Final Examination MATH NOTE TO PRINTER

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1 Final Examination MATH NOTE TO PRINTER (These instructions are for the printer. They should not be duplicated.) This examination should be printed on paper, and stapled with 3 side staples, so that it opens like a long book.

2 McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH THEORY OF INTEREST EXAMINER: Professor W. G. Brown DATE: Thursday, April 24nd, 2003 ASSOCIATE EXAMINER: Prof. N. Sancho TIME: 14:00 17:00 hours SURNAME: MR, MISS, MS, MRS, &c.: GIVEN NAMES: STUDENT NUMBER: 1. Fill in the above clearly. Instructions 2. Do not tear pages from this book; all your writing even rough work must be handed in. 3. Calculators. While you are permitted to use a calculator to perform arithmetic and/or exponential calculations, you must not use the calculator to calculate such actuarial functions as a ni, s ni, (Ia) ni, (Is) ni, (Da) ni, (Ds) ni, etc. without first stating a formula for the value of the function in terms of exponentials and/or polynomials involving n and the interest rate. You must not use your calculator in any programmed calculations. If your calculator has memories, you are expected to have cleared them before the test. 4. This examination booklet consists of this cover, Pages 1 through 8 containing questions; and Pages 9 and 10, which are blank. For all problems you are expected to show all your work, and to simplify algebraic and numerical answers as much as you can. All solutions are to be written in the space provided on the page where the question is printed. When that space is exhausted, you may write on the facing page. Any solution may be continued on the last pages, or the back cover of the booklet, but you must indicate any continuation clearly at the bottom of the page where the question is printed! You may do rough work anywhere in the booklet. 5. You are advised to spend the first few minutes scanning the problems. (Please inform the invigilator if you find that your booklet is defective.) 6. NOT MORE THAN 100 MARKS ARE REQUIRED FOR A PERFECT SCORE. You may attempt as many problems as you wish. PLEASE DO NOT WRITE INSIDE THIS BOX 1(a) 1(b) 1(c) 1(d) 2(a) 2(b) 3(a) 3(b) /3 /3 /4 /5 /8 /7 /2 /2 3(c) 3(d) 3(e) 3(f) 3(g) 4(a) 4(b) 4(c) /2 /2 /2 /2 /3 /2 /3 /4 4(d) 5(a) 5(b) 6(a) 6(b) 6(c) 6(d) 6(e) /6 /7 /8 /3 /2 /3 /4 /3 7(a) 7(b) 8(a) 8(b) 8(c) Total Term /5 /10 /3 /4 /8 /(120)100 /30

3 Final Examination MATH (a) [3 MARKS] The total amount of a loan to which interest has been added is 5,000. The term of the loan was 4 years. If the nominal annual rate of interest was 6% and interest was compounded semi-annually, determine the original amount of the loan, showing all your work. (b) [3 MARKS] Showing all your work, determine the simple interest rate under which a sum of money will double in 5 years. (c) [4 MARKS] Showing all your work, determine the effective annual compound discount rate under which a sum of money will double in 8 years. (d) [5 MARKS] Showing all your work, determine the rate of interest, convertible continuously, that is equivalent to an effective interest rate of 1% per month.

4 Final Examination MATH (a) [8 MARKS] To repay a loan, X is obliged to pay Y 1,000 at the end of December, 2004, and 1,200 at the end of December, He proposes to replace these two payments by a single payment of 2,196 at the end of December, If Y accepts this proposal, what yield rate will he be earning on his loan? Show all your work. (b) [7 MARKS] Showing all your work, determine the value at time t = 0 of a continuous annuity that pays 10,000 per year for 2 years, at an effective annual interest rate of 5%.

5 Final Examination MATH Express each of the following only in terms of l x, and v. (a) [2 MARKS] d 27 (b) [2 MARKS] 4 q 24 (c) [2 MARKS] ä 20:25 (d) [2 MARKS] A 1 20:25 (e) [2 MARKS] A 20:25 (f) [2 MARKS] The probability that a 25-year old will survive 40 years, but will die before reaching age 75. (g) [3 MARKS] 12 a 20:25

6 Final Examination MATH The Wallace Widget Company is planning to borrow 150,000 from the Bank of Antigonish, and to undertake to pay interest annually at a rate of 12%; they plan to contribute equal annual payments to a sinking fund that earns interest at the rate of 9%. The sinking fund will repay the principal at the end of 10 years. Showing all your work, determine (a) [2 MARKS] the annual interest payment, (b) [3 MARKS] the annual payment into the sinking fund At the end of 4 years, when Wallace has made its annual interest payment and its 4th payment to the sinking fund, it proposes that this should be the last payment to the sinking fund. It will apply the balance X accumulated to date in the sinking fund to repay principal, and it will amortize the remainder of the principal by equal annual payments over the next 5 years, at a rate of 10%. (c) [4 MARKS] Determine the annual level payment Y under this proposal. (d) [6 MARKS] Construct an amortization table for this proposal, under the following headings, beginning immediately after the 4th and last payment to the sinking fund; assume also that all outstanding interest on the loan has been made annually to date: Duration Payment Interest Principal Outstanding Repaid Principal X 5 Y =

7 Final Examination MATH Consider a 100 par-value 15-year bond, with semi-annual coupons at the nominal annual interest rate of 4%, convertible every six months. Let t represent time in half-years; assume that the bond is callable at on any coupon date from t = 10 to t = 20 inclusive, at from t = 21 to t = 29 inclusive, but matures at at t = 30. In each of the following cases, determine what price an investor should pay to guarantee himself (a) [7 MARKS] a nominal annual yield rate of 5%, convertible semiannually; (b) [8 MARKS] an effective annual yield rate of 3%.

8 Final Examination MATH In addition to her down payment, Mary s purchase of her new home is financed by a mortgage of 60,000 payable to the vendor; the mortgage is amortized over 20 years, with a level payment at the end of each month, at a nominal annual rate of 6% compounded monthly. (a) [3 MARKS] Determine the monthly payments under this mortgage. (b) [2 MARKS] Divide the first payment into principal and interest. (c) [3 MARKS] Determine the outstanding principal immediately after the 60th payment. (d) [4 MARKS] Divide the 60th payment into principal and interest. (e) [3 MARKS] Determine the payment that Mary could make at the end of each year which would be equivalent to the year s 12 monthly payments.

9 Final Examination MATH (a) [5 MARKS] Define what is meant by (Da) n and (Ia) n, and explain verbally why (Da) 30 + (Ia) 30 = 31a 30. (b) [10 MARKS] Showing all your work, find the present value (using effective annual interest rate i = 6%) of a perpetuity which pays 100 after 1 year, 200 after 2 years, increasing until a payment of 2000 is made, after which payments are level at 2000 per year forever. [For this problem you may assume that (Ia) n = än nv n i (1) (Ia) = ä i (2) (Is) n = s n n.] i (3)

10 Final Examination MATH In order to complete the sale of his home in Vancouver, John accepted, in partial payment, a 200,000 mortgage amortized over 15 years with level semi-annual payments at a nominal annual rate of 5% compounded semi-annually. Fred has cash available, and is prepared to buy the mortgage from John and to invest a fixed portion of the semi-annual payments he receives in a sinking fund that will replace his purchase capital in 15 years. The sinking fund will earn interest at only 4%, compounded semi-annually. Showing all your work, determine the following: (a) [3 MARKS] the amount of the semi-annual mortgage payments (b) [4 MARKS] as a fraction of the purchase price Fred pays for the mortgage, the semi-annual payment into the sinking fund (c) [8 MARKS] the amount that Fred should pay for the mortgage in order to obtain an overall yield rate of 6%, compounded semiannually on his investment. (Note that the sinking fund earns 4% compounded semi-annually.)

11 Final Examination MATH continuation page for problem number You must refer to this continuation page on the page where the problem is printed!

12 Final Examination MATH continuation page for problem number You must refer to this continuation page on the page where the problem is printed!