McGILL UNIVERSITY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH THEORY OF INTEREST

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1 McGILL UNIVERSITY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH THEORY OF INTEREST Information for Students (Winter Term, 2003/2004) Pages 1-8 of these notes may be considered the Course Outline for this course. W. G. Brown April 29, 2004

2 Information for Students in MATH Contents 1 General Information Instructor and Times Course Description Calendar Description Syllabus (in terms of sections of the text-book) Verbal arguments Evaluation of Your Progress Term Mark Assignments Class Test Final Examination Supplemental Assessments Machine Scoring Plagiarism Published Materials Required Text-Book Website Reference Books Other information Prerequisites Calculators Self-Supervision Escape Routes Showing your work; good mathematical form; simplifying answers Timetable 9 3 First Problem Assignment 11 4 Second Problem Assignment 12 5 Solutions, First Problem Assignment 14 6 Third Problem Assignment 18 7 Solutions, Second Problem Assignment 20 8 Solutions, Third Problem Assignment 30 9 Class Tests Class Test, Version Class Test, Version Class Test, Version Class Test, Version Fourth Problem Assignment Solutions to Problems on the Class Tests Fifth Problem Assignment Solutions, Fourth Problem Assignment Solutions, Fifth Problem Assignment References 901 A Supplementary Lecture Notes 2001 A.1 Supplementary Notes for the Lectures of January 5th, 7th, and 9th, A The accumulation and A.1.2 amount functions The effective rate of interest A Simple interest A Compound interest 2003 A Present value A The effective rate of discount A Nominal rates of interest and discount A.2 Supplementary Notes for the Lecture of January 12th, A Forces of interest and discount A Varying interest. 2008

3 Information for Students in MATH A.3.3 A Summary of results 2008 A.3 Supplementary Notes for the Lecture of January 16th, A Introduction A Obtaining numerical results Determining time periods A The basic problem 2011 A.4 Supplementary Notes for the Lecture of January 19th, A The basic problem (continued) A Equations of value 2012 A.4.3 Unknown principal A Unknown time A Unknown rate of interest A Practical examples 2014 A.5 Supplementary Notes for the Lecture of January 21st, A.6 Supplementary Notes for the Lecture of January 23rd, A Introduction A Annuity-immediate 2018 A.7 Supplementary Notes for the Lecture of January 26th, A Annuity-immediate (continued) A Annuity-due A.8 Supplementary Notes for the Lecture of January 28th, A Annuity-due (continued) A.9 Supplementary Notes for the Lecture of January 30th, A Annuity values on any date A.10 Supplementary Notes for the Lecture of February 2nd, A Annuity values on any date (continued) A.11 Supplementary Notes for the Lecture of February 4th, A Perpetuities A Nonstandard terms and interest rates A.12 Supplementary Notes for the Lecture of February 6th, A Nonstandard terms and interest rates A Unknown time A.13 Supplementary Notes for the Lecture of February 9th, A Unknown time (continued) A.14 Supplementary Notes for the Lecture of February 11th, A Unknown rate of interest A Varying interest A Annuities not involving compound interest2047 A.15 Supplementary Notes for the Lecture of February 13th, A Introduction A Annuities payable at a different frequency than interest is convertible A.16 Supplementary Notes for the Lecture of February 16th, A Further analysis of annuities payable less frequently than interest is convertible A Further analysis of annuities payable more frequently than interest is convertible A Continuous annuities 2057 A Basic varying annuities A.17 Supplementary Notes for the Lecture of February 18th,

4 Information for Students in MATH A Basic varying annuities (continued) A More general varying annuities A Continuous varying annuities A Summary of results A.18 Supplementary Notes for the Lecture of February 20th, A Introduction A Discounted cash flow analysis A Uniqueness of the yield rate A Reinvestment rates 2066 A.19 Supplementary Notes for the Lecture of March 1st, A Reinvestment rates (continued) A Interest measurement of a fund A Time-weighted rates of interest A Portfolio methods and investment year methods 2069 A Capital budgeting A More general borrowing/lending models A.20 Supplementary Notes for the Lecture of March 3rd, A Introduction A Finding the outstanding loan balance A.21 Supplementary Notes for the Lecture of March 5th, A Finding the outstanding loan balance (continued) A Amortization schedules A.22 Supplementary Notes for the Lecture of March 8th, A Amortization schedules (continued) A Sinking funds A.23 Supplementary Notes for the Lecture of March 12th, A Sinking funds (continued) A.24 Supplementary Notes for the Lecture of March 15th, A Sinking funds (concluded) A Differing payment periods and interest conversion periods A Varying series of payments A Amortization with continuous payments A Step-rate amounts of principal A.25 Supplementary Notes for the Lecture of March 17th, A Introduction A Types of securities 2090 A.26 Supplementary Notes for the Lecture of March 19th, A Price of a bond A.27 Supplementary Notes for the Lecture of March 22nd, A Premium and discount A.28 Supplementary Notes for the Lecture of March 24th, A Premium and discount (concluded) A.29 Supplementary Notes for the Lecture of March 26th, A Valuation between coupon payment dates A Determination of yield rates A Callable bonds

5 Information for Students in MATH A.30 Supplementary Notes for the Lecture of March 29th, A Callable bonds (continued) A.31 Supplementary Notes for the Lecture of March 31st, A Valuation between coupon payment dates A Serial bonds A Some generalizations 2108 A Other securities A Valuation of securities A.32 Supplementary Notes for the Lecture of April 5th, B Problem Assignments and Tests from Previous Years 3001 B / B.1.1 First 2002/2003 Problem Assignment, with Solutions B.1.2 Second 2002/2003 Problem Assignment, with Solutions B.1.3 Third 2002/2003 Problem Assignment, with Solutions B.1.4 Fourth 2002/2003 Problem Assignment, with Solutions B.1.5 Fifth 2002/2003 Problem Assignment, with Solutions B /2003 Class Tests, with Solutions B.1.7 Final Examination, 2002/

6 Information for Students in MATH General Information Distribution Date: 0th version, Monday, January 5th, 2004 This version, Monday, January 12th, 2004 (All information is subject to change, either by announcements at lectures, on WebCT, or in print.) An updated version may be placed, from time to time, on the Math/Stat website (cf below), and will also be accessible via a link from WebCT.) The Course Outline for MATH can be considered to be pages 1 through 8 of these notes. 1.1 Instructor and Times INSTRUCTOR: Prof. W. G. Brown OFFICE: BURN 1224 OFFICE HRS. W 13:20 14:15 h.; (subject to F h.; change) and by appointment TELEPHONE: BROWN@MATH.MCGILL.CA CLASSROOM: BURN 1B24 CLASS HOURS: MWF 14:35 15:25 h. Table 1: Instructor and Times 1.2 Course Description Calendar Description THEORY OF INTEREST. (3 credits) (Prerequisite: MATH 141.) Simple and compound interest, annuities certain, amortization schedules, bonds, depreciation Syllabus (in terms of sections of the text-book) The central part of the course consists of many of the topics in the first nine chapters of the textbook [1] 1 ; section numbers, where shown, refer to that book. In the list below 1 [n] refers to item n in the bibliography, page 901. UPDATED TO April 29, 2004

7 Information for Students in MATH we show the chapters and appendices of the textbook. Following each is a description as of the date of this revision, of the sections to be excluded. This list will be updated during the semester, as becomes apparent that certain sections are not appropriate to the level of the course or the lecture time available. Chapter 1. The Measurement of Interest Portions of 1.9, will be omitted. For the present 1.10 will also be omitted. Chapter 2. Solution of Problems in Interest [1, pp ] of the method of equated time. In 2.6 You may omit the discussion Chapter 3. Elementary Annuities You may omit [1, 3.6 Nonstandard terms and interest rates], [1, 3.10 Annuities not involving compound interest, pp ] and corresponding exercises. We will also omit [1, 3.9 Varying interest] for the present (possibly to return). Chapter 4. More General Annuities In the following section we shall consider the problems strictly on an ad hoc basis: students are not expected to derive nor to apply the identities obtained: [1, 4.2 Annuities payable at a different frequency than interest is convertible; 4.3 Further analysis of annuities payable less frequently than interest is convertible; 4.4 Further analysis of annuities payable more frequently than interest is convertible]. Omit [1, 4.5 Continuous annuities] for the present. In [1, 4.6 Nonstandard terms and interest rates] we shall consider the derivation of formulæ for (Ia) n, (Is) n, (Da) n, (Ds) n, and their due and perpetual variants, also the question of annuities in geometric progression. Omit [1, 4.7 More general varying annuities, 4.8 Continuous varying annuities, 4.9 Summary of results] for the present, together with their exercises. Chapter 5. Yield Rates Omit [1, 5.2 Discounted cash flow analysis], except for the definition [1, p. 131] of yield rate. Omit [1, 5.3 Uniqueness of the yield rate] except you should read and understand the example [1, p. 133] of a problem where the yield rate is not unique. Omit [1, ] and accompanying exercises; but we will study [1, 5.4 Reinvestment rates] in preparation for Chapter 6. Chapter 6. Amortization Schedules and Sinking Funds In [1, 6.4] omit pages , where the function a n i&j is introduced. Omit [1, ] and accompanying exercises.

8 Information for Students in MATH Chapter 7. Bond and Other Securities Omit [1, 7.6 Determination of yield rates], [1, 7.8 Serial bonds], [1, 7.9 Some generalizations], [1, 7.10 Other securities], [1, 7.11 Valuation of securities]. Chapter 8. Practical Applications Omit this chapter. Chapter 9. More Advanced Financial Analysis Chapter 10. A Stochastic Approach to Interest Omit this chapter. Omit this chapter. Appendix I. Table of compound interest functions be done using calculators, these tables may prove useful. While most calculations will Appendix II. Table numbering the days of the year Appendix III. Basic mathematical review Topics that are beyond the required prerequisites will be explained if, as, and when they are used. Omit this section: no background in prob- Appendix IV. Statistical background ability is prerequisite to Math 329. Appendix V. Iteration methods Appendix VI. Further analysis of varying annuities Omit this Appendix, which is concerned with the formula for Summation by Parts, analogous to integration by parts for functions of a continuous variable. Appendix VII. Illustrative mortgage loan amortization schedule Appendix VIII. Full immunization Omit this Appendix, which is related to [1, 9.9], which is not in the syllabus. Appendix IX. Derivation of the variance of an annuity Appendix X. Derivation of the Black-Scholes formula Omit this Appendix. Omit this Appendix. UPDATED TO April 29, 2004

9 Information for Students in MATH Verbal arguments An essential feature of investment and insurance mathematics is the need to be able to understand and to formulate verbal arguments; that is, explanations of the truth of an identity presented verbally i.e., a proof in words, rather then an algebraic proof. In a verbal argument we seek more than mathematically correctness: we wish to see an explanation that could be presented to a layman who is not competent in the mathematical bases of this subject, but is still possessed of reason, and needs to be assured that he is not being exploited. This facet of the course will be seen, at first, to be quite difficult. When the skill has been mastered it can be used to verify the correctness of statements proved mathematically. Verbal arguments require some care with the underlying language; students who have difficulty with expression in English are reminded that all students have the right to submit any written materials in either English or French Evaluation of Your Progress Term Mark The Term Mark will be computed one-third from the assignment grades, and two-thirds from the class test. The Term Mark will count for 30 of the 100 marks in the final grade, but only if it exceeds 30% of the final examination percentage; otherwise the final examination will be used exclusively in the computation of the final grade Assignments. A total of about 6 assignments will be worth 10 of the 30 marks assigned to Term Work Class Test A class test, will be held on Wednesday, March 10th, 2004, at the regular class time, counting for 20 of the 30 marks in the Term Mark. There will be no make-up test for persons who miss the test. (This date has been changed from the tentative date announced earlier, after discussion with the class at the lecture of Friday, February 13th, 2004.) 2 For a lexicon of actuarial terms in English/French, see The Canadian Institute of Actuaries English- French lexicon [8], at UPDATED TO April 29, 2004

10 Information for Students in MATH Final Examination Written examinations form an important part of the tradition of actuarial mathematics. The final examination in MATH will count for either 70% or 100% of the numerical grade from which the submitted final letter grade will be computed. Where a student s Final Examination percentage is superior to her Term Mark percentage, the Final Examination grade will replace the Term Mark grade in the calculations. A 3-hour-long final examination will be scheduled during the regular examination period for the winter term (April 15th, 2004 through April 30th, 2004). You are advised not to make any travel arrangements that would prevent you from being present on campus at any time during this period. Students who have religious or other constraints that could affect their ability to write examinations at particular times should watch for the Preliminary Examination Timetable, as their rights to apply for special consideration at their faculty may have expired by the time the final examination timetable is published Supplemental Assessments Supplemental Examination. For eligible students who obtain a Final Grade of F or D in the course there will be a supplemental examination. (For information about Supplemental Examinations, see the McGill Calendar, [3].) There is No Additional Work Option. Will students with marks of D, F, or J have the option of doing additional work to upgrade their mark? No. ( Additional Work refers to an option available in certain Arts and Science courses, but not available in this course.) Machine Scoring Will the final examination be machine scored? While there could be Multiple Choice questions on quizzes, and/or the Final Examination, such questions will not be machine scored Plagiarism While students are not discouraged from discussing assignment problems with their colleagues, the work that you submit whether through homework, the class test, or on tutorial quizzes or the final examination should be your own. The Handbook on Student Rights and Responsibilities states in 15(a) 3 that 3

11 Information for Students in MATH No student shall, with intent to deceive, represent the work of another person as his or her own in any academic writing, essay, thesis, research report, project or assignment submitted in a course or program of study or represent as his or her own an entire essay or work of another, whether the material so represented constitutes a part or the entirety of the work submitted. You are also referred to the following URL: 1.4 Published Materials Required Text-Book The textbook for the course this semester is [1] Stephen G. Kellison, The Theory of Interest, Second Edition. Irwin/McGraw-Hill, Boston, etc. (1991), ISBN Website These notes, and other materials distributed to students in this course, will be accessible at the following URL: The notes will be in pdf (.pdf) form, and can be read using the Adobe Acrobat reader, which many users have on their computers. This free software may be downloaded from the following URL: 4 Where revisions are made to distributed printed materials for example these information sheets it is expected that the last version will be posted on the Web. The notes will also be available via a link from the WebCT URL: but not all features of WebCT will be implemented Reference Books The textbook used for may be used as a reference: [5] Michael M. Parmentier, Theory of Interest and Life Contingencies, with Pension Applications: A Problem- Solving Approach, 3rd edition. ACTEX Publications, Winstead, Conn. (1999), ISBN At the time of this writing the current version is 5.1.

12 Information for Students in MATH Other information Prerequisites It is your responsibility as a student to verify that you have the necessary calculus prerequisites. It would be foolish to attempt to take the course without them Calculators The use of non-programmable, non-graphing calculators only will be permitted in homework, tests, or the final examination in this course. Students may be required to convince examiners and invigilators that all memories have been cleared. The use of calculators that are either graphing or programmable will not be permitted during test or examinations, in order to level the playing field Self-Supervision This is not a high-school course, and McGill is not a high school. The monitoring of your progress before the final examination is largely your own responsibility. While the instructor is available to help you, he cannot do so unless and until you identify the need for help. While the significance of the homework assignments and class test in the computation of your grade is minimal, these are important learning experiences, and can assist you in gauging your progress in the course. This is not a course that can be crammed for: you must work steadily through the term if you wish to develop the facilities needed for a strong performance on the final examination. Working Problems on Your Own. You are advised to work large numbers of problems from your textbook. The skills you acquire in solving textbook problems could have much more influence on your final grade than either the homework or the class test Escape Routes At any time, even after the last date for dropping the course, students who are experiencing medical or personal difficulties should not hesitate to consult their advisors or the Student Affairs office of their faculty. Don t allow yourself to be overwhelmed by such problems; the University has resource persons who may be able to help you Showing your work; good mathematical form; simplifying answers When, in a quiz or examination problem, you are explicitly instructed to show all your work, failure to do so could result in a substantial loss of marks possibly even all of the marks; this is the default. The guiding principle should be that you want to be

13 Information for Students in MATH able to communicate your precise reasoning to others and to yourself. You are always expected to simplify any algebraic or numerical expressions that arise in your solutions or calculations. Verbal proofs are expected to be convincing : it will not be sufficient to simply describe mathematical expressions verbally.

14 Information for Students in MATH Timetable Distribution Date: 0th version: Monday, January 5th, 2004 This version, Monday, January 12th, 2004 (Subject to correction and change.) Section numbers refer to the text-book. 5 MONDAY WEDNESDAY FRIDAY JANUARY Problems from , problems 16 Chapter Course changes must be completed on MINERVA by Jan. 18, Chapter 2 21 Chapter , 3.2 Deadline for withdrawal with fee refund = Jan. 25, , FEBRUARY Verification Period: February 2 6, Deadline for withdrawal (with W) from course via MINERVA = Feb. 15, , , , 5.4 Study Break: February 23 27, 2004 No lectures, no regular office hours 23 NO LECTURE 25 NO LECTURE 27 NO LECTURE (Page 10 of the timetable will not be circulated; however, a version is available in the online version of these notes.) 5 Notation: n = Assignment #n due today R = Read Only X = reserved for expansion or review

15 Information for Students in MATH MONDAY WEDNESDAY FRIDAY MARCH , Chapter 6 05 Chapter 6 08 Chapter 6 10 CLASS TEST 12 Chapter 15 Chapter 17 Chapter 4 19 Chapter 22 Chapter 24 Chapter 26 Chapter 29 Chapter 31 Chapter 5 APRIL 02 Chapter 05 Chapter NO LECTURE 12 NO LECTURE 13 (TUESDAY)

16 Information for Students in MATH First Problem Assignment Distribution Date: Thursday, January 15th, 2005 (mounted on the Web) Hard copy distributed on Monday, January 19th, 2004 Solutions are to be submitted by Monday, January 26th, 2004 (This is a short assignment. Subsequent assignments can be expected to be longer.) 1. It is known that the accumulation function a(t) is of the form b (1.1) t + ct 2, where b and c are constants to be determined. (a) If $100 invested at time t = 0 accumulates to $170 at time t = 3, find the accumulated value at time t = 12 of $100 invested at time t = 1. (b) Show that this function satisfies the requirement [1, p. 2, #2] that it be nondecreasing. (c) Determine a general formula for i n, and show that lim i n n = 10%. (Use L Hôpital s Rule.) 2. It is known that 1000 invested for 4 years will earn in interest, i.e., that the value of the fund after 4 years will be Determine the accumulated value of 3500 invested at the same rate of compound interest for 13 years. 3. It is known that an investment of 750 will increase to at the end of 25 years. Find the sum of the present values of payments of 5000 each which will occur at the ends of 10, 15, and 25 years. 4. Find the accumulated value of 1000 at the end of 10 years: (a) if the nominal annual rate of interest is 6% convertible monthly; (b) if the nominal annual rate of discount is 5% convertible every 2 years. 5. Given that i (m) = and d (m) = , find m, the equivalent annual compound interest rate, and the equivalent annual compound discount rate.

17 Information for Students in MATH Second Problem Assignment Distribution Date: Mounted on the Web on Sunday, January 18th, 2004 Distributed in hard copy on Wednesday, January 28th, 2004 Solutions are to be submitted by Monday, February 9th, Find the present value of 1000, to be paid at the end of 37 months under each of the following scenarios: (a) Assume compound interest throughout, and a (nominal) rate of discount of 6% payable quarterly. (b) Assume compound interest for whole years only at a (nominal) rate of discount of 6% payable quarterly, and simple discount at the rate of 1.5% per 3 months during the final fractional period. (c) Assume compound interest throughout, and a nominal rate of interest of 8% payable semi-annually. 2. The sum of 5,000 is invested for the months of April, May, and June at 7% simple interest. Find the amount of interest earned (a) assuming exact simple interest in a non-leap year (b) assuming exact simple interest in a leap year (with 366 days); (c) assuming ordinary simple interest; (d) assuming the Bankers Rule. 3. Find how long 4,000 should be left to accumulate at 5% effective in order that it will amount to 1.25 times the accumulated value of another 4,000 deposited at the same time at a nominal interest rate of 4% compounded quarterly. 4. The present value of two payments of 100 each, to be made at the end of n years and 2n years is If i = 6.25%, find n. 5. (a) Find the nominal rate of interest convertible quarterly at which the accumulated value of 1000 at the end of 12 years is (b) Find the nominal rate of discount convertible semi-annually at which a payment of years from now is presently worth (c) Find the effective annual rate of interest at which the accumulated value of 1000 at the end of 12 years is 3000.

18 Information for Students in MATH An investor deposits 20,000 in a bank. During the first 4 years the bank credits an annual effective rate of interest of i. During the next 4 years the bank credits an annual effective rate of interest of i At the end of 8 years the balance in the account is 22, What would the account balance have been at the end of 10 years if the annual effective rate of interest were i for each of the 10 years? 7. A bill for 1000 is purchased for months before it is due. Find (a) the nominal rate of discount convertible monthly earned by the purchaser; (b) the annual effective rate of interest earned by the purchaser. 8. A signs a 2-year note for 4000, and receives from the bank. At the end of 6 months, a year, and 18 months A makes a payment of If interest is compounded semi-annually, what is the amount outstanding on the note at the time if falls due? 9. The Intermediate Value Theorem for continuous functions tells us that such a function f(x) whose value at x = a has the opposite sign from its value at x = b will assume the value 0 somewhere between a and b. By computing the value of f at the point 1 (a + b), we can infer that there is a 0 of f in an interval half as 2 long as [a, b], and this procedure may be repeated indefinitely to determine a zero of f to any desired accuracy. Assuming that polynomials are continuous, use this idea to determine the nominal quarterly compound interest rate under which the following payments will accumulate to 1000 at the end of 4 years: 300 today 200 at the end of 1 year 300 at the end of 2 years Your answer should be accurate to 3 decimal places, i.e., expressed as a percentage to 1 decimal place.

19 Information for Students in MATH Solutions, First Problem Assignment Distribution Date: Mounted on the Web on Wednesday, 4 February, 2004 Assignment was mounted on the Web on Thursday, January 15th, 2005 Hard copy was distributed on Monday, January 19th, 2004 Solutions were to be submitted by Monday, January 26th, 2004 (This is a short assignment. Subsequent assignments can be expected to be longer.) 1. It is known that the accumulation function a(t) is of the form b (1.1) t + ct 2, where b and c are constants to be determined. (a) If $100 invested at time t = 0 accumulates to $170 at time t = 3, find the accumulated value at time t = 12 of $100 invested at time t = 1. (b) Show that this function satisfies the requirement [1, p. 2, #2] that it be nondecreasing. (c) Determine a general formula for i n, and show that lim i n n = 10%. (Use L Hôpital s Rule.) Solution: [1, Exercise 4, p. 30] Denote the corresponding amount function by A(t). (a) An accumulation function must have the property that a(0) = 1; this implies that 1 = a(0) = b + 0, so b = 1. The given data imply that 170 = 100(a(3)) = 100(1(1.331) + c 3 2 ) (1) which implies that c = We conclude that A(t) A(0) = a(t) = (1.1)t t 2, (2) implying that a(1) = 1.141, a(12) = Then A(12) = A(1) A(12) a(12) = A(1) A(1) a(1) = A(1) = A(1)( ) = so $100 at time t = 1 grows to $ at time t = 12.

20 Information for Students in MATH (b) It follows from (2) that a (t) = (1.1) t ln t, which is positive for positive t; thus a(t) is an increasing function of t for positive t. (This property may also be proved from first principles. Let t 1 t 2. Then a(t 2 ) a(t 1 ) = (1.1) t t 2 2 (1.1) t t 2 1 = (1.1) t 1 ( (1.1) t 2 t 1 1 ) + (0.041) (t 2 t 1 ) (t 2 + t 1 ) (c) where both of the summands are non-negative for 0 t 1 t 1. i n = a(n) a(n 1) a(n 1) = (0.1)(1.1)n 1 + (0.041)(2n 1) (1.1) n 1 (0.041)(n 1) ( ) 2 2n (1.1) = ( n (n 1) 2 (1.1) n 1 ) By L Hôpital s Rule Hence lim x lim x 2x 1 2 = lim (1.1) x 1 x (1.1) x 1 ln 1.1 = 0 (n 1) 2 2(x 1) = lim (1.1) n 1 x (1.1) x 1 ln 1.1 lim i n = n = lim x 2 (1.1) x 1 (ln 1.1) 2 = (0) (0) = 0.1 = 10%. 2. It is known that 1000 invested for 4 years will earn in interest, i.e., that the value of the fund after 4 years will be Determine the accumulated value of 3500 invested at the same rate of compound interest for 13 years. Solution: [1, Exercise 14, p. 30] Let i be the rate of compound interest. Then 1000(1 + i) 4 = The accumulated value of 3500 after 13 years will be ( ) (1 + i) 13 4 = 3500 = It is known that an investment of 750 will increase to at the end of 25 years. Find the sum of the present values of payments of 5000 each which will occur at the ends of 10, 15, and 25 years.

21 Information for Students in MATH Solution: [1, Exercise 21, p. 31] Let i be the interest rate. The known fact is that 750(1 + i) 25 = Hence (1 + i) 25 = 2.797, so v 25 = The present value of three payments of 5000 after 10, 15, and 25 years will, therefore, be 5000(v 10 + v 15 + v 25 ) ( ) = 5000 ( ) ( ) 25 + ( ) 25 = 5000( ) = Find the accumulated value of 1000 at the end of 10 years: (a) if the nominal annual rate of interest is 6% convertible monthly; (b) if the nominal annual rate of discount is 5% convertible every 2 years. Solution: [1, Exercise 32, p. 31] (a) The accumulation factor for each month is 1 + 6% = After 10 years grows to 1000(1.005) = (b) The discount factor for each 2 years is 1 2 5% = 0.09 (moving backwards), 1 corresponding to an accumulation factor of. After 10 years 1000 grows to (0.09) 10 2 = Given that i (m) = and d (m) = , find m, the equivalent annual compound interest rate, and the equivalent annual compound discount rate. Solution: [1, Exercise 30, p. 32] For an mth of a year the relationship between i (m) and d (m) is given by ) ) (1 + (1 i(m) d(m) = 1 m m which is equivalent to or Substituting the given values (m + i (m) )(m d (m) ) = m 2 m = i(m) d (m) i (m) d (m). i (m) = d (m) =

22 Information for Students in MATH gives m = 2. It follows that i = ) 2 (1 + i(2) 1 2 = ( ) 2 1 = % = % = 1 24 ) 2 d = 1 (1 d(2) 2 = = 4% = 1 25

23 Information for Students in MATH Third Problem Assignment Distribution Date: Mounted on the Web on Sunday, February 8th, 2004 Hard copy was distributed on Wednesday, February 11th, 2004 Solutions are to be submitted by Monday, March 1st, 2004 Sketch a time diagram to accompany your solution of all problems except the last. 1. A skier wishes to accumulate 30,000 in a chalet purchase fund fund by the end of 8 years. If she deposits 200 into the fund at the end of each month for the first 4 years, and X at the end of each month for the next 4 years, find X if the fund earns a nominal (annual) rate of 6% compounded monthly. 2. A fund of 2500 is to be accumulated by n annual payments of 50, followed by n + 1 annual payments of 75, plus a smaller final payment of not more than 75 made 1 year after the last regular payment. If the effective annual rate of interest is 5%, find n and the amount of the final irregular payment. 3. On his 30th birthday, a teacher begins to accumulate a fund for early retirement by depositing 5,000 on that day and at the beginnings of the next 24 years as well. Since he expects that his official pension will begin at age 65, he plans that, starting at age 55 he will make an annual level withdrawal at the beginning of each of 10 years. Assuming that all payments are certain to be made, find the amount of these annual withdrawals, if the effective rate of interest is 6% during the first 25 years, and 7% thereafter. 4. At an effective annual interest rate of i it is known that (a) The present value of 5 at the end of each year for 2n years, plus an additional 3 at the end of each of the first n years, is (b) The present value of an n-year deferred annuity-immediate paying 10 per year for n years is Find i. 5. (a) Find ä 12 if the effective rate of discount is 5%. (b) Charles has inherited an annuity-due on which there remain 12 payments of 10,000 per year at an effective discount rate of 5%; the first payment is due immediately. He wishes to convert this to a 25-year annuity-immediate at the same effective rates of interest or discount, with first payment due one year from now. What will be the size of the payments under the new annuity?

24 Information for Students in MATH Give an algebraic proof and a verbal explanation for the formula m a n = a a m v m+n a. 7. A level perpetuity-immediate is to be shared by A, B, C, and D. A receives the first n payments, B the next 2n payments, C payments ##3n + 1,... 5n, and D the payments thereafter. It is known that the present values of B s and D s shares are equal. Find the ratio of the present value of the shares of A, B, C, D. 8. (a) Find an the present value of an annuity which pays 4,000 at the beginning of each 3-month period for 12 years, assuming an effective rate of 2% interest per 4-month period. (b) Suppose that the owner of the annuity wishes to pay now so that payments under his annuity will continue for an additional 10 years. How much should he pay? (c) How much should he pay now to extend the annuity from the present 12 years to a perpetuity? (It is intended that you solve this problem from first principles, not by substitution into formulæ in [1, Chapter 4].) 9. (No time diagram is needed for the solution to this problem.) In Problem 9 of Assignment 2 you were asked to apply the Bisection Method to determine the solution to an interest problem to 3 decimal places. The equation in question was: 300(1 + i) (1 + i) (1 + i) 2 = and the solution given began with the values of f(x) = 3x 4 + 2x 3 + 3x at x = 0 (f(0) = 10), x = 2 (f(2) = 66 > 0), and x = 2 (f( 2) = 34 > 0), and we were interested in the solution between 0 and 2 a solution that is unique because f is positive in this interval. Apply Linear Interpolation 4 times in an attempt to determine the solution we seek. (You are not expected to know the general theory of error estimation.) The intention is that you apply linear interpolation unintelligently, using it to determine a point where you find the function value and thereby confine the zero to a smaller subinterval: the point that you find will replace the midpoint in the bisection method. In some situations, as in the present one, the procedure may not be better than the bisection method. Indeed, in the present example, it could take many more applications than the bisection method to obtain the accuracy you obtained with that method.

25 Information for Students in MATH Solutions, Second Problem Assignment Distribution Date: Mounted on the Web on Friday, February 20th, 2004 Assignment was mounted on the Web on January 19th, Hard copy was distributed on Wednesday, January 28th, 2004 Solutions were due by Monday, February 9th, 2004 (Solutions presented subject to correction of errors and omissions.) 1. Find the present value of 1000, to be paid at the end of 37 months under each of the following scenarios: (a) Assume compound interest throughout, and a (nominal) rate of discount of 6% payable quarterly. (b) Assume compound interest for whole years only at a (nominal) rate of discount of 6% payable quarterly, and simple discount at the rate of 1.5% per 3 months during the final fractional period. (c) Assume compound interest throughout, and a nominal rate of interest of 8% payable semi-annually. Solution: (cf. [1, Exercise 2, p. 53]) ( (a) Present value = ( (b) Present value = ) 37 3 = ) 36 3 ( (c) Present value = (1.04) 37 6 = = ) = = 2. The sum of 5,000 is invested for the months of April, May, and June at 7% simple interest. Find the amount of interest earned (a) assuming exact simple interest in a non-leap year (b) assuming exact simple interest in a leap year (with 366 days); (c) assuming ordinary simple interest; (d) assuming the Bankers Rule. Solution: (cf. [1, Exercise 6, p. 54]) (a) The number of days is = 91; exact simple interest is (0.07) =

26 Information for Students in MATH (b) Exact simple interest is (0.07) = (c) Ordinary simple interest is (d) Interest under the Banker s Rule is (0.07) = (0.07) = Find how long 4,000 should be left to accumulate at 5% effective in order that it will amount to 1.25 times the accumulated value of another 4,000 deposited at the same time at a nominal interest rate of 4% compounded quarterly. Solution: (cf. [1, Exercise 13, p. 55]) The equation of value at n years is so 4000(1.05) n = (1.25)(4000)(1.01) 4n n = ln 1.25 ln ln 101 = The present value of two payments of 100 each, to be made at the end of n years and 2n years is If i = 6.25%, find n. Solution: (cf. [1, Exercise 14, p. 55]) Solving the equation of value, 100v 2n +100v n = 63.57, we obtain v n = 1 ± , 2 in which only the + sign is acceptable, since v n > 0. Taking logarithms gives n = = ln We conclude, to the precision of the problem, that n = 13.5 years. 5. (a) Find the nominal rate of interest convertible quarterly at which the accumulated value of 1000 at the end of 12 years is (b) Find the nominal rate of discount convertible semi-annually at which a payment of years from now is presently worth (c) Find the effective annual rate of interest at which the accumulated value of 1000 at the end of 12 years is Solution: (a) (cf. [1, Exercise 19, p. 55]) The equation of value at time t = 12 is implying that ) (1 + i(4) = 3000, 4 ( ) i (4) = = %. UPDATED TO April 29, 2004

27 Information for Students in MATH (b) The equation of value at time t = 12 is ) (1 d(2) = 1000, 2 implying that ( ) d (2) = = %. (c) The equation of value at time t = 12 is 1000 (1 + i) 12 = 3000, implying that i = = %. 6. An investor deposits 20,000 in a bank. During the first 4 years the bank credits an annual effective rate of interest of i. During the next 4 years the bank credits an annual effective rate of interest of i At the end of 8 years the balance in the account is 22, What would the account balance have been at the end of 10 years if the annual effective rate of interest were i for each of the 10 years? Solution: (cf. [1, Exercise 32, p. 57]) The equation of value is 20000(1 + i) 4 (1 + (i 0.02)) 4 = , which we interpret as a polynomial equation. The equation is of degree 8, and we don t have a simple algebraic method for solving such equations in general. But this equation has the left side a pure 4th power, so we can extract the 4th roots of both sides, obtaining (1 + i)(1 + (i 0.02)) = ( ) 1 4 = , which may be expressed as a quadratic equation in 1 + i: whose only positive solution is (1 + i) (1 + i) = i = (0.02) 2 + 4( ) 2 = from which we conclude that i = 2.25%, and that the account balance after 10 years would be 20000( ) 10 = 20000(1.0325) 10 = 27,

28 Information for Students in MATH A bill for 1000 is purchased for months before it is due. Find (a) the nominal rate of discount convertible monthly earned by the purchaser; (b) the annual effective rate of interest earned by the purchaser. Solution: (cf. [1, Exercise 25, p. 56]) (a) If d (12) be the nominal discount rate, then implying that so d (12) = 15.29%. ) (1 d(12) = d(12) 12 = (b) Let i be the effective annual interest rate. Then 950(1 + i) 1 3 = 1000 implies that i = ( ) = % A signs a 2-year note for 4000, and receives from the bank. At the end of 6 months, a year, and 18 months A makes a payment of If interest is compounded semi-annually, what is the amount outstanding on the note at the time if falls due? Solution: If i be the rate of interest charged semi-annually, then (1 + i ) 4 = 4000 so i = 6.00%; that is i (2) = 12.00%. The value of the 3 payments at the time the note matures is 1000 ( (1.06) 3 + (1.06) 2 + (1.06) 1) = so the amount outstanding before the final payment is

29 Information for Students in MATH The Intermediate Value Theorem for continuous functions tells us that such a function f(x) whose value at x = a has the opposite sign from its value at x = b will assume the value 0 somewhere between a and b. By computing the value of f at the point 1 (a + b), we can infer that there is a 0 of f in an interval half as 2 long as [a, b], and this procedure may be repeated indefinitely to determine a zero of f to any desired accuracy. Assuming that polynomials are continuous, use this idea to determine the nominal quarterly compound interest rate under which the following payments will accumulate to 1000 at the end of 4 years: 300 today 200 at the end of 1 year 300 at the end of 2 years Your answer should be accurate to 3 decimal places, i.e., expressed as a percentage to 1 decimal place. Solution: (We will carry the computations to an accuracy greater than requested in the problem.) (a) Let the effective annual interest rate be i. The equation of value at the end of 4 years is 300(1 + i) (1 + i) (1 + i) 2 = (3) (b) We need a continuous function to which to apply the Intermediate Value Theorem. Some choices may be better than others. We will choose f(x) = 3x 4 + 2x 3 + 3x We observe that f(0) = 10, that f(2) = = 66 > 0, and that f( 2) = = 34 > 0. This tells us that there is a solution to equation (3) for 2 x 0, equivalently for 3 i 1: such a solution is of no interest to us, as it does not fit the constraints of this problem. But the function is a cubic polynomial, and has 2 other zeros. We see that it also has a solution in the interval 0 x 2, and we proceed to progressively halve intervals. (c) The midpoint of interval [0, 2] is 1; f(1) = = 2 < 0 < 66 = f(2), so there must be a root in the interval [1, 2].

30 Information for Students in MATH (d) The midpoint of [1, 2] is 1.5; (e) (f) (g) (h) f(1.5) = 3(1.5) 4 + 2(1.5) 3 + 3(1.5) 2 10 = = > 0 so there must be a zero in the interval [1, 1.5], whose midpoint is f(1.25) = 3(1.25) 4 + 2(1.25) 3 + 3(1.25) 2 10 = = > 0 so there must be a zero in the interval [1, 1.25], whose midpoint is f(1.125) = 3(1.125) 4 + 2(1.125) 3 + 3(1.125) 2 10 = = 1.45 > 0 so there must be a zero in the interval [1, 1.125], whose midpoint is f(1.0625) = 3(1.0625) 4 + 2(1.0625) 3 + 3(1.0625) 2 10 = = < 0 so there must be a zero in the interval [1.0625, 1.125], whose midpoint is f( ) = 3( ) 4 + 2( ) 3 + 3( ) 2 10 = = > 0 so there must be a zero in the interval [1.0625, ], whose midpoint is

31 Information for Students in MATH (i) (j) f( ) = 3( ) 4 + 2( ) 3 + 3( ) 2 10 = = > 0 so there must be a zero in the interval [1.0625, ], whose midpoint is f( ) = 3( ) 4 + 2( ) 3 + 3( ) 2 10 = = < 0 so there must be a zero in the interval [ , ], whose midpoint is (k) f( ) = < 0 so there must be a zero in the interval [ , ], whose midpoint is (l) f( ) = < 0 so there must be a zero in the interval [ , ], whose midpoint is (m) f( ) = > 0 so there must be a zero in the interval [ , ], whose midpoint is (n) f( ) = > 0 so there must be a zero in the interval [ , ], whose midpoint is (o) f( ) = < 0 so there must be a zero in the interval [ , ], whose midpoint is (p) f( ) = > 0 so there must be a zero in the interval [ , ], whose midpoint is (q) f( ) = < 0 so there must be a zero in the interval [ , ], whose midpoint is (r) f( ) = > 0 so there must be a zero in the interval [ , ], whose midpoint is (s) f( ) = > 0 so there must be a zero in the interval [ , ], whose midpoint is (t) f( ) = > 0 so there must be a zero in the interval [ , ], whose midpoint is

32 Information for Students in MATH (u) f( ) = > 0 so there must be a zero in the interval [ , ], whose midpoint is (v) f( ) = > 0 so there must be a zero in the interval [ , ], whose midpoint is (w) f( ) = One zero will be approximately x = Thus the effective annual interest rate is approximately 7.649%. This, however, is not what the problem asked for. The accumulation function for 3-months will then be (1.0749) 1 4 = , so the effective interest rate for a 3-month period will be 1.822%, and the nominal annual interest rate, compounded quarterly, will be 7.288, or 7.3% to the accuracy requested. THE FOLLOWING PROBLEM WAS CONSIDERED FOR INCLUSION IN THE AS- SIGNMENT, BUT WAS (FORTUNATELY) NOT INCLUDED. 10. [1, Exercise 6, p. 88] (a) Show that where 0 < n < m. (b) Show that where 0 < n < m. a m n = a m v m s n = (1 + i) n a m s n s m n = s m (1 + i) m a n = v n s m a n (c) Interpret the results in (a) and (b) verbally. Solution: (a) We prove the first of these identities by technical substitutions in sums, analogous to changes of variables in a definite integral. For the second identity we give a less formal proof. a m n = = = = m n r=1 v r m v r r=1 m v r v m r=1 m v r v m r=1 m r=m n+1 m v r r=m n+1 1 s=n v 1 s v r m

33 Information for Students in MATH = = under the change of variable s = m r + 1 m n v r v m v 1 s r=1 s=1 reversing the order of the 2nd summation m n v r v m (1 + i) s 1 r=1 s=1 = a m v m s n a m n = v + v v m n = v n+1 + v n v 0 + v 1 + v v m n ( v n+1 + v n v 0) = v n ( v 1 + v v n + v n+1 + v n v m) ( (1 + i) n 1 + (1 + i) n (1 + i)v 0) = (1 + i) n ( v 1 + v v n + v n+1 + v n v m) ( (1 + i) 0 + (1 + i) (1 + i)v n 1) = (1 + i) n a m s n (b) These identities could be proved in similar ways to those used above. Instead, we shall show that these identities can be obtained from the preceding simply by multiplying the equations by (1 + i) m n : s m n = (1 + i) m n a m n [1, (3.5), p. 60] = (1 + i) m n ((1 + i) n a m s n ) = (1 + i) m a m (1 + i) m (1 + i) n s n = s m (1 + i) m a n s m n = (1 + i) m n a m n = (1 + i) m n (a m v m s n ) = (1 + i) n (1 + i) m a m (1 + i) n s n = (1 + i) n s m a n (c) i. An (m n)-payment annuity-immediate of 1 has the same present value as an annuity for a total term of m = (m n) + n years minus a correction paid today equal to the value of the deferred n payments. Those n payments are worth s m at time t = m, which amount can be discounted to the present by multiplying by v m. The preceding explanation was based on values at the commencement of the first year of an m-year annuity-immediate. Let us now interpret the