  1  0. 06  100  3. 0; FV  100
                                                           2
     c. cpt I/Y  3. 5
     d. so y  7. 0
     e. second step, compute the duration
                                               Ct   t
     f. formula: D              1
                                 P
                                     ∑ t1
                                       N            2
                                               1y/2 2
                                                           , where cash flow are distributed semiannually.
                                     3.0 1                            3.0100 3
     g. so: D            1               2
                                                      3.01
                                                                                 2
                                                                                       1. 456 4
                       98. 599       10.035        10.035  2       10.035  3



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  187 7.0  162. 79
               1. 02
    c. second step, compute price of second cash flow
    d. PV      200
                   20.0  134. 59
               1. 02
     e.   third step, determine price of barbell
     f.   P  162. 79  134. 59  297. 39
     g.   fourth step, compute duration
     h.   D  297. 39 162. 79  3. 5  134. 59  10. 0  6. 441 8
                   1




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 Δy
                             D

     b. so: Δ%P ≈ − 11. 5  − 100  10. 485
                    1. 02
                               93




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 Δy  1 CXΔy 2
                               D
                                        2
                                                                    2
     b. so: Δ%P ≈ − 1. 025  − 1037 
                     15.0
                                  000
                                          1
                                          2
                                               56. 25  − 1037
                                                              000
                                                                         5. 453 1

				
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