Math 373 Fall 2012 Test 2

Transcription

1 Math 373 Fall 2012 Test 2 October 18, Jordan has the option to purchase either of the two bonds below. Both bonds will be purchased to provide the same yield rate. a. A 20-year zero coupon bond which matures for 100,000 and has a price of 50,000. b. A 20-year bond with a par value of 60,000. The bond matures for par and has a coupon rate of 8% payable semi-annually. Calculate the price of the second bond. For the zero coupon bond: /20 50, 000(1 i) 100, 000 (1 i) 2 i (2) For Bond 2, we need (2) i 2 since the coupons are semi-annual. (2) i 2 1/2 ( ) Using our calculator: FV=60,000;I/Y= ; N 40; PMT 60, 000(0.04) 2400 CPT PV=98,651.09

2 2. Tete is receiving an annuity with payments at the end of each quarter for 10 years. The payments at the end of each quarter during the first year are The payments at the end of each quarter during the second year are The payments continue to increase each year until the payments during 10 th year are 10,000 at the end of each quarter. Calculate the present value of this annuity at an annual effective interest rate of 8%. This is the formula that does not follow the rules (1.08) (1.08) 10(1.08) a 10v PV= , (4) 1/4 i (1.08) 1 4

3 3. Yang Corporation has a loan of 400,000 to be repaid with annual payments for 8 years using the sinking fund method. The annual effective interest rate on the loan is i while the sinking fund earns an annual effective interest rate of 5%. The sinking fund deposit is determined so that the amount in the sinking fund at the end of 8 years will exactly repay the amount of the loan. The total annual payment (interest on the loan plus the sinking fund deposit ) is 66, Calculate i. Total Payment =I+D=66, L il+ s nj 400,000 66, i(400,000) 66, (1.05) i(400,000) 41, , , , i ,000

4 4. Jixiang is receiving an annuity with continuous payments at a rate of 1000t 500 at time t for the next 12 years. Calculate the present value of this continuous annuity at 10% t 0.1t 0.1t PV f ( t) v( t) dt (1000t 500) e dt (1000 t) e dt (500) e dt t 0.1t 1000 te dt 500 e dt 1000( Ia) 500a e e a v 1e e ,

5 5. A loan is being repaid with level annual payments of 1000 followed by a drop payment. The loan has an annual effective interest rate of 3%. The outstanding loan balance right after the 3 rd payment is 20,000. Calculate the amount of interest and the amount of principal in the 5 th payment. Time Payment Interest Principal OLB 3 20, (20,000)(0.03)= =400 20, =19, (19,600)(0.03)= =412 Interest is 588 and principal is 412.

6 6. A 40 year bond is purchased to yield 6% convertible semi-annually. The bond matures for par which is 10,000. The coupon rate is 8% convertible semi-annually. The book value right after the payment of the Calculate k. th k coupon is 12, The Book Value right after a coupon is the present value of future cash flows. Therefore, we can use the calculator to complete this calculation: FV= 10,000 PMT=(10,000)(0.04 )= 400 I/Y = 3% PV=Book Value = 12, CPT N=80 k = 67 Therefore, k = = 13

7 7. Qian is the beneficiary of a trust fund. She receives payments at the end of each month for 20 years. The payment at the end of the first month is 100. The payment at the end of the second month is 200. The payments continue to increase by 100 per month. Calculate the accumulated value of the payments at a nominal interest rate of 6% compounded monthly. Since the payments increase every month, this is the P and Q formula. n Accumulated Value=(Present Value)(1+i) 12 i AV a a (1.005) (1.005) (1.005) (1.005) (1.005) (1.005) , 487,022

8 8. A 40 year bond has level semi-annual coupons of 125. The bond is bought to yield 7.5% convertible semi-annually. The amount of interest in the 10 th coupon is Calculate the amount of interest in the 25 th coupon. I P Fr P 125 P t t The principal is a geometric sequence so P P i (1 ) 2.70(1.035) 4.69 I Fr P

9 9. A 20 year bond has a maturity value of 10,000 and semi-annual coupons of 375. The bond was bought to yield 6% convertible semi-annually. The bond is sold between the 5 th and 6 th coupon. The dirty value using the practical approach is 11, Calculate the clean value using the theoretical approach. Start by finding B5 N 35 <= There are 35 periods remaining FV 10,000; PMT 375; I / Y 6% / 2 3% CPT PV=11, Under Practical Approach, Dirty Value =B (1 fj) 11, , (1 f (0.03)) 5 11, , f (11, )(0.03) 3 f Under Theoretical Approach, Clean Value=B (1 j) Fr( s ) 5 f 2/3 11, (1.03) 375 2/3 (1.03) 1 11,

10 10. A loan is being repaid with level annual payments for n years. The principal in the 5 th payment is The principal in the 10 th payment is The interest in the 20 th payment is Calculate n. The principal is a geometric sequence so ( Principal in 5th Payment) (1 i) ( Principal in 10th Payment) 30.64(1 i) i ( (1 ) ( Principal in 5th Payment) i Principal in 20th Payment)=30.64( ) Q Interest + Principal = = Principal in the 5th payment =Qv n n n4 ln n 4 36 n ln

11 11. Cong Corporation has a loan of 25,000 that is being repaid over 15 years using the sinking fund method. The interest rate on the loan is an annual effective interest rate of 6% and the interest rate earned by the sinking fund is an annual effective interest rate of 4.25%. The sinking fund deposit is calculated so that the amount in the sinking fund at the end of 15 years will equal the amount of the loan. Calculate the amount in the sinking fund after 10 years. L 25, 000 D s 15 (1.0425) 1 nj (1.0425) 1 Amount in the Sinking Fund=D s , kj

12 12. A special 30-year bond matures for 40,000. The bond pays semi-annual coupons that increase every year. Each coupon during the first year is 200. Each coupon during the second year is 400. Coupons continue to increase each year by 200 until each coupon during the 30 th year is Calculate the price of this special bond to produce a yield of 12% convertible semi-annually. Price of the Bond is Present Value of Future Cash Flows. The coupons are use the formula that does not follow the rules. (2) i i (1.06) a Price= (1.1236) , 000(1.1236) 30 ( ) (1.1236) 30(1.1236) , 000(1.1236) 30 27,564.60

13 13. A 10-year bond has a par value of 20,000. The bond pays annual dividends of Fr and matures for 21,000. The bond is bought at a discount of to yield an annual effective interest rate of 8%. Calculate Fr. The Discount =C - P P C Discount 21, , Use the Calculator PV=19, N=10 I/Y=8% FV=21,000 CPT PMT=Fr=1500