5.1 Simple and Compound Interest


 Curtis Barry Stewart
 5 years ago
 Views:
Transcription
1 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? Businesses operate with borrowed money. When a business needs order inventory or expand, it may borrow the money needed for the expansion. The borrower will be charged interest for the opportunity to use the money. The interest on the loan is typically charged at some percentage of the amount borrowed called the interest rate. Not only do businesses borrow, but banks may also borrow money from other banks or even individuals. For instance, you may invest money with a bank and receive interest from the bank. Most consumers borrow money regularly using credit cards. If the balance is not paid off when the lending period is through, you must pay interest to the credit card company for the privilege of borrowing the money. In this section, we ll examine several type of interest. Simple interest is interest where a fixed amount is paid based on the amount borrowed and the length of time the money is borrowed. In compound interest, interest accumulates according to the amount borrowed over time and any interest that has accumulated during that period of time. Both types of interest are used extensively in business and finance. 1
2 Question 1: What is simple interest? In business, individuals or companies often borrow money or assets. The lender charges a fee for the use of the assets. Interest is the fee the lender charges for the use of the money. The amount borrowed is the principal or present value of the loan. Simple interest is interest computed on the original principal only. If the present value PV, in dollars, earns interest at a rate of r for t years, then the interest is I PV rt The future value (also called the accumulated amount or maturity value) is the sum of the principal and the interest. This is the amount the present value grows to after the present value and interest are added. Simple Interest The future value FV at a simple interest rate r per year is FV PV PV rt PV 1rt where PV is the present value that is deposited for t years. The interest rate r is the decimal form of the interest rate written as a percentage. This means an interest rate of 4% per year is equivalent to r In this text, we use the variable names commonly used in finance textbooks. Instead of writing the present value as the single letter P, we use two letters, PV. Be very careful to interpret this as a single variable and not a product of P and V. Similarly, the future value is written FV. This set of letters represents a single quantity, not a product of F and V. This allows us to use groups of letters to represent quantities that suggest their meaning. 2
3 If we know two of the quantities in this formula, we can solve for the other quantity. This formula is also used to calculate simple interest paid on investments or deposits at a bank. In these cases, we think of the deposits or investment as a loan to the bank with the interest paid to the depositor. Example 1 Simple Interest An investment pays simple interest of 4% per year. An investor deposits $500 in this investment and makes no withdrawals for 5 years. a. How much interest does the investment earn over the fiveyear period? Solution Use I PV rt to compute the interest, I Set PV = 500, r = 0.04, and t = b. What is the future value of the investment in 5 years? Solution The future value is computed using FV PV 1 rt, A Set PV = 500, r = 0.04, and t = c. Find an expression for the future value if the deposit accumulates interest for t years. Assume no withdrawals over the period. Solution In this part, the time t is variable, FV t t 3
4 This relationship corresponds to a linear function of t. The vertical intercept is 500 and the slope is 20. This tells us that the initial investment is $500 and the accumulated amount increases by $20 per year. Figure 1 The linear function describing the accumulated amount in Example 1c. Example 2 Simple Interest A small payday loan company offers a simple interest loan to a customer. They will loan the customer $750. The customer promises to repay the company $808 in two weeks. What is the annual interest rate for this loan? Solution Since there are 52 weeks in a year, the length of this loan is 2 52 years. Use the information in the problem in the simple interest formula, FV PV 1 rt, to solve for the rate r: 4
5 r r r r 2 52 Set FV = 808, PV = 750 and Divide both sides by 750 Subtract 1 or Multiply both sides by t 52 from both sides r 2.01 This decimal corresponds to an interest rate of 201% per year. Because of such high rates, many states are passing legislation to limit the interest rates that pay day loan companies charge. 5
6 Question 2: What is compound interest? In a loan or investment earning compound interest, interest is periodically added to the present value. This additional amount earns interest. In other words, the interest earns interest. Let us illustrate this process with a concrete example. Suppose we deposit $500 in an account that earns interest at a rate of 4% compounded annually. This rate is the nominal or stated rate. By saying that interest is compounding annually, we mean that interest is added to the principal at the end of each year. For instance, we use the simple interest formula, FV PV 1 rt value at the end of the first year, FV 500( ) , to compute the future To find the future value at the end of the second year, we let the present value be the future value from the end of the first year in the simple interest formula, A future value from first year Since the present value in this amount includes the interest from the first year, the interest from the first year is earning interest. This is the effect of compounding. To find the future value at the end of the third year, we let the future value at the end of the second year be the present value in the simple interest formula, 2 6
7 2 A future value from first two years Let us summarize these amounts in a table. 3 End of the Calculation for the Future Value Future Value First Year 500(1.04) 520 Second Year Third Year The middle column establishes a simple pattern. At the end of each year, the future value is equal to the present value times several factors of These factors correspond to the compounding of interest. In general, if interest is compounded annually, then the future value is FV PV 1 r where PV is the principal, r is the nominal rate and t is the time in years. If interests compounds more than once a year, finding the future value is more challenging. It is more likely that interest is compounded quarterly (4 times a year), monthly (12 times a year) or daily (365 times a year). The length of time between which interest is earned is the conversion period. The length of time over which the loan or investment earns interest is the term. To account for compounding over shorter conversion periods, we need more factors in the expression for the future value. However, in each of these factors we only earn a fraction of the interest rate. t 7
8 For instance, suppose deposit $500 in an account earning 4% compounded quarterly. To calculate the future value, we multiply the principal by a factor corresponding to onefourth of the interest rate each quarter. The future value after one quarter is FV 500( ) After two quarters, the future value contains two factors corresponding to one percent interest per quarter, FV Continue this pattern for twelve conversion periods (twelve quarters or three years) gives FV If we compare this expression to the expression for compounding quarterly, A , we note several differences. When we compound quarterly, we get four times as many factors in the future value. This is due to the fact compounding quarterly means we need four times as many factors. When we compound quarterly, each factor utilizes a rate that is onefourth the rate for compounding annually. 8
9 Compound Interest The future value FV of the present value PV compounded over n conversion periods at an interest rate of i per period is FV PV 1 i n where r nominal rate i, m number of conversion periods in a year and number of conversion periods in a year term in years nmt. r You may also see compound interest computed from the formula A P1 m mt. This is the exact same formula as the one above except the present value is called the principal P and the future value is called the accumulated amount A. Example 3 Compound Interest A customer deposits $5000 in an account that earns 1% annual interest compounded monthly. If the customer makes no further deposits or withdrawals from the account, how much will be in the account in five years? Solution To utilize the compound interest formula, FV PV 1 i, we must find the present value PV, the interest rate per conversion period i, and the number of conversion periods n. The present value or principal n 9
10 is the amount of the original deposit so P The account earns 1% annual interest, compounded monthly. This means the account earns i percent per month over each conversion period. Since the interest is compounded monthly over 5 years, there are n 12 5 or 60 conversion periods during the time this money is deposited. The future value is FV dollars 12 If the future value, interest rate, and number of conversion periods is known, we can solve for the present value in FV PV 1 i n. In problems like this, we want to know what amount should we start with to grow to a known future value. Example 4 Present Value A couple needs $25,000 for a large purchase in five years. How much must be deposited now in an account earning 2% annual interest compounded quarterly to accumulate this amount? Assume no further deposits or withdrawals during this time period. Solution To find the amount needed today, we must find the present value of $25,000. The interest for each conversion period is i 0.02 percent per period 4 The account earns interest over a total of 45 or 20 conversion periods. Substitute these values into the compound interest formula, n FV PV 1 i, and solve for PV: 10
11 25000 PV Substitute FV 25, 000, i 0.005,and n PV Divide both sides by PV We round the present value in the last step to two decimal places. This ensures the value is accurate to the nearest cent. If the couple invests $22, for five years, it will grow to $25,000 at this interest rate. 11
12 Question 3: What is an effective interest rate? The amount of interest compounded depends on several factors. The nominal rate r and the number of conversion periods m both influence the future value over a predetermined time period. A savings account earning a higher nominal rate over fewer conversion periods might have the same future value as another savings account with a lower nominal rate and a higher number of conversion periods. To help us compare nominal interest rates, we use the effective interest rate. The effective interest rate is the simple interest rate that leads to the same future value in one year as the nominal interest rate compounded m times per year. The effective interest rate is m r re 1 1 m where r is the nominal interest rate, and m is the number of conversion periods per year. Another name for the effective interest rate is the annual percentage yield or APR. Example 5 Best Investment An investor has the opportunity to invest in one of two opportunities. The first opportunity is a certificate of deposit (CD) earning 1.140% compounded daily. The second opportunity is an investment yielding a dividend of 1.141% compounded quarterly. Which investment is best? Solution The better investment is the one with the higher effective interest rate. The nominal rate for the CD is r Interest is earned on a daily basis so m 365. This gives an effective rate of 12
13 r e For the other investment, r and m 4. The effective rate for this investment is r e The effective rate for the CD, %, is higher than the effective rate for the investment, %. Because of this, the CD is the better investment. By law, the effective rate of interest is shown in all transactions involving interest charges. The APR is always prevalent in advertisements, such as the one below for fiveyear CD rates from Bankrate.com on December 29, Institution APR Rate Minimum Deposit Bank of America % 1.19 Compounded monthly $1000 We can also use the APR to compute accumulated amounts. Suppose we want to compute the future value from depositing $1000 in the Bank of America five year CD. We could calculate the future value using the rate, FV Alternatively, we compute the future value using the APR and compound annually, 13
14 FV This gives us another way of computing accumulated amounts. The future value FV compounded at an effective interest rate (APR) of r e is FV PV 1 r e t where PV is the present vvalue or principal, and t is the term in years. Since the APR is always shown in financial transactions, this formula allows us to compute accumulated amounts from the APR. We can also use the compound interest formula to find the rate at which an amount grows. In this case, we think of PV as the original amount and FV as the amount it grows to. Example 6 Growth of Ticket Prices In 2000, the average price of a movies theater ticket was $5.39. In 2010, the average price increased to $7.89. At what annual percentage rate did prices increase over the period from 2000 to 2010 on average? Source: National Association of Theater Owners Solution The original price in 2000 is $ This price grows in ten years to $7.89. Use these values in 1 rate r e : t FV PV r e to find the effective 14
15 r re e 10 To solve for r e, remove the tenth power by raising both sides of the equation to the onetenth power re Multiply exponents, r e Subtract 1 from both sides r e r e Over this period, the price of tickets increased by an average of 3.88% per year. 15
16 Question 4: What is continuous compound interest? As the frequency of compounding increases, the effective interest rate also increases. We can see this by computing the effective interest rate at a specific nominal rate, say r 0.1. Frequency Number of conversion periods per year m Effective interest rate 0.1 m 1 m 1 annually semiannually quarterly monthly daily hourly every minute 525, As the number of conversion periods per year increases, the effective interest rate gets closer and closer to In fact, it is possible to show that the effective interest rate gets closer and closer to the value e as the frequency of computing increases. If this is done at a nominal rate of r 0.1, the accumulated amount is A P 1r 0.1 t 1 e 1 P P e Pe t e t t Set re e Simplify using the fact that a m n a mn 16
17 In general, as the frequency of compounding increases, the effective interest rate gets r closer and closer to e 1. We can express this symbolically by writing, m r r 1 1 e 1 as m m Think of the symbol as meaning approaches. Larger and larger values of m mean that we are compounding interest more and more frequently. When this happens, we say that the interest is term compounded continuously. The future value FV of the present value PV compounded continuously at a nominal interest rate of r per period is FV PV e rt where t is the time in years. Like the compound interest formula, this formula may also be written in several equivalent forms. In a biological context, the size of a population P with an initial amount of P 0 growing at a continuous rate of r % per year over t years grows according to P P e 0 rt In some business applications, an original amount of money or principal P grows to an accumulated amount A at a continuous rate of r % per year over t years according to A Pe rt In each of these applications, some quantity is growing at a continuous rate r. The original amount of the quantity is multiplied by a factor of quantity at some later time. rt e to yield the amount of the 17
18 Example 7 Continuous Interest Third Federal Savings and Loan offers a CD that earns 1.79% compounded quarterly (on February 3, 2012). If $5000 is invested in the CD, how much more money would be in the account in 5 years if the interest is compounded continuously versus quarterly? Solution The future value with interest compounded quarterly is FV The future value with interest compounded continuously is FV e The future value with continuous interest is greater than the future value with interest compounded quarterly by In general, compounding some amount continuously will always yield a larger amount than compounding the same amount at the same rate a finite number of times per year. The greater number of times the amount is compounded in a year, the closer the future value will be to the future value compounded continuously. 18
$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 addon interest) owed on a Principal P (also known as present
More informationAPPENDIX. 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 information3. 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 informationWith 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 informationPresent Value Concepts
Present Value Concepts Present value concepts are widely used by accountants in the preparation of financial statements. In fact, under International Financial Reporting Standards (IFRS), these concepts
More informationPercent, 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 informationCh. 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 lumpsum. Instead, they usually pay it back back gradually over time, in
More information1. 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 informationCalculations for Time Value of Money
KEATMX01_p001008.qxd 11/4/05 4:47 PM Page 1 Calculations for Time Value of Money In this appendix, a brief explanation of the computation of the time value of money is given for readers not familiar with
More informationMGF 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 informationChapter 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 informationWhat 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 informationMathematics. 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 informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More informationIntroduction to the HewlettPackard (HP) 10BII Calculator and Review of Mortgage Finance Calculations
Introduction to the HewlettPackard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Real Estate Division Sauder School of Business University of British Columbia Introduction to the HewlettPackard
More informationPRESENT 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 informationROUND(cell or formula, 2)
There are many ways to set up an amortization table. This document shows how to set up five columns for the payment number, payment, interest, payment applied to the outstanding balance, and the outstanding
More informationCompound 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 informationBasic 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 informationAppendix C 1. Time Value of Money. Appendix C 2. Financial Accounting, Fifth Edition
C 1 Time Value of Money C 2 Financial Accounting, Fifth Edition Study Objectives 1. Distinguish between simple and compound interest. 2. Solve for future value of a single amount. 3. Solve for future
More informationSection 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 informationFinQuiz 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 information380.760: 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 informationWhat 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 informationTime 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. B1) Procedural P1 P2 P3 P4 Apply present value concepts
More informationAbout 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 informationThe Time Value of Money Part 2B Present Value of Annuities
Management 3 Quantitative Methods The Time Value of Money Part 2B Present Value of Annuities Revised 2/18/15 New Scenario We can trade a single sum of money today, a (PV) in return for a series of periodic
More information1 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 twoperiod model with borrowing and
More informationFinance 197. Simple Onetime 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 information2 The Mathematics. of Finance. Copyright Cengage Learning. All rights reserved.
2 The Mathematics of Finance Copyright Cengage Learning. All rights reserved. 2.3 Annuities, Loans, and Bonds Copyright Cengage Learning. All rights reserved. Annuities, Loans, and Bonds A typical definedcontribution
More informationInvestigating 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 informationLO.a: Interpret interest rates as required rates of return, discount rates, or opportunity costs.
LO.a: Interpret interest rates as required rates of return, discount rates, or opportunity costs. 1. The minimum rate of return that an investor must receive in order to invest in a project is most likely
More informationHow 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 informationDISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS
Chapter 5 DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS The basic PV and FV techniques can be extended to handle any number of cash flows. PV with multiple cash flows: Suppose you need $500 one
More informationFinite Mathematics. CHAPTER 6 Finance. Helene Payne. 6.1. Interest. savings account. bond. mortgage loan. auto loan
Finite Mathematics Helene Payne CHAPTER 6 Finance 6.1. Interest savings account bond mortgage loan auto loan Lender Borrower Interest: Fee charged by the lender to the borrower. Principal or Present Value:
More information9. Time Value of Money 1: Present and Future Value
9. Time Value of Money 1: Present and Future Value Introduction The language of finance has unique terms and concepts that are based on mathematics. It is critical that you understand this language, because
More informationFinance 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 informationTIME VALUE OF MONEY (TVM)
TIME VALUE OF MONEY (TVM) INTEREST Rate of Return When we know the Present Value (amount today), Future Value (amount to which the investment will grow), and Number of Periods, we can calculate the rate
More informationTime Value of Money. 2014 Level I Quantitative Methods. IFT Notes for the CFA exam
Time Value of Money 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction...2 2. Interest Rates: Interpretation...2 3. The Future Value of a Single Cash Flow...4 4. The
More informationfirst complete "prior knowlegde"  to refresh knowledge of Simple and Compound Interest.
ORDINARY SIMPLE ANNUITIES first complete "prior knowlegde"  to refresh knowledge of Simple and Compound Interest. LESSON OBJECTIVES: students will learn how to determine the Accumulated Value of Regular
More informationAppendix. Time Value of Money. Financial Accounting, IFRS Edition Weygandt Kimmel Kieso. Appendix C 1
C Time Value of Money C 1 Financial Accounting, IFRS Edition Weygandt Kimmel Kieso C 2 Study Objectives 1. Distinguish between simple and compound interest. 2. Solve for future value of a single amount.
More informationWeek 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 informationPreAlgebra Lecture 6
PreAlgebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals
More informationPowerPoint. to accompany. Chapter 5. Interest Rates
PowerPoint to accompany Chapter 5 Interest Rates 5.1 Interest Rate Quotes and Adjustments To understand interest rates, it s important to think of interest rates as a price the price of using money. When
More informationChapter 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 informationCHAPTER 1. Compound Interest
CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.
More informationChapter 6. Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams
Chapter 6 Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams 1. Distinguish between an ordinary annuity and an annuity due, and calculate present
More informationCHAPTER 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 informationChapter 4: Exponential and Logarithmic Functions
Chapter 4: Eponential and Logarithmic Functions Section 4.1 Eponential Functions... 15 Section 4. Graphs of Eponential Functions... 3 Section 4.3 Logarithmic Functions... 4 Section 4.4 Logarithmic Properties...
More information5. 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 informationInterest 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 informationMAT12X 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 informationTHE 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 informationChapter 6. Discounted Cash Flow Valuation. Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Answer 6.1
Chapter 6 Key Concepts and Skills Be able to compute: the future value of multiple cash flows the present value of multiple cash flows the future and present value of annuities Discounted Cash Flow Valuation
More informationChapter F: Finance. Section F.1F.4
Chapter F: Finance Section F.1F.4 F.1 Simple Interest Suppose a sum of money P, called the principal or present value, is invested for t years at an annual simple interest rate of r, where r is given
More informationCalculating interest rates
Calculating interest rates A reading prepared by Pamela Peterson Drake O U T L I N E 1. Introduction 2. Annual percentage rate 3. Effective annual rate 1. Introduction The basis of the time value of money
More informationMTH 150 SURVEY OF MATHEMATICS. Chapter 11 CONSUMER MATHEMATICS
Your name: Your section: MTH 150 SURVEY OF MATHEMATICS Chapter 11 CONSUMER MATHEMATICS 11.1 Percent 11.2 Personal Loans and Simple Interest 11.3 Personal Loans and Compound Interest 11.4 Installment Buying
More informationTime Value of Money 1
Time Value of Money 1 This topic introduces you to the analysis of tradeoffs over time. Financial decisions involve costs and benefits that are spread over time. Financial decision makers in households
More informationTimeValueofMoney and Amortization Worksheets
2 TimeValueofMoney and Amortization Worksheets The TimeValueofMoney and Amortization worksheets are useful in applications where the cash flows are equal, evenly spaced, and either all inflows or
More informationCheck 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 informationFormulas and Approaches Used to Calculate True Pricing
Formulas and Approaches Used to Calculate True Pricing The purpose of the Annual Percentage Rate (APR) and Effective Interest Rate (EIR) The true price of a loan includes not only interest but other charges
More informationTime 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 informationChapter 4. Time Value of Money
Chapter 4 Time Value of Money Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. 2. Understand the concept of future value
More informationFinding the Payment $20,000 = C[1 1 / 1.0066667 48 ] /.0066667 C = $488.26
Quick Quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? You want to receive $5,000 per month in retirement.
More informationChapter 6. Time Value of Money Concepts. Simple Interest 61. Interest amount = P i n. Assume you invest $1,000 at 6% simple interest for 3 years.
61 Chapter 6 Time Value of Money Concepts 62 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 informationLesson 4 Annuities: The Mathematics of Regular Payments
Lesson 4 Annuities: The Mathematics of Regular Payments Introduction An annuity is a sequence of equal, periodic payments where each payment receives compound interest. One example of an annuity is a Christmas
More informationTime 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 informationMAT116 Project 2 Chapters 8 & 9
MAT116 Project 2 Chapters 8 & 9 1 81: The Project In Project 1 we made a loan workout decision based only on data from three banks that had merged into one. We did not consider issues like: What was the
More informationCurrent Yield Calculation
Current Yield Calculation Current yield is the annual rate of return that an investor purchasing a security at its market price would realize. Generally speaking, it is the annual income from a security
More informationMath 120 Basic finance percent problems from prior courses (amount = % X base)
Math 120 Basic finance percent problems from prior courses (amount = % X base) 1) Given a sales tax rate of 8%, a) find the tax on an item priced at $250, b) find the total amount due (which includes both
More informationLesson 1. Key Financial Concepts INTRODUCTION
Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have
More informationFinance 350: Problem Set 6 Alternative Solutions
Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas
More informationSample problems from Chapter 10.1
Sample problems from Chapter 10.1 This is the annuities sinking funds formula. This formula is used in most cases for annuities. The payments for this formula are made at the end of a period. Your book
More informationChapter The Time Value of Money
Chapter The Time Value of Money PPT 92 Chapter 9  Outline Time Value of Money Future Value and Present Value Annuities TimeValueofMoney Formulas Adjusting for NonAnnual Compounding Compound Interest
More informationAPPENDIX 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 informationSolutions to Time value of money practice problems
Solutions to Time value of money practice problems Prepared by Pamela Peterson Drake 1. What is the balance in an account at the end of 10 years if $2,500 is deposited today and the account earns 4% interest,
More informationMath of Finance Semester 1 Unit 2 Page 1 of 19
Math of Finance Semester 1 Unit 2 Page 1 of 19 Name: Date: Unit 2.1 Checking Accounts Use your book or the internet to find the following definitions: Account balance: Deposit: Withdrawal: Direct deposit:
More informationIt 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 information14 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 informationChapter 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 informationPresent Value (PV) Tutorial
EYK 151 Present Value (PV) Tutorial The concepts of present value are described and applied in Chapter 15. This supplement provides added explanations, illustrations, calculations, present value tables,
More informationMATHEMATICS OF FINANCE AND INVESTMENT
MATHEMATICS OF FINANCE AND INVESTMENT G. I. FALIN Department of Probability Theory Faculty of Mechanics & Mathematics Moscow State Lomonosov University Moscow 119992 g.falin@mail.ru 2 G.I.Falin. Mathematics
More informationIng. Tomáš Rábek, PhD Department of finance
Ing. Tomáš Rábek, PhD Department of finance For financial managers to have a clear understanding of the time value of money and its impact on stock prices. These concepts are discussed in this lesson,
More informationChapter 5 Time Value of Money 2: Analyzing Annuity Cash Flows
1. Future Value of Multiple Cash Flows 2. Future Value of an Annuity 3. Present Value of an Annuity 4. Perpetuities 5. Other Compounding Periods 6. Effective Annual Rates (EAR) 7. Amortized Loans Chapter
More informationBond Price Arithmetic
1 Bond Price Arithmetic The purpose of this chapter is: To review the basics of the time value of money. This involves reviewing discounting guaranteed future cash flows at annual, semiannual and continuously
More informationTIME 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 informationSOCIETY 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 informationChapter 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 informationChapter 4: Time Value of Money
FIN 301 Homework Solution Ch4 Chapter 4: Time Value of Money 1. a. 10,000/(1.10) 10 = 3,855.43 b. 10,000/(1.10) 20 = 1,486.44 c. 10,000/(1.05) 10 = 6,139.13 d. 10,000/(1.05) 20 = 3,768.89 2. a. $100 (1.10)
More informationReal estate investment & Appraisal Dr. Ahmed Y. Dashti. Sample Exam Questions
Real estate investment & Appraisal Dr. Ahmed Y. Dashti Sample Exam Questions Problem 31 a) Future Value = $12,000 (FVIF, 9%, 7 years) = $12,000 (1.82804) = $21,936 (annual compounding) b) Future Value
More informationContinuous Compounding and Discounting
Continuous Compounding and Discounting Philip A. Viton October 5, 2011 Continuous October 5, 2011 1 / 19 Introduction Most realworld project analysis is carried out as we ve been doing it, with the present
More informationICASL  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 informationReview Solutions FV = 4000*(1+.08/4) 5 = $4416.32
Review Solutions 1. Planning to use the money to finish your last year in school, you deposit $4,000 into a savings account with a quoted annual interest rate (APR) of 8% and quarterly compounding. Fifteen
More informationAnnuities 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 informationChapter 2 Applying Time Value Concepts
Chapter 2 Applying Time Value Concepts Chapter Overview Albert Einstein, the renowned physicist whose theories of relativity formed the theoretical base for the utilization of atomic energy, called the
More informationComparing 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 informationFinance Unit 8. Success Criteria. 1 U n i t 8 11U Date: Name: Tentative TEST date
1 U n i t 8 11U Date: Name: Finance Unit 8 Tentative TEST date Big idea/learning Goals In this unit you will study the applications of linear and exponential relations within financing. You will understand
More informationHow 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 informationThe Basics of Interest Theory
Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest
More informationGEOMETRIC 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