Chapter 21. Savings Model

Size: px
Start display at page:

Download "Chapter 21. Savings Model"

Transcription

1 Chapter 21. Savings Model

2 Arithmetic Growth and Simple Interest

3 What s important Bring your calculator. Practice the correct way to use the calculator. Remember the formula and terminology. Write your reasoning( formula you use) in details when you take any test. Answer without any reasoning will get zero point. I won t allow you to share calculator with others. You cannot use your cell phone as a calculator.

4 Terminology Principal The initial balance of the savings account. Example: You are opening a savings account today. You deposited $50. Then, $50 is principal. Interest Money earned on a savings account or a loan. Example: If at the end of the year, your savings account has $51, then the interest is $1.

5 Simple Interest The method of paying interest only on the initial balance in an account, not on any accrued interest.

6 Simple Interest Example You deposited $1000. An annual simple interest rate is 1%. Then, Your Account Balance: After a year: 0.01 X = After two years: 2X(0.01X1000)+1000=1020 After n years: n X (0.01X1000)+1000

7 Simple Interest Example You deposited $1000. An annual simple interest rate is 1%. That is, at the end of each year, you get $10 as an interest no matter how many years you deposit.

8 Simple Interest For a principal P and an annual rate of interest r, after t years, Interest I = P r t Total amount A = P + I = P + P r t = P 1 + r t = P(1 + rt)

9 Use of Simple Interest Private loans between individuals (easy to calculate) Commercial loans for less than a year (Doesn t make much differences from the compound interest)

10 Example Let s suppose that you have exhausted the amount that you can borrow under federal loan programs and need a private direct student loan for $10,000. National City Corporation quoted a rate in May 2008 of 5.7% for the school year. It offers an interest-only repayment option, under which you make monthly interest payments while you are in school and pay on the principal only after graduation. Under this plan, National City earns simple interest from you while you are in school. How much monthly interest would you pay for such a $10,000 loan?

11 Answer Student loan= $10,000. Annual rate= Simple interest 5.7% Monthly interest= Prt = $10, = $

12 Arithmetic Growth(Linear Growth) Growth by a constant amount in each time period Example: simple interest

13 Geometric Growth and Compound Interest

14 Compound Interest Interest that is paid on both the original principal and accumulated interest

15 Compound Interest Suppose that you deposited $10,000 into the savings account with annual compound interest rate of 10%. After.year later Interest 1 10% 10,000 = ,000 = % 11,000 = ,000 = 1,100 Savings total 10, ,000 = 11,000 11, ,100 = 12,100

16 After.year later Interest Compound Interest Savings total 1 10% 10,000 = ,000 = , ,000 = 11, % 11,000 = ,000 = 1,100 11, ,100 = 12, % 12,100 = ,100 = 1,210 12, ,210 = 13,310

17 Nominal Rate Any state rate of interest for a specified length of time. Nominal rate have broad( vague ) meaning. It doesn t indicate or take into account whether or how often interest is compounded.

18 Annual Interest Rate of 10% (Nominal Rate) That might mean Annual interest rate of 10% compounded quarterly. ( every three months, 4 times a year ) Annual interest rate of 10% with simple interest

19 Annual interest rate of 10% compounded quarterly with $1,000 Principal It is compounded every three months, that is, 4 times a year. Annual interest rate of 10% in this case means 2.5% quarterly.

20 Annual interest rate of 10% compounded quarterly with $1,000 Principal After 0 month 3 months 6 months 9 months 12 months Balance 1, , ,000 = 1, , ,025 = 1, , , = 1, , , = 1,103.82

21 Annual interest rate of 10% After a year later, with simple interest 0.1 1, ,000 = 1,100

22 Annual interest rate of 10% with simple interest vs. compound interest with $1,000 a year later A year later Interest amount Effective interest compared to the principal Simple interest $1, % Quarterly Compound interest $1, %

23 Effective Rate and Annual Percentage Rate If we know the effective rate and annual percentage rate for the compound interest, then it is easier to know how much you earn interest. Effective rate: the rate of simple interest that would realize exactly as much interest over the same length of time. Annual percentage yield(apy): Effective rate for a year the money after a year the principal 100 % Principal Annual interest amount = 100(%) Principal

24 Rate per Compounding Period Terminology Meaning Number of compounding per year Rate when nominal annual rate is r. Annually Every year 1 r Bi-annually Twice a year 2 r 2 Quarterly Four times a year 4 r 4 Monthly Every month 12 r 12 Daily Every day 365 r 365

25 Rate per Compounding Period Terminology Periodic rate when nominal annual rate is 12% Annually 12% Bi-annually 12 2 = 6% Quarterly 12 4 = 3% Monthly = 1% Daily = 0.03%

26 Nominal annual interest rate of 10% After years with Principal $1000 Balance in dollars = 1000( ) = = 1000( ) ( ) ( ) = 1000( ) 2 (1+0.1)=1000( ) 3 n 1000( ) n

27 Nominal annual interest rate of 10% with Principal $ ( ) n Number of years Principal Interest rate per year Generalization Interest rate per compounding period Number of compounding

28 Compound Interest Formula An initial principal 𝑃 in an account that pays interest at a periodic interest rate 𝑖 per compounding period grows after 𝑛 compounding periods to 𝒏 𝑨 = 𝑷(𝟏 + π’Š) = 𝑷(𝟏 + 𝒓 𝒏 ) π’Ž Remark: π‘Ÿ, π‘š Periodic interest rate 𝑖 = where r is an annual rate of interest and m is the number of compounding per year.

29 Compound Interest Formula for an annual nominal rate r, t years later 𝒏 𝑨 = 𝑷(𝟏 + π’Š) = 𝑷(𝟏 + 𝒓 𝒏 𝒓 π’Žπ’• ) = 𝑷(𝟏 + ) π’Ž π’Ž 𝑑: Number of years m: Number of compounding per year r: nominal annual interest rate n: Number of compounding

30 Geometric Growth ( Exponential Growth ) Growth proportional to the amount present( Not to the principal )

31 Example Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with annual compounding. How much balance will you have 10 years later?

32 Answer Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with annual compounding. How much balance will you have 10 years later? $1000( )1 10 = $ = $

33 Use of Calculator( Scientific ) (1) Calculate what are in the parenthesis. (2) Calculate the exponent and then raise to the power. (3) Multiply by the principal. However, there are many other correct ways to use the calculator.

34 Example Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with quarterly compounding. How much balance will you have 10 years later?

35 Answer Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with quarterly compounding. How much balance will you have 10 years later? $1000( ) =$1000(1.025)40 = 4 $

36 Example Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with monthly compounding. How much balance will you have 10 years later?

37 Answer Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with monthly compounding. How much balance will you have 10 years later? $1000(1 + ) = $1000(1.0083)120 = $

38 Basic Useful Formula If a = b, then a 1 n = b 1 n. (a m ) n = a mn (a + b) n a n + b n

39 Example A man borrowed $29,000 for two years under simple interest. At the end of the two years his balance due was $31,900. What annual simple interest rate did he pay?

40 Answer A man borrowed $29,000 for two years under simple interest. At the end of the two years his balance due was $31,900. What annual simple interest rate did he pay? 5%

41 Example Suppose you invest $6000 and would like your investment to grow to $8000 in five years. What interest rate, compounded monthly, would you have to earn in order for this to happen?

42 Answer Suppose you invest $6000 and would like your investment to grow to $8000 in five years. What interest rate, compounded monthly, would you have to earn in order for this to happen? 5.77%

43 Group Activity You have $3500 that you invest at 7% simple interest. How long will it take for your balance to reach $4235?

44 Answer You have $3500 that you invest at 7% simple interest. How long will it take for your balance to reach $4235? Three years

45 Group Activity Merrie borrowed $1000 from her parents, agreeing to pay them back when she graduated from college in five years. If she paid interest compounded quarterly at 5%, how much would she owe at the end of the five years?

46 Answer Merrie borrowed $1000 from her parents, agreeing to pay them back when she graduated from college in five years. If she paid interest compounded quarterly at 5%, how much would she owe at the end of the five years? $1282

47 Group Activity Merrie borrowed $500 from her parents, agreeing to pay them back when she graduated from college in four years. If she paid interest compounded daily at 16%, how much would she owe at the end of the four years?

48 Answer Merrie borrowed $500 from her parents, agreeing to pay them back when she graduated from college in four years. If she paid interest compounded daily at 16%, how much would she owe at the end of the four years? $948

49 Effective Rate What percent of interest do you earn after n compounding period compared to the principal? Effective rate: (𝟏 + π’Š)𝒏 𝟏 the money after the principal = π‘·π’“π’Šπ’π’„π’Šπ’‘π’‚π’ 𝑛: number of compounding π‘Ÿ i: interest rate per compounding period ( ) π‘š r: nominal interest rate

50 Annual Percentage Yield (APY) What percent of interest do you get after a year compared to the principal?(i.e. as an annual simple interest rate) Effective rate = (1 + r m )m 1 m: number of compounding per year

51 APY for 4% of nominal interest with $10,000 deposit( Principal) Account type Number of compounding Balance after a year Interest amount % of increase after a year (APY) Simple interest 1 $ % Compounding bi-annually Compounding quarterly 2 $10404= 10000( ) 2 4 $ from %(= ) % Compounding monthly 12 $ from %

52 Example With a nominal annual rate of 6% compounded monthly, what is the APY?

53 Answer With a nominal annual rate of 6% compounded monthly, what is the APY? ( ) 1 = = 6.17%

54 Example Suppose that the monthly statement from the fund reports a beginning balance(p) of $ and a closing balance(a) of $ for 28 days(n). What is the effective daily rate?

55 Example Suppose that the monthly statement from the fund reports a beginning balance(p) of $ and a closing balance(a) of $ for 28 days(n). What is the effective daily rate? Effective rate= (1 + i) n 1 n: number of compounding, i: interest rate per compounding period ( r m ) Difference of money after 28 days = Principal 1 + i n 1 What is i?

56 Example Beginning balance(p): $ , Closing balance(a): $ for 28 days(n). Difference of money after 28 days = Principal 1 + i n = [( 1 + i 28 1] = (1 + i) = (1 + i) (1 + i) 28 = i = ( ) 1 28 = i = So, the daily effective rate is %.

57 Example Angela invests in a savings account that pays 4% interest compounded monthly. What is the APY for this account?

58 Answer Angela invests in a savings account that pays 4% interest compounded monthly. What is the APY for this account? 4.07%

59 Example You have $4300 that you invest at 5% simple interest. How long will it take for your balance to reach $7525?

60 Example You have $4300 that you invest at 5% simple interest. How long will it take for your balance to reach $7525? 15 years

61 Example Suppose you invest in an account that pays 5% interest, compounded quarterly. You would like your investment to grow to $5000 in 16 years. How much would you have to invest in order for this to happen?

62 Answer Suppose you invest in an account that pays 5% interest, compounded quarterly. You would like your investment to grow to $5000 in 16 years. How much would you have to invest in order for this to happen? $2258

63 Simple Interest Versus Compound Interest

64 Continuous Compounding A = Pe rt

65 About Number e e The number (1 + 1 m )m approaches when m gets larger and larger.

66 Continuous Compounding Continuous compounding is the method of calculating interest in which the amount of interest is what compound interest tends toward with more and more frequent compounding.

67 Continuous Interest Formula A = Pe rt A: Balance after t years when a principal P is continuously compounded r: nominal annual rate

68 Example What is the balance after 1 year if the principal is $1000 and 10% continuous compounding?

69 Example What is the balance after 1 year if the principal is $1000 and 10% continuous compounding? 1000 e 0.10 = $

70 Example What is the balance after 5 years if the principal is $1000 and 10% continuous compounding?

71 Example What is the balance after 5 years if the principal is $1000 and 10% continuous compounding? 1000 e (5) (0.10) =

$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be $3500 after 9 months?

$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be $3500 after 9 months? Simple Interest, Compound Interest, and Effective Yield Simple Interest The formula that gives the amount of simple interest (also known as add-on interest) owed on a Principal P (also known as present

More information

Percent, Sales Tax, & Discounts

Percent, Sales Tax, & Discounts Percent, Sales Tax, & Discounts Many applications involving percent are based on the following formula: Note that of implies multiplication. Suppose that the local sales tax rate is 7.5% and you purchase

More information

5.1 Simple and Compound Interest

5.1 Simple and Compound Interest 5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

More information

With compound interest you earn an additional $128.89 ($1628.89 - $1500).

With compound interest you earn an additional $128.89 ($1628.89 - $1500). Compound Interest Interest is the amount you receive for lending money (making an investment) or the fee you pay for borrowing money. Compound interest is interest that is calculated using both the principle

More information

Chapter 4 Nominal and Effective Interest Rates

Chapter 4 Nominal and Effective Interest Rates Chapter 4 Nominal and Effective Interest Rates Chapter 4 Nominal and Effective Interest Rates INEN 303 Sergiy Butenko Industrial & Systems Engineering Texas A&M University Nominal and Effective Interest

More information

Chapter 21: Savings Models

Chapter 21: Savings Models October 16, 2013 Last time Arithmetic Growth Simple Interest Geometric Growth Compound Interest A limit to Compounding Problems Question: I put $1,000 dollars in a savings account with 2% nominal interest

More information

3. Time value of money. We will review some tools for discounting cash flows.

3. Time value of money. We will review some tools for discounting cash flows. 1 3. Time value of money We will review some tools for discounting cash flows. Simple interest 2 With simple interest, the amount earned each period is always the same: i = rp o where i = interest earned

More information

Ch 3 Understanding money management

Ch 3 Understanding money management Ch 3 Understanding money management 1. nominal & effective interest rates 2. equivalence calculations using effective interest rates 3. debt management If payments occur more frequently than annual, how

More information

Compound Interest. Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate:

Compound Interest. Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate: Compound Interest Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate: Table 1 Development of Nominal Payments and the Terminal Value, S.

More information

21.1 Arithmetic Growth and Simple Interest

21.1 Arithmetic Growth and Simple Interest 21.1 Arithmetic Growth and Simple Interest When you open a savings account, your primary concerns are the safety and growth of your savings. Suppose you deposit $1000 in an account that pays interest at

More information

Chapter 22: Borrowings Models

Chapter 22: Borrowings Models October 21, 2013 Last Time The Consumer Price Index Real Growth The Consumer Price index The official measure of inflation is the Consumer Price Index (CPI) which is the determined by the Bureau of Labor

More information

10.6 Functions - Compound Interest

10.6 Functions - Compound Interest 10.6 Functions - Compound Interest Objective: Calculate final account balances using the formulas for compound and continuous interest. An application of exponential functions is compound interest. When

More information

Compounding Quarterly, Monthly, and Daily

Compounding Quarterly, Monthly, and Daily 126 Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly,

More information

Study Questions for Actuarial Exam 2/FM By: Aaron Hardiek June 2010

Study Questions for Actuarial Exam 2/FM By: Aaron Hardiek June 2010 P a g e 1 Study Questions for Actuarial Exam 2/FM By: Aaron Hardiek June 2010 P a g e 2 Background The purpose of my senior project is to prepare myself, as well as other students who may read my senior

More information

Ch. 11.2: Installment Buying

Ch. 11.2: Installment Buying Ch. 11.2: Installment Buying When people take out a loan to make a big purchase, they don t often pay it back all at once in one lump-sum. Instead, they usually pay it back back gradually over time, in

More information

Week 2: Exponential Functions

Week 2: Exponential Functions Week 2: Exponential Functions Goals: Introduce exponential functions Study the compounded interest and introduce the number e Suggested Textbook Readings: Chapter 4: 4.1, and Chapter 5: 5.1. Practice Problems:

More information

Chapter Two. THE TIME VALUE OF MONEY Conventions & Definitions

Chapter Two. THE TIME VALUE OF MONEY Conventions & Definitions Chapter Two THE TIME VALUE OF MONEY Conventions & Definitions Introduction Now, we are going to learn one of the most important topics in finance, that is, the time value of money. Note that almost every

More information

Check off these skills when you feel that you have mastered them.

Check off these skills when you feel that you have mastered them. Chapter Objectives Check off these skills when you feel that you have mastered them. Know the basic loan terms principal and interest. Be able to solve the simple interest formula to find the amount of

More information

Comparing Simple and Compound Interest

Comparing Simple and Compound Interest Comparing Simple and Compound Interest GRADE 11 In this lesson, students compare various savings and investment vehicles by calculating simple and compound interest. Prerequisite knowledge: Students should

More information

About Compound Interest

About Compound Interest About Compound Interest TABLE OF CONTENTS About Compound Interest... 1 What is COMPOUND INTEREST?... 1 Interest... 1 Simple Interest... 1 Compound Interest... 1 Calculations... 3 Calculating How Much to

More information

Time Value of Money 1

Time Value of Money 1 Time Value of Money 1 This topic introduces you to the analysis of trade-offs over time. Financial decisions involve costs and benefits that are spread over time. Financial decision makers in households

More information

Chapter 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

More information

MGF 1107 Spring 11 Ref: 606977 Review for Exam 2. Write as a percent. 1) 3.1 1) Write as a decimal. 4) 60% 4) 5) 0.085% 5)

MGF 1107 Spring 11 Ref: 606977 Review for Exam 2. Write as a percent. 1) 3.1 1) Write as a decimal. 4) 60% 4) 5) 0.085% 5) MGF 1107 Spring 11 Ref: 606977 Review for Exam 2 Mr. Guillen Exam 2 will be on 03/02/11 and covers the following sections: 8.1, 8.2, 8.3, 8.4, 8.5, 8.6. Write as a percent. 1) 3.1 1) 2) 1 8 2) 3) 7 4 3)

More information

Finding Rates and the Geometric Mean

Finding Rates and the Geometric Mean Finding Rates and the Geometric Mean So far, most of the situations we ve covered have assumed a known interest rate. If you save a certain amount of money and it earns a fixed interest rate for a period

More information

Future Value of an Annuity Sinking Fund. MATH 1003 Calculus and Linear Algebra (Lecture 3)

Future Value of an Annuity Sinking Fund. MATH 1003 Calculus and Linear Algebra (Lecture 3) MATH 1003 Calculus and Linear Algebra (Lecture 3) Future Value of an Annuity Definition An annuity is a sequence of equal periodic payments. We call it an ordinary annuity if the payments are made at the

More information

Topics Covered. Compounding and Discounting Single Sums. Ch. 4 - The Time Value of Money. The Time Value of Money

Topics Covered. Compounding and Discounting Single Sums. Ch. 4 - The Time Value of Money. The Time Value of Money Ch. 4 - The Time Value of Money Topics Covered Future Values Present Values Multiple Cash Flows Perpetuities and Annuities Effective Annual Interest Rate For now, we will omit the section 4.5 on inflation

More information

Time Value of Money CAP P2 P3. Appendix. Learning Objectives. Conceptual. Procedural

Time Value of Money CAP P2 P3. Appendix. Learning Objectives. Conceptual. Procedural Appendix B Time Value of Learning Objectives CAP Conceptual C1 Describe the earning of interest and the concepts of present and future values. (p. B-1) Procedural P1 P2 P3 P4 Apply present value concepts

More information

MAT12X Intermediate Algebra

MAT12X Intermediate Algebra MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions

More information

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM FM SAMPLE QUESTIONS

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM FM SAMPLE QUESTIONS SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Copyright 2005 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions

More information

APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS

APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS CHAPTER 8 Current Monetary Balances 395 APPENDIX Interest Concepts of Future and Present Value TIME VALUE OF MONEY In general business terms, interest is defined as the cost of using money over time. Economists

More information

CHAPTER 8 INTEREST RATES AND BOND VALUATION

CHAPTER 8 INTEREST RATES AND BOND VALUATION CHAPTER 8 INTEREST RATES AND BOND VALUATION Solutions to Questions and Problems 1. The price of a pure discount (zero coupon) bond is the present value of the par value. Remember, even though there are

More information

Interest Rate and Credit Risk Derivatives

Interest Rate and Credit Risk Derivatives Interest Rate and Credit Risk Derivatives Interest Rate and Credit Risk Derivatives Peter Ritchken Kenneth Walter Haber Professor of Finance Weatherhead School of Management Case Western Reserve University

More information

380.760: Corporate Finance. Financial Decision Making

380.760: Corporate Finance. Financial Decision Making 380.760: Corporate Finance Lecture 2: Time Value of Money and Net Present Value Gordon Bodnar, 2009 Professor Gordon Bodnar 2009 Financial Decision Making Finance decision making is about evaluating costs

More information

14 ARITHMETIC OF FINANCE

14 ARITHMETIC OF FINANCE 4 ARITHMETI OF FINANE Introduction Definitions Present Value of a Future Amount Perpetuity - Growing Perpetuity Annuities ompounding Agreement ontinuous ompounding - Lump Sum - Annuity ompounding Magic?

More information

How To Calculate A Balance On A Savings Account

How To Calculate A Balance On A Savings Account 319 CHAPTER 4 Personal Finance The following is an article from a Marlboro, Massachusetts newspaper. NEWSPAPER ARTICLE 4.1: LET S TEACH FINANCIAL LITERACY STEPHEN LEDUC WED JAN 16, 2008 Boston - Last week

More information

Solutions to Exercises, Section 4.5

Solutions to Exercises, Section 4.5 Instructor s Solutions Manual, Section 4.5 Exercise 1 Solutions to Exercises, Section 4.5 1. How much would an initial amount of $2000, compounded continuously at 6% annual interest, become after 25 years?

More information

What is the difference between simple and compound interest and does it really matter?

What is the difference between simple and compound interest and does it really matter? Module gtf1 Simple Versus Compound Interest What is the difference between simple and compound interest and does it really matter? There are various methods for computing interest. Do you know what the

More information

Dick Schwanke Finite Math 111 Harford Community College Fall 2013

Dick Schwanke Finite Math 111 Harford Community College Fall 2013 Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of

More information

4 Annuities and Loans

4 Annuities and Loans 4 Annuities and Loans 4.1 Introduction In previous section, we discussed different methods for crediting interest, and we claimed that compound interest is the correct way to credit interest. This section

More information

Dick Schwanke Finite Math 111 Harford Community College Fall 2013

Dick Schwanke Finite Math 111 Harford Community College Fall 2013 Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of

More information

Section 8.1. I. Percent per hundred

Section 8.1. I. Percent per hundred 1 Section 8.1 I. Percent per hundred a. Fractions to Percents: 1. Write the fraction as an improper fraction 2. Divide the numerator by the denominator 3. Multiply by 100 (Move the decimal two times Right)

More information

Preparing cash budgets

Preparing cash budgets 3 Preparing cash budgets this chapter covers... In this chapter we will examine in detail how a cash budget is prepared. This is an important part of your studies, and you will need to be able to prepare

More information

PRESENT VALUE ANALYSIS. Time value of money equal dollar amounts have different values at different points in time.

PRESENT VALUE ANALYSIS. Time value of money equal dollar amounts have different values at different points in time. PRESENT VALUE ANALYSIS Time value of money equal dollar amounts have different values at different points in time. Present value analysis tool to convert CFs at different points in time to comparable values

More information

Logarithmic and Exponential Equations

Logarithmic and Exponential Equations 11.5 Logarithmic and Exponential Equations 11.5 OBJECTIVES 1. Solve a logarithmic equation 2. Solve an exponential equation 3. Solve an application involving an exponential equation Much of the importance

More information

Simple and Compound Interest

Simple and Compound Interest 8 Simple and Compound Interest Interest is the fee paid for borrowed money. We receive interest when we let others use our money (for example, by depositing money in a savings account or making a loan).

More information

THE TIME VALUE OF MONEY

THE TIME VALUE OF MONEY QUANTITATIVE METHODS THE TIME VALUE OF MONEY Reading 5 http://proschool.imsindia.com/ 1 Learning Objective Statements (LOS) a. Interest Rates as Required rate of return, Discount Rate and Opportunity Cost

More information

SEQUENCES ARITHMETIC SEQUENCES. Examples

SEQUENCES ARITHMETIC SEQUENCES. Examples SEQUENCES ARITHMETIC SEQUENCES An ordered list of numbers such as: 4, 9, 6, 25, 36 is a sequence. Each number in the sequence is a term. Usually variables with subscripts are used to label terms. For example,

More information

1 Present and Future Value

1 Present and Future Value Lecture 8: Asset Markets c 2009 Je rey A. Miron Outline:. Present and Future Value 2. Bonds 3. Taxes 4. Applications Present and Future Value In the discussion of the two-period model with borrowing and

More information

Finance 197. Simple One-time Interest

Finance 197. Simple One-time Interest Finance 197 Finance We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for

More information

Credit Card Loans. Student Worksheet

Credit Card Loans. Student Worksheet Student Worksheet Credit Card Loans Name: Recall the formula for simple interest where, I is the interest owed P is the principal amount outstanding r is the interest rate t is the time in years. Note:

More information

CARMEN VENTER COPYRIGHT www.futurefinance.co.za 0828807192 1

CARMEN VENTER COPYRIGHT www.futurefinance.co.za 0828807192 1 Carmen Venter CFP WORKSHOPS FINANCIAL CALCULATIONS presented by Geoff Brittain Q 5.3.1 Calculate the capital required at retirement to meet Makhensa s retirement goals. (5) 5.3.2 Calculate the capital

More information

FinQuiz Notes 2 0 1 5

FinQuiz Notes 2 0 1 5 Reading 5 The Time Value of Money Money has a time value because a unit of money received today is worth more than a unit of money to be received tomorrow. Interest rates can be interpreted in three ways.

More information

Compound Interest Formula

Compound Interest Formula Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt $100 At

More information

What You ll Learn. And Why. Key Words. interest simple interest principal amount compound interest compounding period present value future value

What You ll Learn. And Why. Key Words. interest simple interest principal amount compound interest compounding period present value future value What You ll Learn To solve problems involving compound interest and to research and compare various savings and investment options And Why Knowing how to save and invest the money you earn will help you

More information

4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS

4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally 4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4th Edition,

More information

5. Time value of money

5. Time value of money 1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned

More information

Annuities and Sinking Funds

Annuities and Sinking Funds Annuities and Sinking Funds Sinking Fund A sinking fund is an account earning compound interest into which you make periodic deposits. Suppose that the account has an annual interest rate of compounded

More information

2.6 Exponents and Order of Operations

2.6 Exponents and Order of Operations 2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated

More information

Payroll Accruals: Wages, Taxes and More!

Payroll Accruals: Wages, Taxes and More! Payroll Accruals: Wages, Taxes and More! Presented by: Daniel Dycus, CPP / Daniel.Dycus@intelsat.com / 703.559.7692 1 DEFINITIONS Accrual A revenue or expense that has occurred but has not yet been recorded

More information

2. In solving percent problems with a proportion, use the following pattern:

2. In solving percent problems with a proportion, use the following pattern: HFCC Learning Lab PERCENT WORD PROBLEMS Arithmetic - 11 Many percent problems can be solved using a proportion. In order to use this method, you should be familiar with the following ideas about percent:

More information

GEOMETRIC SEQUENCES AND SERIES

GEOMETRIC SEQUENCES AND SERIES 4.4 Geometric Sequences and Series (4 7) 757 of a novel and every day thereafter increase their daily reading by two pages. If his students follow this suggestion, then how many pages will they read during

More information

How Does Money Grow Over Time?

How Does Money Grow Over Time? How Does Money Grow Over Time? Suggested Grade & Mastery Level High School all levels Suggested Time 45-50 minutes Teacher Background Interest refers to the amount you earn on the money you put to work

More information

EXPONENTIAL FUNCTIONS 8.1.1 8.1.6

EXPONENTIAL FUNCTIONS 8.1.1 8.1.6 EXPONENTIAL FUNCTIONS 8.1.1 8.1.6 In these sections, students generalize what they have learned about geometric sequences to investigate exponential functions. Students study exponential functions of the

More information

Finance CHAPTER OUTLINE. 5.1 Interest 5.2 Compound Interest 5.3 Annuities; Sinking Funds 5.4 Present Value of an Annuity; Amortization

Finance CHAPTER OUTLINE. 5.1 Interest 5.2 Compound Interest 5.3 Annuities; Sinking Funds 5.4 Present Value of an Annuity; Amortization CHAPTER 5 Finance OUTLINE Even though you re in college now, at some time, probably not too far in the future, you will be thinking of buying a house. And, unless you ve won the lottery, you will need

More information

How to calculate present values

How to calculate present values How to calculate present values Back to the future Chapter 3 Discounted Cash Flow Analysis (Time Value of Money) Discounted Cash Flow (DCF) analysis is the foundation of valuation in corporate finance

More information

Semester Exam Review ANSWERS. b. The total amount of money earned by selling sodas in a day was at least $1,000. 800 4F 200 F

Semester Exam Review ANSWERS. b. The total amount of money earned by selling sodas in a day was at least $1,000. 800 4F 200 F Unit 1, Topic 1 P 2 1 1 W L or P2 L or P L or P L 2 2 2 2 1. 2. A. 5F 160 C 9 3. B. The equation is always true, because both sides are identical.. A. There is one solution, and it is x 30. 5. C. The equation

More information

For additional information, see the Math Notes boxes in Lesson B.1.3 and B.2.3.

For additional information, see the Math Notes boxes in Lesson B.1.3 and B.2.3. EXPONENTIAL FUNCTIONS B.1.1 B.1.6 In these sections, students generalize what they have learned about geometric sequences to investigate exponential functions. Students study exponential functions of the

More information

Also, compositions of an exponential function with another function are also referred to as exponential. An example would be f(x) = 4 + 100 3-2x.

Also, compositions of an exponential function with another function are also referred to as exponential. An example would be f(x) = 4 + 100 3-2x. Exponential Functions Exponential functions are perhaps the most important class of functions in mathematics. We use this type of function to calculate interest on investments, growth and decline rates

More information

Introduction to Real Estate Investment Appraisal

Introduction to Real Estate Investment Appraisal Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has

More information

It Is In Your Interest

It Is In Your Interest STUDENT MODULE 7.2 BORROWING MONEY PAGE 1 Standard 7: The student will identify the procedures and analyze the responsibilities of borrowing money. It Is In Your Interest Jason did not understand how it

More information

Lesson Plan -- Simple and Compound Interest

Lesson Plan -- Simple and Compound Interest Lesson Plan -- Simple and Compound Interest Chapter Resources - Lesson 4-14 Simple Interest - Lesson 4-14 Simple Interest Answers - Lesson 4-15 Compound Interest - Lesson 4-15 Compound Interest Answers

More information

Example 1 - Solution. Since the problΓ©m is of the form "find F when given P" the formula to use is F = P(F/P, 8%, 5) = $10,000(1.4693) = $14,693.

Example 1 - Solution. Since the problΓ©m is of the form find F when given P the formula to use is F = P(F/P, 8%, 5) = $10,000(1.4693) = $14,693. Example 1 Ms. Smith loans Mr. Brown $10,000 with interest compounded at a rate of 8% per year. How much will Mr. Brown owe Ms. Smith if he repays the loan at the end of 5 years? Example 1 - Solution Since

More information

ICASL - Business School Programme

ICASL - Business School Programme ICASL - Business School Programme Quantitative Techniques for Business (Module 3) Financial Mathematics TUTORIAL 2A This chapter deals with problems related to investing money or capital in a business

More information

APPENDIX 3 TIME VALUE OF MONEY. Time Lines and Notation. The Intuitive Basis for Present Value

APPENDIX 3 TIME VALUE OF MONEY. Time Lines and Notation. The Intuitive Basis for Present Value 1 2 TIME VALUE OF MONEY APPENDIX 3 The simplest tools in finance are often the most powerful. Present value is a concept that is intuitively appealing, simple to compute, and has a wide range of applications.

More information

Pre-Session Review. Part 2: Mathematics of Finance

Pre-Session Review. Part 2: Mathematics of Finance Pre-Session Review Part 2: Mathematics of Finance For this section you will need a calculator with logarithmic and exponential function keys (such as log, ln, and x y ) D. Exponential and Logarithmic Functions

More information

Basic Concept of Time Value of Money

Basic Concept of Time Value of Money Basic Concept of Time Value of Money CHAPTER 1 1.1 INTRODUCTION Money has time value. A rupee today is more valuable than a year hence. It is on this concept the time value of money is based. The recognition

More information

1. Annuity a sequence of payments, each made at equally spaced time intervals.

1. Annuity a sequence of payments, each made at equally spaced time intervals. Ordinary Annuities (Young: 6.2) In this Lecture: 1. More Terminology 2. Future Value of an Ordinary Annuity 3. The Ordinary Annuity Formula (Optional) 4. Present Value of an Ordinary Annuity More Terminology

More information

Chapter 2 Balance sheets - what a company owns and what it owes

Chapter 2 Balance sheets - what a company owns and what it owes Chapter 2 Balance sheets - what a company owns and what it owes SharePad is packed full of useful financial data. This data holds the key to understanding the financial health and value of any company

More information

NPV I: Time Value of Money

NPV I: Time Value of Money NPV I: Time Value of Money This module introduces the concept of the time value of money, interest rates, discount rates, the future value of an investment, the present value of a future payment, and the

More information

Mathematics. Rosella Castellano. Rome, University of Tor Vergata

Mathematics. Rosella Castellano. Rome, University of Tor Vergata and Loans Mathematics Rome, University of Tor Vergata and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings

More information

Personal Financial Literacy

Personal Financial Literacy Personal Financial Literacy 7 Unit Overview Being financially literate means taking responsibility for learning how to manage your money. In this unit, you will learn about banking services that can help

More information

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and

More information

8.1 Simple Interest and 8.2 Compound Interest

8.1 Simple Interest and 8.2 Compound Interest 8.1 Simple Interest and 8.2 Compound Interest When you open a bank account or invest money in a bank or financial institution the bank/financial institution pays you interest for the use of your money.

More information

To Evaluate an Algebraic Expression

To Evaluate an Algebraic Expression 1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum

More information

Student Loans. The Math of Student Loans. Because of student loan debt 11/13/2014

Student Loans. The Math of Student Loans. Because of student loan debt 11/13/2014 Student Loans The Math of Student Loans Alice Seneres Rutgers University seneres@rci.rutgers.edu 1 71% of students take out student loans for their undergraduate degree A typical student in the class of

More information

Chapter 6. Time Value of Money Concepts. Simple Interest 6-1. Interest amount = P i n. Assume you invest $1,000 at 6% simple interest for 3 years.

Chapter 6. Time Value of Money Concepts. Simple Interest 6-1. Interest amount = P i n. Assume you invest $1,000 at 6% simple interest for 3 years. 6-1 Chapter 6 Time Value of Money Concepts 6-2 Time Value of Money Interest is the rent paid for the use of money over time. That s right! A dollar today is more valuable than a dollar to be received in

More information

CHAPTER 4 DISCOUNTED CASH FLOW VALUATION

CHAPTER 4 DISCOUNTED CASH FLOW VALUATION CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. Assuming positive cash flows and interest rates, the future value increases and the present value

More information

Which home loan is for me?

Which home loan is for me? Which home loan is for me? Home loans made easy Home Loans About this booklet At ING DIRECT, we try to make finding the right home loan as easy as possible. That s what this booklet is all about. All our

More information

Investigating Investment Formulas Using Recursion Grade 11

Investigating Investment Formulas Using Recursion Grade 11 Ohio Standards Connection Patterns, Functions and Algebra Benchmark C Use recursive functions to model and solve problems; e.g., home mortgages, annuities. Indicator 1 Identify and describe problem situations

More information

Time Value Conepts & Applications. Prof. Raad Jassim

Time Value Conepts & Applications. Prof. Raad Jassim Time Value Conepts & Applications Prof. Raad Jassim Chapter Outline Introduction to Valuation: The Time Value of Money 1 2 3 4 5 6 7 8 Future Value and Compounding Present Value and Discounting More on

More information

TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION

TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION Professor Peter Harris Mathematics by Sharon Petrushka Introduction This problem will focus on calculating mortgage payments. Knowledge of Time Value

More information

Time Value of Money. Appendix

Time Value of Money. Appendix 1 Appendix Time Value of Money After studying Appendix 1, you should be able to: 1 Explain how compound interest works. 2 Use future value and present value tables to apply compound interest to accounting

More information

Chapter 07 Interest Rates and Present Value

Chapter 07 Interest Rates and Present Value Chapter 07 Interest Rates and Present Value Multiple Choice Questions 1. The percentage of a balance that a borrower must pay a lender is called the a. Inflation rate b. Usury rate C. Interest rate d.

More information

CE 314 Engineering Economy. Interest Formulas

CE 314 Engineering Economy. Interest Formulas METHODS OF COMPUTING INTEREST CE 314 Engineering Economy Interest Formulas 1) SIMPLE INTEREST - Interest is computed using the principal only. Only applicable to bonds and savings accounts. 2) COMPOUND

More information

Unique Living s guide to purchasing a luxury property in Cyprus

Unique Living s guide to purchasing a luxury property in Cyprus Unique Living s guide to purchasing a luxury property in Cyprus Choice of Property We strongly recommend that the buyer select a property first, but should not pay any money until a meeting with a lawyer

More information

Engineering Economy. Time Value of Money-3

Engineering Economy. Time Value of Money-3 Engineering Economy Time Value of Money-3 Prof. Kwang-Kyu Seo 1 Chapter 2 Time Value of Money Interest: The Cost of Money Economic Equivalence Interest Formulas Single Cash Flows Equal-Payment Series Dealing

More information

Stock and Bond Valuation: Annuities and Perpetuities

Stock and Bond Valuation: Annuities and Perpetuities Stock and Bond Valuation: Annuities and Perpetuities Lecture 3, slides 3.1 Brais Alvarez Pereira LdM, BUS 332 F: Principles of Finance, Spring 2016 February 23, 2016 Important Shortcut Formulas Present

More information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

More information

Untangling F9 terminology

Untangling F9 terminology Untangling F9 terminology Welcome! This is not a textbook and we are certainly not trying to replace yours! However, we do know that some students find some of the terminology used in F9 difficult to understand.

More information

Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations

Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Real Estate Division Sauder School of Business University of British Columbia Introduction to the Hewlett-Packard

More information