MATH 373 Test 4 Spring 2017 May 5, 2017

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1 MATH 373 Tes 4 Spring 017 May 5, The Bell Life Insurance Company has a wo year annuiy where i has promised o pay Elizabeh 5,000 a he end of each year for he nex wo years. Bell wans o absoluely mach he annuiy paymens using he following bonds: a. Bond A is a one year bond wih annual coupons of 100 and a mauriy value of 000. b. Bond B is a wo year bond wih annual coupons of 00 and a mauriy value of Deermine he number of Bond A ha Bell should buy. Assume ha you can purchase parial bonds. End of Year 1 End of Year Annuiy 5,000 5,000 Bond A 100 Bond B , B 5, 000 B B 100A 5, ( ) 100 A 5, 000 5, ( ) A of Bond A 100

2 . An annuiy due makes annual paymens a he beginning of each year for four years. Using an ineres rae of 7%, calculae he Macaulay Convexiy of his annuiy. MacCon C ( ) v P(0 )(1.07) P(1 )(1.07) P( )(1.07) P(3 )(1.07) C v P(1.07) P(1.07) P(1.07) P(1.07) (1.07) 4(1.07) 9(1.07) (1.07) (1.07) (1.07)

3 3. A 0 year bond issued by Talbo Indusries has a par value of 10,000. The bond maures for is par value and pays semi-annual coupons a a rae of 8% converible semi-annually. Calculae he Modified Duraion of his bond using an annual effecive rae of 10.5%. Coupon (10, 000)(0.08 / ) 400 ModDur v C () v C v (1.105) 1 400(0.5)(1.105) 400(1)(1.105) 400(1.5)(1.105) (0)(1.105) (10, 000)(0)(1.105) (1.105) 400(1.105) 400(1.105) (1.105) (10, 000)(1.105) (1.105) 1 00(1.05) 400(1.05) 600(1.05) (1.05) (10, 000)(0)(1.105) (1.05) 400(1.05) 400(1.05) (1.05) (10, 000)(1.05) 40 since (1.05) The numeraor excep for he las erm is he P&Q formula wih P=00 and Q=00. = (1.105) (1.05) 00 1 (1.05) (40)(1.05) (10, 000) (0)(1.105) (1.05) (10, 000)(1.05) , (1.105) 8.519

4 4. A hree year bond pays annual coupons of 500 and has a mauriy value of Using an ineres rae of 6%, calculae he Modified Convexiy of he bond. ModCon v C ( )( 1) v C v (1.06) 500(1)()(1.06) 500()(3)(1.06) 6500(3)(4)(1.06) (1.06) 500(1.06) 6500(1.06) , (1.06)

5 5. Using an ineres rae of 5%, calculae he Macaulay duraion of a perpeuiy immediae wih paymens of 500 a he end of each year C () v 500(1) v 500() v 500(3) v (0.05) MacDur 1 3 C v v v v 0.05

6 6. A bond has a price of 100,000 using an ineres rae of 6%. The bond also has a modified duraion of and a modified convexiy of 400 using an ineres rae of 6%. 1s Order E MAC is he esimaed price of he bond using he firs order Macaulay approximaion if he ineres rae changes o 8%. nd Order E MOD is he esimaed price of he bond using he second order Modified approximaion if he ineres rae changes o 8%. Calculae E 1s Order MAC E. nd Order MOD E 1 i P( i ) 1 i 1s Order 0 MAC 0 MacaulayDuraion Modified Duraion = v(macaulay Duraion)==> 1 (1.06) (Macaulay Duraion) Macaulay Duraion = ()(1.06) 3.3 E MAC 1.06 (100,000) , nd Order ( i i0 ) EMOD P( i0) 1 ( i i0)( ModDur) ( ModCon) ( ) =(100,000) 1 ( )() (400) 64, 000 E 1s Order MAC E 64, , nd Order MOD

7 7. The Trou Life Insurance Company owns he following bonds: a. Bond A is a zero coupon bond mauring for 100,000 a he end of 6 years. This bond can be purchased o yield an annual effecive rae of 10%. b. Bond B is a 0 year bond wih a Macaulay duraion of 1 and a Macaulay convexiy of 100. This bond has a price of 90,000. The duraion, convexiy, and price are calculaed a an annual effecive ineres rae of 10%. Deermine he Modified Convexiy of his porfolio. MacCon MacDur 66 Modified Convexiy of Bond A = (1 i) 1.1 MacCon MacDur Modified Convexiy of Bond A = (1 i) 1.1 P C P C Modified Convexiy of Porfolio = A B P P A A B B 6 (100, 000)(1.10) ( ) (90, 000)( ) (100, 000)(1.10) 90, 000

8 8. William mus pay Zai 1,000,000 a he end of 1 years. William has used Reddingon immunizaion using he following o bonds o proec himself from ineres rae changes. a. Bond 1 is a zero coupon wih a mauriy value of 5,000 a he end of X years. b. Bond is a zero coupon bond mauring for 15,000 a he end of 19 years. Based on an ineres rae of 7.5%, William purchased bonds idenical o Bond. Deermine X. 19 Price of Bond = (15,000)(1.075) Toal Spend on Bond = (Number)(Price) = ( )( ) = 174, Presen Value of Paymen = (1,000,000)(1.075) 419, Toal Amoun Spend on Bond 1 + Toal Amoun Spend on Bond = Presen Value of Paymen Toal Amoun Spend on Bond 1 419, , , (7)(419,854.13) Using he shorcu = (419,854.13) 44, X 19 X 44, X (7)(419,854.13) , If you do no use he shorcu, hen you se he duraion of he asses equal o he duraion of he liabiliies: ( X )(44,914.76) (19)(174,939.36) (1)(419,854.13) (19)(174,939.36) 1 X 7 419, ,914.76

9 9. Lauren purchased a wo year bond wih semi annual coupons of 00 and a mauriy value of The bond can be purchased o yield 4.% converible semi-annually. The bond was priced based on he following spo ineres rae curve: Time Spo Rae r r r.5 Deermine r. We can calculae he price of he bond as r 1 r 1 r 1 r or 4 00a 6000 v where he ineres rae is he yield rae (1.01) ==> r r r (1.01) % 1 r

10 10. Using he following spo ineres rae curve, calculae he presen value of an annuiy due wih annual paymens of 40,000 for hree years: Time Spo Rae r , , 000 PV 40, , (1.06) (1.06)

11 11. Lauren can purchase he following hree bonds: a. Bond 1 is a one year bond wih annual coupons of 500 and a mauriy value of The bond sells for b. Bond is a wo year bond wih annual coupons of 1000 and a mauriy value of This bond has a price of c. Bond 3 is a zero coupon bond wih a mauriy value of 10,000 and a price of 805. Insead, Lauren decides o purchase a hree year annuiy immediae wih paymens of 13,000 a he end of each year for hree years. Deermine he presen value of Lauren s annuiy. Firs, we need o find he spo ineres raes using boosrapping: Using Bond 1==> 4700 r r Using Bond ==> r (1 r ) (1 r ) r , , 000 Using Bond 3==> 805 r r /3 13, , , 000 PV 34, ( ) ( )

12 1. You are given he following spo ineres raes: Time Spo Rae r Calculae f [,4]. (1 r ) (1 r ) (1 f ) (1.054) (1.046) (1 f ) [,4] [,4] f (1.054) (1.046) [,4]

13 You are given he following spo ineres raes and informaion for problems 13-14: Time Spo Rae r Kaarina has a loan from Hemenway Bank. The loan is for 300,000 he firs year. A he end of one year, Kaarina repays 100,000 leaving a loan of 00,000 during he second year. A he end of he second year, Kaarina repays anoher 100,000 leaving a loan of 100,000 during he hird year. Kaarina will pay Hemenway Bank a variable ineres rae equal o he one year spo ineres rae a he beginning of each year. Kaarina would like o have a fixed ineres rae so she eners ino an ineres rae swap wih Lily. Under he ineres rae swap, Kaarina will pay a fixed rae o Lily and Lily will pay a variable rae o Kaarina. The variable rae will be he same rae ha Kaarina is paying o Hemenway Bank. The oher erms of he swap will mirror he loan ha Kaarina has. 13. Quesions a. hrough d. are considered one quesion: a. This is an accreing swap. True or False This is false. I is an amorizing swap. b. Wha is he selemen period for his swap? The selemen period is one year c. Sae he noional amoun of his swap? The noional amoun of his swap changes each year. I is 300,000 for he firs year, 00,000 for he second year and 100,000 for he hird year. d. Lis he counerparies o he swap. The counerparies are Kaarina and Lily.

14 14. Calculae he swap ineres rae for Kaarina s swap. R n i1 Q f n i1 * i [ i1, i ] i QP i i P Q f P Q f P Q f P Q P Q P Q P * * * 1 [0,1] 1 [1,] 3 [,3] * 1r (1.046) * 1r3 (1.051) f [1,] 1 [,3] f and r r (1.046) 1 (300, 000)(0.043)(1.043) (00, 000)( )(1.046) (100, 000)( )(1.051) 1 3 (300, 000)(1.043) (00, 000)(1.046) (100, 000)(1.051) %

15 15. You are given he following spo ineres raes: Time Spo Rae r Tommy purchases a deferred ineres rae swap wih a erm of five years. Under he swap, here is no swapping of ineres raes during he firs wo years. During he las hree years, he selemen period will be one year. Under his swap, Tommy will be he payer. The variable ineres rae will be based on he one year spo rae a he sar of each selemen period. The noional amoun of his swap is 500,000. Calculae he swap rae for his swap. R P P P P (1.046) (1.056) 5 0 n 5 n P3 P4 P5 (1.051) (1.054) (1.056) P i i %

16 16. Miaoqi and Nui enered ino a four year ineres rae swap on May 5, 015. The noional amoun of he swap was a level 50,000 for all four years. The swap has annual selemen periods wih he firs period saring on May 5, 015. Under he swap, Miaoqi agreed o pay a variable rae based on he one year spo rae a he beginning of each selemen period. Nui will pay Miaoqi he fixed rae of 4% on each selemen dae. On May 5, 017, he spo ineres rae curve was as follows: Time Spo Rae r Miaoqi decides ha she wans o sell he swap on May 5, 017. Calculae he marke value of he swap on May 5, 017 from Miaoqi s posiion in he swap. Miaoqi is receiving he fixed ineres rae and paying he variable rae. This means ha he firs year she will receive he fixed ineres rae of 4% and pay he variable rae of 3.8%. During he second year, Miaoqi will receive he fixed ineres rae of 4% and pay he variable rae of f. [1,] (1.041) f[1,] Marke Value = Presen Value of Expeced Cash Flows = (50, 000)( ) (50, 000)( ) (1.041)

17 17. Beckley Farms has a 500,000 loan from Bailey Bank. Under he erms of he loan, Beckley will pay ineres annually o Bailey Bank based on LIBOR plus 10 basis poins. Addiionally, Beckley will pay he principal of 500,000 a he end of five years. Beckley would prefer o know he annual ineres cos ha will be incurred. To fix he ineres rae on he loan, Beckley eners ino a five-year ineres rae swap wih a noional amoun of 500,000 and annual selemen daes. The erms of he swap are ha Beckley will make swap paymens based on a fixed rae of 5.35% and will receive swap paymens based on a variable rae of LIBOR plus 50 basis poins. During he hird year of he loan, LIBOR is 5.6%. Deermine he ne ineres paymen made by Beckley a he end of he hird year. The ne ineres paymen is he ineres paid on he loan plus any ne swap paymen made by he loan holder less any ne swap paymen received by he loan holder. The ineres paid on he loan (500,000)( LIBOR 0.01) (500,000)( ) 34,000 Ne swap paymen received by Beckley (500,000)( LIBOR ) (500, 000)( ) 3750 Ne ineres paymen 34, ,50 Noe ha his can also be calculaed as he noional amoun mulipled by he sum of he ineres rae paid by he loan holder under he swap plus any spread beween he loan and he swap. This means he ne ineres paymen is: (500, 000)( ) 30, 50

18 18. The curren spo ineres rae curve is as follows: r % % %.00.48% %.5.80% % % % % % % r Rafael has a one year loan for 1,000,000 which has a variable ineres rae ha reses a he beginning of each hree monh period. The ineres rae will be he spo ineres rae a he beginning of each hree monh period. Rafael eners ino an ineres rae swap where he is he payer wih he characerisics of he swap exacly mach he loan. Deermine he quarerly swap rae ha Rafael will pay. 1 P 1 (1.019) 1 1 R P P 0.5 P 0.75 P 1 (1.015) (1.0165) (1.0179) (1.019) Noe ha his is a quarerly effecive ineres rae.

19 19. You are given he following zero coupon bond prices for a zero coupon bond ha maures for 1 on he mauriy dae: Mauriy Dae Price 1 Year Years Years Years Years Josh and Phillip ener ino a four year swap wih a noional amoun of 00,000. The swap has annual selemen periods. Under he swap, Josh will pay Phillip he fixed swap rae a he end of each year while Phillip will pay Josh he variable rae where he variable rae is he one year spo rae a he beginning of each year. Deermine he ne swap paymen a he end of he firs year. Be sure o sae who makes he paymen and who receives he paymen. Soluions: Firs we need o find he swap rae. 1 P R P 1 P P 3 P 4 A he end of he firs year, Josh owes he fixed rae so he owes (00,000)( ) = A he end of he firs year, Phillip owes he variable rae so (00,000)(1/ ) = Josh pays

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