Financial Mathematics

FINANCIAL

MATHEMATICS BY

CLARENCE

H.

RICHARDSON,

PH.D.

Professor of Mathematics, Bucknell University

AND

ISAIAH LESLIE MILLER Late Professor of Mathematics, South Dakota State College of Agriculture and Mechanic Arts

D.

NEW YORK VAN NOSTRAND COMPANY, 250 FOURTH 1946

AVENUE

INC.

COPY RIGHT, 1946 BY D.

VAN NOSTHAND COMPANY,

INC.

All Rights Reserved

Thin book, or any parts thereof, may not be reproduced in any form without written permission from the authors and the publishers.

Based on Business \fathematics,

I.

L. Miller, copyright 1935; second edition copyright 1939;

and Commercial Algebra and Mathematics of Finance, I. L. Miller and C. H. Richardson, copyright 1939 by D. Van Nostrand Company, Inc.

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE This text is designed for a three-hour, one-year course for students desire a knowledge of the mathematics of modern business and finance. While the vocational aspects of the subject should be especially

who

attractive to students of

commerce and business administration, yet an

understanding of the topics that are considered nuities,

bond valuation,

depreciation, insurance

interest, discount, an-

may

well be desirable

information for the educated layman.

To

live intelligently in this complex age requires more than a superknowledge of the topics to which we have just alluded, and it is palpably absurd to contend that the knowledge of interest, discount, bonds, and insurance that one acquires in school arithmetic is sufficient to understand modern finance. Try as one may, one cannot escape questions of ficial

finance.

The

real issue is: shall

we

deal with

them with understanding and

effectiveness or with superficiality and ineffectiveness? While this text presupposes a knowledge of elementary algebra, we have listed for the student's convenience, page x, a page of important

formulas from Miller and Richardson, Algebra: Commercial Statistical that should be adequate for the well-prepared student. Although we make frequent reference to this Algebra in this text on Financial Mathematics, the necessary formulas are found in this reference list.

In the writing of this text the general student and not the pure mathematician has been kept constantly in mind. The text includes those techniques and artifices that many years of experience in teaching the subject

have proved to be pedagogically fruitful. Some general features may be enumerated here: (1) The illustrative examples are numerous and are worked out in detail, many of them having been solved by more than one method in order that the student may compare the respective methods of attack. (2) Line diagrams, valuable in the analysis and presentation of problem material, have been given emphasis. (3) Summaries of important formulas occur at strategic points. (4) The exercises and problems are nufrierous, and they are purposely selected to show the applications of the theory to the many fields of activity. These exercises and problems are abundant, and no class will hope to do more than half of them. (5) Sets

Preface

iv

of review problems are

found at the ends of the chapters and the end of the

book.

A

few special features have also been included: (1) Interest and discount have been treated with unusual care, the similarities and differences having been pointed out with detail. (2) The treatment of annuities is pedagogical and logical. This treatment has been made purposely flexible so that, if it is desired, the applications may be made to depend upon two No new formulas are developed for the solution of general formulas. problems involving annuities due and deferred annuities, and these special annuities are analyzed in terms of ordinary annuities. (3) The discussion of probability and its application to insurance is more extended than that

found in many texts. In this edition we are including Answers to the exercises and problems. While we have exercised great care in the preparation of this book, it is

too

much

to expect that

tion of such errors,

we

it is

shall

entirely free

from

errors.

For the

notifica-

be truly grateful. C. H. RICHARDSON.

Buckneli University, Lewisburg, Pennsylvania, 1946.

CONTENTS CHAPTER

ART.

I,

SIMPLE INTEREST AND DISCOUNT

. PAGE

1.

Interest

2.

Simple Interest Relations Ordinary and Exact Interest Methods of Counting Time The Six Per Cent Method of Computing Ordinary Interest Present Value and True Discount Bank Discount Summary and Extension Comparison of Simple Interest and Simple Discount Hates Rates of Interest Corresponding to Certain Discount Rates in the Terms of Settlement

3.

4.

5. 6. 7.

8. 9.

10.

11. 12.

13.

1

Exchanging Debts To Find the Date When the Various Sums (Debts) Due at Different Times May Be Paid in One Sum To Find the Equated Date of an Account

CHAPTER

1

3 4 7 9 12 15 17

20 22 25 28

COMPOUND INTEREST AND COMPOUND

II,

DISCOUNT Compound Compound

16.

Nominal and Effective Rates

14.

35 36 38 42 45 48 50 53 54

Interest

15.

Interest

Formula of Interest

Present Value at Compound Interest 18. Other Problems Solved by the Compound Interest Formulas

17.

19.

20. 21. 22.

Equation of Value Equated Time

Compound Discount at a Discount Rate Summary of Interest and Discount CHAPTER

III,

ANNUITIES CERTAIN 57 58 63

23. Definitions 24.

25.

Amount

an Annuity Present Value of an Annuity

26. Relation

of

and

between 0rfl

66 8 n\

Formulas of an Ordinary Annuity of Annual Rent Annually for n Years 28. Other Derivations of a^i and 8*\ 27.

Summary.

v

R

Payable

67 67

Contents

vi

PAGE

AKT. 29.

Amount

of

an Annuity, Where the Annual Rent, R,

is

Payable in

p Equal 69

Installments

Value of an Annuity ments

30. Present

31.

Summary

in

Install-

p Equal

78 79

Ordinary Annuity Formulas

Due

32. Annuities 33.

of

Annual Rent, R, Payable

of

83

89

Deferred Annuities

Finding the Interest Rate of an Annuity 35. The Term of an Annuity 36. Finding the Periodic Payment 37. Perpetuities and Capitalized Cost

92 95 97

34.

38. Increasing

CHAPTER IV, SINKING 39. Sinking

100

and Decreasing Annuities

105

FUNDS AND AMORTIZATION

Funds

Ill

Amortization 41. Book Value

Ill

40.

113

114

44.

Amount in the Sinking Fund at Any Time Amount Remaining Due After the kth Payment lias Been Made The Amortization and Sinking Fund Methods Compared

45.

Retirement of a Bonded Debt

118

42.

43.

CHAPTER V,

113 116

DEPRECIATION 122

46. Definitions 47.

Methods

48.

The

123

of Treating Depreciation

Straight Line

Method

123

49. Fixed-Percentage-on-Decreasing- Value 50. 51.

125

The Sinking Fund Method The Unit Cost Method

52. Depreciation of 53.

Method

128

130 134

Mining Property

Composite Life of a Plant

CHAPTER

136

VI,

VALUATION OF BONDS

54. Definitions

141

Purchase Price 56. Premium and Discount 57. Amortization of Premium and Accumulation of Discount 58. Bonds Purchased Between Dividend Dates

141

55.

59.

Annuity Bonds Bonds Use of Bond Tables Determining the Investment Rate Given

144 147 150 152 153

60. Serial 61. 62.

154

When

the Purchase Price of a

Bond

is

155

Contents CHAPTER

VII,

vii

PROBABILITY AND ITS APPLICATION IN LIFE INSURANCE

ART. 63.

The History

64.

Meaning

of

161 162

of Probabilities

a

BA/SW PAGE

priori Probability

Empirical Probability

164

66. Permutations.

165

67.

Number of Permutations of Things All Different Combinations. Number of Combinations of Things All Different Some Elementary Theorems in Probability

167

Mathematical Expectation Repeated Trials Meaning of Mortality Table

172

65. Relative

68.

69. 70.

71.

Frequency.

169 173

176

178

72. Probabilities of Life

CHAPTER

VIII,

LIFE ANNUITIES

73.

Pure Endowments

182

74.

Whole

185

75.

76. 77. 78. 79.

80. 81.

Life

Annuity Present Value (Cost) of a Life Annuity Life Annuity Due Deferred Life Annuity Temporary Life Annuity Forborne Temporary Life Annuity Due Summary of Formulas of Life Annuities. Annuities Payable m Times a Year

185

186

186

187 189

Examples

CHAPTER IX, LIFE INSURANCE,

(SINGLE

1

90

193

NET PREMIUMS

AND ANNUAL)

82. Definitions

198

83.

Whole

199

84.

87.

Term Insurance Endowment Insurance Annual Premium Payable by m Equal Installment/a Summary of Formulas of Life Insurance Premiums

202 204 205 207

88.

Combined Insurance and Annuity

208

85.

86.

Life Policy

CHAPTER X, 89. 90. 91. 92. 93.

Policies

VALUATION OF POLICIES. RESERVES

Meaning of Reserves Computing Reserves, Numerical Illustration Fackler's Accumulation Formula Prospective Method of Valuation Retrospective Method of Valuation

212 213 214 216 218

OTHER METHODS OF VALUATION, POLICY OPTIONS AND PROVISIONS, SURPLUS AND DIVIDENDS

CHAPTER XI, GROSS PREMIUMS,

94.

Gross Premiums and Dividends

95. Surplus

221 222

Contents

viii

PAGE

ART.

223 223 224 225 227

96. Policy Options 97. Surrender or Loan Value

Extended Insurance Paid-up Insurance 100. Preliminary Term Valuation 101. Modified Preliminary Term Valuation 98. 99.

102. Concluding

231

Remarks

235

Review Problems Tables

237

T-I-1

T-XIII-77

Answers

245

Index

261

USEFUL FORMULAS

From I.

Miller and Richardson, Algebra: Commercial

PAGE

= N, logaN = x MN = log a M + log a N

M=

47

M - log N = N loga M loga M*

3. log a

4.

46

If a*

2. Iog

N

47

a

loga

47

Arithmetical Progression

=

+

a

1.

I

2.

Sn = -

(a

3.

Sn = -

[2a

-

(n

84

l)d

+

84

+

2 III.

*

Logarithms 1.

II.

Statistical

(n

-

84

Geometrical Progression = n ~L 1.

ar

i

a

ar

2. 1

a

-

when

4.

(a

+

n

b)

(8)

87

Article

60

(9)

87

r

1