SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS

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1 SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS This set of sample questios icludes those published o the iterest theory topic for use with previous versios of this examiatio. I additio, the followig have bee added to reflect the revised syllabus begiig Jue 07: Questios o iterest rate swaps have bee added. Questios are from the previous set of fiacial ecoomics questios. Questio 58 is ew. Questios 66, 78, 87-9 relate to the study ote o approximatig the effect of chages i iterest rates. Questios ad 9-95 relate to the study ote o determiats of iterest rates. Questios 96-0 o iterest rate swaps were added. March 08 Questio 57 has bee deleted. April 08 Questios 4, 4, 80, 08, 6, ad 6 were deleted. Effective October 08 they do ot relate to the syllabus. Some of the questios i this study ote are take from past SOA examiatios. These questios are represetative of the types of questios that might be asked of cadidates sittig for the Fiacial Mathematics (FM) Exam. These questios are iteded to represet the depth of uderstadig required of cadidates. The distributio of questios by topic is ot iteded to represet the distributio of questios o future exams. The followig model solutios are preseted for educatioal purposes. Alterative methods of solutio are, of course, acceptable. I these solutios, s is the m-year spot rate ad m m f t is the m-year forward rate, deferred t years. Copyright 08 by the Society of Actuaries. FM-0-7

2 . Solutio: C Give the same pricipal ivested for the same period of time yields the same accumulated value, the two measures of iterest i () = 0.04 ad δ must be equivalet, which meas: () i + = e δ over a oe-year period. Thus, () i e δ = + = = δ = l(.0404) = Solutio: E From basic priciples, the accumulated values after 0 ad 40 years are + i + + i i = [( ) ( ) ( ) ] 00 ( + i) ( + i) 4 ( + i) 4 4 ( + i) ( + i) ( + i) [( + i) + ( + i) + + ( + i) ] = The ratio is 5, ad thus (settig x 4 = ( + i) ) ( + i) ( + i) x x 5 = = ( + i) ( + i) x x 5x 5x = x x x = x x x 5 0 5x + 4= 0 x = 5 5 ( )( 4) 0. Oly the secod root gives a positive solutio. Thus x 5 = 4 x = X = 00 =

3 Auity symbols ca also be used. Usig the aual iterest rate, the equatio is s s 00 = 5(00) a a i + i = 5 i i + i + i + = 40 0 ( ) ( ) 40 0 ( ) 5( ) i = 0 ( ) 4 ad the solutio proceeds as above. 3. Solutio: C 5 i i Eric s (compoud) iterest i the last 6 moths of the 8th year is i Mike s (simple) iterest for the same period is 00. Thus, 5 5 i i i 00 + = 00 i + = i + = i = = 9.46%. 4. Deleted 3

4 5. Solutio: E The begiig balace combied with deposits ad withdrawals is 75 + (0) = 50. The edig balace of 60 implies 0 i iterest was eared. The deomiator is the average fud exposed to earig iterest. Oe way to calculate it is to weight each deposit or withdrawal by the remaiig time: () = The rate of retur is 0/ = =.0%. 6. Solutio: C + v 77. = v( Ia) + i a + v v = v + i i + + a v v = + i i i a v v = = = i i = v.05 = l(0.4997) = = 9. l(.05) To obtai the preset value without rememberig the formula for a icreasig auity, cosider the paymets as a perpetuity of startig at time, a perpetuity of startig at time 3, up to a perpetuity of startig at time +. The preset value oe period before the start of each perpetuity is /i. The total preset value is ( / i)( v+ v + + v ) = ( / ia ). 4

5 7. Solutio: C The iterest eared is a decreasig auity of 6, 5.4, etc. Combied with the aual deposits of 00, the accumulated value i fud Y is 6( Ds) + 00s ( ) s = = ( ) = Deleted 9. Solutio: D For the first 0 years, each paymet equals 50% of iterest due. The leder charges 0%, therefore 5% of the pricipal outstadig will be used to reduce the pricipal. At the ed of 0 years, the amout outstadig is 000( 0.05) 0 = Thus, the equatio of value for the last 0 years usig a compariso date of the ed of year 0 is = Xa = 6.446X X = % 0. Solutio: B The book value at time 6 is the preset value of future paymets: BV = 0, 000v + 800a = = 0, The iterest portio is 0,693(0.06) = Solutio: A The value of the perpetuity after the fifth paymet is 00/0.08 = 50. The equatio to solve is: = X( v+.08v v ) = X( v+ v+ + v) = X(5) /.08 X = 50(.08) = 54. 5

6 . Solutio: C Equatio of value at ed of 30 years: ( / 4) (.03) 0(.03) 00 d + = ( d / 4) [00 0(.03) ] / = = /40 / d = = d = 4( ) = = 4.53%. 3. Solutio: E The accumulatio fuctio is The accumulated value of 00 at time 3 is a t s ds t t 3 ( ) = exp ( /00) = exp( / 300) exp(3 / 300) = The amout of iterest eared from time 3 to time 6 equals the accumulated value at time 6 mius the accumulated value at time 3. Thus ( ) X [ a(6) / a(3) ] = X ( X)( / ) = X ( X) = X = 0.387X X = Solutio: A 5 ( + k) = 0a + 0(.09) 5 9.% t.09 = ( + k) / = ( + k ) /.09 ( )[ ( + k) /.09] = ( + k) / = ( + k) /.09 + k =.0400 k = K = 4.0%. t 6

7 5. Solutio: B Optio : 000 = Pa P = 99 Total paymets = 990 Optio : Iterest eeds to be = = i[ ] =, 000i i = 0.09 = 9.00% 6. Solutio: B Mothly paymet at time t is 000(0.98) t. Because the loa amout is ukow, the outstadig balace must be calculated prospectively. The value at time 40 moths is the preset value of paymets from time 4 to time 60: [ ] OB = v + + v v 0.98 v = 000, v = / (.0075) 0.98v = 000 = Solutio: C The equatio of value is 98S + 98S = 8000 ( i) 3 3 ( + i) ( + i) 8.63 i + = + = i = 8.63 i i 0 = 8.63 i i =.5% 7

8 8. Solutio: B Covert 9% covertible quarterly to a effective rate of j per moth: ( + j) = + or j = The a v ( Ia) = = = Solutio: C For Accout K, the amout of iterest eared is 5 00 X + X = 5 X. The average amout exposed to earig iterest is 00 (/)X + (/4)X = 00. The i = 5 X 00. For Accout L, examie oly itervals separated by deposits or withdrawals. Determie the iterest for the year by multiplyig the ratios of edig balace to begiig balace. The i = X Settig the two equatios equal to each other ad solvig for X, 5 X 3, 5 = 00 00(5 X ) (5 X)(5 X) = 3, 5 00(5 X) 3,5 50 3, 5, X X + X = + X 50X +, 400 = 0 X = 0. The i = (5 0)/00 = 0.5 = 5%. 8

9 0. Solutio: A Equatig preset values: v + 300v = 600v (0.76) + 300(0.76) = 600v 45.8 = 600v = v = v.0350 = + i i = = 3.5%. 0. Solutio: A The accumulatio fuctio is: t t dr l 0 8 ( 8 ) 8 r + r + t + 0 () = = =. at e e 8 Usig the equatio of value at ed of 0 years: 0 a(0) 0 8 / 8 0 0, 000 = ( 8 k + tk ) dt = k (8 t) dt k 8dt 0 at ( ) + = 0 (8 + t) / 8 0 0, 000 = 80k k = =. 80. Solutio: D Let C be the redemptio value ad v= / ( + i). The X = 000ra + Cv i v = 000r i = + 000(.035)( ) = Solutio: D Equate et preset values: v v = v Xv X 000 = X =

10 4. Deleted 5. Solutio: D The preset value of the perpetuity = X/i. Let B be the preset value of Bria s paymets. X B = Xa = 0.4 i 0.4 a = 0.4 = v v = 0.6 i X K = v i X K = 0.36, i Thus the charity s share is 36% of the perpetuity s preset value. 6. Solutio: D The give iformatio yields the followig amouts of iterest paid: 0 0. Seth = = = Jaice = 5000(0.06)(0) = Lori = P(0) 5000 = where P= = a The sum is % 0

11 7. Solutio: E For Bruce, 0 0 X 00[( i) ( i) ] 00( i) i = + + = +. Similarly, for Robbie, 6.Dividig the secod equatio by the first gives = 0.5( + i) which implies /6 i = = Thus 0 X = 00(.46) (0.46) = X = 50( + i) i 8. Solutio: D t Year t iterest is ia = v +. + t i t t Year t+ pricipal repaid is ( v ) = v. X = v + v = + v v = + v d + t t t t ( ). 9. Solutio: B For the first perpetuity, ( ) 0 / ( ) 3 3v = 0v v 3 = v + v + = v v 3 3 = 3 / 4. For the secod perpetuity, /3 /3 /3 /3 /9 /9 X = v + v + = v / ( v ) = (3 / 4) / [ (3 / 4) ] = Solutio: D Uder either sceario, the compay will have 8,703(0.05) = 4,35 to ivest at the ed of each of the four years. Uder Sceario A these paymets will be ivested at 4.5% ad accumulate to 4,35s = 4,35(4.78) = 75,984. Addig the maturity value produces 998,687 for a loss of,33. Note that oly aswer D has this value. The Sceario B calculatio is 4,35s = 4,35(4.343) = 78, 6+ 8, 703, 000, 000 =, Solutio: D. The preset value is v+ v + + v [ ].07v.07 v = 5000 = 5000 =, v.0905

12 3. Solutio: C. The first cash flow of 60,000 at time 3 ears 400 i iterest for a time 4 receipt of 6,400. Combied with the fial paymet, the ivestmet returs,400 at time 4. The preset value is 4, 400(.05) = 00, 699. The et preset value is Solutio: B. Usig spot rates, the value of the bod is: 3 60 / / /.09 = Solutio: E. Usig spot rates, the value of the bod is: + + = The aual effective rate is the solutio to = + +. Usig a calculator, the solutio is 8.9% / / / a 000( i) 3i 35. Solutio: C. Duratio is the egative derivative of the price multiplied by oe plus the iterest rate ad divided by the price. Hece, the duratio is ( 700)(.08)/00 = Solutio: C The size of the divided does ot matter, so assume it is. The the duratio is t= t t= tv v t ( Ia) a / i / ( di). = = = = = =. a / i / i d Solutio: B Duratio = t t t tv Rt tv.0 t= t= t t t vrt v.0 t= t= ( Ia) a / j j j = = = =. a / j d j The iterest rate j is such that ( + j) =.0v=.0 /.05 j = 0.03 /.0. The the duratio is / d = ( + j) / j = (.05 /.0) / (0.03 /.0) =.05 / 0.03 = 35.

13 45. Solutio: A For the time weighted retur the equatio is: X + 0 = 0 + 0X = X 0 = X X = X The the amout of iterest eared i the year is = 0 ad the weighted amout exposed to earig iterest is 0() + 60(0.5) = 40. The Y = 0/40 = 5%. 46. Solutio: A The outstadig balace is the preset value of future paymets. With oly oe future paymet, that paymet must be 559.(.08) = The amout borrowed is a = 000. The first paymet has 000(0.08) = 60 i iterest, thus the pricipal repaid is = Alteratively, observe that the pricipal repaid i the fial paymet is the outstadig loa balace at the previous paymet, or Pricipal repaymets form a geometrically 3 decreasig sequece, so the pricipal repaid i the first paymet is 559. /.08 = Solutio: B Because the yield rate equals the coupo rate, Bill paid 000 for the bod. I retur he receives 30 every six moths, which accumulates to 30s where j is the semi-aual iterest rate. The 0 j 0 equatio of value is 000(.07) = 30s s = Usig a calculator to solve for the iterest rate produces j = ad so 0 j 0 j i =.0476 = = 9.75%. 48. Solutio: A To receive 3000 per moth at age 65 the fud must accumulate to 3,000(,000/9.65) = 30, The equatio of value is 30, = Xs = X / 49. Solutio: D (A) The left-had side evaluates the deposits at age 0, while the right-had side evaluates the withdrawals at age 7. (B) The left-had side has 6 deposits, ot 7. (C) The left-had side has 8 deposits, ot 7. (D) The left-had side evaluates the deposits at age 8 ad the right-had side evaluates the withdrawals at age 8. (E) The left-had side has 8 deposits, ot 7 ad 5 withdrawals, ot 4. 3

14 50. Deleted 5. Solutio: D Because oly Bod II provides a cash flow at time, it must be cosidered first. The bod provides 05 at time ad thus 000/05 = uits of this bod provides the required cash. This bod the also provides (5) = at time 0.5. Thus Bod I must provide = at time 0.5. The bod provides 040 ad thus /040 = uits must be purchased. 5. Solutio: C Because oly Mortgage II provides a cash flow at time two, it must be cosidered first. The mortgage provides Y / a = Y at times oe ad two. Therefore, Y = for Y = Mortgage I must provide = 000 at time oe ad thus X = 000/.06 = The sum is Solutio: A Bod I provides the cash flow at time oe. Because 000 is eeded, oe uit of the bod should be purchased, at a cost of 000/.06 = Bod II must provide 000 at time three. Therefore, the amout to be reivested at time two is 000/.065 = The purchase price of the two-year bod is /.07 = The total price is Solutio: C Give the coupo rate is greater tha the yield rate, the bod sells at a premium. Thus, the miimum yield rate for this callable bod is calculated based o a call at the earliest possible date because that is most disadvatageous to the bod holder (earliest time at which a loss occurs). Thus, X, the par value, which equals the redemptio value because the bod is a par value bod, must satisfy = 0.04Xa + Xv0.03 =.96X X = 440. Price = Solutio: B Because 40/00 is greater tha 0.03, for early redemptio the earliest redemptio should be 30 evaluated. If redeemed after 5 years, the price is 40a + 00 /.03 = If the bod is redeemed at maturity, the price is 40a + 00 /.03 = The smallest value should be selected, which is

15 56. Solutio: E Give the coupo rate is less tha the yield rate, the bod sells at a discout. Thus, the miimum yield rate for this callable bod is calculated based o a call at the latest possible date because that is most disadvatageous to the bod holder (latest time at which a gai occurs). Thus, X, the par value, which equals the redemptio value because the bod is a par value bod, must satisfy = 0.0Xa + Xv0.03 = 0.855X X = 00. Price = Solutio: B Give the price is less tha the amout paid for a early call, the miimum yield rate for this callable bod is calculated based o a call at the latest possible date. Thus, for a early call, the 9 effective yield rate per coupo period, j, must satisfy Price = 0.50 = a + 00v j. Usig the calculator, j =.86%. We also must check the yield if the bod is redeemed at maturity. The 0 equatio is 0.50 = a + 00v j. The solutio is j =.46% Thus, the yield, expressed as a 0 j omial aual rate of iterest covertible semiaually, is twice the smaller of the two values, or 4.9%. 9 j 58. Moved to Derivatives sectio 59. Solutio: C First, the preset value of the liability is PV = 35, 000a = 335, % The duratio of the liability is: t 5 tv Rt 35, 000v+ (35, 000) v + + 5(35, 000) v,3,5.95 d = = = = t vr 335, , t Let X deote the amout ivested i the 5 year bod. X X (5) + (0) X 08,556. The, 335, = => = 335,

16 60. Solutio: A The preset value of the first eight paymets is: v 000(.03) v PV = 000v + 000(.03) v (.03) v = = 3, v The preset value of the last eight paymets is: PV = 000(.03) 0.97v + 000(.03) (0.97) v (.03) (0.97 ) v (.03) 0.97v 000(.03) (0.97) v = = 7, v Therefore, the total loa amout is L = 0, Solutio: E 000 = 500 exp 0 r 00 dr 3 r r t 3 4 exp 0.5 t exp 0.5l 3 r = dr = + r 50 0 t t 4 = exp 0.5l + = t 6 = t = t Solutio: E Let F, C, r, ad i have their usual iterpretatios. The discout is ( Ci Fr a ad the discout i the coupo at time t is ( Ci Fr) v + t. The, 94.8 = ( Ci Fr) v = ( Ci Fr) v 6 = = = v v i = = 6 ( Ci Fr) 94.8(.095) Discout = 06.53a =, ) 6

17 63. Solutio: A = Pv 85 + P = (aual paymet) P = = I = = L = = 5500 (loa amout) Total iterest = 84.39(8) 5500 = Solutio: D OB 8 8 s =, 000(.007) = 6, ,337.0 = Pa P = Solutio: C If the bod has o premium or discout, it was bought at par so the yield rate equals the coupo rate, ( v+ v + + v + v ) (90) (90) 4(90) 4(5000) d = v+ 90v v v ( Ia) (5000) v d = a v d = Or, takig advatage of a shortcut: d = a =.07. This is i half years, so dividig by two, d = = Solutio: A v = = P(0.08) = P(0.07) ( iv ) [ ] [ ] P(0.08) = 000 (0.008)(7.45) =

18 67. Solutio: E 3 3 ( + s ) = ( + s ) ( + f ) =, s 3 3 = 0.05 (+ s ) =, s = ( + s ) =.050 ( + f ) f = Solutio: C Let d 0 be the Macaulay duratio at time 0. d d d d d = a = = d = = a = = = This solutio employs the fact that whe a coupo bod sells at par the duratio equals the preset value of a auity-due. For the duratio just before the first coupo the cash flows are the same as for the origial bod, but all occur oe year sooer. Hece the duratio is oe year less. Alteratively, ote that the umerators for d ad d are idetical. That is because they differ oly with respect to the coupo at time (which is time 0 for this calculatio) ad so the paymet does ot add aythig. The deomiator for d is the preset value of the same bod, but with 7 years, which is The deomiator for d has the extra coupo of 50 ad so is 550. The desired ratio is the 5000/550 = Solutio: A Let N be the umber of shares bought of the bod as idicated by the subscript. N N N C B A (05) = 00, N = C (00) = (5), N = (07) = (5), N = A B 8

19 70. Solutio: B All are true except B. Immuizatio requires frequet rebalacig. 7. Solutio: D Set up the followig two equatios i the two ukows: A(.05) + B(.05) = A(.05) B(.05) = 0. Solvig simultaeously gives: A = 7.09 B = A B = Solutio: A Set up the followig two equatios i the two ukows. 3 () 5000(.03) + B(.03) =, B(.03) =, 000 B(.03) = () 3(5000)(.03) bb(.03) = 0 6, b = 0 b =.5076 B = B = 807. b b b b b 9

20 73. Solutio: D 9 P = A( + i) + B( + i) P A L = 95, 000( + i) P = A( + i) 9 B( + i) A P = 5(95, 000)( + i) L 6 Set the preset values ad derivatives equal ad solve simultaeously A B= 78, A 6.080B= 375, , 083(.7780 / ) 375, 400 B = = 47, (.7780 / ) A = [78, (47, 630)] / = 48, 59 A =.03 B 74. Solutio: D Throughout the solutio, let j = i/. For bod A, the coupo rate is (i )/ = j For bod B, the coupo rate is (i 0.04)/ = j 0.0. The price of bod A is 0 A = 0, 000( + 0.0) + 0, 000( + ). 0 j P j a j The price of bod B is 0 B = 0, 000( 0.0) + 0, 000( + ). 0 j P j a j Thus, P P = 5,34. = [00 ( 00)] a = 400a a A B 0 j 0 j 0 j = 5,34. / 400 = Usig the fiacial calculator, j = 0.04 ad i =(0.04)=

21 75. Solutio: D The iitial level mothly paymet is 400, , 000 R = = = 4, a a / The outstadig loa balace after the 36th paymet is B = Ra = 4, a = 4, (87.87) = 356, The revised paymet is 4, = 3, Thus, 356, = 3, 647.9a a 44 j/ 44 j/ = 356, / 3, = Usig the fiacial calculator, j/ = 0.575%, for j = 6.9%. 76. Solutio: D The price of the first bod is 000(0.05 / ) a + 00( / ) = 5a + 00(.05) = =, / The price of the secod bod is also, The equatio to solve is = a + + j 60, ( / ). 60 j/ The fiacial calculator ca be used to solve for j/ =.% for j = 4.4%. 77. Solutio: E Let = years. The equatio to solve is 000(.03) = (000)(.005) l.03 + l000 = l l = = This is 85.3 moths. The ext iterest paymet to Lucas is at a multiple of 6, which is 88 moths. 78. Solutio: B The edig balace is 5000(.09) + 600sqrt(.09) = The time-weighted rate of retur is (500/5000) x [864.08/( )] =

22

23 79. Solutio: A Equatig the accumulated values after 4 years provides a equatio i K. 4 K 4 + = dt 5 0 K 0.5t 0 0 exp K + 4l( K) = dt = 4l(K+ 0.5 t) = 4l( K + ) 4l( K) = 4l 0 0 K + 0.5t K K K = K 0.04K = K = 5. Therefore, 4 X = 0( + 5 / 5) = Deleted 8. Solutio: D 5 a5 The outstadig balace at time 5 is 00( Da) = 00. The priciple repaid i the 6th 5 i 5 a5 paymet is X = 500 i(00) = a = 00 a. The amout borrowed 5 5 i 5 is the preset value of all 50 paymets, 500a + v 00( Da). Iterest paid i the first paymet is the i 500a + v 00( Da) ( v ) 00 v (5 a ) v 500v v 00a5 = Xv = + =

24 8. Solutio: A The exposure associated with i produces results quite close to a true effective rate of iterest as log as the et amout of pricipal cotributed at time t is small relative to the amout i the fud at the begiig of the period. 83. Solutio: E The time-weighted weight of retur is j = (0,000 / 00,000) x (30,000 / 50,000) x (00,000 / 80,000) = 30.00%. Note that 50,000 = 0, ,000 ad 80,000 = 30,000 50, Solutio: C The accumulated value is 000 s = 50,38.6. This must provide a semi-aual auitydue of Let be the umber of paymets. The solve 3000a = 50,38.6 for = Therefore, there will be 6 full paymets plus oe fial, smaller, paymet. The equatio is 6 50,38.6 = 3000 a + X(.04) with solutio X = 430. Note that the while the fial paymet is the 7th paymet, because this is a auity-due, it takes place 6 periods after the auity begis. 85. Solutio: D For the first perpetuity, + = 7. + ( i) = ( + i) 6. i = For the secod perpetuity, R + 3 = ( ).8639R = 7.(.0875) R =.74. (.0875) 7. ( ) 4

25 86. Solutio: E a 5v 0, 000 = 00( Ia) + Xv a = 00 + Xv a , 000 = X 075 = X Solutio: C 5000 = Xs (.05) X = = (.763) Solutio: E The mothly paymet o the origial loa is 65, 000 a 80 8/% = 6.7. After paymets the outstadig balace is 6.7a = 6, The revised paymet is 68 8/% 6, = a 68 6/% 89. Solutio: E At the time of the fial deposit the fud has 750s = 5, This is a immediate auity because the evaluatio is doe at the time the last paymets is made (which is the ed of 7 the fial year). A tuitio paymet of 6000(.05) = 3, 75. is made, leavig, It ears 7%, so a year later the fud has,747.6(.07) =, Tuitio has grow to 3,75.(.05) = 4, The amout eeded is 4,439.7, =, Solutio: B The coupos are 000(0.09)/ = 45. The preset value of the coupos ad redemptio value at 40 5% per semiaual period is P= 45a + 00(.05) = Solutio: A For a bod bought at discout, the miimum price will occur at the latest possible redemptio 0 date. P= 50a + 000(.06) =

26 9. Solutio: C =.5% 93. Solutio: D The accumulated value of the first year of paymets is 000s = 4, 67.. This amout icreases at % per year. The effective aual iterest rate is.005 = The preset value is the 5 5 k k.0 P = 4,67..0 (.06678) = 4,67. k=.0 k= = 4,87.37 = 374, This is 56 less tha the lump sum amout. k 94. Solutio: A The mothly iterest rate is 0.07/ = five years from today has value (.006) = The equatio of value is (.006) 3400(.006). Let = + x =.006. The, solve the quadratic equatio 3400x + 700x = (3400)( ) x = = (3400) The,.006 = l(.006) = l(0.935) =.73. To esure there is 6500 i five years, the deposits must be made earlier ad thus the maximum itegral value is. 6

27 95. Solutio: C ( d ) 4 ( d /4) d ( ) d 38 = = 39 39( d ) = 38 38( d 4) 38 d = d = / ( ) = 0. ( ) + i = d / =.95 =.08 i = 0.8%. 96. Solutio: C 3 The mothly iterest rate is 0.04/ = The quarterly iterest rate is.0035 = The ivestor makes 4 quarterly deposits ad the edig date is 4 moths from the start. Usig Jauary of year y as the compariso date produces the followig equatio: X X + = k k = Substitutig =.0035 gives aswer (C). 97. Solutio: D Covert the two aual rates, 4% ad 5%, to two-year rates as.04 = ad, 05 = The accumulated value is 00 s (.05) + 00 s = 00(3.5678)(.55) + 00(.380) = With oly five paymets, a alterative approach is to accumulate each oe to time te ad add them up. The two-year yield rate is the solutio to 00 s = Usig the calculator, the two-year 5i 0.5 rate is The aual rate is = which is 4.58%. 98. Solutio: C ( ).08 = , 000ä = Xä 5 4 8% %.08 5, 000(3.5770) X = = (7.790) 7

28 99. Solutio: B PV perp. = + (5, 000) 5, = 64, , 000 = 79, a X a + 79, = X = 79, X = 7, Solutio: A 4 a 4(.03) = (.50 + X) a + X (.03) = (.50 + X).96+ X => 79.3X = 598 => X = Solutio: D The amout of the loa is the preset value of the deferred icreasig auity: 4 30 a 30(.05) a Ia a (.05) ( ) = (.05 )(500) + = 64, / Solutio: C ( + i) (.03) ( + i) (.03) 50, 000 ( + i) = 5, ( + i) ( i 0.03) i ,000 / ( ) 5,000 + i = 9 ( ) 0 i = = + i = / The accumulated amout is (.08637) (.03) 50, 000 (.08637) = 797, (.08637) ( ) 8

29 03. Solutio: D The first paymet is,000, ad the secod paymet of,00 is.005 times the first paymet. Sice we are give that the series of quarterly paymets is geometric, the paymets multiply by.005 every quarter. Based o the quarterly iterest rate, the equatio of value is 3 3,000 00, 000 =, 000 +, 000(.005) v+, 000(.005) v +, 000(.005) v + =.005v..005v=, 000 /00, 000 v= 0.98 / The aual effective rate is ( ) 4 v = 0.98 /.005 = = 0.6%. 04. Solutio: A.06 Preset value for the first 0 years is l.06 ( ) ( ) Preset value of the paymets after 0 years is 0 s ( ) ( ) ( ) 0 ( ) ( ) = 7.58 s ds = = l.06 l.03 Total preset value = Solutio: C 0 dt 5 t+ 5 0, 000(.06) X (.06) e + = 75, 000 ( 3, X ) = 75, X = 7,56.83 X = 4, Solutio: D The effective aual iterest rate is i = d = = ( ) ( 0.055) 5.8% The balace o the loa at time is 9 5, 000, 000(.058) = 6, 796,809. The umber of paymets is give by, 00, 000a = 6, 796,809 which gives = => 9 paymets of,00,000. The fial equatio of value is 9 30, 00, 000 a + X(.058) = 6, 796,809 X = (6, 796,809 6, 6, 0)( ) = 959, 490.

30 07. Solutio: C ( ) 0.55( ) v = v = + v v = v= 0.47( ) v ( ) / v = v = = v = = l( ) / l(0.959) = 08. Deleted 09. Solutio: C The mothly paymet is 00, 000 / a = Usig the equivalet aual effective rate of 6.7%, the preset value (at time 0) of the five extra paymets is 4,99.54 which reduces the origial loa amout to 00,000 4,99.54 = 58, The umber of moths required is the solutio to 58, = 99.0a. Usig calculator, = 5.78 moths are eeded to pay off this amout. So there are 5 full paymets plus oe fractioal paymet at the ed of the 6th moth, which is December 3, Solutio: D The aual effective iterest rate is 0.08/( 0.08) = The level paymets are 500, 000 / a = 500, 000 / = 7,535. This rouds up to 8,000. The equatio of value for X is a X = 5 8, 000 (.08696) 500, X = (500, , )(.579) = 5, 7.. Solutio: B The accumulated value is the reciprocal of the price. The equatio is X[(/0.94)+(/0.95)+(/0.96)+(/0.97)+(/0.98)+(/0.99)] = 00,000. X= 6,078 30

31 . Solutio: D Let P be the aual paymet. The fifth lie is obtaied by solvig a quadratic equatio. P 0 ( ) 3600 Pv v = = 487 v = 5 v 487 v = v v v 0 5 = = i = = v 3600 X = P = = 48, 000 i Solutio: A Let j = periodic yield rate, r = periodic coupo rate, F = redemptio (face) value, P = price, = umber of time periods, ad v = j + j. I this problem, j = (.0705 ) = , r = 0.035, P = 0,000, ad = 50. The preset value equatio for a bod is yields P = Fv + Fra ; solvig for the redemptio value F P 0, 000 0, 000 F = = = = 9,98.. v + ra + a + j j 50 (.03465) (3.6044) j j 4. Solutio: B Jeff s mothly cash flows are coupos of 0,000(0.09)/ = 75 less loa paymets of 000(0.08)/ = 3.33 for a et icome of At the ed of the te years (i additio to the 6.67) he receives 0,000 for the bod less a,000 loa repaymet. The equatio is 8000 = 6.67a ( + i /) i () () 0 i / / = i = = = () %. 3

32 5. Solutio: B The preset value equatio for a par-valued aual coupo bod is the coupo rate r yields P Fvi P vi r = = Fa a F a i i i. P = Fv + Fra ; solvig for All three bods have the same values except for F. We ca write r = x(/f) + y. From the first two bods: = x/000 + y ad = x/00 + y. The, = x(/000 /00) for x = 96.8 ad y = /000 = For the third bod, r = 96.8/ = =.93%. i i 6. Solutio: A () () i i The effective semi-aual yield rate is.04 = + => =.9804%. The, = c(.0) v+ c(.0 v) + + c(.0 v) + 50v.0 v (.0 v) = c + 50v =.05c => c= v 3.0 v (.0 v) = c + 50v =.05c => c= v 7. Solutio: E Book values are liked by BV3( + i) Fr = BV4. Thus 54.87(.06) Fr = Therefore, the coupo is Fr = The prospective formula for the book value at time 3 is ( 3) = (.06) 0.06 ( 3) = 00.97(.06) l( /00.97) 3 = = 7. l(.06) ( 3) Thus, = 0. Note that the fiacial calculator ca be used to solve for 3. 3

33 8. Solutio: A Book values are liked by BV3( + i) Fr = BV4. Thus BV3(.04) 500(0.035) = BV Therefore, BV3 = [500(0.035) ]/0.04 = The prospective formula for the book value at time 3 is, where m is the umber of six-moth periods. ( m 3) = 500(0.035) + 500(.04) 0.04 ( m 3) = 3.5(.04) l(/ 3.5) m 3 = = 0. l(.04) ( m 3) Thus, m = 3 ad = m/ = 6.5. Note that the fiacial calculator ca be used to solve for m Solutio: C s = f = f ( + s ) ( + s ) = 0.06 = s = (.06)(.04) = f ( + s ) ( + s ) 3 = 0.08 = s = [(.08)(.04995) ] = = 6%. 3 / Solutio: D Iterest eared is 55,000 50,000 8, ,000 = 7,000. Equatig the two iterest measures gives the equatio 7, = = , (6, 000 / 3) 0, 000( t) , 000 = (55, , , 000 t) t = [7, (45,333.33)] /,366.7 = Solutio: B 0() + ( v) + ( v ) v+ v The Macaulay duratio of Auity A is 0.93 = =, which leads to the + v+ v + v+ v quadratic equatio.07v v 0.93 = 0. The uique positive solutio is v = 0.9. The Macaulay duratio of Auity B is 3 0() ( v) ( v ) 3( v ) v+ v + v =

34 34

35 . Solutio: D With v =/.07, 3 4 (40, 000) v + 3(5, 000) v + 4(00, 000) v D = = , 000v + 5, 000v + 00, 000v 3. Deleted 4. Solutio: C v Ia = 0 di + i i 30 = MacD = = = = = a / d ( + i) / i i v = 0 / ( ) ( ) / MacD 30 The, ModD = = = i + 30 ad so i = / Solutio: D Let D be the ext divided for Stock J. The value of Stock F is 0.5D/(0.088 g). The value of Stock J is D/( g). The relatioship is 0.5D D = g g 0.5 D( g) = D(0.088 g).5g = 0.3 g = = 5.3%. 6. Solutio: B I) False. The yield curve structure is ot relevat. II) True. III) False. Matchig the preset values is ot sufficiet whe iterest rates chage. 35

36 7. Solutio: A The preset value fuctio ad its derivatives are Pi ( ) = X+ Y( + i) 500( + i) 000( + i) 3 4 P ( i) = 3 Y( + i) + 500( + i) ( + i) 4 5 P i = Y + i + i + i ( ) ( ) 000( ) 0, 000( ). The equatios to solve for matchig preset values ad duratio (at i = 0.0) ad their solutio are P(0.) = X Y = 0 P (0.) =.0490Y = 0 Y = 896.9/.0490 = 43.8 X = (43.8) = The secod derivative is P = = (0.) (43.8)(.) 000(.) 0, 000(.) Redigto immuizatio requires a positive value for the secod derivative, so the coditio is ot satisfied. 8. Solutio: D This solutio uses time 8 as the valuatio time. The two equatios to solve are Pi i X i 8 y ( ) = 300, 000( + ) + ( + ), 000, 000 = 0 Pi = + i+ yx + i = 7 y ( ) 600,000( ) (8 ) ( ) 0. Isertig the iterest rate of 4% ad solvig: 8 y 300, 000(.04) X (.04), 000, y 600, 000(.04) (8 yx ) (.04) 0 X + = + = = = y 8 (.04) [, 000, , 000(.04) ] / , = 7 64, 000 (8 y)(.04) (493,595.85) 0 y = + = , 000 / [493,595.85(.04) ] X = 493,595.85(.04) = 70,

37 9. Solutio: A This solutio uses Macaulay duratio ad covexity. The same coclusio would result had modified duratio ad covexity bee used. The liabilities have preset value have a preset value of / /.07 = 000. Oly portfolios A, B, ad E 5 The duratio of the liabilities is [(573) / (70) /.07 ] /000 = 3.5. The duratio of a zero coupo bod is its term. The portfolio duratio is the weighted average of the terms. For portfolio A the duratio is [500() + 500(6)]/000 = 3.5. For portfolio B it is [57() + 48(6)]/000 = 3.4. For portfolio E it is 3.5. This elimiates portfolio B. 5 The covexity of the liabilities is [4(573) / (70) /.07 ] /000 = 4.5. The covexity of a zero-coupo bod is the square of its term. For portfolio A the covexity is [500() + 500(36)]/000 = 8.5 which is greater tha the covexity of the liabilities. Hece portfolio A provides Redigto immuizatio. As a check, the covexity of portfolio E is.5, which is less tha the liability covexity. 30. Solutio: D The preset value of the liabilities is 000, so that requiremet is met. The duratio of the 3 liabilities is 40.[. + (.) + 3(.) ] /000 = Let X be the ivestmet i the oeyear bod. The duratio of a zero-coupo is its term. The duratio of the two bods is the [X + (000 X)(3)]/000 = X. Settig this equal to.9365 ad solvig yields X =

38 3. Solutio: A Let x, y, ad z represet the amouts ivested i the 5-year, 5-year, ad 0-year zero-coupo bods, respectively. Note that i this problem, oe of these three variables is 0. The preset value, Macaulay duratio, ad Macaulay covexity of the assets are, respectively, x+ y+ z,, x+ y+ z x+ y+ z x y z x y z. We are give that the preset value, Macaulay duratio, ad Macaulay covexity of the liabilities are, respectively, 9697, 5.4, ad Sice preset values ad Macaulay duratios eed to match for the assets ad liabilities, we have the two equatios 5x+ 5y+ 0z x+ y+ z = 9697, = 5.4. x+ y+ z Note that 5 ad 5 are both less tha the desired Macaulay duratio 5.4, so z caot be zero. So try either the 5-year ad 0-year bods (i.e. y = 0), or the 5-year ad 0-year bods (i.e. x = 0). I the former case, substitutig y = 0 ad solvig for x ad z yields (0 5.4)9697 (5.4 5)9697 x = = ad z = = We eed to check if the Macaulay covexity of the assets exceeds that of the liabilities. The Macaulay covexity of the assets is 5 (3077.8) + 0 (669.8) = 8.00, which exceeds 9697 the Macaulay covexity of the liabilities, The compay should ivest 3077 for the 5-year bod ad 660 for the 0-year bod. Note that settig x = 0 produces y = ad z = ad the covexity is 33.40, which is less tha that of the liabilities. 3. Solutio: E The correct aswer is the lowest cost portfolio that provides for $,000 at the ed of year oe ad provides for $,00 at the ed of year two. Let H, I, ad J represet the face amout of each purchased bod. The time oe paymet ca be exactly matched with H + 0.J =,000. The time two paymet ca be matched with I +.J =,00. The cost of the three bods is H/. + I/.3 + J. This fuctio is to be miimized uder the two costraits. Substitutig for H ad I gives (,000 0.J)/. + (,00.J)/.3 + J = 9, J. This is miimized by purchasig the largest possible amout of J. This is,00/. = 0, The, H =,000 0.(0,803.57) = The cost of Bod H is /. = 8,

39 33. Solutio: C The strategy is to use the two highest yieldig assets: the oe-year bod ad the two-year zerocoupo bod. The cost of these bods is 5, 000 / , 000 /.05 = 4, Solutio: E Let P be the aual iterest paid. The preset value of Joh s paymets is Pa. The preset X 0.05 X X value of Kare s paymets is P(.05) a = P(.05) / The, 0.05 X P(.05) / 0.05 =.59Pa X = X.59 =.59(.05) X l.59 = l.59 X l.05 X = 0. X Solutio: A Cheryl s force of iterest at all times is l(.07) = Gomer s accumulatio fuctio is from time 3 is + yt ad the force of iterest is y/( + yt). To be equal at time, the equatio is = y/( + y), which implies y = y for y = Gomer s accout value is 000( + x0.0785) = Solutio: D Oe way to view these paymets is as a sequece of level immediate perpetuities of that are deferred -,, +, years. The preset value is the + 3 v / i+ v / i+ v / i+ = ( v / i)( v+ v + v + ) = v / i. Notig that oly aswers C, D, ad E have this form ad all have the same umerator, v / i = v /( vi) = v / d. 37. Solutio: B / The mothly iterest rate is j = (.08) = 0.643%. The, 0, 000 s = Xs, 90,.4 = X, X =

40 38. Solutio: D 0δ 0δ e e 0δ 0δ a =.5 a, =.5, e.5e = 0. Let 0 δ X = e. We the have 0 0 δ δ the quadratic equatio X.5X = 0 with solutio X = 0.5 for δ = l 0.5 / ( 0) = The, the accumulated value of a 7-year cotiuous auity of is s 7 7( ) e = = Solutio: B The preset value is v + v + v + + v v v ( v ) ( v ) a = = = 7 7 v v a 7 a. 40. Solutio: C.09 From the first auity, X =.8s =.8 = 00[.09 ] From the secod auity, Hece, 00[.09 ] = 9, [.09 ] = 9, 08 / 00 = = 9.8 X = 00(9.8) = 960. v X = 9, 08( v + v + ) = 9, 08 = 9, 08. v Solutio: C 60 a 60v ( Ia) = = =, %

41 4. Solutio: E Let j be the semi-aual iterest rate. The, j 475, 000 = a + ( + j ) 00( Ia ) = / j + 00 / j j 474, 700 j 300 j 00 = (474, 700)( 00) j = = (474, 700) i = ( + j) = 0.04 = 4.%. 43. Solutio: B The preset value is 4a + ( Ia) = 4 / (.06) / 0.06 = Solutio: A The preset value of the icome is 00a = 00 / 0.05 = The preset value of the 0.05 ivestmet is X +.05 /.05 + (.05 /.05) + (.05 /.05) + (.05 /.05) + (.05 /.05) = X[ ] = X = X..05 The 975.6=5.395X for X = Solutio: A The preset value of the te level paymets is remaiig paymets is Xa = X. The preset value of the v /.05 X( v.05 + v.05 + ) = X X X. v.05 =.05 /.05 = The, 45,000 = 8.078X X = X for X =

42 46. Solutio: D The equatio of value is v e 0, 000 = X( v+ v v ) = X = X = 5.89 X. v e The solutio is X = 0,000/5.89 = Solutio: D Discoutig at 0%, the et preset values are 4.59,.36, ad 9.54 for Projects A, B, ad C respectively. Hece, oly Project A should be fuded. Note that Project C s et preset value eed ot be calculated. Its cash flows are the same as Project B except beig 50 less at time ad 50 more at time 4. This idicates Project C must have a lower et preset value ad therefore be egative. 48. Solutio: D The loa balace after 0 years is still 00,000. For the ext 0 paymets, the iterest paid is 0% of the outstadig balace ad therefore the pricipal repaid is 5% of the outstadig 0 balace. After 0 years the oustadig balace is 00, 000(0.95) = 59,874. The, X = 59,874 / a = 59,874 / = 9, Solutio: B First determie umber of regular paymets: 4 4 a = 600 va, = (4000 / 600).06 = Usig the calculator, =.07 ad thus there are regular paymets. The equatio for the balloo paymet, X, is: = 600v a + Xv = X, X = Solutio: C 5 ( ) 0, 000 = X a +. a = X( /.68506) = X X = 0, 000 / =

43 5. Solutio: A The pricipal repaid i the first paymet is 00 il. The outstadig pricipal is L 00 + il = L + 5. Hece, il = 5. Also, ( v ) 00( v ) L= 300a 00a = 6 8 i 5 = il = v 300v 300v 00v + 5 = 0 v 00 ± 00 4(300)(5) 00 ± 00 = = = The larger of the two values is used due to the value beig kow to exceed 0.3. The outstadig valace at time eight is the preset value of the remaiig paymets: a = 300 = /8 5. Solutio: E Let j be the mothly rate ad X be the level mothly paymet. The pricipal repaid i the first paymet is 400 = X 60,000j. The pricipal repaid i the secod paymet is 44 = X (60, )j. Substitutig X = ,000j from the first equatio gives 44 = ,000j 58,600j or 4 = 400j ad thus j = 0.0 ad X = 000. Let be the umber of paymets. The 60,000 = 000a ad the calculator (or algebra) gives = The 0.0 equatio for the drop paymet, P, is = , 000 = 000a + Pv = 58, P for P 53. Solutio: C The accumulated value is 4 ( s s / / ) 000 ( /) + = 000(5.430(.79) ) = 55, Deleted 43

44 55. Solutio: E The otioal amout ad the future -year LIBOR rates (ot give) do ot factor ito the calculatio of the swap s fixed rate. Required quatities are () Zero-coupo bod prices: = ,.045 = ,.055 = ,.065 = ,.075 = () -year implied forward rates: ,.045 /.04 = ,.055 /.045 = , /.055 = ,.075 /.065 = The fixed swap rate is: (0.04) (0.0500) ( ) ( ) (0.649) = The calculatio ca be doe without the implied forward rates as the umerator is = Solutio: C I the secod year of the swap cotract, Compay ABC has the followig iterest paymet outflows: Existig debt:,000,000 (LIBOR + 0.5%) =,000,000 (4.0% + 0.5%) = 90,000. Swap cotract, fixed rate, to the swap couterparty:,000, % = 60,000. Also, i the secod year of the swap cotract, ABC has the followig iterest paymet iflow: Swap cotract, variable rate, to the swap couterparty:,000,000 LIBOR =,000, % = 80,000. Thus, the combied et paymet that Compay ABC makes is: (90, ,000) (80,000) = 70,000, which is a outflow. 57. DELETED 44

45 58. Solutio: D First, the implied forward rates are: Year Implied forward rate.5% 3.7% 4.0% 4.% 5.6% 5.% PV (floatig paymets) = (.034) (.036) (.04) = = r r r PV (fixed paymets) = = ( )r =.59457r. (.034) (.036) (.04) Equatig floatig to fixed paymets: =.59457r for 0.83 r = = 4.6% Solutio: C / Each moth the pricipal paid icreases by.. Thus, the amout of pricipal paid icreases / 30 6 to 500(. ) = 500(.) = Solutio: C It = i 900 a + 300a = 900( v ) + 300( v ) = v 900v 0 i 0 i 0 It = i 900 a = 900( v ) 0 i It = It v 900v = v / 3 v v + = v = 0 It = 900( v ) = Deleted 45

46 6. Deleted 63. Solutio: C The origial mothly paymet is 85, 000 / a = 85, 000 / = O July, there has bee 4 years of paymets, hece 6x = 9 remaiig paymets. The outstadig balace is a = (3.380) = 75, The umber of remaiig paymets after refiacig is determied as , = 500a = =.0045 = l(0.3457) / l(.0045) = Thus the fial paymet will be 5 moths from Jue 30, 009. This is 0 years ad moths ad so the fial paymet is May 3, Solutio: B Just prior to the extra paymet at time 5, the outstad balace is 300a = 300(0.5940) = 3, After the extra paymet it is,7.0. Payig this off i 5 years requires aual paymets of,7.0 / a =,7.0 / = Solutio: C Durig the first redemptio period the modified coupo rate is 000(0.035)/50 =.80% which is larger tha the desired yield rate. If redeemed durig this period, bod sells at a premium ad so the worst case for the buyer is the earliest redemptio. The price if called at that time is 0 35a + 50(.05) = 35(5.589) = Durig the secod redemptio period the modified coupo rate is 000(0.035)/5 = 3.% which is also larger tha the desired yield rate ad the worst case for the buyer is agai the earliest redemptio. The price if 40 called at that time is 35a + 5(.05) = 35(5.08) = Fially, if the bod is ot called, its value is 35a + 000(.05) = 35( ) = The appropriate price is the lowest of these three, which relates to the bod beig called after the 40th coupo is paid. 46

47 66. Solutio: B Because the yield is less tha the coupo rate, the bod sells at a premium ad the worst case for the buyer is a early call. Hece the price should be calculated based o the bod beig called at 6 time 6. The price is 00a + 000(.05) = 00(0.0378) = Solutio: A All calculatios are i millios. For the te-year bod, at time te it is redeemed for 0 (.08) = After beig reivested at % it matures at time twety for (.) = The thirty-year bod has a redemptio value of For the buyer to ear 0%, it is sold for = (.) (.08) = =. The gai is Solutio: A The book value after the third coupo is (0.037) a + C(.065) = C ad after the fourth coupo it is (0.037) 36 (.065) a + C = + C. The, C ( C) = C = 8.3 C = Solutio: C / The semiaual yield rate is. = Assumig the bod is called for 900 after four 8 years, the purchase price is 50a + 900(.0488) = 50(6.4947) = With a call after the first coupo, the equatio to solve for the semi-aual yield rate (j) ad the the aual effective rate (i) is = ( ) / ( + j) + j =.054 i = =

48 70. Solutio: C 34 The book value after the sixth coupo is 000( r/ ) a + 000(.036) = 976.0r After the seveth coupo it is 000( r/ ) a + 000(.036) = r The, = r+ 3.6 (976.0r ) = r r = ( ) /50. = Solutio: B The two equatios are: P= ra + = r+ 5 (0, 000 ) 9, 000(.04) 44,58. 7, P= [0, 000( r+ 0.0)] a +, 000(.04) = 44,58.r+ 9, Subtractig the first equatio from the secod gives 0.P = for P = 0, Isertig this i the first equatio gives r = (0, ,397.34)/44,58. = Solutio: C Whe the yield is 6.8% < 8%, the bod is sold at a premium ad hece a early call is most 0 disadvatageous. Therefore, P= 40a + 000(.034) = Whe the yield is 8.8% > 8%, the bod is sold at discout. Hece, Q < 000 < P. ad thus Q = = Also, because the bod is sold at a discout, the latest call is the most disadvatageous. Thus, = 40a + 000(.044) = + (.044) (.044) = = 90.90(.044) = l(7.70 / 90.90) / l(.044) = 38 = 9. 48

49 73. Solutio: B The fud will have = after four years. After returig 75% to the 4 500(.05) 00s isured, the isurer receives 0.5(76.74) = So the isurer s cash flows are to pay 00 at time 0, receive 5 at time, ad receive 44.9 at time four. The equatio of value ad the solutio are: 4 00( ) 5( ) ( 5) 4(00)( 44.9) ( + i) = = i =.399 i = 4%. + i + i = ± Solutio: A If the value of X icreases, the 9% rate from July to December 3 couts more heavily tha the ( )/4000 = 8% rate from Jauary to Jue 30. So the aual effective yield rate icreases. The time-weighted rate depeds oly o percetage icreases i each sub-period ad thus it remais uchaged. 75. Solutio: B The amout of iterest eared is 00, ,000 30,000 00,000 = 0,000. The amout ivested for the year is 00,000 + ( 5/) 30,000 ( 3/4) 50,000 = 05,000. The dollarweighted rate of retur is 0,000/05,000 = 9.05%. 76. Solutio: B The Macaulay duratio of the perpetuity is v ( Ia) = ( + i) / i + i v a = / i i = = = = + / i = 7.6. This implies that i = /6.6. With i = i = /6.6, the duratio is + 6.6/ = Solutio: A Because the iterest rate is greater tha zero, the Macaulay duratio of each bod is greater tha its modified duratio. Therefore, the bod with a Macaulay duratio of c must be the bod with a modified duratio of a ad a = c/( + i) which implies + i = c/a. The Macaulay duratio of the other bod is b( + i) =bc/a. 49

50 78. Solutio: B.0 P(0.05) P(0.0) = P(0.0)..05 chage is 00( ) =.47%. Therefore, the approximate percetage price 79. Solutio: C The preset value of the divideds is: = = ( ) 80. Solutio: B Cash-flow matchig limits the umber of ivestmet choices available to the portfolio maager to a subset of the choices available for immuizatio. 8. Solutio: C Optios for full immuizatio are: J (cost is 3000), K+L (cost is 500), ad M (cost is 4000). The lowest possible cost is 500. Aother way to view this is that the prices divided by total cash flows are 0.6, 0.5, 0.5, ad 0.8. The cheapest optio will be to use K ad L, if possible. 8. Solutio: B The preset value of the assets is 5, ,000 = 60,000 which is also the preset value of the liability. The modified duratio of the assets is the weighted average, or 0.5(.80) Dmod. The modified duratio of the liability is 3/. ad so Dmod = (3/. 0.45)/0.75 = Solutio: C Let A be the redemptio value of the zero-coupo bods purchased ad B the umber of twoyear bods purchased. The total preset value is: = A/.05 + B(00 / /.06 ) = A B. To exactly match the cash flow at time oe, A + 00B = 000. Substitutig B = 0 0.0A i the first equatio gives = A A for A = / = 95. The amout ivested is the 95/.05 =