Math 373 Test 2 Fall 2013 October 17, 2013 1. You are given the following table of interest rates: Year 1 Year 2 Year 3 Portfolio Year 2007 0.060 0.058 0.056 0.054 2010 2008 0.055 0.052 0.049 0.046 2011 2009 0.053 0.051 0.048 0.044 2012 2010 0.050 0.049 0.047 0.042 2013 2011 0.047 0.046 0.045 2012 0.045 0.044 2013 0.043 Deepa invests 1000 on January 1, 2008 and another 2000 on January 1, 2011 in a fund that pays interest using the Investment Year Method. How much does Deepa have on December 31, 2012? 1000(1.055)(1.052)(1.049)(1.046)(1.044) 1271.381451 2000(1.047)(1.046) 2190.324 1271.3814512190.324 3461.705451
2. Sarah has a sinking fund loan of 500,000. She is repaying the loan with monthly interest payments using an interest rate of 12% compounded monthly and a monthly sinking fund deposit for the next 10 years. The sinking fund earns an interest rate of 8% compounded monthly. The sinking fund deposits are determined so that the amount in the sinking fund at the end of 10 years will exactly equal the amount of the loan. Sarah decides to repay the loan balance of 500,000 at the end of 8 years using the amount in the sinking fund plus a lump sum payment of P. Calculate P..08 0.006666 12 500, 000 500, 000 D 2733.046386 s 120 (1.006666) 1 120 i.666% 0.006666 2733.046386s P 500,000 96 i.666% P 500, 000 365,869.0468 134,130.9532
3. JT is receiving monthly annuity payments for four years while he is in college. The payments increase each month and are payable at the end of the month. The first payment is 200. The second payment is 400. Each subsequent payment is 200 larger than the previous payment. JT invests each payment in an account that pays a nominal interest rate of 9% compounded monthly. Calculate the amount in the account at the end of 4 years..09 0.0075 12 P&Q Formula PV 200 48 200 a ( 48 ) 48.75% 48.75% 0.0075 a v i i PV 200(40.18478189) 26, 666.6666(40.18478189 33.53347852) PV 185, 405.0462 AV 185, 405.0462(1.0075) AV 265,389.772 48
4. Qinyu has the choice of the following two bonds: a. Bond A is a zero coupon bond which has a purchase price of 20,000 and matures in 10 years for 42,000. b. Bond B is a 10 year bond with semi annual coupons. Bond B can be bought at a discount of 3000 for a price of 20,000. Bond A and Bond B has the same yield rate. Calculate the amount of the semi-annual coupon on Bond B. Bond A: zero coupons 20, 000 42, 000v v 10 10 0.476190476 1i 1.07701544 i 0.07701544 0.5 We need the semi-annual rate for Bond B = (1.0770144) 1 3.77793544% Bond B: semi-annual coupons, discount of 3000 20, 000 Ca 23, 000v 20i3.7793544% Calculator : N 20 I / Y 3.7793544 PV 20, 000 FV 23, 000 CPTPMT 652.80 20
5. Yishen is receiving an annuity with payments of 2500 at the end of each year for the next 10 years. Yishen invests the payments in Fund A which earns an annual effective interest rate of 8%. At the end of each year, Yishen takes the interest earned in Fund A and invests it in Fund B which earns an annual effective interest rate of 10%. Calculate the total amount (in both Funds) that Yishen will have at the end of 10 years. Fund A AV 10* 2500 25, 000 Fund B 200 9 PV 200 a ( a 9 v ) 9i10% 9i10%.10 PV 200(5.759023816) 2000(5.759023816 3.816878565) PV 5036.095265 AV 5036.095265(1.10) AV 11,874.8492 9 TotalAmt 25, 000 11,874.8492 36,874.8492
6. A 10 year continuous annuity pays at an annual rate of t If the discount function is (1 0.05 t) 2 3t at time t., calculate the present value of this annuity. 10 2 ( 3 )(1 0.05 ) 0 10 3 2 ( 0.05t 0.85t 3 t)dt 0 t t t dt 4 3 2 t t t 0.05( ) 0.85( ) 3( ) 4 3 2 308.333 0,10
7. Kexin buys a special 20 year bond which matures for 20,000 and has annual coupons that increase each year. The first coupon is 1000. The second coupon is 1000(1.08). The third coupon is 1000(1.08) 2. The coupons continue to increase such that each coupon is 1.08 times the prior coupon. The bond is bought to yield an annual effective return of 6%. Calculate the price of the bond. Price=1000(1.08) v 1000(1.08) v 1000(1.08) v... 1000(1.08) v 20, 000v 0 1 2 2 3 19 20 20 1000v1000(1.08) 1.08 1 1.06 v 20 21 20, 000v 20 427.6494993 6236.094538 0.018867925 28,901.51743
8. A loan of 60,000 is being repaid with 20 level annual payments. The principal in the 5 th payment is 1995.13. The principal in the 20 th payment is 5095.14. Calculate the principal in the 15 th payment. Prin (1 i) Prin 15 5 20 15 1995.13(1 i) 5095.14 15 (1 i) 2.553788475 1i 1.0645 Prin (1 i) Prin 10 5 15 10 1995.13(1.0645) 3727.587586
9. A loan is being repaid with 30 annual payments. The payments in odd numbered years (years 1, 3, 5,..., 29) are 10,000. The payments in the even numbered years (years 2, 4, 6,..., 30) are 20,000. The loan has an annual effective interest rate of 10%. Calculate the interest in the 10,000 payment made at the end of the 29 th year. OLB (1 i) Q OLB k1 OLB * i INT k1 k k OLB OLB OLB OLB OLB 30 29 29 28 28 0 (1.10) 20, 000 0 18181.82 (1.10) 10, 000 18181.82 25619.835 INT 29 25619.385(0.10) 2561.98
10. A 30 year bond which matures for par has a coupon rate of 4% convertible semi-annually. The bond is bought to 5% convertible semi-annually. The book value of this bond is 106,892.51 right after the 15 th coupon is paid. The book value of this bond is 107,095.70 right after the 16 th coupon is paid. Determine the price of this bond. B B P t t1 t 107, 095.7 106,892.51 P P 16 203.19 16 T PMT INT PRIN BV 15 106,892.51 16 106892.51(.025) -203.19 107,095.51 106892.51(.025) 203.19 PMT PMT 2469.12275 The book value represents the present value of future cash flows. To get the price, we can treat that as our future value in the calculator and calculate the present value at time zero. Calculator N 15 represents the number of coupons from time zero until the book value right after the 15th coupon I / Y 2.5 PMT 2469.12275 FV 106,892.51 CPT PV 104,376.74 Price
11. Zach s grandparents have set up a trust fund that will pay him annuity payments at the beginning of each month for the next 25 years. The payments are 25 per month in the first year, 50 per month in the second year, 75 per month in the third year, etc. This pattern continues until 625 is paid each month during the 25 th year. Calculate the present value of these payments at an annual effective interest rate of 6%. annual effective i=6% (12) (12) i 12 i (1.06)=(1+ ) 0.004867551 12 12 Formula that does not follow the rules: 25 (12) (12) a 25v 25i 6% i6% i i PV (FirstPmt) 1 <==The 1 is becau (12) se the payments i 12 12 12 are at the beginning of the year. 13.55035753 5.824965763 PV (25) (1.004867551) 0.004867551 PV 39,871. 16
12. Samantha has a sinking fund loan of 154,000. The term of the loan is 8 years. Samantha will make annual payments of interest to the lender based on an interest rate of i. She will also make annual payments into a sinking fund based on an annual interest rate of 5%. The amount in the sinking fund at the end of 8 years will exactly repay the loan. The total amount that Samantha will pay (interest on the loan plus sinking fund deposit) each year is 25,000. Calculate i. I il 154, 000i L 154,000 D 16127.1593 s s n j 8i5% I D25, 000 I 8872.840701 154,000i i 0.057615849
13. Jordan has a loan of 40,000 that is being repaid with level annual payments of 5000 plus a balloon payment of B. The annual effective interest rate on this loan is 9%. Determine B. Calculator I / Y 9 PV 40, 000 PMT 5000 FV 0 CPT N 14.77140526 2 nd AMORT P1=1 P2=14 BAL=3573.135132 B BAL + Normal Payment =3573.135132 5000 8573.14