For each year, we equate the cash flows of the liabilities and the bonds. (500)(13.75) (10, 000) num 45, 000 num

Transcription

1 1. Qian Corporaion owns Hu Company he following wo paymens: i. 45,000 a he end of 1 year; and ii. 110,000 a he end of 2 years. Qian wans o absoluely mach hese paymens using he following wo bonds: iii. Bond 1 is a zero coupon bond mauring for 10,000 a he end of 1 year; and iv. Bond 2 is a 2 year bond wih annual coupons of 500 and a mauriy value of Deermine he number of Bond 1 ha Qian should purchase. (Assume ha you can purchase parial bonds.) Soluion: Cash Flows Year 1 2 Liabiliy 45, ,000 Bond 1 10,000 0 Bond =8000 For each year, we equae he cash flows of he liabiliies and he bonds. 110, 000 ( )( num ) num (500)(13.75) (10, 000) num 45, 000 num Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

2 2. Trena has agreed o pay Jayme he following paymens: a. 30,000 a he end of 2 years; b. 50,000 a he end of 5 years; and c. 10,000 a he end of 8 years. You are given ha v 0.9. Calculae he Modified Duraion of Trena s liabiliy. Soluion: v ModDur C () v Cv 30, 000(2)(0.9) 50, 000(5)(0.9) 10, 000(8)(0.9) 30, 000(0.9) 50, 000(0.9) 10, 000(0.9) (0.9) , (0.9)( ) , Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

3 3. Trena has agreed o pay Jayme he following paymens: d. 30,000 a he end of 1 years; e. 50,000 a he end of 2 years; and f. 10,000 a he end of 3 years. You are given he following hree bonds: a. A one year bond wih annual coupons of 100 and a mauriy value of 2000 wih a price of b. A wo year bond wih annual coupons of 80 and a mauriy value of 1000 wih a price of c. A hree year bond wih annual coupons of 50 and a mauriy value of 1800 wih a price of Deermine he amoun of he wo year bond ha should be purchased o exacly mach he cash flows of Trena s liabiliy. (Assume ha you can purchase parial bonds.) Soluion: Cash Flows Time Benefi 30,000 50,000 10,000 Bond A 2100 Bond B Bond C The cash flows from he bonds mus equal he cash flows o Treena each year. A(2100) B(80) C(50) 30, 000 B(1080) C(50) 50, 000 C(1850) 10, , 000 C of Bond C , ( ) B of Bond B , ( ) 80( ) A of Bond A 2100 We wan o purchase of he wo year bond. Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

4 4. A 20 year bond has annual coupons of 400. The bond maures for 13,000. Calculae he Macaulay Duraion of his bond a an annual effecive ineres rae of 5.5%. Soluion: MacDur Cv (400)(1) v (400)(2) v... (400)(20) v (13,000)(20) v C v (400) v (400) v... (400) v (13,000) v (1.055) 400 (13, 000)(1.055) (1.055) (1.055) (1.055) (13, 000)(2 20 0)(1.055) Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

5 5. Jieyu is he recipien of an annuiy ha will pay he following: i. 100,000 oday; ii. 200,000 a he end of wo years; and iii. 300,000 a he end of four years. Calculae he Modified Duraion of Jieyu s paymens a an ineres rae of 4%. Soluion: C v 1 (100, 000)(0)(1.04) (200, 000)(2)(1.04) (300, 000)(4)(1.04) ModDur v (1.04) 2 4 C v 100, 000 (200, 000)(1.04) (300, 000)(1.04) 1 1,395, , Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

6 6. Jieyu is he recipien of an annuiy ha will pay he following: i. 100,000 oday; ii. 200,000 a he end of wo years; and iii. 300,000 a he end of four years. Calculae he Macaulay Convexiy of Jieyu s paymens a an ineres rae of 4%. Soluion: MacCon C v (100, 000)(0)(1.04) (200, 000)(2 )(1.04) (300, 000)(4 )(1.04) 2 4 C v 100, , 000(1.04) 300, 000(1.04) ,842, , Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

7 7. Omer is receiving an eigh year annuiy immediae wih semi-annual paymens of 100. Calculae he Modified Duraion of his annuiy a an annual effecive ineres rae of 10.25%. Soluion: Cv (100)(0.5) v (100)(1.0) v (100)(1.5) v... (100)(8) v Cv (100) v (100) v (100) v... (100) v Mod Dur v v 2 i We noe ha (1 i) v (1.05) and v (1.05) ec. 50(1.05) (100)(1.03)... (800)(1.03) 100a (1.1025) a ( a 16(1.05) ) 16 5% 16 5% 1 (1.1025) a 16 5% a 16 5% ModDur 3.39 Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

8 8. Yong is receiving an annuiy due wih quarerly paymens of 1000 for 30 years. Calculae he modified duraion of Yong s annuiy a an annual effecive ineres rae of %. Soluion: Cv (1000)(0) v (1000)(0.25) v (1000)(0.50) v... (1000)(29.75) v Cv (1000) v (1000) v (1000) v... (1000) v Mod Dur v v 4 i We noe ha (1 i) v (1. 03) and v (1.03) ec. ( ) (1.03) (500)(1.03)... (29,750)(1.03) 1000a ( ) a a a a and a Mod Dur Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

9 9. Calculae he modified convexiy for a 3 year bond wih annual coupons of 300 and a mauriy value of 5000 using an annual effecive ineres rae of 8%. ModCon v 2 C ( 1) v Cv 300(1)(2)(1.08) 300(2)(3)(1.08) 5300(3)(4)(1.08) 300(1.08) 300(1.08) 5300(1.08) (1.08) Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

10 10. Alicia has a scholarship from Purdue ha pays: a now; b in one year; and c in wo years. Purdue hires Brad o calculae he Modified Convexiy of hese paymens a an annual effecive ineres rae of 8%. Deermine Brad s answer. Soluion: Mod Con 2 v Cv C 1 v Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

11 11. Tong LTD owns wo bonds. The bonds have he following characerisics: Price Macaulay Duraion Macaulay Convexiy Bond A 40,000 8 X Bond B 60, The Modified Convexiy of he porfolio of bonds a 6% is Calculae X. Soluion: MacDur MacCon 8 X ModCon ModConForA and ModConForB (1 i) (1.06) (1.06) 8 X X (40, 000) (60, 000) (40, 000) 5,339, (1.06) (1.06) (1.06) ModConPor , , , , 624, , x x Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017

12 12. The Modified Convexiy of a zero coupon bond is a an annual effecive ineres rae of 10%. Calculae he erm of he bond in years. (In oher words, when does he bond maure in years.) Soluion: MacDur MacCon 2 2 MacDur MacCon ModCov (1 i) (1.10) (1.10) (4)(1)( ) Copyrigh Jeffrey Beckley 2013, 2014, 2015, 2016, 2017