MATH 373 Tes 4 Fall 017 December 1, 017 1. A hree year bond has an annual coupon of 40 and a mauriy value of 1100. Calculae he Modified Convexiy for his bond a an annual effecive ineres rae of 6.5%. ModCon v C ( )( 1) v Cv (1.065) 40(1)()(1.065) 40()(3)(1.065) 1140(3)(4)(1.065) 3 40a 1100(1.065) 1 3 3 10.07065 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
. Sanley Insurance Company has agreed o pay Jacqueline 400,000 a he end of each year for hree years. Sanley wans o use he following hree bonds o exacly mach he paymens o Jacqueline. a. Bond A is a one year bond wih a mauriy value of 1000, annual coupons of 70, and a price of 1005. b. Bond B is a wo year bond wih annual coupons of 00 and a mauriy value of 1000. I sells o yield an annual effecive ineres rae of 8%. c. Bond C is a hree year bond wih annual coupons of 70. The mauriy value and he price of his bond is 3000. Calculae he cos o buy he bonds o exacly mach he paymens. Assume ha we can purchase parial bonds. Soluions: Time Year 1 Year Year 3 Paymens 400,000 400,000 400,000 Bond A 1070 0 0 Bond B 00 100 0 Bond C 70 70 370 Le A be he number of Bond A purchased. Le B be he number of Bond B purchased. Le C be he number of Bond C purchased. 400, 000 370( C) 400, 000 C 1.34 370 400, 000 (70)(1.34) 100( B) 70( C) 400, 000 B 305.8104 100 1070( A) 00( B) 70( C) 400, 000 400, 000 00(305.8104) 70(1.34) A 85.8041 1070 Toal Cos ( A)(1005) ( B)(Price of B) ( C)(3000) 1 (1.08) Price of B = 00a 1000v 00 1000(1.08) 113.99 0.08 (85.8041)(1005) (305.8104)(113.99) (1.34)(3000) 1, 05, 456.49 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
3. A 30 year bond maures for 10,000 and has semi-annual coupons of 300. Calculae he Modified duraion a an annual effecive ineres rae of 10.5%. i () 0.5 (1.105) 1 0.05 1 (1.05) 0.05 60 60 Price 300 10, 000(1.05) 614.14 Modified Duraion = 0.5 1.5 8 30 Cv 300(0.5) v 300(1) v 300(1.5) v... 300(30) v (10, 000)(30) v 1 v Cv 614.14 1.105 0.5 1 1.5 30 30 150(1.105) 300(1.105) 450(1.105)... 9000(1.105) (10, 000)(30) v 1 614.14 1.105 1 3 60 30 150(1.05) 300(1.05) 450(1.05)... 9000(1.05) 10, 000(30) v 1 614.14 1.105 a (614.14) 1.105 150 150a 60 1.05 (300, 000) 1.105 60 0.05 60 0.05 0.05 60 30 60 60 1 (1.05) 150 1 (1.05) 150 0.05 60 60(1.05) 300, 0001.105 0.05 0.05 (614.14) 1.105 30 9.641 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
4. A hree year bond wih annual coupons of 300 maures for 4000. Calculae he Macaulay Convexiy of his bond a an annual effecive rae of 5%. MacCon C ( ) v Cv 300(1 )(1.05) 300( )(1.05) 4300(3 )(1.05) 3 300a 4000(1.05) 1 3 3 34,804.6647 8.14654 47.34803 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
5. The Moses Life Insurance Company owns he following wo bonds: Price Macaulay Duraion Macaulay Convexiy Bond A 60,000 A 90 Bond B 40,000 1 15 Using an ineres rae of 8%, he Modified Convexiy of his bond porfolio is 98.654. Deermine A. Bond A MacDur MacCon A 90 ModCon (1.08) (1.08) Bond B MacDur MacCon 1 15 ModCon 117.4554 (1.08) (1.08) A 90 (60, 000) (40, 000)(117.4554) A A B B (1.08) P C P C ModConPor 98.654 A B P P 60, 000 40, 000 A 90 98.654(100, 000) (40, 000)(117.4554) 86.1197 (1.08) 60, 000 A 86.1197(1.08) 90 10.45 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
6. Billie has promised o make a paymen of 1,000,000 o Bokun a he end of 7.5 years. Billie wans o immunize his paymen using Reddingon Immunizaion and he following wo zero coupon bonds: a. Bond 1 is a zero coupon bond wih a mauriy value of 10,000 a he end of 4 years. b. Bond is a zero coupon bond wih a mauriy value of 15,000 a he end of 8 years. Assuming an ineres rae of 8%, deermine he number of Bond 1 which Billie should purchase. Assume ha we can purchase parial bonds. We can use he shorcu o do his problem. D D D Bond Liabiliy Amoun of Money o be used o Purchase Bond 1 = ( PVofLiabiliy) D Bond Bond1 8 7.5 (1,000,000)(1.08) 7.5 70,18.98683 8 4 4 Price of Bond 1 = (10,000)(1.08) 7350.9857 Amoun of Money 70,18.98683 Number of Bonds = 9.5483 Price of Bond 1 7350.9857 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
7. A perpeuiy due pays 100 a he sar of each year. Calculae he Macaulay Duraion a an ineres rae of 6%. MacDur C v v v v 0 1 () (100)(0) (100)(1) (100)()... 0 1 Cv (100) v (100) v (100) v... 100 100 1 1 1 (100)(1) v (100)() v... i i i 0 1 (100) v (100) v (100) v... 100 (1 i) (1 i) i 1 1 0.06 16.67 1.06 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
8. Graham owns a bond wih a price of 130,000 a an annual effecive yield rae of 7%. The bond has a Modified Duraion of 14.0187 and a Modified Convexiy of 100 a an annual effecive ineres rae of 7%. Graham esimaes ha price of his bond a an ineres rae of i using he firs order Macaulay approximaion o be 117,886.61. Deermine i. Firs Order Macaulay Approximaion MacDur 1 i0 P i0 P( i) ( ) 1 i ModDur V MacDur MacDur MacDur 1 ( ) 14.0187 (1.07) ( ) (14.0187)(1.07) 15 1.07 1.07 117,886.61 Pi ( ) 117,886.61 (130,000) 1 i 1 i 130,0 00 15 1 15 0.993500466 1.07 1 i 1.077 i 0.077 0.993500466 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
9. You are given he following spo ineres raes: Time () r 0.5.1% 1.0.3% 1.5.6%.0 3.0%.5 3.3% 3.0 3.7% 3.5 4.% 4.0 4.8% 4.5 5.5% 5.0 6.0% Megan purchases a wo year par value bond wih semi-annual coupons a a rae of 8% converible semi-annually. The par value of he bond is 10,000. Calculae he price of he bond. Coupon (10, 000)(0.08 / ) 400 400 400 400 10,400 P 0.5 1 1.5 10,974.76 (1.01) (1.03) (1.06) (1.03) May 3, 018 Copyrigh Jeffrey Beckley 017, 018
10. The following hree bonds are priced using he same spo ineres raes: a. Bond A is a one year bond wih a mauriy value of 1000, annual coupons of 70, and a price of 1005. b. Bond B is a wo year bond wih annual coupons of 00 and a mauriy value of 1000. I sells o yield an annual effecive ineres rae of 8%. c. Bond C is a hree year bond wih annual coupons of 70. The mauriy value and he price of he bond are 3000. Krisin has a hree year annuiy immediae wih annual paymens of 1000. Using he same spo ineres raes, deermine he accumulaed value of Krisin s annuiy. Firs we need o find he spo raes using boosrapping. Then, using he spo raes, we will find he presen value of he annuiy. Finally, we will find he accumulaed value of he annuiy. 1070 1070 1005 r1 1 0.0646766 1 r 1005 1 00 100 1 (1.08) Price of B = 00 1000 00 1000(1.08) 113.99 a v 1 r1 (1 r) 0.08 100 00 100 113.99 106.14 r 1 0.08140 (1 r ) 1.0646766 106.14 0.5 70 70 370 370 70 70 3000 3000 515.50 1 r (1 r ) (1 r ) (1 r ) 1.0646766 (1.08140) 3 3 1 3 3 370 r3 515.50 1/3 1 0.091373 PV 1000 1000 1000 1000 1000 1000 1 r (1 r ) (1 r ) 1.0646766 (1.08140) (1.091373) 3 1 3 3 563.64 AV PV r 3 3 (1 3) (563.64)(1.091373) 333.55 May 3, 018 Copyrigh Jeffrey Beckley 017, 018
11. You are given: a. f[0,1] 0.05 b. f[1,] 0.06 c. f[,3] 0.07 d. f[3,4] 0.08 Calculae he price of a zero coupon bond ha maures for 100,000 a he end of four years. 100, 000 P 77, 749.45 (1.05)(1.06)(1.07)(1.08) May 3, 018 Copyrigh Jeffrey Beckley 017, 018
You are given he following spo ineres raes and informaion for Quesions 1-15: Time Spo Rae r 1 0.053 0.057 3 0.063 4 0.071 5 0.080 The F&G Corporaion borrows 700,000 from Abbo Bank. The erms of he loan are such ha F&G will pay a variable ineres rae on he loan each year and will repay he 700,000 a he end of four years. The variable ineres rae will be he one year spo rae a he beginning of each selemen period. F&G is uncomforable wih he risk of having a variable loan ineres rae. Therefore, F&G purchases a four year ineres rae swap from M&J Invesmen Bank. A he end of each year, F&G will pay a fixed rae o M&J Invesmen Bank. A he end of each year, M&J will pay he variable rae o F&G. The amoun of he paymen will be based on he amoun of he loan of 700,000. 1. Answer he quesions below. These four pars are worh one quesion. a. Lis he Payer under his agreemen. F&G Corporaion b. Lis he Receiver under his agreemen. M&J Invesmen Bank c. Sae he Selemen Period under he agreemen. The selemen period is one year. d. Lis he noional amoun in he firs year under his agreemen. The noional amoun in he firs year is 700,000. 13. Calculae he swap rae under his agreemen. 1 P 1 (1.071) 0.069807645 4 4 R P 1 3 4 1 P P 3 P 4 (1.053) (1.057) (1.063) (1.071) May 3, 018 Copyrigh Jeffrey Beckley 017, 018
You are given he following spo ineres raes and informaion for Quesions 1-15: Time Spo Rae r 1 0.053 0.057 3 0.063 4 0.071 5 0.080 The F&G Corporaion borrows 700,000 from Abbo Bank. The erms of he loan are such ha F&G will pay a variable ineres rae on he loan each year and will repay he 700,000 a he end of four years. The variable ineres rae will be he one year spo rae a he beginning of each selemen period. F&G is uncomforable wih he risk of having a variable loan ineres rae. Therefore, F&G purchases a four year ineres rae swap from M&J Invesmen Bank. A he end of each year, F&G will pay a fixed rae o M&J Invesmen Bank. A he end of each year, M&J will pay he variable rae o F&G. The amoun of he paymen will be based on he amoun of he loan of 700,000. 14. Based on he spo ineres rae curve a he ime of he loan, wha is he implied rae ha F&G would pay o Abbo Bank in he hird year of he loan? The implied rae is f. [,3] (1 r ) (1.063) (1 r ) (1 r ) (1 f ] f 1 1 0.07510 3 3 3 3 3 [,3] [,3] (1 r ) (1.057) 15. Calculae he ne swap paymen a he end of he firs year. Sae who receives he paymen and who makes he paymen. F&G Owes (noional amoun)(fixed rae) (700,000)(0.069807645) 48,865.35 M&J Owes (noional amoun)(variable rae) (700,000)(0.053) 37,100.00 F&G Pays 11,765.35 o M&J May 3, 018 Copyrigh Jeffrey Beckley 017, 018
16. You are given he following spo ineres raes: Time Spo Rae r 1 0.053 0.057 3 0.063 4 0.071 5 0.080 Thomas and Bre ener ino a five year deferred ineres rae swap. There will be no swap during he firs hree years. The selemen periods will be one year. Under he swap, Thomas agrees o pay a variable ineres rae o Bre a he end of he fourh year and a he end of he fifh year. The variable ineres rae is based on he one year spo ineres rae a he sar of each year. In reurn, Bre agrees o pay a fixed rae o Thomas a he end of he fourh year and he fifh year. The noional swap amoun is 100,000 for boh selemen periods. Calculae he swap rae for his swap. P P (1.063) (1.080) 0.10547 3 5 3 5 R P 4 5 4 P 5 (1.071) (1.080) May 3, 018 Copyrigh Jeffrey Beckley 017, 018
17. The curren spo ineres rae curve is as follows: r 0.5.50% 1.75 3.40% 0.50.65%.00 3.48% 0.75.79%.5 3.80% 1.00.9%.50 4.10% 1.5 3.10%.75 4.35% 1.50 3.5% 3.00 4.50% r Cai has a wo year loan for 500,000 which has a variable ineres rae ha reses a he beginning of each six monh period. The ineres rae will be he six monh spo ineres rae a he beginning of each six monh period. Cai eners ino an ineres rae swap where she is he payer. The characerisics of he swap mirror hose of he loan. Deermine he six monh swap rae ha Cai will pay. 1 P 1 (1.0348) 0.0171956 R P 0.5 1 1.5 0.5 P 1 P 1.5 P (1.065) (1.09) (1.035) (1.0348) May 3, 018 Copyrigh Jeffrey Beckley 017, 018
18. Sue and Gavin ener ino a hree year swap. Under he swap, Sue is he payer and Gavin is he receiver. The swap has annual selemen periods. The variable rae o be paid is he one year spo rae a he beginning of each year for he nex hree years. The following is he spo ineres rae curve: Time Spo Rae r 1 0.060 0.055 3 0.050 4 0.048 5 0.046 The noional amoun for he swap changes each year. The noional amoun during he firs year is 500,000. The noional amoun during he second year is 300,000. The noional amoun during he hird year is 100,000. Calculae he swap rae for his swap. Q f P Q f P Q f P R Q P Q P Q P * * * 1 [0,1] 1 [1,] 3 [,3] 3 1 1 3 3 f r 0.06 [0,1] 1 f [1,] (1.055) 1.06 1 0.05003585 f 3 (1.05) (1.055) [,3] 1 0.040070978 1 (500, 000)(0.06)(1.06) (300, 000)(0.05003585)(1.055) (100, 000)(0.040070978)(1.05) 1 3 (500, 000)(1.06) (300, 000)(1.055) (100, 000)(1.05) 3 0.054671 May 3, 018 Copyrigh Jeffrey Beckley 017, 018