YORK UNIVERSITY MATH MATHEMATICAL THEORY OF INTEREST FINAL EXAM DECEMBER 14TH, 2011, 2:00 p.m. 5:00 p.m. DO NOT WRITE IN THIS AREA

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1 YORK UNIVERSITY MATH MATHEMATICAL THEORY OF INTEREST FINAL EXAM DECEMBER 14TH, 2011, 2:00 p.m. 5:00 p.m. Last Name: Given Names: Student Number: Signature : DO NOT WRITE IN THIS AREA INSTRUCTIONS: 1. You are allowed to have a BA-II Plus (or Professional) calculator. An official formula sheet will also be provided for you. No other aids are allowed. 2. All work must be done in the free space provided below each problem or on the back of the preceding page. No additional sheets will be accepted or marked. Scrap paper will be provided for you. 3. This exam consists of 17 pages in total. There are three parts to this exam: a TRUE/FALSE section with 5 questions (worth 20 points), a MULTIPLE CHOICE section with 10 questions (worth 80 points), and a SHORT ANSWER section with 5 questions (worth a total of 90 points). Make sure to use your time wisely, and do not spend too much time on any one question. 4. Show all work for the short answer section. Answers without sufficient supporting work will receive *no credit*, even if the answers are correct. Question Max Score Your Score PART I 20 PART II TOTAL 190 GGGGGGGGGOOD LUCK! 1

2 PART I: TRUE OR FALSE Circle the correct answer: true (T) or false (F). Also, please write the letter answer beside the question. If there is any ambiguity in your answer you will receive no credit. 1). For a loan repaid with level payments, the principal repaid decreases as time increases under the amortization method. T F 2). (Ia) = 1 i 2 T F 3). Under the net present value method, the choice of interest rate doesn t matter. That is, regardless of the interest rate chosen, you will always make the same decision on your investment options. T F 4). If the annual effective interest rate is fixed, then the equivalent discount rate convertible m-thly increases as m increases. T F 5). a n i v n = a 2n i a n i T F 2

3 PART II: MULTIPLE CHOICE For each question, choose (circle) one answer which is the closest to the correct answer. Also, please write the letter answer beside the question. If there is any ambiguity in your answer you will receive no credit. 1). Which of the following is NOT an expression for s n i? [ ] (A) v (1+i) n 1 d (B) 1 + (1 + i) + (1 + i) (1 + i) n 1 (C) (1 + i) n 1 [ 1 v n d (D) (1 + i) n 1 a n i ] (E) v [ (1+i) n 1 1 v ] 3

4 2). Xin borrows 100,000 for 10 years at an annual effective interest rate of 10%, and repays the loan with annual level payments at the end of each year. The total interest paid by Xin over the 10 year period is X. Calculate X. (A) 5,582 (B) 16,275 (C) 54,235 (D) 62,745 (E) 81,375 4

5 3). Calculate the nominal rate of discount convertible monthly that is equivalent to a nominal rate of interest of 10% per year convertible semi-annually. Hint: don t do any rounding in your calculations. (A) 9.52% (B) 9.72% (C) 9.76% (D) 9.92% (E) 9.80% 5

6 4). The accumulated value at the time of the last payment of a series of 50 annual payments starting at 1000 and increasing by 100 each year thereafter is equal to X. The annual effective interest rate is 1%. Calculate X. (A) 196,590 (B) 209,095 (C) 215,541 (D) 221,010 (E) 243,105 6

7 5). Steven pays off a loan of $10,000 with annual payments of size $1000 at the end of each year, plus a final smaller payment. The interest charged on the loan is 3% per year. The size of the final smaller payment is X. Calculate X. (A) 1.03 (B) (C) (D) (E)

8 6). A 20-year 10% bond with face value 100 and redemption value of 150 is priced at $123. Calculate the yield rate. (A) 2.03% (B) 4.28% (C) 8.21% (D) 8.56% (E) 8.68% 8

9 7). A 10% bond with face value 1000 pays semi-annual coupons. If the present value of the redemption amount is 300 at a yield rate of 15%, what is the price of the bond? (A) (B) (C) (D) (E)

10 8). A customer is offered an investment where interest is calculated according to the force of interest δ t = 0.02t for t 0. The customer invests 1000 at time t =0. What average annual rate of interest does the customer earn over the first four-year period? Hint: again, don t round in your calculations. (A) 4.00% (B) 4.04% (C) 4.08% (D) 8.67% (E) 17.35% 10

11 9). Peter takes out a mortgage loan for $750,000 amortized over 30 years at i (2) =5%. Find Peter s monthly mortgage payment. (A) 2, (B) 4, (C) 4, (D) 4, (E) 48,

12 10). Helen invests $100,000 and receives $75,000 in one year and $50,000 in two years. What was Helen s internal rate of return? (A) 5.00 (B) 9.08 (C) (D) (E)

13 PART III: SHORT ANSWER Note that not all questions are worth the same amount. Don t forget to explain your reasoning. 1). Assume that you take a 250,000 mortgage to be amortized over 25 years at i (12) =12%. (a). [5 points] Find the size of the monthly payment and the outstanding principal after 5 years. (b). [15 points] After 10 years you will need to re-finance your mortgage and you are afraid that interest rates will be much higher by this point in time. You therefore decide to buckle down and make additional monthly payments of X in years 6 through 10 so that after 10 years your outstanding balance is only 125,000. Find X. 13

14 2). [20 points] John borrows 1000 for 10 years at an annual effective interest rate of 10%. He can repay this loan using the amortization method with payments P at the end of each year. Instead, John repays the 1000 using a sinking fund that pays an annual effective rate of 14%. The deposits to the sinking fund are equal to P minus the interest on the loan and are made at the end of each year for 10 years. Determine the balance in the sinking fund immediately after the repayment of the loan. 14

15 3). (10 points) Justify each of the following equalities using only a timeline and your own words. Do not show any algebra! (a). ä n = a n + 1 v n (b). (Ia) n i + (Da) n i = (n + 1) a n i 15

16 4). [20 points] A 20-year 5% bond with face amount 100 is callable on a coupon date in the 10th to 20th years. In the 10th through 15th years the bond is callable with a redemption amount of 90, while in the 16th through 20th years the bond is callable with a redemption amount of 110. What price should an investor pay in order to ensure a minimum nominal annual yield to maturity of 4%? 16

17 5). [20 points] Lara is saving for her retirement and estimates that she will need X in savings by the time that she is 65. Her estimate, X, is the present value at i (12) =12% of a perpetuity which pays 5000 per month at the beginning of each month in the first year. The monthly payment then increases by 3% each year. Find X. 17