YORK UNIVERSITY MATH MATHEMATICAL THEORY OF INTEREST FINAL EXAM APRIL 9, 2010, 9:00 a.m. 12:00 p.m. DO NOT WRITE IN THIS AREA

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1 YORK UNIVERSIY MAH MAHEMAICAL HEORY O INERES INAL EXAM APRIL 9, 2010, 9:00 a.m. 12:00 p.m. Last Name: Given Names: Student Number: Signature : DO NO WRIE IN HIS AREA INSRUCIONS: 1. he only aids allowed are a calculator and one single-sided hand-written aidsheet (8.5 by 11 ). You may NO borrow a calculator or a formula sheet from another student. 2. All work must be done in the free space provided below each problem or on the back of the preceding page. No additional aid sheets will be accepted or marked. Scrap paper will be provided for you. 3. Show all work. Answers without sufficient supporting work will receive *no credit*, even if the answers are correct. Round answers to 2 decimal places and all interest rates to 4 decimal places, unless specified otherwise. 4. his exam consists of 17 pages in total. here are three parts to this exam: a SHOR ANSWER section with 4 questions (worth a total of 60 points), a RUE/ALSE section with 10 questions (worth 30 points), and a MULI- PLE CHOICE section with 10 questions (worth 60 points). Make sure to use your time wisely, and do not spend too much time on any one question. Question Max Score Your Score PAR II 30 PAR III 60 OAL 155 GGGGGGGGGOOD LUCK! 1

2 PAR I: SHOR 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] ind 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 and 7, and additional monthly payments of 2X in years 8-10 so that after 10 years your outstanding balance is only 125,000. ind X. 2

3 2). [10 points] A customer is offered an investment where interest is calculated according to the following force of interest. { 0.02t 0 t 3, δ t = t > 3. he customer invests 1000 at time t =0. What nominal rate of interest, compounded monthly, is earned over the first four-year period? 3

4 3). [15 points] ill in the blanks of the first few lines of the amortization schedule for a loan with level payments. Don t forget to show your work! t OB t K t I t P R t

5 4). [15 points] A 15-year 10% bond with face amount 1000 is callable (at the option of the issuer) on a coupon date in the 10th to 15th years. In the 10th and 11th years the bond is callable at par, while in the 12th through 15th years the bond is callable with a redemption amount of What price should an investor pay in order to ensure a minimum nominal annual yield to maturity of 9.5%? 5

6 PAR II: RUE OR ALSE Circle the correct answer: true () or false (). 1). or a level payment perpetuity, each payment is equal to the interest earned on the prinicipal. 2). (Ia) n i + (Da) n i = n a n i 3). he dollar-weighted rate of return is always higher than the time-weighted rate of return. 4). Under the net present value method, the choice of interest rate doesn t matter. hat is, regardless of the interest rate chosen, you will always make the same decision on your investment options. 5). If the annual effective interest rate is fixed, then the equivalent discount rate convertible m-thly decreases as m increases. 6

7 6). s n i (1 + i) n = s 2n i s n i 7). When the time period is smaller than one, the interest charged using simple interest of i = 5% is greater than the interest charged using compound interest with i = 5%. 8). or i > 0, a n i increases as n increases. 9). or any financial transaction, the internal rate of return is greater than -100%. 10). Alexey is re-paying a loan with level payments using the amortization methods. he principal he repays with each payment increases as time increases. 7

8 PAR III: MULIPLE CHOICE or each question, choose (circle) one answer which is the closest to the correct answer (choosing more than one answer will receive no credit). Also, please write the letter answer beside the question. 1). Which of the following expressions does NO represent an expression for ä n i? (A) v n 1 [ (1+i) n 1 i (B) 1 + v + v v n (C) 1 vn d s n i (D) (1+i) n 1 (E) 1 vn 1 v ] 8

9 2). Eddy borrows 50,000 for 5 years at an annual effective interest rate of 5%. At the end of year, he pays the interest on the loan and deposits the level amount necessary to repay the principal to a sinking fund earning an annual effective interest rate of 3%. he total interest paid by Eddy over the 5 year period is X. Calculate X. (A) 6,945 (B) 7,744 (C) 8,143 (D) 8,940 (E) 9,589 9

10 3). Jane receives cash flows of 50 today, 200 in one year, and 100 in two years. he present value of these cash flows is at an annual effective interest rate i. Calculate i. (A) 10% (B) 11% (C) 12% (D) 13% (E) 14% 10

11 4). he present value of a series of 100 payments starting at 1000 at the end of the first year and decreasing by 5 each year thereafter is equal to X. he annual effective interest rate is 5%. Calculate X. (A) 8,015 (B) 9,924 (C) 13,650 (D) 17,939 (E) 19,848 11

12 5). Calculate the nominal rate of discounted convertible monthly that is equivalent to a nominal rate of interest of 12% per year convertible monthly. (A) 10.91% (B) 11.26% (C) 11.65% (D) 11.88% (E) 11.94% 12

13 6). Derek pays off a loan of $1000 payments with annual payments of size $100 at the end of each year, plus a final smaller payment. he interest charged on the loan is 3% per year. he size of the final smaller payment is X. Calculate X. (A) 4.60 (B) 6.75 (C) (D) (E)

14 7). A 15-year bond with face value 1000 pays 10% coupons semi-annually. he bond is priced at 950 to yield an annual nominal rate of 12% convertible semi-annually. Calculate the redemption value of the bond. (A) 699 (B) 1503 (C) 1678 (D) 2460 (E)

15 8). You are given the following information about an investment account: Value Immediately Date Before Deposit Deposit January 1 10 August 1 12 X December 31 X he time-weighted return is 0%, and the dollar-weighted rate of return is Y. Calculate Y. (A) % (B) % (C) 0.00% (D) (E)

16 9). Andrew takes out a mortgage loan for $275,000 amortized over 30 years at i (2) =4%. ind Andrew s monthly mortgage payment. (A) (B) (C) (D) (E) 15,

17 10). or 17,982, Perrin purchases an annuity-immediate that pays 400 monthly at an interest rate of 12% compouned monthly. ind the number of years during which Perrin receives payments. (A) 5 (B) 10 (C) 20 (D) 45 (E) 60 17