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

Transcription

1 SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS Interest Theory Ths page ndcates changes made to Study Note FM January 14, 014: Questons and solutons were added. June, 014 Queston 58 was moved to the Dervatves Markets set of sample questons. Questons were added. Many of the questons were re-worded to conform to the current style of queston wrtng. The substance was not changed. December, 014: Questons were added. January, 015: Questons were added. May, 015: Questons were added. Some of the questons n ths study note are taken from past SOA examnatons. These questons are representatve of the types of questons that mght be asked of canddates sttng for the Fnancal Mathematcs (FM) Exam. These questons are ntended to represent the depth of understandng requred of canddates. The dstrbuton of questons by topc s not ntended to represent the dstrbuton of questons on future exams. The followng model solutons are presented for educatonal purposes. Alternatve methods of soluton are, of course, acceptable. Copyrght 015 by the Socety of Actuares. FM PRINTED IN U.S.A. 1

2 1. Soluton: C Gven the same prncpal nvested for the same perod of tme yelds the same accumulated value, the two measures of nterest () 0.04 and must be equvalent, whch means: () 1 e over a one-year perod. Thus, () e ln(1.0404) Soluton: E From basc prncples, the accumulated values after 0 and 40 years are [(1 ) (1 ) (1 ) ] 100 (1 ) (1 ) 4 1 (1 ) 4 4 (1 ) (1 ) 1 (1 ) [(1 ) (1 ) (1 ) ] The rato s 5, and thus (settng x 4 (1 ) ) (1 ) (1 ) x x 5 (1 ) (1 ) x x 5x 5x x x x 1 x x x ( x 1)( x 4) 0. Only the second root gves a postve soluton. Thus x 5 4 x X

3 Annuty symbols can also be used. Usng the annual nterest rate, the equaton s s s 100 5(100) a a (1 ) 1 (1 ) (1 ) 5(1 ) (1 ) 4 and the soluton proceeds as above. 3. Soluton: C 15 Erc s (compound) nterest n the last 6 months of the 8th year s Mke s (smple) nterest for the same perod s 00. Thus, %. 4. Soluton: A The perodc nterest s 0.10(10,000) = Thus, deposts nto the snkng fund are = Then, the amount n snkng fund at end of 10 years s ,133 s. After repayng the loan, the fund has,133, whch rounds to,130. 3

4 5. Soluton: E The begnnng balance combned wth deposts and wthdrawals s (10) = 50. The endng balance of 60 mples 10 n nterest was earned. The denomnator s the average fund exposed to earnng nterest. One way to calculate t s to weght each depost or wthdrawal by the remanng tme: (1) The rate of return s 10/ = = 11.0%. 6. Soluton: C n1 nv 77.1 via n n n1 a nv n nv v n1 n1 an nv nv n n an 1v 1v n v n ln( ) n 19. ln(1.105) To obtan the present value wthout rememberng the formula for an ncreasng annuty, consder the payments as a perpetuty of 1 startng at tme, a perpetuty of 1 startng at tme 3, up to a perpetuty of 1 startng at tme n + 1. The present value one perod before the start of each perpetuty s 1/. The total present value s (1/ )( v v v n ) (1/ ) a n. 4

5 7. Soluton: C The nterest earned s a decreasng annuty of 6, 5.4, etc. Combned wth the annual deposts of 100, the accumulated value n fund Y s 6( Ds) 100s s Deleted 9. Soluton: D For the frst 10 years, each payment equals 150% of nterest due. The lender charges 10%, therefore 5% of the prncpal outstandng wll be used to reduce the prncpal. At the end of 10 years, the amount outstandng s Thus, the equaton of value for the last 10 years usng a comparson date of the end of year 10 s Xa X X % 10. Soluton: B The book value at tme 6 s the present value of future payments: BV 10,000v 800a , The nterest porton s 10,693(0.06) = Soluton: A The value of the perpetuty after the ffth payment s 100/0.08 = 150. The equaton to solve s: X ( v 1.08v 1.08 v ) X ( v v v) X (5) /1.08 X 50(1.08) 54. 5

6 1. Soluton: C Equaton of value at end of 30 years: (1 d / 4) (1.03) 0(1.03) (1 d / 4) [100 0(1.03) ] / /40 1 d / d 4( ) %. 13. Soluton: E The accumulaton functon s The accumulated value of 100 at tme 3 s a t s ds t t 3 ( ) exp ( /100) exp( / 300) exp(3 / 300) The amount of nterest earned from tme 3 to tme 6 equals the accumulated value at tme 6 mnus the accumulated value at tme 3. Thus X [ a(6) / a(3) 1] X ( X)( / ) X ( X) X X X Soluton: A 5 (1 k) a 10(1.09) 5 9.% t (1 k) / (1 k ) /1.09 ( )[1 (1 k) /1.09] (1 k) / (1 k) /1.09 1k k K 3. 99%. t 6

7 15. Soluton: B Opton 1: 000 Pa P 99 Total payments 990 Opton : Interest needs to be [ ] 11, % 16. Soluton: B Monthly payment at tme t s (0.98) t. Because the loan amount s unknown, the outstandng balance must be calculated prospectvely. The value at tme 40 months s the present value of payments from tme 41 to tme 60: OB v v [ ] v 0.98 v 1000, v 1/ (1.0075) v Soluton: C The equaton of value s 98S 98S n n 3n n (1 ) 1 (1 ) n % 7

8 18. Soluton: B Convert 9% convertble quarterly to an effectve rate of j per month: (1 j) 1 or j = Then 60 a 60v ( Ia) Soluton: C For Account K, the amount of nterest earned s X + X = 5 X. The average amount exposed to earnng nterest s 100 (1/)X + (1/4)X = 100. Then 5 X 100. For Account L, examne only ntervals separated by deposts or wthdrawals. Determne the nterest for the year by multplyng the ratos of endng balance to begnnng balance. Then X 1. Settng the two equatons equal to each other and solvng for X, 5 X 13, (15 X ) (5 X )(15 X ) 13, 5 100(15 X ) 3, , 5 1, X X X X 50X, X 10. Then = (5 10)/100 = 0.15 = 15%. 8

9 0. Soluton: A Equatng present values: n v 300v 600v n (0.76) 300(0.76) 600v v v v % Soluton: A The accumulaton functon s: 1 dr t 8 ln8 r 8 r t 8 t 0 0 ( ). a t e e Usng the equaton of value at end of 10 years: 10 a(10) / 8 0, k tk dt k (8 t) dt k 10 18dt 0 a( t) 0 (8 t) / 8 0 0, k k Soluton: D Let C be the redempton value and v1/ (1 ). Then X 1000ra Cv n n n 1 v 1000r (1.0315)( )

10 3. Soluton: D Equate net present values: v 4000v v Xv 4000 X X Soluton: E For the amortzaton method, the payment s determned by 0,000 Xa , X For the snkng fund method, nterest s 0.08(000) = 1600 and total payment s gven as X, the same as for the amortzaton method. Thus the snkng fund depost = X 1600 = = The snkng fund, at rate j, must accumulate to 0000 n 0 years. Thus, 15.13s 0,000, 0 j whch yelds (usng calculator) j = 14.18%. 5. Soluton: D The present value of the perpetuty = X/. Let B be the present value of Bran s payments. X B Xa 0.4 n 0.4 n n a v v 0.6 n n X K v X K 0.36, Thus the charty s share s 36% of the perpetuty s present value. 10

11 6. Soluton: D The gven nformaton yelds the followng amounts of nterest pad: Seth Jance 5000(0.06)(10) Lor P(10) where P = a The sum s % 7. Soluton: E For Bruce, X 100[(1 ) (1 ) ] 100(1 ). Smlarly, for Robbe, 6.Dvdng the second equaton by the frst gves 10.5(1 ) whch mples 1/ Thus 10 X 100(1.146) (0.146) X 50(1 ) 8. Soluton: D n t 1 Year t nterest s a 1 v. n t 1 nt nt Year t+1 prncpal repad s 1 (1 v ) v. X v v v v v d nt1 nt nt nt 1 1 (1 ) Soluton: B For the frst perpetuty, ( v v ) 10 v / (1 v ) 3 3v 10v v / 4. For the second perpetuty, 1/3 /3 1/3 1/3 1/9 1/9 X v v v / (1 v ) (3 / 4) /[1 (3 / 4) ]

12 30. Soluton: D Under ether scenaro, the company wll have 8,703(0.05) = 41,135 to nvest at the end of each of the four years. Under Scenaro A these payments wll be nvested at 4.5% and accumulate to 41,135 s 41,135(4.78) 175,984. Addng the maturty value produces 998,687 for a loss of 1,313. Note that only answer D has ths value. The Scenaro B calculaton s 41,135 s 41,135(4.343) 178,61 8,703 1,000,000 1, Soluton: D. The present value s [1.07 v 1.07 v 1.07 v ] v1.07 v , v Soluton: C. The frst cash flow of 60,000 at tme 3 earns 400 n nterest for a tme 4 recept of 6,400. Combned wth the fnal payment, the nvestment returns 1,400 at tme 4. The present value s 4 1, 400(1.05) 100,699. The net present value s Soluton: B. Usng spot rates, the value of the bond s: 3 60 / / / Soluton: E. Usng spot rates, the value of the bond s: 3 60 / / / a 1000(1 ) 3 The annual effectve rate s the soluton to. Usng a calculator, the soluton s 8.9%. 35. Soluton: C. Duraton s the negatve dervatve of the prce multpled by one plus the nterest rate and dvded by the prce. Hence, the duraton s ( 700)(1.08)/100 =

13 36. Soluton: C The sze of the dvdend does not matter, so assume t s 1. Then the duraton s t1 t t1 tv v t ( Ia) a / 1/ ( d) a 1/ 1/ d Soluton: B Duraton = t t t tv Rt tv 1.0 t1 t1 t t t v Rt v 1.0 t1 t1 ( Ia) a / j j j 1. a 1/ j d j 1 The nterest rate j s such that (1 j) 1.0v 1.0 /1.05 j 0.03/1.0. Then the duraton s 1/ d (1 j) / j (1.05/1.0) / (0.03/1.0) 1.05/ Soluton: A For the tme weghted return the equaton s: 1 X X 1X 10 X X X Then the amount of nterest earned n the year s = 10 and the weghted amount exposed to earnng nterest s 10(1) + 60(0.5) = 40. Then Y = 10/40 = 5%. 46. Soluton: A The outstandng balance s the present value of future payments. Wth only one future payment, that payment must be 559.1(1.08) = The amount borrowed s a 000. The frst payment has 000(0.08) = 160 n nterest, thus the prncpal repad s = Alternatvely, observe that the prncpal repad n the fnal payment s the outstandng loan balance at the prevous payment, or Prncpal repayments form a geometrcally 3 decreasng sequence, so the prncpal repad n the frst payment s /

14 47. Soluton: B Because the yeld rate equals the coupon rate, Bll pad 1000 for the bond. In return he receves 30 every sx months, whch accumulates to 30s where j s the sem-annual nterest rate. The 0 j 10 equaton of value s 1000(1.07) 30 s 1000 s Usng a calculator to solve for the nterest rate produces j = and so 0 j 0 j %. 48. Soluton: A To receve 3000 per month at age 65 the fund must accumulate to 3,000(1,000/9.65) = 310, The equaton of value s 310, Xs X /1 49. Soluton: D (A) The left-hand sde evaluates the deposts at age 0, whle the rght-hand sde evaluates the wthdrawals at age 17. (B) The left-hand sde has 16 deposts, not 17. (C) The left-hand sde has 18 deposts, not 17. (D) The left-hand sde evaluates the deposts at age 18 and the rght-hand sde evaluates the wthdrawals at age 18. (E) The left-hand sde has 18 deposts, not 17 and 5 wthdrawals, not Deleted 51. Soluton: D Because only Bond II provdes a cash flow at tme 1, t must be consdered frst. The bond provdes 105 at tme 1 and thus 1000/105 = unts of ths bond provdes the requred cash. Ths bond then also provdes (5) = at tme 0.5. Thus Bond I must provde = at tme 0.5. The bond provdes 1040 and thus /1040 = unts must be purchased. 5. Soluton: C Because only Mortgage II provdes a cash flow at tme two, t must be consdered frst. The mortgage provdes Y / a Y at tmes one and two. Therefore, Y = for Y = Mortgage I must provde = 1000 at tme one and thus X = 1000/1.06 = The sum s

15 53. Soluton: A Bond I provdes the cash flow at tme one. Because 1000 s needed, one unt of the bond should be purchased, at a cost of 1000/1.06 = Bond II must provde 000 at tme three. Therefore, the amount to be renvested at tme two s 000/1.065 = The purchase prce of the two-year bond s / The total prce s Soluton: C Gven the coupon rate s greater than the yeld rate, the bond sells at a premum. Thus, the mnmum yeld rate for ths callable bond s calculated based on a call at the earlest possble date because that s most dsadvantageous to the bond holder (earlest tme at whch a loss occurs). Thus, X, the par value, whch equals the redempton value because the bond s a par value bond, must satsfy Xa Xv X X Prce = Soluton: B Because 40/100 s greater than 0.03, for early redempton the earlest redempton should be 30 evaluated. If redeemed after 15 years, the prce s 40a 100 / If the bond s redeemed at maturty, the prce s should be selected, whch s a 1100 / The smallest value 56. Soluton: E Gven the coupon rate s less than the yeld rate, the bond sells at a dscount. Thus, the mnmum yeld rate for ths callable bond s calculated based on a call at the latest possble date because that s most dsadvantageous to the bond holder (latest tme at whch a gan occurs). Thus, X, the par value, whch equals the redempton value because the bond s a par value bond, must satsfy Xa Xv X X 100. Prce =

16 57. Soluton: B Gven the prce s less than the amount pad for an early call, the mnmum yeld rate for ths callable bond s calculated based on a call at the latest possble date. Thus, for an early call, the 19 effectve yeld rate per coupon perod, j, must satsfy Prce = a 100v j. Usng the calculator, j =.86%. We also must check the yeld f the bond s redeemed at maturty. The 0 equaton s a 1100v j. The soluton s j =.46% Thus, the yeld, expressed as a 0 j nomnal annual rate of nterest convertble semannually, s twce the smaller of the two values, or 4.9%. 19 j 58. Moved to Dervatves secton 59. Soluton: C Frst, the present value of the lablty s PV 35,000a 335, % The duraton of the lablty s: t 15 tv Rt 35, 000v (35, 000) v 15(35, 000) v,31,51.95 d t vr 335, , t Let X denote the amount nvested n the 5 year bond. X X (5) 1 (10) X 08,556. Then, 335, , Soluton: A The present value of the frst eght payments s: v 000(1.03) v PV 000v 000(1.03) v (1.03) v 13, v The present value of the last eght payments s: PV 000(1.03) 0.97v 000(1.03) (0.97) v 000(1.03) (0.97 ) v (1.03) 0.97v 000(1.03) (0.97) v v Therefore, the total loan amount s L = 0, ,

17 61. Soluton: E exp 0 r 100 dr 3 r r t 3 4 exp 0.5 t exp 0.5ln 3 r dr r t t 4 exp 0.5ln t t t 1 6. Soluton: E Let F, C, r, and have ther usual nterpretatons. The dscount s ( C Fr a and the dscount n 1 the coupon at tme t s ( C Fr) v n t. Then, ( C Fr) v ( C Fr) v v v ( C Fr) 194.8(1.095) Dscount 06.53a 1, ) n 63. Soluton: A Pv P (annual payment) P I L 5500 (loan amount) Total nterest = 84.39(8)

18 64. Soluton: D OB s , 000(1.007) , , Pa P Soluton: C If the bond has no premum or dscount, t was bought at par so the yeld rate equals the coupon rate, (190) v (190) v 14(190) v 14(5000) v d v 190v 190v 5000v 95 Ia 7(5000) v 14 d a 5000v d Or, takng advantage of a shortcut: d a Ths s n half years, so dvdng by two, d Soluton: A v P(0.08) P(0.07) 1 ( ) v P(0.08) (0.008)(7.45) Soluton: E (1 s ) (1 s ) (1 f ) , s (1 s ) , s (1 s ) (1 f ) f

19 68. Soluton: C Let d 0 be the Macaulay duraton at tme 0. d d d d d a d a Ths soluton employs the fact that when a coupon bond sells at par the duraton equals the present value of an annuty-due. For the duraton just before the frst coupon the cash flows are the same as for the orgnal bond, but all occur one year sooner. Hence the duraton s one year less. Alternatvely, note that the numerators for d 1 and d are dentcal. That s because they dffer only wth respect to the coupon at tme 1 (whch s tme 0 for ths calculaton) and so the payment does not add anythng. The denomnator for d s the present value of the same bond, but wth 7 years, whch s The denomnator for d 1 has the extra coupon of 50 and so s 550. The desred rato s then 5000/550 = Soluton: A Let N be the number of shares bought of the bond as ndcated by the subscrpt. N N N C B A (105) 100, N C (100) (5), N (107) (5), N A B 70. Soluton: B All are true except B. Immunzaton requres frequent rebalancng. 19

20 71. Soluton: D Set up the followng two equatons n the two unknowns: A(1.05) B(1.05) 6000 A(1.05) B(1.05) 0. Solvng smultaneously gves: A B AB Soluton: A Set up the followng two equatons n the two unknowns. 3 (1) 5000(1.03) B(1.03) 1, B(1.03) 1, 000 B(1.03) () 3(5000)(1.03) bb(1.03) 0 16, b b.5076 B B b b b b b 73. Soluton: D 0 9 P A(1 ) B(1 ) P A L 95, 000(1 ) P A(1 ) 9 B(1 ) A P 5(95, 000)(1 ) L 6 Set the present values and dervatves equal and solve smultaneously A0.7059B78, A B 375, , 083( / ) 375, 400 B 47, ( / ) A [78, (47, 630)] / , 59 A B

21 74. Soluton: D Throughout the soluton, let j = /. For bond A, the coupon rate s ( )/ = j For bond B, the coupon rate s ( 0.04)/ = j 0.0. The prce of bond A s P j a j 10,000( 0.0) 10,000(1 ) 0 A. 0 j The prce of bond B s P j a j 10,000( 0.0) 10,000(1 ) 0 B. 0 j Thus, P P 5,341.1 [00 ( 00)] a 400a a A B 0 j 0 j 0 j 5,341.1 / Usng the fnancal calculator, j = 0.04 and =(0.04)= Soluton: D The ntal level monthly payment s 400, , 000 R 4, a a / The outstandng loan balance after the 36th payment s B Ra 4, a 4,057.07( ) 356, The revsed payment s 4, = 3, Thus, 356, , a a 144 j/1 144 j/1 356, / 3, Usng the fnancal calculator, j/1 = 0.575%, for j = 6.9%. 76. Soluton: D The prce of the frst bond s 1000(0.05 / ) a 100( / ) 5a 100(1.05) , / The prce of the second bond s also 1, The equaton to solve s 60 1, a 800(1 j/ ). 60 j/ The fnancal calculator can be used to solve for j/ =.% for j = 4.4%. 1

22 77. Soluton: E Let n = years. The equaton to solve s 1000(1.03) (1000)(1.005) n 1n nln1.03 ln1000 1nln1.005 ln n n Ths s 85.3 months. The next nterest payment to Lucas s at a multple of 6, whch s 88 months. 78. Soluton: B The endng balance s 5000(1.09) + 600sqrt(1.09) = The tme-weghted rate of return s (500/5000) x [ /( )] 1 = Soluton: A Equatng the accumulated values after 4 years provdes an equaton n K. 4 K 4 1 dt 5 0 K 0.5t exp K 1 4ln( K) dt 4ln(K 0.5t) 4ln( K 1) 4ln( K) 4ln K K K 0.04K 1 K 5. Therefore, 0 0 K 0.5t K 4 X 10(1 5/ 5) Soluton: C To repay the loan, the snkng fund must accumulate to The depost s (1000). Therefore, s (1 0.8 ) (1 0.8 ) /

23 81. Soluton: D 5 a5 The outstandng balance at tme 5 s 100( Da) 100. The prncple repad n the 6th 5 5 a5 payment s X 500 (100) a 100 a. The amount borrowed s the present value of all 50 payments, 500a v 100( Da). Interest pad n the frst payment s then 500a v 100( Da) (1 v ) 100 v (5 a ) v 500v v 100a Xv Soluton: A The exposure assocated wth produces results qute close to a true effectve rate of nterest as long as the net amount of prncpal contrbuted at tme t s small relatve to the amount n the fund at the begnnng of the perod. 83. Soluton: E The tme-weghted weght of return s j = (10,000 / 100,000) x (130,000 / 150,000) x (100,000 / 80,000) 1 = 30.00%. Note that 150,000 = 10, ,000 and 80,000 = 130,000 50, Soluton: C The accumulated value s 1000s 50, Ths must provde a sem-annual annutydue of Let n be the number of payments. Then solve 3000a 50,38.16 for n = n Therefore, there wll be 6 full payments plus one fnal, smaller, payment. The equaton s 6 50, a X(1.04) wth soluton X = Note that the whle the fnal payment s the 7th payment, because ths s an annuty-due, t takes place 6 perods after the annuty begns. 3

24 85. Soluton: D For the frst perpetuty, For the second perpetuty, 1 R R 7.1(1.0875) R (1.0875) Soluton: E a 5v 10, ( Ia) Xv a 100 Xv a , X 1075 X Soluton: C 5000 Xs (1.05) X (1.763) Soluton: E The monthly payment on the orgnal loan s 65, 000 a 180 8/1% After 1 payments the outstandng balance s 61.17a 6, The revsed payment s 168 8/1% 6, a 168 6/1% 4

25 89. Soluton: E At the tme of the fnal depost the fund has 750s 5, Ths s an mmedate annuty because the evaluaton s done at the tme the last payments s made (whch s the end of 17 the fnal year). A tuton payment of 6000(1.05) 13,75.11 s made, leavng 11, It earns 7%, so a year later the fund has 11,747.16(1.07) = 1, Tuton has grown to 13,75.11(1.05) = 14, The amount needed s 14, , = 1, Soluton: B The coupons are 1000(0.09)/ = 45. The present value of the coupons and redempton value at 40 5% per semannual perod s P 45a 100(1.05) Soluton: A For a bond bought at dscount, the mnmum prce wll occur at the latest possble redempton 0 date. P 50a 1000(1.06) Soluton: C % 93. Soluton: D The accumulated value of the frst year of payments s 000s 4, Ths amount ncreases at % per year. The effectve annual nterest rate s The present value s then 5 5 k1 k P 4, ( ) 4,671.1 k1 1.0 k , , Ths s 56 less than the lump sum amount. k 5

26 94. Soluton: A The monthly nterest rate s 0.07/1 = fve years from today has value (1.006) The equaton of value s n n (1.006) 3400(1.006). Let x n. Then, solve the quadratc equaton 3400x 1700x x (3400) Then, (3400)( ) n nln(1.006) ln(0.935) n To ensure there s 6500 n fve years, the deposts must be made earler and thus the maxmum ntegral value s Soluton: C d d 39 1 d ( d ) 38 38( d 4) 4 1 d / d d 1/ ( ) d / %. 96. Soluton: C Thee monthly nterest rate s 0.04/1 = The quarterly nterest rate s The nvestor makes 41 quarterly deposts and the endng date s 14 months from the start. Usng January 1 of year y as the comparson date produces the followng equaton: X 41 k1 Substtutng X k gves answer (C). 6

27 97. Soluton: D Convert the two annual rates, 4% and 5%, to two-year rates as and 1, The accumulated value s 100 s (1.05) 100s 100( )(1.1551) 100(.31801) Wth only fve payments, an alternatve approach s to accumulate each one to tme ten and add them up. The two-year yeld rate s the soluton to 100s Usng the calculator, the two-year rate s The annual rate s whch s 4.58%. Solve for the -year yeld and then convert to annual yeld: 98. Soluton: C , 000ä Xä % % , 000( ) X ( ) 99. Soluton: B PV perp. (15, 000) 15, , , , a X a 179, X , X 17,384 7

28 100. Soluton: A 14 a 14(1.03) (.50 X ) a X (1.03) (.50 X ) X X 598 X Soluton: D The amount of the loan s the present value of the deferred ncreasng annuty: 30 a 30(1.03) a Ia a (1.03) ( ) (1.03 )(500) 64, / Soluton: C (1 ) (1.03) (1 ) (1.03) 50, 000 (1 ) 5, (1 ) ( 0.03) ,000 / (1 ) 5,000 9 (1 ) 10 1/ The accumulated amount s ( ) (1.03) 50, 000 ( ) 797, ( ) ( 0.03) 103. Soluton: D The frst payment s,000, and the second payment of,010 s tmes the frst payment. Snce we are gven that the seres of quarterly payments s geometrc, the payments multply by every quarter. Based on the quarterly nterest rate, the equaton of value s 3 3, , 000, 000, 000(1.005) v, 000(1.005) v, 000(1.005) v v v, 000 /100, 000 v 0.98 / The annual effectve rate s v / %. 8

29 104. Soluton: A Present value for the frst 10 years s ln 1.06 Present value of the payments after 10 years s 10 s s ds ln 1.06 ln 1.03 Total present value = Soluton: C 10 1 dt 5 t1 5 1, X 1.06 e 75, , X 75, X 7,56.83 X 4, Soluton: D The effectve annual nterest rate s d 1 1 (1 ) 1 ( ) 1 5.8% The balance on the loan at tme s 15,000,000(1.058) 16,796,809. The number of payments s gven by 1, 00,000a n 16,796,809 whch gves n = => 9 payments of 1,00,000. The fnal equaton of value s , 00, 000 a X(1.058) 16, 796,809 X (16, 796,809 16, 61, 01)( ) 959, Soluton: C 4 1 v 0.55(1 v ) (1 v ) v v n n n 1 v 0.147(1 v ) 1 v ( ) / v n ln( ) / ln( ) 9

30 108. Soluton: B Let X be the annual depost on the snkng fund. Because the snkng fund deposts must accumulate to the loan amount, L Xs X. At tme 7 the fund has Xs X. Ths s 641 short of the loan amount, so a second equaton s L = X Combnng the two equatons gves X X = 641 wth mples X= Soluton: C The monthly payment s 00,000 / a Usng the equvalent annual effectve rate of 6.17%, the present value (at tme 0) of the fve extra payments s 41,99.54 whch reduces the orgnal loan amount to 00,000 41,99.54 = 158, The number of months requred s the soluton to 158, a n whch s Usng calculator, n = months are needed to pay off ths amount. So there are 15 full payments plus one fractonal payment at the end of the 16th month, whch s December 31, Soluton: D The annual effectve nterest rate s 0.08/(1 0.08) = The level payments are 500,000 / a 500,000 / ,535. Ths rounds up to 18,000. The equaton of value for X s , 000 a X( ) 500, X (500, , )(1.5179) 15, Soluton: B The accumulated value s the recprocal of the prce. The equaton s X[(1/0.94)+(1/0.95)+(1/0.96)+(1/0.97)+(1/0.98)+(1/0.99)] = 100,000. X= 16,078 30

31 11. Soluton: D Let P be the annual payment. The ffth lne s obtaned by solvng a quadratc equaton. P 10 (1 v ) 3600 Pv v v v v v v v 3600 X P 48, Soluton: A Let j = perodc yeld rate, r = perodc coupon rate, F = redempton (face) value, P = prce, n = 1 number of tme perods, and v j 1 j. In ths problem, j , r = 0.035, P = 10,000, and n = 50. The present value equaton for a bond s yelds n j nj P Fv Fra ; solvng for the redempton value F P 10, , 000 F 9,918.. v ra a 50 ( ) (3.6044) n j nj 114. Soluton: B Jeff s monthly cash flows are coupons of 10,000(0.09)/1 = 75 less loan payments of 000(0.08)/1 = for a net ncome of At the end of the ten years (n addton to the 61.67) he receves 10,000 for the bond less a,000 loan repayment. The equaton s a 8000(1 /1) (1) (1) 10 /1 / (1) %. 31

32 115. Soluton: B The present value equaton for a par-valued annual coupon bond s the coupon rate r yelds n P Fv P 1 v r Fa a F a n n n n. P Fv Fra ; solvng for All three bonds have the same values except for F. We can wrte r = x(1/f) + y. From the frst two bonds: = x/ y and = x/ y. Then, = x(1/1000 1/1100) for x = 96.8 and y = /1000 = For the thrd bond, r = 96.8/ = =.93%. n n 116. Soluton: A () () The effectve sem-annual yeld rate s %. Then, c(1.0) v c(1.0 v) c(1.0 v) 50v 1.0 v (1.0 v) 1 c 50v 1.015c c v v (1.0 v) c 50v 1.015c c v 117. Soluton: E Book values are lnked by BV3(1 + ) Fr = BV4. Thus (1.06) Fr = Therefore, the coupon s Fr = The prospectve formula for the book value at tme 3 s ( n3) (1.06) 0.06 ( n3) (1.06) ln( / ) n ln(1.06) ( n3) Thus, n = 0. Note that the fnancal calculator can be used to solve for n 3. 3

33 118. Soluton: A Book values are lnked by BV3(1 + ) Fr = BV4. Thus BV3(1.04) 500(0.035) = BV Therefore, BV3 = [500(0.035) ]/0.04 = The prospectve formula for the book value at tme 3 s, where m s the number of sx-month perods. ( m3) (0.035) 500(1.04) 0.04 ( m3) (1.04) ln(11/ 31.5) m ln(1.04) ( m3) Thus, m = 13 and n = m/ = 6.5. Note that the fnancal calculator can be used to solve for m Soluton: C S 1 0, S so S (1.06)(1.04) , 1 S1 3 1 S so S [(1.08)( ) ] %. 1 S 3 1/3, Soluton: D Interest earned s 55,000 50,000 8, ,000 = 7,000. Equatng the two nterest measures gves the equaton 7, , 000 (16, 000 / 3) 10, 000(1 t) , (55, , , 000 t) t [7, (45,333.33)] /1,

34 11. Soluton: B 0(1) 1( v) ( v ) v v The Macaulay duraton of Annuty A s 0.93, whch leads to the 1 v v 1 v v quadratc equaton 1.07v 0.07v The unque postve soluton s v = 0.9. The Macaulay duraton of Annuty B s 3 0(1) 1( v) ( v ) 3( v ) 1 v v v Soluton: D Wth v =1/1.07, (40, 000) v 3(5, 000) v 4(100, 000) v D , 000v 5, 000v 100, 000v Soluton: C A The Macaulay duraton of Bond A s MacD a % A A MacD The modfed duraton of Bond A s ModD The modfed duraton of Bond B s also The Macaulay duraton of Bond B s B B MacD ModD (1 0.1/ ) Soluton: C n nv Ia n0 d 30 MacD n a 1/ d (1 ) / v n0 1/ ( ) (1 ) / 1 MacD 30 Then, ModD and so = 1/30. 34

35 15. Soluton: D Let D be the next dvdend for Stock J. The value of Stock F s 0.5D/(0.088 g). The value of Stock J s D/( g). The relatonshp s 0.5D D g g 0.5 D(0.088 g) D(0.088 g).5g 0.13 g %. 16. Soluton: B I) False. The yeld curve structure s not relevant. II) True. III) False. Matchng the present values s not suffcent when nterest rates change. 17. Soluton: A The present value functon and ts dervatves are P( ) X Y(1 ) 500(1 ) 1000(1 ) P( ) 3 Y(1 ) 500(1 ) 4000(1 ) 4 5 P Y ( ) 1 (1 ) 1000(1 ) 0, 000(1 ). The equatons to solve for matchng present values and duraton (at = 0.10) and ther soluton are P(0.1) X Y P(0.1).0490Y Y / X (1413.8) The second dervatve s P (0.1) 1(1413.8)(1.1) 1000(1.1) 0,000(1.1) Redngton mmunzaton requres a postve value for the second dervatve, so the condton s not satsfed. 35

36 18. Soluton: D Ths soluton uses tme 8 as the valuaton tme. The two equatons to solve are P X 8 y ( ) 300, 000(1 ) (1 ) 1, 000, P y X 7 y ( ) 600,000(1 ) (8 ) (1 ) 0. Insertng the nterest rate of 4% and solvng: 8 y 300, 000(1.04) X (1.04) 1, 000, y 600, 000(1.04) (8 yx ) (1.04) 0 X y 8 (1.04) [1, 000, , 000(1.04) ] / , , 000 (8 y)(1.04) (493,595.85) 0 y , 000 / [493,595.85(1.04) ] X 493,595.85(1.04) 701, Soluton: A Ths soluton uses Macaulay duraton and convexty. The same concluson would result had modfed duraton and convexty been used. The labltes have present value have a present value of / / Only portfolos A, B, and E 5 The duraton of the labltes s [(573) /1.07 5(701) /1.07 ]/ The duraton of a zero coupon bond s ts term. The portfolo duraton s the weghted average of the terms. For portfolo A the duraton s [500(1) + 500(6)]/1000 = 3.5. For portfolo B t s [57(1) + 48(6)]/1000 = For portfolo E t s 3.5. Ths elmnates portfolo B. 5 The convexty of the labltes s [4(573) /1.07 5(701) /1.07 ]/ The convexty of a zero-coupon bond s the square of ts term. For portfolo A the convexty s [500(1) + 500(36)]/1000 = 18.5 whch s greater than the convexty of the labltes. Hence portfolo A provdes Redngton mmunzaton. As a check, the convexty of portfolo E s 1.5, whch s less than the lablty convexty Soluton: D The present value of the labltes s 1000, so that requrement s met. The duraton of the 1 3 labltes s 40.11[1.1 (1.1) 3(1.1) ]/ Let X be the nvestment n the oneyear bond. The duraton of a zero-coupon s ts term. The duraton of the two bonds s then [X + (1000 X)(3)]/1000 = X. Settng ths equal to and solvng yelds X =

37 131. Soluton: A Let x, y, and z represent the amounts nvested n the 5-year, 15-year, and 0-year zero-coupon bonds, respectvely. Note that n ths problem, one of these three varables s 0. The present value, Macaulay duraton, and Macaulay convexty of the assets are, respectvely, x y z,, x y z x y z x y z x y z. We are gven that the present value, Macaulay duraton, and Macaulay convexty of the labltes are, respectvely, 9697, 15.4, and Snce present values and Macaulay duratons need to match for the assets and labltes, we have the two equatons 5x 15y 0z x y z 9697, x y z Note that 5 and 15 are both less than the desred Macaulay duraton 15.4, so z cannot be zero. So try ether the 5-year and 0-year bonds (.e. y = 0), or the 15-year and 0-year bonds (.e. x = 0). In the former case, substtutng y = 0 and solvng for x and z yelds (0 15.4)9697 (15.4 5)9697 x and z We need to check f the Macaulay convexty of the assets exceeds that of the labltes. The Macaulay convexty of the assets s 5 ( ) 0 (6619.8) 81.00, whch exceeds 9697 the Macaulay convexty of the labltes, The company should nvest 3077 for the 5-year bond and 660 for the 0-year bond. Note that settng x = 0 produces y = and z = and the convexty s 33.40, whch s less than that of the labltes. 13. Soluton: E The correct answer s the lowest cost portfolo that provdes for $11,000 at the end of year one and provdes for $1,100 at the end of year two. Let H, I, and J represent the face amount of each purchased bond. The tme one payment can be exactly matched wth H + 0.1J = 11,000. The tme two payment can be matched wth I + 1.1J = 1,100. The cost of the three bonds s H/1.1 + I/ J. Ths functon s to be mnmzed under the two constrants. Substtutng for H and I gves (11, J)/1.1 + (1, J)/ J = 19, J. Ths s mnmzed by purchasng the largest possble amount of J. Ths s 1,100/1.1 = 10, Then, H = 11, (10,803.57) = The cost of Bond H s /1.1 = 8,

38 133. Soluton: C The strategy s to use the two hghest yeldng assets: the one year zero coupon bond and the two year zero coupon bond. The cost of these bonds s 5,000 / ,000 / ,