1. (S09T3) John must pay Kristen 10,000 at the end of 1 year. He also must pay Ahmad 30,000 at the end of year 2.

Transcription

1 Chaper 9, Secion 1 1. (S09T3) John mus pay Krisen 10,000 a he end of 1 year. He also mus pay Ahmad 30,000 a he end of year 2. John wans o exacly mach his liabiliies by purchasing he following wo bonds: a. Bond A is a one year zero coupon bond mauring for b. Bond B is a wo year bond wih annual coupons of 200 and a mauriy value of Calculae he amoun of each bond ha John should buy. John needs o make he cash flows going ou equal o he cash flows coming in. To do his, we can se up wo equaions where a is he amoun of bond A and b is he amoun of bond B. We know ha a he end of year 1 John will pay 10,000 and he will receive 1000 from every bond A and 200 from every bond B. Then a he end of year wo when he needs o pay 30000, he will be receiving 1200 for every bond B a200b b Simply solve his sysem algebraically b b a 200(25) a 5

2 2. (S08T3) Yvonne mus make a paymen of 80,000 a he end of one year. Addiionally, she mus make a paymen of 40,000 a he end of wo years. Finally, she mus make a paymen of 60,000 a he end of 3 years. She wans o purchase bonds o exacly mach her paymens. She can purchase he following hree bonds: Bond Number Term of Bond Annual Coupon Mauriy Value 1 1 Year Years Years Calculae he amoun of Bond 2 which Yvonne should purchase. Once again, we jus need o se up equaions ha will mach he cash flows a 60b 70c b70c c Now we need o solve, saring wih c c b b

3 3. (F11HW) Rivera Insurance Company has commied o paying 10,000 a he end of one year and 40,000 a he end of wo years. I s Chief Financial Officer, Miguel, wans o exacly mach his obligaion using he following wo bonds: Bond A is a one year bond which maures a par of 1000 and pays an annual dividend a a rae of 6%. This bond can be bough o yield 6% annually. Bond B is a wo year bond which maures a par of 1000 and pays an annual dividend a a rae of 10%. This bond can be bough o yield 7% annually. Calculae he amoun of each bond ha Rivera should purchase. Calculae he cos of Rivera o exacly mach his obligaion a100b b Solving, b a Now, le s find he price of each bond in order o find he oal cos o mach. Price of Bond 1: N 1 I / Y 6% PMT 60 FV 1000 CPT PV 1000 Noe: We could also know his inuiively because he coupon rae is he same as he yield. Price of Bond 2: N 2 I / Y 7 PMT 100 FV 1000 CPT PV Now we can find he oal price (1000) ( )

4 4. (F11HW) Wang Life Insurance Company issues a hree year annuiy ha pays 40,000 a he end of each year. Wang uses he following hree bonds o absoluely mach he cash flows under his annuiy: a. A zero coupon bond which maures in one year for b. A wo year bond which maures for 1200 and pays an annual coupon of 100. This bond is priced using an annual yield of 7%. c. A hree year bond which maures for 2000 and pays annual coupons of 75. This bond has a price of 1,750. I cos Wang 104,000 o purchase all hree bonds o absoluely mach his annuiy. Calculae he one year spo ineres rae. Absolue maching means ha he cash flow from he bonds a any given ime should be exacly he same as he cash flow from he annuiy. In his case, we need he bonds o produce a cash flow of $40,000 a imes 1,2, and 3. In order o do his we will need o buy a cerain amoun of each bond. These values will be x, y, and z. Someimes i is easies o se up a able ha summarizes cash flows: Bond Amoun Price CF a =1 CF a =2 CF a =3 A X 1000/(1+r 1 ) B Y C Z In he able above, he Price for bon B was found using he calculaor. N 2 PMT 100 I / Y 7 FV 1200 CPT PV Now we can se up a few equaions in order o solve for he unknown variables. Firs, since here is only one bond wih a cash flow a ime 3 we can find z very easily: 2075Z Z Then we can find Y: 1300Y 75Z (19.277) Y Finally we can find X.

5 1000X 100Y 75Z (19.277) 100(29.657) X Now ha we know he amoun of each bond we are purchasing and we are given our oal cos, we jus need o find he price of he one year bond which will in urn give us he one year spo rae X Y 1750Z 1 r Noe ha he price of he firs bond = 1 r. Now jus plug in he values we found above o ge r Chaper 9, Secion 2 5. (S12HW) Ace is receiving an annuiy immediae wih level annual paymens of 500 for 18 years. Calculae he Macaulay duraion and he Modified duraion a an annual effecive ineres rae of 6%. Use for formula for Macaulay Duraion firs: 2 18 Cv 500v 500(2) v (18) v 500Ia Di (, ) 2 18 C v 500v 500 v v 500a Ia 18 a 18v i 0.06 a a Now ha we have he Macaulay Duraion, i is easy o find he modified/ Modified D( i, ) v

6 6. (S12HW) Hopkins Life Insurance Company is paying Keih an annuiy due of 234 per year for he nex 10 years. Calculae he Modified duraion of Keih s annuiy a an annual effecive ineres rae of 10%. C v v v v 234Ia Di (, ) Ia (2) (9) Cv 234v 234 v v 234a10 9 a 9v i 0.10 a a Now ha we have he Macaulay Duraion, i is easy o find he modified/ Modified D( i, ) v (S08T3) Kyle purchases a 10 year bond. The bond maures for 1300 and has annual coupons of 80. Calculae he Macaulay duraion of Kyle s bond a an ineres rae of 8%. Use your calculaor o find he price of he bond. PMT 80 FV 1300 N 10 I / Y 8 CPT PV Macaulay Duraion= Cv 80(1) v 80(2) v... 80(10) v 1300(10) v Cv v v a 10v 80 Ia

7 8. (S12HW) Wenda owns an 8 year bond wih a par value of The bond maures for par and pays semi-annual coupons a a rae of 6% converible semi-annually. Calculae he Modified duraion of his bond a an annual effecive ineres rae of 8.16%. Use your calculaor o find he price of he bond. PMT 30 FV 1000 N 16 I / Y 4 CPT PV Modified Duraion = Cv 30(0.5) v 30(1) v 30(1.5) v... 30(8) v 1000(8) v 1 v Cv a a (F11HW) A five year bond maures for 20,000. The bond pays coupons of: d a he end of he firs year, e a he end of he second year, f a he end of he hird year, g. 750 a he end of he fourh year, and h. 600 a he end of he fifh year. Calculae he Macaulay Duraion of his bond a 5% Macaulay Duraion= Di (, ) Cv 3000v 1500(2) v 1000(3) v 750(4) v 600(5) v 20000(5) v C v 3000v 1500v 1000v 750v 600v 20000v v v v v v v 3000v 1500v 1000v 750v 600v 20000v

8 10. (F11HW) James has a loan of 10,000 which is o be repaid wih 10 level annual paymens a an annual effecive ineres rae of 12%. Calculae he Macaulay duraion of he loan using he 12% ineres rae. Firs use your calculaor o find he paymens: PV 1000 N 10 I / Y 12 CPT PMT Cv v (2) v (10) v Cv a 10v Cv.12 Cv Cv Ia (F11HW) Tokoly Invesmens owns a preferred sock which pays a quarerly dividend of $5 per quarer wih he nex dividend paid in 3 monhs. Calculae he modified duraion of his sock a an annual effecive rae of % Firs, le s find he proper i: C 2 v 5(.25) v 5(.5) v v ( v) Cv (F11HW) The Macaulay Duraion of a perpeuiy immediae wih level annual paymens is 26. Deermine he ineres rae ha was used o calculae he Macaulay Duraion. 1 1 P(1 i) i C 2 v PIa i i C P(1 i) v P(1 i) i i i 1 i.04 25

9 13. (F11HW) Sparks-Norris Asse Parners (SNAP) manages he following porfolio of bonds: Bond Price Macaulay Duraion 1 5, , , ,000 2 The duraion is calculaed a an annual effecive ineres rae of 7%. Calculae he modified duraion of SNAP s porfolio. D PORT DP P where D is Modified Duraion. In order o ge Modified Duraion from Macaulay Duraion, we muliply Macaulay by v. DP P v 5.607

10 Chaper 9, Secion (F11HW) An annuiy immediae pays 100 a he end of each year for 5 years. Calculae he Macaulay convexiy and he Modified convexiy of his annuiy a an annual effecive rae of 6%. Firs find he Price Funcion and i s 1 s and 2 nd Derivaives. P( i) 100 (1 i) (1 i) (1 i) (1 i) (1 i) P '( i) 100 (1 i) 2(1 i) 3(1 i) 4(1 i) 5(1 i) P ''( i) 100 2(1 i) 6(1 i) 12(1 i) 20(1 i) 30(1 i) Modified Convexiy: P''( i) 2(1.06) 6(1.06) 12(1.06) 20(1.06) 30(1.06) Pi ( ) (1.06) (1.06) (1.06) (1.06) (1.06) Macaulay Convexiy: (1.06) 2 (1.06) 3 (1.06) 4 (1.06) 5 (1.06) (1.06) (1.06) (1.06) (1.06) (1.06) 1(1.06) 4(1.06) 9(1.06) 16(1.06) 25(1.06) (1.06) (1.06) (1.06) (1.06) (1.06)

11 15. (S09T3) A 3 year bond has annual coupons. The coupon a he end of he firs year is 100. The coupon a he end of he second year is 300. The coupon a he end of he hird year is 500. The bond maures for 700. Calculae he modified convexiy of his bond a an annual effecive rae of ineres of 6%. Firs find he Price Funcion and is derivaives P( i) 100(1 i) 300(1 i) 1200(1 i) P '( i) 100(1 i) 600(1 i) 3600(1 i) P ''( i) 200(1 i) 1800(1 i) 14400(1 i) Modified Convexiy: P''( i) 200(1.06) 1800(1.06) 14400(1.06) Pi ( ) 100(1.06) 300(1.06) 1200(1.06)

12 16. (F11HW) A bond has a Macaulay Duraion of and a Macaulay Convexiy of when calculaed using an annual effecive ineres rae of 8%. The price of he bond is A. Esimae he price of he bond if he annual ineres rae increases o 8.5% using only he duraion. B. Esimae he price of he bond if he annual ineres rae increases o 8.5% using boh he duraion and he convexiy. C. The values in his problem are based on a 5 year bond wih annual coupons of 70 and a mauriy value of Wha would he acual price be a 8.5%. A: Di ( 0, ) P P( i0 )( i) ( )(.005) i P(8.5%) P(8%) P B: We need o find he Modified Convexiy: C( i, ) D( i, ) C( i, m) (1 i) (1.08) Now we can find he change in price: 2 D( i0, ) P( i0)( i) P P( i0 ) i C( i, m) 1 i (.005) ( )(.005) P(8.5%) P(8%) P C: Use your calculaor: PMT 70 N 5 I / Y 8.5 FV 1200 CPT PV

13 17. (S09T3) Jenna owns he following porfolio. Asse Price Macaulay Macaulay Duraion Convexiy Bond 1 25, Bond 2 30, Bond 3 45, The price, Macaulay Duraion, and Macaulay Convexiy were calculaed a an annual effecive rae of 5%. Esimae he price of he porfolio a an annual effecive rae of ineres of 7% using boh he duraion and convexiy. Firs find he Convexiy and he Duraion of he Porfolio: Porfolio 25, , , 000 D ( i, ) (6) (4.5) (3) , , , 000 Porfolio 25, , , 000 C ( i, ) (40) (25) (12) , , , 000 Then find he Modified Convexiy: C( i, ) D( i, ) C( i, m) (1 i) (1.05) Now we can find he change in price: 2 D( i0, ) P( i0)( i) P P( i0 ) i C( i, m) 1 i , 000(.02) 100, 000(.02) P(7%) P(5%) P 100, 000 7, ,

14 Chaper 9, Secion (F11PP) Sun wans o fully immunize a fuure paymen of 100,000 a ime 10 using he following wo bonds: a. A zero coupon bond mauring in 5 years; and b. A zero coupon bond mauring in 20 years. Deermine he amoun ha Sun should spend on each bond a an annual effecive ineres rae of 10%. Presen Value Maching: Le A= Presen value of 5 year bond Le B= Presen value of 20 year bond PV(Asses)=PV(Liabiliies) 10 AB 100,000(1.1) Duraion Maching: Duraion(Asses)=Duraion(Liabiliies) 10 5A20B 10(100,000(1.1) ) Now we have wo equaions wih wo unknowns. We can solve for A and B. 10 A100, 000(1.1) B A38, B 5(38, B) 20B10(38, ) 15B 385, , B 12, A 38, , A 25,

15 19. (F11PP) Lauren wans o fully immunize a fuure paymen of X a ime Y using he following wo bonds: a. Bond A is a zero coupon bond mauring in 2 years; and b. Bond B is a zero coupon bond mauring in 10 years. Lauren pays 13, for Bond A and 6, for Bond B. Deermine X and Y if he annual effecive ineres rae of 5%. Presen Value Maching: y x(1.05) Duraion Maching: y yx(1.05) 2( ) 10( ) Now we have wo equaions wih wo unknowns, so we can solve for x and y: y x (1.05) y y y( (1.05) )(1.05) y y x (1.05) x

16 Answers 1. A => 5 B => A => B=> Cos => 44, % 5. Macaulay => Modified => % Macaulay = and Modified = A B C , , on five year zero and 12, on weny year zero. 19. X = 24,680 and Y = 4.5