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[Pages:277]Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics

A Semester Course in Finite Mathematics for Business and Economics

Marcel B. Finan c All Rights Reserved

August 10, 2012

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Contents

Preface

4

Mathematics of Finance

5

1. Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Discrete and Continuous Compound Interest . . . . . . . . . . . 12

3. Ordinay Annuity, Future Value and Sinking Fund . . . . . . . . 19

4. Present Value of an Ordinay Annuity and Amortization . . . . . 26

Matrices and Systems of Linear Equations

34

5. Solving Linear Systems Using Augmented Matrices . . . . . . . . 34

6. Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . . 42

7. The Algebra of Matrices . . . . . . . . . . . . . . . . . . . . . . 53

8. Inverse Matrices and their Applications to Linear Systems . . . . 62

Linear Programming

69

9. Solving Systems of Linear Inequalities . . . . . . . . . . . . . . . 69

10. Geometric Method for Solving Linear Programming Problems . 77

11. Simplex Method for Solving Linear Programming Problems . . 86

12. The Dual Problem: Minimization with Constraints . . . . . . 97

Counting Principles, Permuations, and Combinations

106

13. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

14. Counting Principles . . . . . . . . . . . . . . . . . . . . . . . . 117

15. Permutations and Combinations . . . . . . . . . . . . . . . . . 123

Probability

129

16. Sample Spaces, Events, and Probability . . . . . . . . . . . . . 129

17. Probability of Unions and Intersections; Odds . . . . . . . . . . 141

18. Conditional Probability and Independent Events . . . . . . . . 148

19. Conditional Probability and Bayes' Formula . . . . . . . . . . . 154

20. Random Variable, Probabiltiy Distribution, and Expected Value 160

Statistics

168

21. Graphical Representations of Data . . . . . . . . . . . . . . . . 168

22. Measures of Central Tendency . . . . . . . . . . . . . . . . . . . 183

23. Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . 196

24. Binomial Distributions . . . . . . . . . . . . . . . . . . . . . . . 205

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25. Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . 217

Answer Key

236

Index

276

3

Preface

This book is a collection of lecture notes for a freshmen level course in mathematics designated for students in Business, Economics, Life Sciences and Social Sciences. The content is suitable for a one semester course. A college algebra background is required for this course. Marcel B. Finan Russellville, Arkansas

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Mathematics of Finance

1. Simple Interest

Interest is a change of value of money. For example, when you deposit money into a savings account, the interest will increase your money based on the interest rate paid by your bank. In contrast, when you get a loan, the interest will increase the amount you owe based upon the interest rate charged by your bank. We can look at interest as the fee for using money. There are two types of interest: Simple interest and compound interest. In this section we discuss the former one and postpone the discussion of the later to the next section. Interest problems generally involve four quantities: principal(s), investment period length(s), interest rate(s), amount value(s). The money invested in financial transactions will be referred to as the principal or the present value, denoted by P. The amount it has grown to will be called the amount value or the future value and will be denoted by A. The difference I = A - P is the amount of interest earned during the period of investment. Interest expressed as a percent of the principal will be referred to as an interest rate. The unit in which time of investment is measured is called the measurement period. The most common measurement period is one year but may be longer or shorter (could be days, months, years, decades, etc.) Let r denote the annual fee per $100 and P denote the principal. We call r the simple interest rate. Thus, if one deposits P = $500 in a savings account that pays r = 5% interest then the interest earned for the two years period will be 500 ? 0.05 ? 2 = $50. In general, if I denotes the interest earned on an investment of P at the annual interest rate r for a period of t years then

I = P rt

and the amount of the investment at the end of t years is

A(t) = principal + interest = P (1 + rt).

Example 1.1 John borrows $1,500 from a bank that charges 10% annual simple interest rate. He plans to make weekly payments for two years to repay the loan. (a) Find the total interest paid for this loan.

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(b) Find the total amount to be paid back. (c) Find the weekly installment. Through out this document we shall round all amounts to the nearest cent.

Solution. (a) The total interest paid is I = P rt = 1, 500 ? 0.10 ? 2 = $300. (b) The total amount to be paid back is A(2) = 1, 500 + 300 = $1, 800. (c) The weekly installment is

1, 800 = $17.31

2 ? 52

Example 1.2 NBA bank is offering its customers an annual simple interest of 3% for deposits in savings accounts. How long will take for a deposit of $1,500 to earn total interest of $225?

Solution. We are given I = $225, P = $1, 500, and r = 0.03. We are asked to find t. From the formula I = P rt we find

I

225

t= =

= 5 years

P r 1, 500(0.03)

Example 1.3 A new bank is offering its customers the option to double their investments in eight years with an annual simple interest rate r. Calculate r.

Solution. We are given that

2P = P (1 + 8r).

Solving this equation for r, we find

P (1 + 8r) =2P 1 + 8r =2 8r =1 1 r = = 12.5% 8

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Example 1.4 You deposit P dollars into a savings account that pays 12% annual simple interest. At the end of nine months, you have $109 in your account. Calculate P.

Solution.

We

are

given:

r

= 0.12,

A = $109,

t=

9 12

= 0.75. We

are

asked

to

find

P.

We have

A

109

P=

=

= $100

1 + rt 1 + 0.12(0.75)

Example 1.5 Treasury bills (or T-bills) are financial instruments that the US government uses to finance public debt. Suppose you buy a 90-day T-bill with a maturity value of $10,000 for $9,800. Calculate the annual simple interest rate earned for this transaction. Round your answer to three decimal places. In all problems involving days, we assume a year has 360 days.

Solution. We are given P = $9, 800, A = $10, 000, t = 0.25 and we are asked ti find r. We have

9, 800(1 + 0.25r) =10, 000

50 1 + 0.25r =

49

50

1

0.25r = - 1 =

49

49

1

r=

8.163%

49(0.25)

Example 1.6 The table below provides the commission that a brokerage firm charges for selling or buying stocks.

Transaction Amount $0 - $2,499 $2,500 - $9,999 $10, 000

Commission $29 + 1.6% of transaction amount $49 + 0.8% of transaction amount $99 + 0.3% of transaction amount

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Suppose you purchase 45 shares of a stock at $40.25 a share. After 180 days, you sell the stock for $50.75 a share. Using the table above, find the annual simple interest rate earned by this investment. Round your answer to three decimal places.

Solution. The amount of money needed for buying the stock is

45(40.25) = $1, 811.25.

The commission paid on this amount is

29 + 0.016(1, 811.25) = $57.98.

Thus, the total cost for this investment is

1, 811.25 + 57.98 = $1, 869.23.

The return from selling the stocks is

50.75(45) = $2, 283.75.

The commission paid for this transaction is

29 + 0.016(2, 283.75) = $65.54.

The total return for this transaction is

2, 283.75 - 65.54 = $2, 218.21.

Now, using the simple interest formula with P = $1, 869.23, A = $2, 218.21,

and

t

=

180 360

=

0.5,

we

find

1, 869.23(1 + 0.5r) =2, 218.21

2, 218.21 1 + 0.5r =

1, 869.23

2, 218.21

348.98

0.5r =

-1=

1, 869.23

1, 869.23

348.98

r=

37.339%

1, 869.23(0.5)

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