# Investigating Investment Formulas Using Recursion Grade 11

Size: px
Start display at page:

Transcription

1 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 involving an iterative process that can be represented as a recursive function; e.g., compound interest. Mathematical Processes Benchmarks A. Construct algorithms for multi-step and nonroutine problems. J. Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of a solution within a model, and validation to original problem situation. Lesson Summary: In this lesson, students derive the formula for the balance of a loan after a given number of payments. The lesson illustrates that an annuity is simply a loan situation with a reverse of the payment (a loan pays off an amount while an annuity accumulates to an amount). Estimated Duration: Two to two and one-half hours Commentary: This lesson provides an excellent opportunity for students to reflect upon and discuss real life financial mathematics with others. The development of the formula and the use of technology provide students relevant skills they can use as investors and borrowers in the future. Students should be familiar with the formulas used to calculate simple and compound interest prior to this lesson. Pre-Assessment: Distribute Attachment A, Pre-Assessment. Allow the class five to 10 minutes to complete these problems. Scoring Guidelines: Informally assess the students abilities to evaluate or simplify expressions and problem situations through observation while students work and during the class discussion of solutions. Provide review of the formulas and opportunities for students to apply the formula to relevant situations. Post-Assessment: Distribute Attachment C, Investigating Investment Formulas Post-Assessment. Have students use estimation to rank the items A through H (before making computations) from the one that generates the greatest value to the one that generates the least value. Have students solve each problem situation, showing the expression used to solve the problem. Group students and have them compare their expressions and solution. After students reach consensus for the correct answers, instruct each group to make a display of their rankings. Conduct a class discussion debating any discrepancies among the groups. 1

2 Have the class determine the correct order. Have groups compare the pre-rankings and the final rankings. Scoring Guidelines: Use the rubric to determine progress toward expectations. Meets Expectations Shows complete understanding of using appropriate formulas correctly and accurately computing the solution. Approaching Shows partial understanding of Expectations Using appropriate formulas with few minor errors and accurately computing the solution based on the errors. OR Using appropriate formulas correctly and computing the solutions with few Intervention Needed minor errors. Shows minimal understanding of using inappropriate formulas or is unable to derive an appropriate formula and may have several computational errors. Instructional Procedures: Instructional Tips: This lesson may be approached in a variety of ways. a. Have students do the steps in the problem and discuss as the class progresses. b. Have students work at their desks in pairs to discuss ideas as they solve the problems. c. Form groups of three to four students and have the groups share their answers as the class works through the problems. Start with using recursion to find values for specific problem situations. Sample problems and the progression of the problems are given. 1. Pose the following problem situation. Luanne takes out a loan for \$1000. The loan has a 1% interest rate that is compounded monthly. Payments are made at the end of each month. She will take one year to repay the loan (1 payments). The payment size is \$ Ask students questions about the scenario. What is the balance after the first month? = \$91.14 What is the balance after two months? = \$ a. Explain to the students that the solution for determining the balance after two months uses the answer for the balance at the end of one month. This process is called recursion, using one or more previous terms to generate the current term. b. Explain to the students that this expression is identical to the previous one except the \$1000 starting balance has been replaced by the \$91.14 balance.

3 3. Have students calculate the solutions for three, four and five months. What is the balance after five months? Discussion point what do you have to do to calculate this value? Month three = \$ Month four = \$ Month five = \$ Instructional Tip: This is a time-consuming drawback to using recursion. A program or spreadsheet can alleviate this problem. The sequence mode on some calculators can also calculate this solution. 4. Discuss how banks handle the issue of rounding money. Ask questions, such as: Do the banks round correctly or truncate? Do they always round up? Does rounding up make a large difference in the amounts paid? Instructional Tip: Students may be interested in investigating this. (These solutions always round up, so the bank does not lose even a fraction of a cent owed it). This would require a larger loan amount and a larger payment period to see the real effect. Technology is required to adequately investigate this in a time efficient manner. 5. Have students determine the pay-off or final amount to be paid in month 1. Ask, What is the pay-off amount or in other words the final payment? (\$87.83, plus the interest earned that last month 87.83(1.01) = \$88.71) a. Inform students that the payments are made at the end of the month, so interest has accrued on the balance that month. b. Have students use the answer key on a calculator to make this a quick process. Use ANS (1.01) repeatedly for most calculators. This may need to be demonstrated. 6. Provide other problems of this type that will show the need to create a closed-form equation for loans. Use Deriving the Formula, Attachment F with the students to go through the procedure for deriving the formula. The formula is bal 0 (1 + r) n PMT (1 + r) n 1 = balance at a given payment n. r Variables are defined in Attachment E. Instructional Tip: Complete as a group activity or step by step together in class. The ability of the students to handle the detailed symbol manipulations will determine what the appropriate course of action is. 3

4 6. Provide this problem situation to the class. The Emanis family borrowed \$150,000 at 5.% for thirty years to buy a house. Their payment is \$83.67 per month, paid at the end of the month. What is the balance at year ten? a. Have students determine the balance at year 10. Observe the formulas students use and provide assistance to students as necessary. b. Discuss the high cost of interest on a large loan since students may be surprised by the large amount of interest. c. Have students determine the balance after 359 payments? Ask students why the balance after 359 is relevant for the situation? Have to determine the final amount to pay before calculating any interest on the final balance. r PV 7. Provide and discuss the formula for determining payments. PMT = -N 1 ( 1+ r) 8. Have students calculate the payment on the previous problem to compare and verify the payment. 9. Explain that an annuity is like a loan, except the buyer makes payments into a fund that starts at zero and builds to a given amount. So, you have a deposit instead of a payment. In the loan formula all remains the same except the interest is added to the deposit and balance of the previous month, not subtracted. The formula becomes: bal n (1 + r) n + PMT (1 + r) n 1 r 10. Provide additional problem situations, such as: Ahmed deposits \$100 into an insurance annuity at the end of each month. It pays 5% and is compounded monthly. Determine the amount of money is in Ahmed s account after two years. Point out bal 0 is zero, so the first term bal 0 (1 + r) n is also zero. a. Have students solve the problem situation. b. Allow students to share and compare answers and make necessary corrections. 4

5 11. Provide students opportunities to practice using the formula for determining the amount of interest earned on annuity. a. Place enough slips of paper in a bowl for each student to pick one with a possible regular annual deposit amount, i.e., \$100, \$00, \$500, \$1,000, \$5,000 etc. Each value is repeated two or three times. b. In another bowl, put slips of paper for each student with interest rates on them, i.e. 3%, 3.5%, 4%, etc. Again, each rate is repeated two or three times. c. Instruct students to use the amount of the regular annual deposit and the interest rate chosen to calculate how much money they will have in an annuity after 5 years. 1. Assign the Post Assessment, Attachment C. Differentiated Instruction: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). Allow the use of technology to calculate solutions to problem situations. Use advertisements from newspapers and magazines regarding annuity, home purchases, credit card payments, etc. Have students determine interest and payments for the situations. Provide problems with smaller terms or provide formula sheets as a guide. Extension: Have students write in their journals their expectations for retirement regarding the amount of money they would need and how long they expect to work. Have them include when they would start saving for retirement and how much they would save? Also, have them consider employer contributions towards their retirement, and how that would influence their personal saving habits toward retirement. Home Connections: Have students investigate different retirement plans and investments available, such as Roth IRAs, 401(k) plans, Cash Balance plans, 403(b) plans, and proposals for changing Social Security. These may seem to be high-level concepts, but remind students these are options that are being discussed in Congress that everyone needs to understand. Have students investigate different types of formulas for finance, including the one for determining an interest rate. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. 5

6 For the teacher: Overhead projector or blackboard, appropriate technology For the student: Calculator or computer software application Vocabulary: Annuity Compounding Future Value Present Value Simple interest Technology Connections: Allow students to use recursion in spreadsheets (and writing formulas with subscripted variables). Also, programming on a calculator or in Java or any language will emphasize the role of recursion). Calculators and computer software applications (either spreadsheets or business specific software) have business functions as part of their standard menus. Review the instructions for the appropriate application pertaining to this lesson. Use technology tools under most circumstances. There are specific finance calculators available, as well as finance functions on many graphing calculators. Students may explore these, as well. Attachments: Attachment A, Pre-Assessment Attachment B, Pre-Assessment Answer Key Attachment C, Investigating Investment Formulas Post-Assessment Attachment D, Investigating Investment Formulas Post-Assessment Answer Key Attachment E, Defining Variables Attachment F, Deriving the Formula Attachment G, Deriving the Formula Answer Key 6

7 Attachment A Pre-Assessment Name Date Compute the following. 1. Find 4% of \$700 is invested for one year at a simple interest rate of 4.5%. How much is the investment now worth? ( )= ( ) = ( 1.05) 8 6. \$5000 times 4% 7. How much is in an account if \$1000 is invested for one year at an annual rate of 3.7% 8. How much is in an account if \$5000 is invested for 5 years at 4% compounded: Annually? Monthly? Daily? 7

8 Attachment B Pre-Assessment Answer Key 1. Find 4% of 1500 = = ( 1.045)= ( )= ( ) = ( 1.05) 8 = \$5000 times 4% = \$1000 invested for one year at 3.7% 1000( 1.037)= 1444 or 1000( )= \$5000 invested for 5 years at 4% compounded annually: 5000( 1.04) 5 \$30, or 5000( ) 5 \$30, monthly: daily: (1 5) (365 5) = \$ = \$

9 Attachment C Investigating Investment Formulas Post-Assessment Name Date Directions: List the following Future Annuity Values from greatest to least. Round all answers to the nearest dollar. 1. Starting today, contribute \$5,000 a year for 16 years, earning 7% a year. Calculate the value immediately after the 16 th contribution.. Starting today, contribute \$500 a year for 35 years, earning 10% a year. Calculate the future value for the full 35 years. 3. Starting a year from now, contribute \$,800 a year for 5 years, earning 5% a year. Calculate the future value for the full 5 years after the 5 th contribution. 4. Starting today, contribute \$1,000 a year for eight years, earning 10% a year. Calculate the future value immediately after the eighth contribution. 5. Starting today, contribute \$1,000 a year for eight years, earning 8% a year. Calculate the future value for the full eight years. 6. Starting a year from now, contribute \$6,500 a year for 0 years, earning 1% a year. Calculate the future value for the full 0 years after the 0 th contribution. 7. Starting today, contribute \$,500 a year for 30 years, earning 4% a year. Calculate the future value immediately after the 30 th contribution. 8. Starting today, contribute \$4,500 a year for 18 years, earning 6% a year. Calculate the future value for the full 18 years. 9

10 Attachment D Investigating Investment Formulas Post-Assessment Answer Key 1. Starting today, contribute \$5,000 a year for 16 years, earning 7% a year. Calculate the value immediately after the 16 th contribution. FV = (( ) / 0.07) * \$5,000 = \$139,440. Starting today, contribute \$500 a year for 35 years, earning 10% a year. Calculate the future value for the full 35 years. FV = (( ) / 0.10) * \$500 -\$500= \$149, Starting a year from now, contribute \$,800 a year for 5 years, earning 5% a year. Calculate the future value for the full 5 years (after the 5 th contribution). FV = (( ) / 0.05) * \$,800 = \$133, Starting today, contribute \$1,000 a year for eight years, earning 10% a year. Calculate the future value immediately after the eighth contribution. FV = (( ) / 0.10) * \$1,000 = \$137,31 5. Starting today, contribute \$1,000 a year for eight years, earning 8% a year. Calculate the future value for the full eight years. FV = ((( ) / 0.08) * \$1, ) = \$137, Starting a year from now, contribute \$6,500 a year for 0 years, earning 1% a year. Calculate the future value for the full 0 years (after the 0 th contribution). FV = (( ) / 0.01) * \$6,500 = \$143,14 7. Starting today, contribute \$,500 a year for 30 years, earning 4% a year. Calculate the future value immediately after the 30 th contribution. FV = (( ) / 0.04) * \$,500 = \$140,1 8. Starting today, contribute \$4,500 a year for 18 years, earning 6% a year. Calculate the future value for the full 18 years. FV = (( ) / 0.06) * \$4,500 - \$4,500 = \$147,40 From greatest to least:, 8, 6, 7, 1, 5, 4, and 3 10

11 Defining the variables: N = Total number of payments n = payments that have been made Investigating Investment Formulas Using Recursion Attachment E Defining Variables C/Y = compounding periods per year P/Y = payments per year C/Y and P/Y are generally the same I% = the total interest rate I r = interest rate per compounding period C/Y PV = Present Value in a loan problem, the principal. FV = Future Value bal n = balance after n payment These variables are based on common variable names from financial calculators. Other names may be used to reflect a text being used, etc. 11

12 Attachment F Deriving the Formula Name Date Bob purchased a new car for \$1000. He has a five-year loan at an annual percentage rate of 5.%. He will make monthly payments of \$ What is the interest rate per month?. What is bal 0, that is, the balance at time zero? 3. What is the balance after one month, or bal 1? 4. What is the balance after two months, or bal? Rewrite your steps using the variables defined in class. 5. What is the balance after three months, or bal 3? Rewrite your steps using the variables defined in class. 6. Generalize the expression for bal n. 1

13 Attachment G Deriving the Formula Answer Key 1. What is the interest rate per month? r =.05 1 = What is bal 0, that is, the balance at time zero? bal 0 = \$ What is the balance after one month, or bal 1? bal 1 = = = bal 1 + bal 1 () r PMT = bal 1 ( 1+ r) PMT 4. What is the balance after two months, or bal? bal = = =bal 1 + bal = bal PMT ( 1+r) -PMT( 1+r+1) =

14 Attachment G (Continued) Deriving the Formula Answer Key 5. What is the balance after three months, or bal 3? bal 3 = = =bal + bal 7.56 = bal = bal = bal r 3 3 ( ) 3 -PMT 1+r = PMT ( ) +1+r ( )

15 Attachment G (Continued) Deriving the Formula Answer Key 6. Generalize the expression for bal n. bal n (1 + r) n PMT (1 + r) n 1 + (1 + r) n (1 + r) n PMT[ sum of a geometric series] a1 rn Sum of a geometric series with n terms = S n =, where a is the first term and r is the common ratio. ( ) 1 (1 + r) 11 (1 + r)n So, S n = Therefore, = 1 (1 + r)n r = (1 + r)n 1 r ( ) 1 r bal n (1 + r) n PMT (1 + r) n 1 + (1 + r) n (1 + r) n PMT (1 + r)n 1 r (1 + r) n PMT r (1 + r) n 1 15

### Counting Money and Making Change Grade Two

Ohio Standards Connection Number, Number Sense and Operations Benchmark D Determine the value of a collection of coins and dollar bills. Indicator 4 Represent and write the value of money using the sign

### Comparing Sets of Data Grade Eight

Ohio Standards Connection: Data Analysis and Probability Benchmark C Compare the characteristics of the mean, median, and mode for a given set of data, and explain which measure of center best represents

### Time-Value-of-Money and Amortization Worksheets

2 Time-Value-of-Money and Amortization Worksheets The Time-Value-of-Money and Amortization worksheets are useful in applications where the cash flows are equal, evenly spaced, and either all inflows or

### 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

### 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

### Financial Literacy in Grade 11 Mathematics Understanding Annuities

Grade 11 Mathematics Functions (MCR3U) Connections to Financial Literacy Students are building their understanding of financial literacy by solving problems related to annuities. Students set up a hypothetical

### 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,

### first 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

### 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?

Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using

### 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

### Activity 3.1 Annuities & Installment Payments

Activity 3.1 Annuities & Installment Payments A Tale of Twins Amy and Amanda are identical twins at least in their external appearance. They have very different investment plans to provide for their retirement.

### Models for Dividing Fractions Grade Six

Ohio Standards Connection Number, Number Sense and Operations Benchmark H Use and analyze the steps in standard and nonstandard algorithms for computing with fractions, decimals and integers. Indicator

### 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

### Writing Simple Stories Grade One

Ohio Standards Connections Writing Applications Benchmark A Compose writings that convey a clear message and include well-chosen details. Indicator 1 Write simple stories with a beginning, middle and end

### Chapter 3 Mathematics of Finance

Chapter 3 Mathematics of Finance Section 3 Future Value of an Annuity; Sinking Funds Learning Objectives for Section 3.3 Future Value of an Annuity; Sinking Funds The student will be able to compute the

Ohio Standards Connection Geometry and Spatial Sense Benchmark E Use proportions to express relationships among corresponding parts of similar figures. Indicator 1 Use proportional reasoning to describe

Ohio Standards Connection Geometry and Spatial Sense Benchmark I Describe, identify and model reflections, rotations and translations, using physical materials. Indicator 7 Identify, describe and use reflections

Ohio Standards Connection Number, Number Sense and Operations Benchmark C Represent commonly used fractions and mixed numbers using words and physical models. Indicator 5 Represent fractions and mixed

### 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

### 10. Time Value of Money 2: Inflation, Real Returns, Annuities, and Amortized Loans

10. Time Value of Money 2: Inflation, Real Returns, Annuities, and Amortized Loans Introduction This chapter continues the discussion on the time value of money. In this chapter, you will learn how inflation

### 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

Ohio Standards Connection: Government Benchmark A Identify the responsibilities of the branches of the U.S. government and explain why they are necessary. Indicator 2 Explain the structure of local governments

### Main TVM functions of a BAII Plus Financial Calculator

Main TVM functions of a BAII Plus Financial Calculator The BAII Plus calculator can be used to perform calculations for problems involving compound interest and different types of annuities. (Note: there

### This lesson plan is from the Council for Economic Education's publication: Mathematics and Economics: Connections for Life 9-12

This lesson plan is from the Council for Economic Education's publication: Mathematics and Economics: Connections for Life 9-12 To purchase Mathematics and Economics: Connections for Life 9-12, visit:

### Chapter 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

### Drawing Lines of Symmetry Grade Three

Ohio Standards Connection Geometry and Spatial Sense Benchmark H Identify and describe line and rotational symmetry in two-dimensional shapes and designs. Indicator 4 Draw lines of symmetry to verify symmetrical

### Some Mathematics of Investing in Rental Property. Floyd Vest

Some Mathematics of Investing in Rental Property Floyd Vest Example 1. In our example, we will use some of the assumptions from Luttman, Frederick W. (1983) Selected Applications of Mathematics of Finance

### The 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

### Scarcity and Choices Grade One

Ohio Standards Connection: Economics Benchmark A Explain how the scarcity of resources requires people to make choices to satisfy their wants. Indicator 1 Explain that wants are unlimited and resources

### 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

### 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

### 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

### Bar Graphs with Intervals Grade Three

Bar Graphs with Intervals Grade Three Ohio Standards Connection Data Analysis and Probability Benchmark D Read, interpret and construct graphs in which icons represent more than a single unit or intervals

### Chapter F: Finance. Section F.1-F.4

Chapter F: Finance Section F.1-F.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

Ohio Standards Connection: Number, Number Sense and Operations Standard Benchmark B Use models and pictures to relate concepts of ratio, proportion and percent. Indicator 1 Use models and visual representation

### 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

### 2. How would (a) a decrease in the interest rate or (b) an increase in the holding period of a deposit affect its future value? Why?

CHAPTER 3 CONCEPT REVIEW QUESTIONS 1. Will a deposit made into an account paying compound interest (assuming compounding occurs once per year) yield a higher future value after one period than an equal-sized

### TIME VALUE OF MONEY #6: TREASURY BOND. Professor Peter Harris Mathematics by Dr. Sharon Petrushka. Introduction

TIME VALUE OF MONEY #6: TREASURY BOND Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction This problem assumes that you have mastered problems 1-5, which are prerequisites. In this

### 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

### 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

Ohio Standards Connection Data Analysis and Probability Benchmark C Represent data using objects, picture graphs and bar graphs. Indicators 3. Read and construct simple timelines to sequence events. 5.

### Review Page 468 #1,3,5,7,9,10

MAP4C Financial Student Checklist Topic/Goal Task Prerequisite Skills Simple & Compound Interest Video Lesson Part Video Lesson Part Worksheet (pages) Present Value Goal: I will use the present value formula

### Chapter 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

### International Financial Strategies Time Value of Money

International Financial Strategies 1 Future Value and Compounding Future value = cash value of the investment at some point in the future Investing for single period: FV. Future Value PV. Present Value

### Casio 9860 Self-Guided Instructions TVM Mode

Using TVM: Casio 9860 Self-Guided Instructions TVM Mode Instructions Screenshots TVM stands for 'Time, Value, Money'. TVM is the Financial Mode on the calculator. However, Financial Mathematics questions

### 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?

### Lesson 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

### The values in the TVM Solver are quantities involved in compound interest and annuities.

Texas Instruments Graphing Calculators have a built in app that may be used to compute quantities involved in compound interest, annuities, and amortization. For the examples below, we ll utilize the screens

### Lesson 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

### 1. If you wish to accumulate \$140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%?

Chapter 2 - Sample Problems 1. If you wish to accumulate \$140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%? 2. What will \$247,000 grow to be in

### 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

Ohio Standards Connection Geometry and Spatial Sense Benchmark C Specify locations and plot ordered pairs on a coordinate plane. Indicator 6 Extend understanding of coordinate system to include points

### Sample 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

### 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

### The explanations below will make it easier for you to use the calculator. The ON/OFF key is used to turn the calculator on and off.

USER GUIDE Texas Instrument BA II Plus Calculator April 2007 GENERAL INFORMATION The Texas Instrument BA II Plus financial calculator was designed to support the many possible applications in the areas

### TIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY

TIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction In this assignment we will discuss how to calculate the Present Value

### Teaching Pre-Algebra in PowerPoint

Key Vocabulary: Numerator, Denominator, Ratio Title Key Skills: Convert Fractions to Decimals Long Division Convert Decimals to Percents Rounding Percents Slide #1: Start the lesson in Presentation Mode

### 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

### 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

### Chapter 5 & 6 Financial Calculator and Examples

Chapter 5 & 6 Financial Calculator and Examples Konan Chan Financial Management, Spring 2016 Five Factors in TVM Present value: PV Future value: FV Discount rate: r Payment: PMT Number of periods: N Get

Ohio Standards Connection Data Analysis and Probability Benchmark F Determine and use the range, mean, median and mode to analyze and compare data, and explain what each indicates about the data. Indicator

### THE VALUE OF MONEY PROBLEM #3: ANNUITY. Professor Peter Harris Mathematics by Dr. Sharon Petrushka. Introduction

THE VALUE OF MONEY PROBLEM #3: ANNUITY Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction Earlier, we explained how to calculate the future value of a single sum placed on deposit

### Time 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

### 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

### Section 5.1 - Compound Interest

Section 5.1 - Compound Interest Simple Interest Formulas If I denotes the interest on a principal P (in dollars) at an interest rate of r (as a decimal) per year for t years, then we have: Interest: Accumulated

### Activity 5 Calculating a Car Loan

Teaching Notes/Lesson Plan Objective Within this lesson, the participant will be able to use the Casio calculator to determine such information as monthly payment, interest rate, and total cost of the

### Module 5: Interest concepts of future and present value

file:///f /Courses/2010-11/CGA/FA2/06course/m05intro.htm Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present

### Using Percents in the Real-World Grade Six

Ohio Standards Connection Number, Number Sense and Operations Benchmark I Use a variety of strategies including proportional reasoning to estimate, compute, solve and explain solutions to problems involving

### Chapter 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

### DISCOUNTED 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

### Fin 5413 CHAPTER FOUR

Slide 1 Interest Due Slide 2 Fin 5413 CHAPTER FOUR FIXED RATE MORTGAGE LOANS Interest Due is the mirror image of interest earned In previous finance course you learned that interest earned is: Interest

### KENT FAMILY FINANCES

FACTS KENT FAMILY FINANCES Ken and Kendra Kent have been married twelve years and have twin 4-year-old sons. Kendra earns \$78,000 as a Walmart assistant manager and Ken is a stay-at-home dad. They give

### REVIEW MATERIALS FOR REAL ESTATE ANALYSIS

REVIEW MATERIALS FOR REAL ESTATE ANALYSIS 1997, Roy T. Black REAE 5311, Fall 2005 University of Texas at Arlington J. Andrew Hansz, Ph.D., CFA CONTENTS ITEM ANNUAL COMPOUND INTEREST TABLES AT 10% MATERIALS

### Appendix 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

### CHAPTER 4. The Time Value of Money. Chapter Synopsis

CHAPTER 4 The Time Value of Money Chapter Synopsis Many financial problems require the valuation of cash flows occurring at different times. However, money received in the future is worth less than money

### 2016 Wiley. Study Session 2: Quantitative Methods Basic Concepts

2016 Wiley Study Session 2: Quantitative Methods Basic Concepts Reading 5: The Time Value of Money LESSO 1: ITRODUCTIO, ITEREST RATES, FUTURE VALUE, AD PREST VALUE The Financial Calculator It is very important

### Chapter 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

### Systems of Transportation and Communication Grade Three

1 Ohio Standards Connection: Geography Benchmark D Analyze ways that transportation and communication relate to patterns of settlement and economic activity. Indicator 8 Identify systems of transportation

### Time Value of Money Practice Questions Irfanullah.co

1. You are trying to estimate the required rate of return for a particular investment. Which of the following premiums are you least likely to consider? A. Inflation premium B. Maturity premium C. Nominal

### Problem Set: Annuities and Perpetuities (Solutions Below)

Problem Set: Annuities and Perpetuities (Solutions Below) 1. If you plan to save \$300 annually for 10 years and the discount rate is 15%, what is the future value? 2. If you want to buy a boat in 6 years

### \$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

### 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

### Regular Annuities: Determining Present Value

8.6 Regular Annuities: Determining Present Value GOAL Find the present value when payments or deposits are made at regular intervals. LEARN ABOUT the Math Harry has money in an account that pays 9%/a compounded

### Decision Making in Finance: Future Value of an Investment VI.A Student Activity Sheet 1: You Have to Get Money to Make Money

Decision Making in Finance: Future Value of an Investment VI.A Student Activity Sheet 1: You Have to Get Money to Make Money 1. Kafi is considering three job offers in educational publishing. One is a

### LO.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

### Finance 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

### Bank: The bank's deposit pays 8 % per year with annual compounding. Bond: The price of the bond is \$75. You will receive \$100 five years later.

ü 4.4 lternative Discounted Cash Flow Decision Rules ü Three Decision Rules (1) Net Present Value (2) Future Value (3) Internal Rate of Return, IRR ü (3) Internal Rate of Return, IRR Internal Rate of Return

### Chapter 6 Contents. Principles Used in Chapter 6 Principle 1: Money Has a Time Value.

Chapter 6 The Time Value of Money: Annuities and Other Topics Chapter 6 Contents Learning Objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate present and future values

### 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

### Change Number Stories Objective To guide children as they use change diagrams to help solve change number stories.

Number Stories Objective To guide children as they use change diagrams to help solve change number stories. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game

### Reducing balance loans

Reducing balance loans 5 VCEcoverage Area of study Units 3 & 4 Business related mathematics In this chapter 5A Loan schedules 5B The annuities formula 5C Number of repayments 5D Effects of changing the

### 300 Chapter 5 Finance

300 Chapter 5 Finance 17. House Mortgage A couple wish to purchase a house for \$200,000 with a down payment of \$40,000. They can amortize the balance either at 8% for 20 years or at 9% for 25 years. Which

### 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

### A = P [ (1 + r/n) nt 1 ] (r/n)

April 23 8.4 Annuities, Stocks and Bonds ---- Systematic Savings Annuity = sequence of equal payments made at equal time periods i.e. depositing \$1000 at the end of every year into an IRA Value of an annuity

### Lesson Description. Texas Essential Knowledge and Skills (Target standards) Skills (Prerequisite standards) National Standards (Supporting standards)

Lesson Description The students are presented with real life situations in which young people have to make important decisions about their future. Students use an online tool to examine how the cost of