SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE SOLUTIONS Interest Theory

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1 SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS Ineres Theory Ths page ndcaes changes made o Sudy Noe FM January 4, 04: Quesons and soluons were added. June, 04 Queson 58 was moved o he Dervaves Markes se of sample quesons. Quesons 6-73 were added. Many of he quesons were re-worded o conform o he curren syle of queson wrng. The subsance was no changed. December, 05: Quesons were added. Some of he quesons n hs sudy noe are aken from pas SOA/CAS examnaons. These quesons are represenave of he ypes of quesons ha mgh be asked of canddaes sng for he Fnancal Mahemacs (FM) Exam. These quesons are nended o represen he deph of undersandng requred of canddaes. The dsrbuon of quesons by opc s no nended o represen he dsrbuon of quesons on fuure exams. The followng model soluons are presened for educaonal purposes. Alernae mehods of soluon are, of course, accepable. Copyrgh 04 by he Socey of Acuares. FM-09-4 PRINTED IN U.S.A.

2 . Soluon: C Gven he same prncpal nvesed for he same perod of me yelds he same accumulaed value, he wo measures of neres () 0.04 and mus be equvalen, whch means: () e over a one-year perod. Thus, () e ln(.0404) Soluon: E From basc prncples, he accumulaed values afer 0 and 40 years are [( ) ( ) ( ) ] 00 ( ) ( ) 4 ( ) 4 4 ( ) ( ) ( ) [( ) ( ) ( ) ] The rao s 5, and hus (seng x 4 ( ) ) ( ) ( ) x x 5 ( ) ( ) x x 5x 5x x x x x x 5 0 5x ( x )( x 4) 0. Only he second roo gves a posve soluon. Thus x 5 4 x X

3 Annuy symbols can also be used. Usng he annual neres rae, he equaon s s s 00 5(00) a a ( ) ( ) ( ) 5( ) ( ) 4 and he soluon proceeds as above. 3. Soluon: C 7 Erc s (compound) neres n he las 6 monhs of he 8h year s 00. Mke s (smple) neres for he same perod s 00. Thus, %. 4. Soluon: A The perodc neres s 0.0(0,000) = 000. Thus, deposs no he snkng fund are = Then, he amoun n snkng fund a end of 0 years s ,33 s. Afer repayng he loan, he fund has,33, whch rounds o,30. 3

4 5. Soluon: E The begnnng balance combned wh deposs and whdrawals s 75 + (0) = 50. The endng balance of 60 mples 0 n neres was earned. The denomnaor s he average fund exposed o earnng neres. One way o calculae s o wegh each depos or whdrawal by he remanng me: () The rae of reurn s 0/ = =.0%. 6. Soluon: C n nv 77. via n n n a nv n nv v n n an nv nv n n an v v n v n ln(0.4997) n 9. ln(.05) To oban he presen value whou rememberng he formula for an ncreasng annuy, consder he paymens as a perpeuy of sarng a me, a perpeuy of sarng a me 3, up o a perpeuy of sarng a me n +. The presen value one perod before he sar of each perpeuy s /. The oal presen value s (/ )( v v v n ) (/ ) a n. 4

5 7. Soluon: C The neres earned s a decreasng annuy of 6, 5.4, ec. Combned wh he annual deposs of 00, he accumulaed value n fund Y s 6( Ds) 00s s Deleed 9. Soluon: D For he frs 0 years, each paymen equals 50% of neres due. The lender charges 0%, herefore 5% of he prncpal ousandng wll be used o reduce he prncpal. A he end of 0 years, he amoun ousandng s Thus, he equaon of value for he las 0 years usng a comparson dae of he end of year 0 s Xa 6.446X X % 0. Soluon: B The book value a me 6 s he presen value of fuure paymens: BV 0,000v 800a , The neres poron s 0,693(0.06) = Soluon: A The value of he perpeuy afer he ffh paymen s 00/0.08 = 50. The equaon o solve s: X ( v.08v.08 v ) X ( v v v) X (5) /.08 X 50(.08) 54. 5

6 . Soluon: C Equaon of value a end of 30 years: ( d / 4) (.03) 0(.03) ( d / 4) [00 0(.03) ] / /40 d / d 4( ) %. 3. Soluon: E The accumulaon funcon s The accumulaed value of 00 a me 3 s a s ds 3 ( ) exp ( /00) exp( / 300) exp(3 / 300) The amoun of neres earned from me 3 o me 6 equals he accumulaed value a me 6 mnus he accumulaed value a me 3. Thus X [ a(6) / a(3) ] X ( X)( / ) X ( X) X X X Soluon: A 5 ( k) a 0(.09) 5 9.%.09 ( k) / ( k ) /.09 ( )[ ( k) /.09] ( k) / ( k) /.09 k.0399 k K 3. 99%. 6

7 5. Soluon: B Opon : 000 Pa P 99 Toal paymens 990 Opon : Ineres needs o be [ ], % 6. Soluon: B Monhly paymen a me s 000(0.98). Because he loan amoun s unknown, he ousandng balance mus be calculaed prospecvely. The value a me 40 monhs s he presen value of paymens from me 4 o me 60: OB v v [ ] v 0.98 v 000, v / (.0075) 0.98v Soluon: C The equaon of value s 98S 98S n n 3n n ( ) ( ) 8.63 n % 7

8 8. Soluon: B Conver 9% converble quarerly o an effecve rae of j per monh: ( j) or j = Then 60 a 60v ( Ia) Soluon: C For Accoun K, he amoun of neres earned s 5 00 X + X = 5 X. The average amoun exposed o earnng neres s 00 (/)X + (/4)X = 00. Then 5 X. 00 For Accoun L, examne only nervals separaed by deposs or whdrawals. Deermne he neres for he year by mulplyng he raos of endng balance o begnnng balance. Then X. Seng he wo equaons equal o each oher and solvng for X, 5 X 3, (5 X ) (5 X )(5 X ) 3, 5 00(5 X ) 3,5 50 3, 5, X X X X 50X, X 0. Then = (5 0)/00 = 0.5 = 5%. 8

9 0. Soluon: A Equang presen values: n 00 00v 300v 600v 0 0 3n (0.76) 300(0.76) 600v v v v %. 0. Soluon: A The accumulaon funcon s: dr 8 ln8 r 8 r ( ). a e e Usng he equaon of value a end of 0 years: 0 a(0) 0 8 / 8 0, k k d k (8 ) d k 0 8d 0 a( ) 0 (8 ) / 8 0 0, k k. 80. Soluon: D Le C be he redempon value and v/ ( ). Then X 000ra Cv n n n v 000r (.035)( )

10 3. Soluon: D Equae ne presen values: v 4000v v Xv 4000 X X Soluon: E For he amorzaon mehod, he paymen s deermned by 0,000 Xa.085, X For he snkng fund mehod, neres s 0.08(000) = 600 and oal paymen s gven as X, he same as for he amorzaon mehod. Thus he snkng fund depos = X 600 = = 5.3. The snkng fund, a rae j, mus accumulae o 0000 n 0 years. Thus, 5.3s 0,000, 0 j whch yelds (usng calculaor) j = 4.8%. 5. Soluon: D The presen value of he perpeuy = X/. Le B be he presen value of Bran s paymens. X B Xa 0.4 n 0.4 n n a 0.4 v v 0.6 n n X K v X K 0.36, Thus he chary s share s 36% of he perpeuy s presen value. 0

11 6. Soluon: D The gven nformaon yelds he followng amouns of neres pad: 0 0. Seh Jance 5000(0.06)(0) Lor P(0) where P = a The sum s % 7. Soluon: E For Bruce, 0 0 X 00[( ) ( ) ] 00( ). Smlarly, for Robbe, 6.Dvdng he second equaon by he frs gves 0.5( ) whch mples / Thus 0 X 00(.46) (0.46) X 50( ) 8. Soluon: D n Year neres s a v. n n n Year + prncpal repad s ( v ) v. X v v v v v d n n n n ( ). 9. Soluon: B For he frs perpeuy, ( v v ) 0 v / ( v ) 3 3v 0v v / 4. For he second perpeuy, /3 /3 /3 /3 /9 /9 X v v v / ( v ) (3 / 4) /[ (3 / 4) ]

12 30. Soluon: D Under eher scenaro, he company wll have 8,703(0.05) = 4,35 o nves a he end of each of he four years. Under Scenaro A hese paymens wll be nvesed a 4.5% and accumulae o 4,35 s 4,35(4.78) 75,984. Addng he maury value produces 998,687 for a loss of,33. Noe ha only answer D has hs value. The Scenaro B calculaon s 4,35 s 4,35(4.343) 78,6 8,703,000,000, Soluon: D. The presen value s [.07 v.07 v.07 v ].07v.07 v , v Soluon: C. The frs cash flow of 60,000 a me 3 earns 400 n neres for a me 4 recep of 6,400. Combned wh he fnal paymen, he nvesmen reurns,400 a me 4. The presen value s 4, 400(.05) 00,699. The ne presen value s Soluon: B. Usng spo raes, he value of he bond s: 3 60 / / / Soluon: E. Usng spo raes, he value of he bond s: 3 60 / / / a 000( ) 3 The annual effecve rae s he soluon o. Usng a calculaor, he soluon s 8.9%. 35. Soluon: C. Duraon s he negave dervave of he prce mulpled by one plus he neres rae and dvded by he prce. Hence, he duraon s ( 700)(.08)/00 = 7.56.

13 36. Soluon: C The sze of he dvdend does no maer, so assume s. Then he duraon s v v ( Ia) a / / ( d).. a / / d Soluon: B v R v.0 ( Ia) a / j j j Duraon =. a / j d v R v.0 j The neres rae j s such ha ( j).0v.0 /.05 j 0.03/.0. Then he duraon s / d ( j) / j (.05/.0) / (0.03/.0).05/ Soluon: A For he me weghed reurn he equaon s: X 0 0 0X X 0 X X X Then he amoun of neres earned n he year s = 0 and he weghed amoun exposed o earnng neres s 0() + 60(0.5) = 40. Then Y = 0/40 = 5%. 46. Soluon: A The ousandng balance s he presen value of fuure paymens. Wh only one fuure paymen, ha paymen mus be 559.(.08) = The amoun borrowed s a 000. The frs paymen has 000(0.08) = 60 n neres, hus he prncpal repad s = Alernavely, observe ha he prncpal repad n he fnal paymen s he ousandng loan balance a he prevous paymen, or Prncpal repaymens form a geomercally 3 decreasng sequence, so he prncpal repad n he frs paymen s 559. /

14 47. Soluon: B Because he yeld rae equals he coupon rae, Bll pad 000 for he bond. In reurn he receves 30 every sx monhs, whch accumulaes o 30s where j s he sem-annual neres rae. The 0 j 0 equaon of value s 000(.07) 30 s 000 s Usng a calculaor o solve for he neres rae produces j = and so 0 j 0 j %. 48. Soluon: A To receve 3000 per monh a age 65 he fund mus accumulae o 3,000(,000/9.65) = 30, The equaon of value s 30, Xs X / 49. Soluon: D (A) The lef-hand sde evaluaes he deposs a age 0, whle he rgh-hand sde evaluaes he whdrawals a age 7. (B) The lef-hand sde has 6 deposs, no 7. (C) The lef-hand sde has 8 deposs, no 7. (D) The lef-hand sde evaluaes he deposs a age 8 and he rgh-hand sde evaluaes he whdrawals a age 8. (E) The lef-hand sde has 8 deposs, no 7 and 5 whdrawals, no Deleed 5. Soluon: D Because only Bond II provdes a cash flow a me, mus be consdered frs. The bond provdes 05 a me and hus 000/05 = uns of hs bond provdes he requred cash. Ths bond hen also provdes (5) = a me 0.5. Thus Bond I mus provde = a me 0.5. The bond provdes 040 and hus /040 = uns mus be purchased. 5. Soluon: C Because only Morgage II provdes a cash flow a me wo, mus be consdered frs. The morgage provdes Y / a Y a mes one and wo. Therefore, Y = for Y = Morgage I mus provde = 000 a me one and hus X = 000/.06 = The sum s

15 53. Soluon: A Bond I provdes he cash flow a me one. Because 000 s needed, one un of he bond should be purchased, a a cos of 000/.06 = Bond II mus provde 000 a me hree. Therefore, he amoun o be renvesed a me wo s 000/.065 = The purchase prce of he wo-year bond s / The oal prce s Soluon: C Gven he coupon rae s greaer han he yeld rae, he bond sells a a premum. Thus, he mnmum yeld rae for hs callable bond s calculaed based on a call a he earles possble dae because ha s mos dsadvanageous o he bond holder (earles me a whch a loss occurs). Thus, X, he par value, whch equals he redempon value because he bond s a par value bond, mus sasfy 30 Prce = Xa Xv.96 X X Soluon: B Because 40/00 s greaer han 0.03, for early redempon he earles redempon should be 30 evaluaed. If redeemed afer 5 years, he prce s 40a 00 / If he bond s redeemed a maury, he prce s should be seleced, whch s a 00 / The smalles value 56. Soluon: E Gven he coupon rae s less han he yeld rae, he bond sells a a dscoun. Thus, he mnmum yeld rae for hs callable bond s calculaed based on a call a he laes possble dae because ha s mos dsadvanageous o he bond holder (laes me a whch a gan occurs). Thus, X, he par value, whch equals he redempon value because he bond s a par value bond, mus sasfy 0 Prce = Xa Xv X X Soluon: B Gven he prce s less han he amoun pad for an early call, he mnmum yeld rae for hs callable bond s calculaed based on a call a he laes possble dae. Thus, for an early call, he 9 effecve yeld rae per coupon perod, j, mus sasfy Prce = 0.50 a 00v j. Usng he calculaor, j =.67%. We also mus check he yeld f he bond s redeemed a maury. The 0 equaon s 0.50 a 00v j. The soluon s j =.46% Thus, he yeld, expressed as a 0 j nomnal annual rae of neres converble semannually, s wce he smaller of he wo values, or 4.9%. 5 9 j

16 58. Moved o Dervaves secon 59. Soluon: C Frs, he presen value of he lably s PV 35,000a 335, % The duraon of he lably s: 5 v R 35, 000v (35, 000) v 5(35, 000) v,3,5.95 d vr 335, , Le X denoe he amoun nvesed n he 5 year bond. X X Then, (5) (0) X 08, , , Soluon: A The presen value of he frs egh paymens s: v 000(.03) v PV 000v 000(.03) v (.03) v 3, v The presen value of he las egh paymens s: PV 000(.03) 0.97v 000(.03) (0.97) v 000(.03) (0.97 ) v (.03) 0.97v 000(.03) (0.97) v 0.97v Therefore, he oal loan amoun s L = 0, ,55.. 6

17 6. Soluon: E exp 0 r 00 dr 3 r r 3 4 exp exp 0.5ln 3 r dr r exp 0.5ln Soluon: E Le F, C, r, and have her usual nerpreaons. The dscoun s ( C Fr a and he dscoun n he coupon a me s ( C Fr) v n. Then, 94.8 ( C Fr) v ( C Fr) v v v ( C Fr) 94.8(.095) Dscoun 06.53a, ) n 63. Soluon: A Pv 85 P (annual paymen) P I L 5500 (loan amoun) Toal neres = 84.39(8)

18 64. Soluon: D OB 8 8 s , 000(.007) , ,337.0 Pa P Soluon: C If he bond has no premum or dscoun, was bough a par so he yeld rae equals he coupon rae, (90) v (90) v 4(90) v 4(5000) v d v 90v 90v 5000v 95 Ia 4(5000) v 4 d 4 90a 5000v d Or, akng advanage of a shorcu: d a.07. Ths s n half years, so dvdng by wo,.07 d Soluon: A v P(0.08) P(0.07) ( ) v P(0.08) 000 (0.008)(7.45) Soluon: E 3 3 ( s ) ( s ) ( f ) , s ( s ) , s ( s ) ( f ) f

19 68. Soluon: C Le d 0 be he Macaulay duraon a me 0. d a d d d d d a Ths soluon employs he fac ha when a coupon bond sells a par he duraon equals he presen value of an annuy-due. For he duraon jus before he frs coupon he cash flows are he same as for he orgnal bond, bu all occur one year sooner. Hence he duraon s one year less. Alernavely, noe ha he numeraors for d and d are dencal. Tha s because hey dffer only wh respec o he coupon a me (whch s me 0 for hs calculaon) and so he paymen does no add anyhng. The denomnaor for d s he presen value of he same bond, bu wh 7 years, whch s The denomnaor for d has he exra coupon of 50 and so s 550. The desred rao s hen 5000/550 = Soluon: A Le N be he number of shares bough of he bond as ndcaed by he subscrp. N N N C B A (05) 00, N C (00) (5), N (07) (5), N A B 70. Soluon: B All are rue excep B. Immunzaon requres frequen rebalancng. 7. Soluon: D Se up he followng wo equaons n he wo unknowns: A(.05) B(.05) 6000 A(.05) B(.05) 0. Solvng smulaneously gves: A 7.09 B AB

20 7. Soluon: A Se up he followng wo equaons n he wo unknowns. 3 () 5000(.03) B(.03), B(.03), 000 B(.03) () 3(5000)(.03) bb(.03) 0 6, b b.5076 B B b 807. b b b b 73. Soluon: D 9 P A( ) B( ) P A L 95, 000( ) P A( ) 9 B( ) A P 5(95, 000)( ) L 6 Se he presen values and dervaves equal and solve smulaneously A0.7059B78, A 6.080B 375, , 083(.7780 / ) 375, 400 B 47, (.7780 / ) A [78, (47, 630)] / , 59 A.03 B 0

21 74. Soluon: D Throughou he soluon, le j = /. For bond A, he coupon rae s ( )/ = j For bond B, he coupon rae s ( 0.04)/ = j 0.0. The prce of bond A s P j a j 0,000( 0.0) 0,000( ) 0 A. 0 j The prce of bond B s P j a j 0,000( 0.0) 0,000( ) 0 B. 0 j Thus, P P 5,34. [00 ( 00)] a 400a a A B 0 j 0 j 0 j 5,34. / Usng he fnancal calculaor, j = 0.04 and =(0.04)= Soluon: D The nal level monhly paymen s 400, , 000 R 4, a a / The ousandng loan balance afer he 36h paymen s B Ra 4, a 4,057.07(87.87) 356, The revsed paymen s 4, = 3, Thus, 356, , 647.9a a 44 j/ 44 j/ 356, / 3, Usng he fnancal calculaor, j/ = 0.575%, for j = 6.9%. 76. Soluon: D The prce of he frs bond s 000(0.05 / ) a 00( 0.05 / ) 5a 00(.05) , / The prce of he second bond s also, The equaon o solve s 60, a 800( j/ ). 60 j/ The fnancal calculaor can be used o solve for j/ =.% for j = 4.4%.