Math 373 Fall 2016 Test 3 November 15, 2016

Transcription

1 Mah 373 Fall 016 Tes 3 November 15, The sock of Riad Corporaion pays a quarerly dividend of 6. The nex dividend is due in monhs. The sock is bough by Alya who wans o earn an annual effecive yield of 13%. Using he dividend discoun mehod, deermine he price ha Alya should pay. Soluion: Since he dividends are quarerly, we need he quarerly effecive ineres rae. i 4 (4) 0.5 (1.13) /3 Price = Presen Value of Fuure Dividends ( ) /3 We need o muliply by ( ) o adjus for he fac ha he nex dividend is paid in monhs. Copyrigh Jeffrey Beckley 017, 018

2 . You are given he following hree bonds: a. Bond 1 is a zero coupon bond mauring for 10,000 a he end of one year. The bond has a price of b. Bond is a wo year bond wih annual coupons of 400 and a mauriy value of This bond is priced o yield 7%. c. Bond 3 is a hree year bond wih annual coupons of 1000 and a mauriy value of 5,000. You are also given ha f[1,3] 0.1. Calculae he price of Bond 3. Soluion: We need r, r, and r o find he price of Bond 3. We can use boosrapping o find r and r We can hen use f o find r. [1,3] 3 10, , 000 Bond 1==> P 1 r1 r r Bond ==> P r (1 r ) 1.07 (1.07) (1 r ) r r (1.07) Bond , 000 P 1 r (1 r ) (1 r ) bu (1 r ) (1 r )(1 f ) [1,3] , 000 P 1, ( ) ( )(1.1) Copyrigh Jeffrey Beckley 017, 018

3 3. A 0 year bond has semi-annual coupons of 50 and a mauriy value of Calculae he Modified Duraion of his bond a an annual effecive ineres rae of 1.36%. Soluion: ModDur v C () v C v (1.136) 1 50(0.5)(1.136) 50(1)(1.136)... 50(0)(1.136) 1600(0)(1.136) (1.136) 50(1.136)... 50(1.136) 1600(1.136) Since (1.136) , we can subsiue (1.06) for (1.136) which leads o he following: (1.136) 1 5(1.06) 50(1.06) (1.06) 1600(0)(1.06) 40 50( a ) 1600(1.06) (1.136) 1 5 5( a ) ( a 40( 1.06) ) 1600(0)(1.06) ( a ) 1600(1.06) (1.136) Copyrigh Jeffrey Beckley 017, 018

4 4. A perpeuiy due makes annual paymens of 50. Calculae he Macaulay Duraion of his perpeuiy a an annual effecive ineres rae of 10%. Soluion: C v v v v MacDur C v v v v 0 1 () (50)(0) 50(1) 50() (50) i i 0.10 (0.10) (1 i) (1.10) i Copyrigh Jeffrey Beckley 017, 018

5 5. The Whiney Life Insurance Company is paying an annuiy o Shannon. The annuiy is a hree year annuiy immediae wih level annual paymens of 100,000. Whiney wans o absoluely mach his annuiy using he following hree bonds: a. Bond 1 is a one year bond wih an annual coupon of 150 and a mauriy value of b. Bond is a wo year bond wih annual coupons of 50 and a mauriy value of c. Bond 3 is a hree year bond wih annual coupons of 500 and a mauriy value of Calculae he number of each bond ha Whiney should purchase. (Assume ha you can buy parial bonds.) Soluions: Cash Flows Year 1 Year Year Annuiy 100, , ,000 Bond Bond Bond , 000 Bond 3 0 Bonds 5000 Bond 100,000 (0)(500) ( Bond)(550) 100, 000 (0)(500) Bonds 550 Bond 1 100,000 (0)(500) ( )(50) ( Bond1)(5150) 100, 000 (0)(500) ( )(50) Bonds 5150 Copyrigh Jeffrey Beckley 017, 018

6 6. The Purdue Life Insurance Company has wo liabiliies: a. Liabiliy 1 is a conrac in which Purdue agrees o pay 500,000 a he end of 7 years. b. Liabiliy is an annuiy wih a Macaulay Duraion of 16 and Macaulay Convexiy of 145. The presen value of he paymens under his annuiy is 300,000. The curren ineres rae is 6.5% and all values above are deermined a an ineres rae of 6.5%. Esimae he presen value of his porfolio of liabiliies using he second order modified esimae if he ineres rae decreases o 5.6%. Soluion: Liabiliy Presen Value Macaulay Duraion Macaulay Convexiy 1 500,000(1.065) =49 300, MacDur P D P D (500, 000)(1.065) (7) (300, 000)(16) of Porfolio P P (500, 000)(1.065) (300, 000) ModDur 1 of Porfolio ( MacDur)( v) ( )(1.065) P C P C (500, 000)(1.065) (49) (300, 000)(145) MacCon of Porfolio P P (500, 000)(1.065) (300, 000) ModDur ( MacCon MacDur) of Porfolio (1 i) (1.065) P( i) P( i ) 1 ( i i ( i i ) 0 0 0)( ModDur) ( ModCon) 1 ( )( ) ( ) ( ) 7 (500, 000)(1.065) (300, 00) 671, Copyrigh Jeffrey Beckley 017, 018

7 7. An annuiy due makes a level paymen a he sar of each year for 17 years. Calculae he Macaulay Duraion for his annuiy using an annual effecive ineres rae of 7%. Soluion: C () v ( P)(0) v ( P)(1) v... ( P)(16) v MacDur C v ( P) v ( P) v... ( P) v 1 16 (1) v... (16) v v v... v a17 a 1 a 16(1.07) Copyrigh Jeffrey Beckley 017, 018

8 8. Malik Manufacuring jus paid a dividend of 3.40 earlier oday. The dividends are quarerly and he nex dividend will be paid in hree monhs. Malik is a rapidly growing business and expecs he dividends o increase in he fuure. The nex dividend is expeced o be 3.40(1.0). The dividend o be paid in six monhs is expeced o be expeced o be 10% of he prior dividend. 3.40(1.0). Each dividend hereafer is Using he dividend discoun mehod, calculae he price of Malik Manufacuring sock o yield a rae of 1% compounded quarerly. Soluion: Since he dividends are quarerly, we need he quarerly effecive ineres rae. (4) i Price = Presen Value of Fuure Dividends 3.40(1.0)(1.03) (1.0)(1.03) (1.0)(1.03) 3.40(1.0) (1.03)... 1 Copyrigh Jeffrey Beckley 017, 018

9 9. Iverson Insurance Company has agreed o pay John 150,000 a he end of 18 years. Iverson wans o fully immunize his paymen using he following wo bonds: a. A zero coupon bond ha maures for 10,000 a he end of 6 years. b. A zero coupon bond ha maures for 0,000 a he end of 0 years. Using an annual effecive rae of ineres of 8.5%, calculae he number of 0 year zero coupon bonds ha Iverson should purchase. (Assume ha you can buy parial bonds.) Soluion: DLiabiliy DBonda Amoun of Money o Be used o Purchase Bond b = ( PVofLiabiliy) DBondb DBonda 18 6 (150,000)(1.085) 18 30, Price of Bond b = (0,000)(1.085) Amoun of Money 30,863.9 Number of Bonds = Price of Bond b Copyrigh Jeffrey Beckley 017, 018

10 10. An annuiy makes a paymen of 100 oday, a paymen of 500 a ime 3, and a paymen of 300 a ime 8. Calculae he Modified Convexiy of his annuiy a an ineres rae of 5%. Soluion: ModCon v C ( )( 1) v C v (100)(0)(1)(1.05) (500)(3)(4)(1.05) (300)(8)(9)(1.05) (100)(1.05) (500)(1.05) (300)(1.05) (1.05) Copyrigh Jeffrey Beckley 017, 018

11 11. You are given he following spo ineres rae curve: r % % %.0 4.5%.5 5.0% % A wo year bond wih semi-annual coupons of 100 and a mauriy value of 500 is priced using he above spo ineres raes. Deermine he yield rae converible semi-annually on his bond. Soluion: P (1.03) (1.034) (1.039) (1.045) Then o ge he yield, we mus use he BA-II Plus N 4; PV ; PMT 100; FV 500; CPT I / Y This is he rae for a six monh period. To ge he rae converible semi-annually, muliply by. ( )() 4.399% Copyrigh Jeffrey Beckley 017, 018

12 1. A hree year annuiy immediae makes annual paymens of Calculae he Macaulay Convexiy a an annual effecive ineres rae of 9%. Soluion: MacCon C ( ) v v C (1000)(1 )(1.09) (1000)( )(1.09) (1000)(3 )(1.09) 1 3 (1000)(1.09) (1000)(1.09) (1000)(1.09) Copyrigh Jeffrey Beckley 017, 018