Mathematics Of Investment Pdf William Hart

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1. . THE MATHEMATICS OF ' INVESTMENT BY WILLIAM L. HART, Pn.D, AS3O01ATH PBOTESSOR OF MATHEMATICS I2T THH OF MINWBSOTA D. 0. HEATH AND COMPANY BOSTON NEW YORK CHICAGO LONDON
2. COPYRIGHT, 1934, BY D. C. HEATH AND COMPANY 2B0 PBINTBD D? U.S.A.
3. PREFACE THIS book provides an elementary course in the theory and the application of annuities certain and in the mathematical aspects of life insurance. The book is particularly adapted to the needs of students in colleges of business administration, but it is also fitted for study by college students of mathematics who are not specializing in business. Annuities certain and their applications are considered in Part I, life insurance is treated in Part II, and a treatment of logarithms and of progressions IB given in Part III. The prerequisites for the study -of the book are three semesters of high school algebra and an acquaintance with progressions and logarithmic computation. Very complete preparation would be furnished by three semesters ofhigh school algebra and a course in college algebra including logarithms. The material in the book has been thoroughly tested by the author through the teaching of it, in mimeographed form, for two years in ; classes at the University of Minnesota. It has been aimed in this book to present the subject in such a way that its beautiful simplicity & and great' usefulness will be thoroughly appreciated by all of thestudents to whom it is taught. Features of the book which will appeal to teachers of the subject are as follows : 1. Illustrative examples are consistently used throughout to illuminate new theory, to illustrate new methods, and to supply models for the solution of problems by the students. 2. Large groups of problems are supplied to illustrate each topic, and, in addition, sets of miscellaneous problems are given at the close of each important chapter, while a review set is placed at the end of 6ach of the major parts of the book. 3. Flexibility in the length, of the course is provided for ; the teacher can conveniently choose from this book either a one or a two semester, three-hour course, on account of the latitude afforded by (a) the large number of problems, (6) the segregation of optional methods and difficult topics into Supplementary Sections whose omission does not break the continuity of the remainder of the book, and (c) the possi-' bility of the omission of all of Part II, where life insurance is con- sidered. ui
4. IV PREFACE 4. The concept of an equation of value is emphasized as a unifying principle throughout. 6. Formulas are simplified and reduced to as small a number as the author considers possible, if the classical notation of the subject is to be preserved. In Part I, a simplification is effected by the use of the interest period instead of the year as a time unit in a final pair of formulas for the amount and for the present value of an annuity certain. By use of these two formulas, the present values and the amounts of most annuities met in practice can be conveniently com- puted with the aid of the standard tables. In the applications of annuities certain, very few new formulas are introduced. The student is called upon to recognize all usual problems involving amortization, sinking funds, bonds, etc., as merely different instances of a single algebraic problem; that is, the finding of one unknown quantity by the solution of one of the fundamental pair of annuity formulas. 6. Interpolation methods are used to a very great extent and then1 logical and practical completeness is emphasized. Some problems solved by interpolation are treated by other methods, as well, and such optional methods are found segregated into Supplementary Sections. 7. Practical aspects of the subject are emphasized throughout. 8. Very complete tables are provided, including a five-place table of logarithms, the values of the interest and annuity functions for twenty-five interest rates, tables of the most essential insurance functions, and a table of squares, square roots, and reciprocals. The tables may be obtained either bound with the book or bound separately. . 9. In the discussion of life annuities and life insurance, the em- phasis is placed on methods and on principles rather than on manip- ulative proficiency. It is aimed to give the student a clear conception of the mathematical foundations of the subject. No attempt is made to prepare the student as an insurance actuary, but the treatment in this book is an excellent introduction to more advanced courses in actuarial science. The interest and annuity tables prepared in connection with this book make possible the solution of most problems accurately to cents, if ordinary arithmetic is used. Results can be obtained with sufficient accuracy for most class purposes if the five-place table of logarithms is used in the computations. If the teacher considers it desirable to use seven-place logarithms, the author recommends the use of Glover's Tables from Applied Mathematics. These excellent tables contain the
5. PREFACE V values, and the seven-place logarithms of the values, of the interest and annuity functions, a standard seven-place table of'logarithms, and a variety of other useful tables dealing with insurance and statistics. The author acknowledges his indebtedness to Professor James Glover for his permission to publish in the tables of this book certain extracts from Glover's Tables which were published, for the first time, in that book. UNIVERSITY OP MINNESOTA, January 1, 1924.
6. CONTENTS PART I ANNUITIES CERTAIN CHAPTER I SIMPLE INTEREST AND SIMPLE DISCOUNT IEOTION ' PAQH 1. Definition of interest 1 2. Simple interest 1 3. Ordinary and exact interest 3 4. Algebraic problems in simple interest 5 5. Simple discount 6 6. Banking use of simple discount ...... 9 7. Discounting notes 10 CHAPTER II COMPOUND INTEREST vu
7. viii CONTENTS CHAPTER III ANNUITIES CERTAIN SECTION FAGH 19. Definitions 39 20. Special cases 41 21. Formulas in the most simple case 42 22. Annuities paid p times per interest period .... 45 23. Most general formulas 49 24. Summary of annuity formulas .50 25. Annuities due 56 26. Deferred annuities 59 27.* Continuous annuities , .62 28.* Computations of high accuracy 63 Miscellaneous problems 64 CHAPTER IV PROBLEMS IN ANNUITIES .29. Outline of problems 67 30. Determination of payment 67 31. Determination of term 70 32. Determination of interest rate 71 33.* Difficult cases and exact methods 74 Miscellaneous problems 76 CHAPTER V PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 34. Amortization of a debt 78 35. Amortization of a bonded debt 81 36. Problems where the payment is known 83 37. Sinking fund method 85 38. Comparison of the amortization and the sinking fund methods 88 Miscellaneous problems 89 39.* Funds invested with building and loan associations . . 92 40.* Retirement of loans made by building and loan associations 94 * Supplementary section.
8. CONTENTS k CHAPTER VI DEPRECIATION, PERPETUITIES, AND CAPITALIZED COST PAOH 41. Depreciation, sinking fund plan ...... 96 42. Straight line method ........ 98 43. Composite life .......... 99 44. Valuation of a mine ........ 101 45. Perpetuities .......... 103 46. Capitalized cost ......... 105 47.* Difficult cases under perpetuities .' ..... 108 48.* Constant percentage method of depreciation .... 109 Miscellaneous problems ....... Ill CHAPTER Vn BONDS 49. Terminology .......... 113 50. Meaning of the investment rate ...... 113 51. Purchase price of a bond at a given yield .... 114 52. Changes in book value ........ 117 53. Price at a given yield between interest dates .... 121 54. Professional practices in bond transactions .... 124 55. Approximate bond yields ....... 126 66. Yield on a dividend date by interpolation .... 128 57. Special types of bonds ........ 130 58.* Yield of a bond bought between interest dates . . . .131 Miscellaneous problems ........ 133 Review problems on Part I ....... 135 PART II LIFE INSURANCE CHAPTER LIFE ANNUITIES 59. Probability .......... 147 60. Mortality Table ......... 148 61. Formulas for probabilities of living and dying ... 150 * Supplementary section.
9. X CONTENTS SECTION PAGE 62. Mathematical expectation ....... 152 63. Present value of pure endowment 153 64. Whole life annuity 155 65. Commutation symbols 157 66. Temporary and deferred life annuities 159 67. Annuities due 162 Miscellaneous problems 163 CHAPTER IX LIFE INSURANCE 68. Terminology 165 69. Net single premium, whole life insurance .... 166 70. Term insurance 168 71. Endowment insurance 170 72. Annual premiums .171 73.* Net single premiums as present values of expectations . . 175 74.* Policies of irregular type 176 CHAPTER X POLICY RESERVES 75. Policy reserves 178 76. Computation of the reserve 180 Supplementary Exercise 183 Miscellaneous Problems on Insurance 184 PART III AUXILIARY SUBJECTS CHAPTER XI LOGARITHMS 77. Definition of logarithms ... ..... 187 78. Properties of logarithms .188 79. Common logarithms 190 80. Properties of the mantissa and the characteristic , . .191 81. Use of tables of mantissas 193 82. Logarithms of numbers with five significant digits , 194 * Supplementary section.
10. CONTENTS xi SECTION . PAGH 83. To find the number when the logarithm is given . . . 196 84. Computation of products and of quotients .... 197 86. Computation of powers and of roots 198 86. Problems in computation 200 87. Exponential equations 201 88.* Logarithms to bases different from 10 203 CHAPTER XII PROGRESSIONS 89. Arithmetical progressions 204 90. Geometrical progressions 205 91. Infinite geometrical progressions 207 APPENDIX Note 1. Approximate determination of the time to double money at compound interest 211 Note 2. Approximate determination of the equated time . . 211 Note 3. Solution of equations by interpolation . ". . . 212 Note 4. Abridged multiplication 213 Note