MATH 373 Test 3 Fall 2015 November 17, 2015

Transcription

1 MATH 7 Test Fall 015 November 17, A three year bond with annual coupons of 800 matures for 1,000. The price of this bond is P at an annual effective yield rate of 6%. The current spot interest rate curve is: t r t r Using the spot interest rate curve, the price of the bond is also P. Determine r to five decimal places , 000 a v PV N I/Y=6 PMT=800 FV=1000 CPT PV=1, (1.04) 800(1.05) 1,800(1 r ) 1, r %

2 . You are given the following two bonds: a. A one year zero coupon bond that matures for 10,000 with a price of b. A two year bond with annual coupons of 000 and a maturity value of 10,000. The price of the bond is 1,745. Using these bonds, determine the forward rate for time 1 to which is f [1,]. a) , 000(1 r ) r % b) 000( ) 1, 000(1 r ) 1, , 000(1 r ) 10,85 r % (1 r )(1 f ) (1 r ) ( )(1 f ) ( ) 1 [1,] [1,] f [1,] 6. 40%

3 . Ian is receiving annual payments of 100 at the beginning of each year for the next 0 years under an annuity due. Calculate the Modified Duration of Ian s annuity at an annual effective interest rate of 9%. 100(0) v 100(1) v100() v (19) v MacDur 100a a a ( a 19 v ) a 19 ==> PMT=1 N =19 I =9 CPT PV= a ( a 19 v ) a ==> ND BGN ND SET ND QUIT 0 PMT=100 N=0 I=9 CPT PV= MacDur ModDur (1.09)

4 4. RAN Bank has agreed to pay Yue a non-level annuity over the next three years. The first payment at the end of one year is 100,000. The second payment at the end of two years is 00,000. The last payment at the end of three years is 500,000. RAN Bank wants to protect itself from interest rate changes using exact matching with the following three bonds: a. Bond A is a one year bond with annual coupons of 100 and a maturity value of b. Bond B is a two year bond with annual coupons of 00 and a maturity value of c. Bond C is a three year bond with annual coupons of 400 and a maturity value of 1,000. Assuming the RAN Bank can purchase partial bonds, calculate the number of Bond A that RAN should purchase. 1 A 1100 B C , 400C 500, 000 C B 00, (40.58) B A 100, (4.01) 00(40.58) A

5 5. Anderson Bank has agreed to make payments to Madison. The first payment is 1000 at the end of years. The second payment is 500 at the end of years. The final payment is 4000 at the end of 6 years. Calculate the Modified Convexity of Madison s payments at an annual effective interest rate of 4.5%. ModCon v 1000()() ()(4) (6)(7) v 500v 4000v 6 160, (1.045).87 Or MacCon () () (6) v 500v 4000v () v 500() v 4000(6) v MacDur ModCon (1.045) ( ).87

6 6. Lyu Life Insurance Company has a bond portfolio consisting of the following two bonds: a. A zero coupon bond that matures for 100,000 at the end of 5 years. b. A 10 year bond with a price of 110,000. The bond has a Macaulay duration of 7 and a Macaulay convexity of 40. These values are based on an annual effective interest rate of 8%. Calculate the Modified Convexity of Lyu s portfolio of bonds at an annual effective interest rate of 8%. For the zero coupon bond: MacDur 5 MacCon Price 100, 000(1.08) 68, For the 10 year bond: MacDur :7 MacCon 40 Price 110, 000 5(68, ) 7(110, 000) MacDur , , 000 5(68, ) 40(110, 000) MacCon , , 000 ModCon (1.08) ( ) 4.741

7 7. A 0 year bond has annual coupons of 500 and a maturity value of Calculate the Macaulay Duration of the bond using an annual effective interest rate of 4%. 500(1) v 500() v (0) v 7000(0) v MacDur Price of Bond a ( a 0 v ) 7000(0) v a 7000 v a 0 =+> PMT=1 N =0 I =4 CPT PV= , a ( a 0 v ) 7000(0) v Price of Bond: PMT=500 N =0 I =4 FV=7000 Cpt PV=10, ,9.857 MacDur ,804.47

8 8. A 15 year bond has a price of 100,000. The bond has a Modified duration of 11 and a Macaulay convexity of 100. These values are based on an annual effective interest rate of 8%. Estimate the price of this bond at an interest rate of 6% using both duration and convexity. MacDur 11(1.08) ModCon (1.08) ( ) ( ) 100, 000[1 11( ) ] 1,918.8

9 9. The stock of Raya Inc pays an annual dividend. The next dividend is 5 and is payable later today. The dividends are expected to increase in the future. The dividend paid at the end of one year is expected to be 7. The dividend paid at the end of years is expected to be 9. The dividends will continue to increase by each time until a dividend of 5 is paid. All dividends thereafter are expected to remain at 5. Using the dividend discount method and an annual effective interest rate of 8%, calculate the price of the stock. The price is the present value of expected future cash flows. The first dividend is today and the present value is (1.08) 1 (1.08) The next 14 payments are 7 14(1.08) The level payments of 5 has a present value of (1.08) (1.08) 1 (1.08) Total (1.08) (1.08)

10 10. Nathan will make payments of 000 at the end of 5 years and 4000 at the end of 10 years. Calculate the Macaulay convexity of Nathan s payments using an annual effective interest rate of 7.% (5 ) v 4000(10 ) v 4, v 4000v MacCov

11 11. The Purdue Life Insurance Company has promised to pay 1,000,000 to Hannah at the end of 6 years. Purdue wants to protect itself from interest rate changes using Reddington Immunization at an annual effective interest rate of 5.5% using the following two bonds: a. Zero coupon bonds maturing for 10,000 at the end of years; and b. Zero coupon bonds maturing for 5000 at the end of 8 years. Assuming that Purdue can buy partial bonds, determine the number of 8 year zero coupon bonds that Purdue should buy. A total price of bond a B total price of bond b A+B=1,000,000v 6 A 8B 6 1,000,000 v 6 B 490, Price of B=5000v , nb

12 1. The stock of Jones Mining Company pays a quarterly dividend of 6 with the next dividend payable in 1 month. Calculate the price of Jones stock using the dividend discount method with an annual effective interest rate of 1%. (4) i i 4 (4).877% 6 Price (1.0877)