Enhanced Instructional Transition Guide

Transcription

1 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Unit 13: Financial Planning (5 days) Possible Lesson 01 (5 days) POSSIBLE LESSON 01 (5 days) This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing with districtapproved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and districts may modify the time frame to meet students needs. To better understand how your district is implementing CSCOPE lessons, please contact your child s teacher. (For your convenience, please find linked the TEA Commissioner s List of State Board of Education Approved Instructional Resources and Midcycle State Adopted Instructional Materials.) Lesson Synopsis: Students investigate, analyze, and compare investment options, including savings, stocks, bonds, mutual funds, annuities, and other retirement plans. TEKS: The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas law. Any standard that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit. The TEKS are available on the Texas Education Agency website at M.1 Mathematical Models with Applications/Knowledge and Skills. The student uses a variety of strategies and approaches to solve both routine and non-routine problems. The student is expected to: M.1A M.1B M.1C Compare and analyze various methods for solving a real-life problem. Use multiple approaches (algebraic, graphical, and geometric methods) to solve problems from a variety of disciplines. Select a method to solve a problem, defend the method, and justify the reasonableness of the results. M.7 Mathematical Models with Applications/Knowledge and Skills. The student uses algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning. The student is expected to: M.7C Investigate and compare investment options including stocks, bonds, annuities, and retirement plans. Performance Indicator(s): page 1 of 32

2 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days High School Mathematics Mathematical Models with Applications Unit 13 PI 01 Analyze and compare various investment options, including savings, annuities, stocks, bonds, and mutual funds, in problem situations such as the following: Elvira is planning on putting $5,000 into a savings investment for five years. The bank has offered her three different options for savings investment: simple interest at 6%, compounded quarterly at 6%, and compounded daily at 6%. Create tables and graphs to compare the options. What option would give Elvira a higher yield? Explain your reasoning. Christine will be retiring in one year and is preparing by analyzing her retirement plan. Although her pension and Social Security will be enough for her basic needs, Christine would like an additional $10,000 per year with which she can travel. She wants to invest an amount into an annuity now that will pay her $10,000 per year for the next 15 years. If the annuity is compounded annually at an annual percent rate of 6.5%, how much will Christine need to invest up front? How does this compare to the amount she will receive after 15 years? The following is an example of a section of the stock market table: What was the price of a share of DOW stock on the previous day? What was the percent increase between the high and low of DOW stock over the past 52-week period? If you owned 50 shares of DOW stock, how much will you receive in dividends for the quarter? Suppose you purchased 150 shares of DOW at its lowest price in the last 52-week period and sold it on this day. You paid $12.00 for an online fee for the each transaction. Determine if you made or lost money. Show calculations to back up your conclusion. Julie has $20,000 she wants to invest in savings and mutual funds. The savings account will pay an APR of 4%. The mutual fund will pay an APR of 6%. To remain in the same tax bracket, Julie does not want to earn over $950 in interest for the year. What amount should she invest in savings and what amount should she invest in the mutual fund? Create a graphic organizer for each problem situation, including appropriate tables, mathematical models, and calculations. Write a summary explaining why it is important to have a diversity of investment options in a retirement plan. Standard(s): M.1A, M.1B, M.1C, M.7C ELPS ELPS.c.1E, ELPS.c.5G Key Understanding(s): Future financial security and retirement can depend on making the most of investment options (e.g., savings, stocks, bonds, annuities, mutual funds, and other retirement options) during the years of employment. page 2 of 32

3 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Vocabulary of Instruction: 401K annuity annuity due bond compounded interest dividend mutual fund National Association of Securities Dealer Automated Quotations (NASDAQ) New York Stock Exchange (NYSE) ordinary annuity present value of an annuity Roth IRA share of stock simple interest social security stock Materials List: chart markers (1 set per 2 students) chart paper (1 sheet per 2 students) graphing calculator (1 per student) graphing calculator with display (1 per teacher) Attachments: All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website. Stowing Away Savings Annuities, Retirement Plans, and Life Insurance KEY Annuities, Retirement Plans, and Life Insurance Stocks, Bonds, and Mutual Funds KEY page 3 of 32

4 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Stocks, Bonds, and Mutual Funds Feeling Financially Secure Evaluating Financial Planning KEY Evaluating Financial Planning PI GETTING READY FOR INSTRUCTION Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using the Content Creator in the Tools Tab. All originally authored lessons can be saved in the My CSCOPE Tab within the My Content area. Suggested Day Suggested Instructional Procedures Notes for Teacher 1 Topics: ATTACHMENTS Savings options Interest rates Compounding periods Engage 1 Students analyze and compare different savings options by interest rate and compounding periods. Instructional Procedures: 1. Place students in pairs. Distribute handout: Stowing Away Savings, 1 sheet of chart paper, and 1 set of chart markers to each pair of students. Display teacher resource: Stowing Away Savings, and facilitate a discussion of the activity. Teacher Resource: Stowing Away Savings (1 per teacher) Handout: Stowing Away Savings (1 per 2 students) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) chart paper (1 sheet per 2 students) page 4 of 32

5 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Suggested Day Suggested Instructional Procedures 2. Assign each pair a different annual percent interest rate (APR), such as 2.75%, 3.5%, 4%, 5.25%, 6.35%, etc. 3. Instruct students to work with their partner to complete procedures on handout: Stowing Away Savings, using their assigned interest rate and create a display of their results. When groups have completed the displays, instruct them to post them around the room. 4. Facilitate a class discussion of student results, comparing and contrasting effects of varied interest rates on returns. Ask: Which interest rate gave the best return on the investment of $20,000? (The higher interest rate, the better the returns on the investment.) How did the return on simple interest compare to the return on compounded interest? (Compounded interest always had better returns, but the longer the amount of time the greater the difference.) Which method of paying interest gave the best return on the investment of $20,000? (Compounding daily gave the best return on the investment.) Notes for Teacher chart markers (1 set per 2 students) TEACHER NOTE Answer keys are not given for the activity since the assigned interest rates will vary. However, higher interest rates will yield better profits and more compounding periods will yield greater profits. The most impactful should be the higher interest rate, as long as it is a large enough percent difference. 2 Topics: Annuities Retirement plans Life insurance Explore/Explain 1 ATTACHMENTS Teacher Resource: Annuities, Retirement Plans, and Life Insurance KEY (1 per teacher) Teacher Resource: Annuities, Retirement Plans, and Life page 5 of 32

6 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Suggested Day Suggested Instructional Procedures Notes for Teacher Students analyze and compare different types of annuities and retirement plans. Instructional Procedures: 1. Place students in pairs. Distribute handout: Annuities, Retirement Plans, and Life Insurance to each student. Refer students to the top of page 1. Display teacher resource: Annuities, Retirement Plans, and Life Insurance, and facilitate a class discussion of the definition of annuity at the top of page Instruct students to work with their partner to fill in the tables on Sample problems 1 and 2 on handout: Annuities, Retirement Plans, and Life Insurance to investigate models of ordinary annuity and annuity due. Allow students time to complete the problems. Monitor to check for understanding. Using teacher resource: Annuities, Retirement Plans, and Life Insurance, facilitate a class discussion of student results. 3. Refer students to the top of page 2. Using teacher resource: Annuities, Retirement Plans, and Life Insurance, facilitate a discussion of the definitions and formulas at the top of page Instruct students to work with their partner to complete Samples 3 and 4 on handout: Annuities, Retirement Plans, and Life Insurance. Allow students time to complete the problems. Monitor and assess students to check for understanding. Using teacher resource: Annuities, Retirement Plans, and Life Insurance, facilitate a class discussion of student results. Insurance (1 per teacher) Handout: Annuities, Retirement Plans, and Life Insurance (1 per student) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE Handout: Annuities, Retirement Plans, and Life Insurance refers to the TVM Solver on the TI graphing calculator to calculate amortization. If you are using another type of calculator, refer to the calculator reference materials to determine how to use the finance capabilities of that calculator. 5. Refer students to the top of page 3. Using teacher resource: Annuities, Retirement Plans, and Life Insurance, facilitate a discussion of present value of annuities and life insurance at the top of page 3, modeling Sample problems 5 and 6. page 6 of 32

7 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Suggested Day Suggested Instructional Procedures Notes for Teacher 6. Instruct students to work with their partner to complete the Practice Problems on handout: Annuities, Retirement Plans, and Life Insurance. This may be completed as homework, if necessary. 3 Topics: Stocks Mutual funds Bonds Explore/Explain 2 Students analyze and compare optional forms of investment that carry more risk, such as stocks, mutual funds, and bonds. Instructional Procedures: 1. Facilitate a class discussion to debrief Practice Problems on handout: Annuities, Retirement Plans, and Life Insurance to check for understanding. 2. Place students in pairs. Distribute handout: Stocks, Bonds, and Mutual Funds to each student. Refer students to the top of page 1. Display teacher resource: Stocks, Bonds, and Mutual Funds, and facilitate a class discussion of stocks, modeling Sample problem 1. ATTACHMENTS Teacher Resource: Stocks, Bonds, and Mutual Funds KEY (1 per teacher) Teacher Resource: Stocks, Bonds, and Mutual Funds (1 per teacher) Handout: Stocks, Bonds, and Mutual Funds (1 per student) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) 3. Refer students to the bottom of page 2. Display teacher resource: Stocks, Bonds, and Mutual Funds, and facilitate a class discussion of mutual funds and bonds, modeling Sample problem Instruct students to complete the Practice Problems on handout: Stocks, Bonds, and page 7 of 32

8 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Suggested Day Suggested Instructional Procedures Notes for Teacher Mutual Funds. This may be completed as homework, if necessary. 4 Topics: Building a retirement plan Elaborate 1 Students invest an amount of money into at least three different options to help finance a retirement plan. Instructional Procedures: 1. Place students in pairs. Distribute handout: Feeling Financially Secure to each pair of students. Display teacher resource: Feeling Financially Secure, and facilitate a class discussion of the problem situation and procedures. 2. Instruct students to work with their partner to design a financial investment plan for the money, and write a written report on their calculations, results, and conclusions. ATTACHMENTS Teacher Resource: Feeling Financially Secure (1 per teacher) Handout: Feeling Financially Secure (1 per 2 students) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) 5 Evaluate 1 Instructional Procedures: 1. Assess student understanding of related concepts and processes by using the Performance Indicator(s) aligned to this lesson. Performance Indicator(s): High School Mathematics Mathematical Models with Applications Unit 13 PI 01 Analyze and compare various investment options, including savings, annuities, stocks, bonds, and mutual ATTACHMENTS Teacher Resource (optional): Evaluating Financial Planning KEY (1 per teacher) Handout (optional): Evaluating Financial Planning PI (1 per student) MATERIALS page 8 of 32

9 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Suggested Day Suggested Instructional Procedures Notes for Teacher funds, in problem situations such as the following: Elvira is planning on putting $5,000 into a savings investment for five years. The bank has offered her three different options for savings investment: simple interest at 6%, compounded quarterly at 6%, and compounded daily at 6%. Create tables and graphs to compare the options. What option would give Elvira a higher yield? Explain your reasoning. graphing calculator (1 per student) TEACHER NOTE As an optional assessment tool, use handout (optional): Evaluating Financial Planning PI. Christine will be retiring in one year and is preparing by analyzing her retirement plan. Although her pension and Social Security will be enough for her basic needs, Christine would like an additional $10,000 per year with which she can travel. She wants to invest an amount into an annuity now that will pay her $10,000 per year for the next 15 years. If the annuity is compounded annually at an annual percent rate of 6.5%, how much will Christine need to invest up front? How does this compare to the amount she will receive after 15 years? The following is an example of a section of the stock market table: What was the price of a share of DOW stock on the previous day? What was the percent increase between the high and low of DOW stock over the past 52-week period? If you owned 50 shares of DOW stock, how much will you receive in dividends for the quarter? Suppose you purchased 150 shares of DOW at its lowest price in the last 52-week period and sold it on this day. You paid $12.00 for an online fee for the each transaction. Determine if you made or lost money. Show calculations to back up your conclusion. page 9 of 32

10 Enhanced Instructional Transition Guide High School Courses/Mathematical Models with Applications Unit 13: Suggested Duration: 5 days Suggested Day Suggested Instructional Procedures Notes for Teacher Julie has $20,000 she wants to invest in savings and mutual funds. The savings account will pay an APR of 4%. The mutual fund will pay an APR of 6%. To remain in the same tax bracket, Julie does not want to earn over $950 in interest for the year. What amount should she invest in savings and what amount should she invest in the mutual fund? Create a graphic organizer for each problem situation, including appropriate tables, mathematical models, and calculations. Write a summary explaining why it is important to have a diversity of investment options in a retirement plan. Standard(s): M.1A, M.1B, M.1C, M.7C ELPS ELPS.c.1E, ELPS.c.5G 04/25/13 page 10 of 32

11 Stowing Away Savings Mathematical Models w ith Applications How savvy are you about what savings options produce the most financial security? If you get a windfall of $20,000, which of the following savings options would bring the greatest return? Simple interest Compounded quarterly Compounded monthly Compounded daily Show the relationship between future value and number of years using tables and graphs. Write a summary of your conclusions. Make a display of your results and conclusions using the model below. Variables Formulas Calculations Tables Graphs Conclusions 2012, TESCCC 12/06/12 page 1 of 1

12 Mathematical Models with Applications Annuities, Retirement Plans, and Life Insurance KEY Good financial planning involves looking at your goals for the future and determining investments that will help you meet those goals. Two of the more secure methods are savings investments and annuities. An annuity is an account in which a fixed amount of money is paid in over a specified amount of time at a set interest rate. Sample 1 A deposit of $500 is made into an annuity account at the beginning of every six-month period for three years. The account earns 6% compounded semi-annually. a. How does this differ from the savings accounts analyzed previously? In the savings account, one deposit was added in at the beginning which accrued interest. In the annuity account, money is added in periodically and the account also accrues interest. b. Fill in the table below to model the growth of the annuity account over the three years. Interest Amount Deposited Beginning Earned in Earning Amount Balance Compounding Interest Period Number of Compounding Periods Ending Balance 1 (6 months) $500 $ $ $15.00 $ (12 months) $500 $ $ $30.45 $ (18 months) $500 $ $ $46.36 $ (24 months) $500 $ $ $62.75 $ (30 months) $500 $ $ $79.64 $ (36 months) $500 $ $ $97.03 $ Sample 2 A deposit of $500 is made into an annuity account at the end of every six-month period for three years. The account earns 6% compounded semi-annually. a. Fill in the table below to model the growth of the annuity account over the three years. Number of Compounding Periods Deposited Amount Beginning Balance Amount Earning Interest Interest Earned in Compounding Period Ending Balance 1 (6 months) $500 $0 $0 $0 $500 2 (12 months) $500 $500 $500 $15.00 $ (18 months) $500 $ $ $30.45 $ (24 months) $500 $ $ $46.36 $ (30 months) $500 $ $ $62.75 $ (36 months) $500 $ $ $79.64 $ b. How did changing from adding the deposit at the beginning to the end affect the ending balance? Answers will vary. Sample: The first annuity payment made at the beginning allows for one more period of interest accrued ($97.03). 2012, TESCCC 04/25/13 page 1 of 4

13 Mathematical Models with Applications Annuities, Retirement Plans, and Life Insurance KEY An ordinary annuity is one in which the payment or deposit is made at the end of each period. An annuity due is one in which the payment or deposit is made at the beginning of each period. Formulas can simplify calculations and be used to find the future value of either type of annuity. Future Value of an Ordinary Annuity Future Value of an Annuity Due FV n 1r 1 P r FV P 1r n 1r 1 r P = annuity payment or deposit r = interest rate per period in decimal form n = number of compounding periods Sample 3 Annuity payments of $200 are made monthly for ten years at 8% compounded monthly. What are the predicted values at the end of ten years if using an ordinary annuity? What are the predicted values at the end of ten years if using an annuity due? How do they compare? Future Value of an Ordinary Annuity Future Value of an Annuity Due FV FV FV = $36, FV = $36, The annuity due method would result in a higher yield, because it would have one additional interest payment. Sample 4 How could the TVM Solver be used to calculate future values of annuities? Check your answers to Sample 3 and see if your conclusions are accurate. N = 10*12 = 120 I% = 8 PV = 0 (Since the annuity begins with no money until the first deposit is made) PMT = -200 FV = (Amount to be solved for) P/Y = 12 C/Y = 12 PMT:END (use when calculating ordinary annuity), PMT:BEGIN (use when calculating annuity due) Answers do check. 2012, TESCCC 04/25/13 page 2 of 4

14 Mathematical Models with Applications Annuities, Retirement Plans, and Life Insurance KEY Up to this point, the annuities investigated have started with a zero balance and increased in value. In some cases, annuities will begin with a present value and will decrease due to fixed payments that are made at specified time intervals while the account still collects a set interest. Sample 5 Pat will be retiring from his job at the end of the year. He estimates that in addition to his social security and 401K at work, he will need $15,000 per year. He received an inheritance from his grandfather and would like to determine how much money he would have to put in an ordinary annuity at 5% compounded annually to receive payments of $15,000 per year for the next 20 years. Calculate the present value that he would have to deposit. How does it compare to how much money he will receive over the 20 years? N = 20 I% = 5 PV = (Amount to be solved for) = $186, PMT = FV = 0 (At the end of 20 years, no money will be left in the account.) P/Y = 1 C/Y = 1 PMT:END (use when calculating ordinary annuity) He will have to put $186, into the annuity at the present. Over the next twenty years he will receive $300,000. That is approximately 160% of the original investment. If Pat has the money to invest, it would be well worth it. In order to provide for the family in the event of death (terminal or critical illness), many people purchase life insurance. Life insurance is a contract between the insurer and the policy owner whereby a benefit is paid to the designated beneficiary of the insured. Different types of life insurance are available including term, whole life, universal life, and variable life. Sample 6 A 35-year old woman wants to purchase a 20-year life insurance policy with a face value of $150,000. The annual rate is $6.25 per $1,000 of face value. What would be the annual premium? (150000/1000)6.25 = $ annual premium If premiums are paid in intervals less than one year, a percentage of the annual premium is paid for each specified interval. Semiannually 54% Quarterly 28% Monthly 10% What would be the premium for each semiannual payment? Quarterly payment? Monthly payment? How do these compare to the cost of one annual payment? Semiannual premium (937.50)(.54) = $ Over annual by 2(506.25) = $75 Quarterly premium (937.50)(.28) = $ Over annual by 4(262.50) = $ Monthly premium (937.50)(.10) = $93.75 Over annual by 12(93.75) = $ , TESCCC 04/25/13 page 3 of 4

15 Mathematical Models with Applications Annuities, Retirement Plans, and Life Insurance KEY Practice Problems 1. Paul s company will match him dollar-for-dollar up to 6% of his monthly salary to invest into an annuity. Paul makes $4,000 per month and invests the full 6% of his salary plus the company s matching fund. In 8 years, what will be the future value of an annuity due, if the annuity is at 4.5% annual interest compounded monthly? $55, When Jackson was born, his parents established a college fund annuity in which they deposit $150 at the end of each month. The annual interest is 5.5% compounded monthly. How much will be in his college account when he turns eighteen? $55, Marcia invests $2,500 into an IRA at the beginning of each year as part of her retirement plan. If she plans to retire in 20 years, how much will be in the IRA, if it earns 7.5% compounded annually? $116, The state must keep sufficient funds to cover all lottery winners. If a winner is to receive $50,000 per year for 20 years from the beginning day they win, how much money must the state have on deposit at 6% compounded annually in order to make the payments? $607, Tomas purchased a $500,000, 20-year life insurance policy. The rate is $0.825 per $100 of face value. Tomas has decided to pay monthly at 8.75% of the annual rate. What will be his monthly payments? How much could he have saved by paying one annual premium? Annual premium: (500000/100).825 = $4125 Monthly premium: (4125)(0.0875) = $ Saved by paying one annual premium: 12(360.94) 4125 = $ , TESCCC 04/25/13 page 4 of 4

16 Annuities, Retirement Plans, and Life Insurance Mathematical Models with Applications Good financial planning involves looking at your goals for the future and determining investments that will help you meet those goals. Two of the more secure methods are savings investments and annuities. An is an account in which a fixed amount of money is paid in over a specified amount of time at a set interest rate. Sample 1 A deposit of $500 is made into an annuity account at the beginning of every six-month period for three years. The account earns 6% compounded semi-annually. a. How does this differ from the savings accounts analyzed previously? b. Fill in the table below to model the growth of the annuity account over the three years. Number of Compounding Periods Deposited Amount Beginning Balance Amount Earning Interest Interest Earned in Compounding Period Ending Balance 1 (6 months) $500 $ $ $15.00 $ (12 months) $500 $ $ (18 months) $500 $ (24 months) 5 (30 months) 6 (36 months) Sample 2 A deposit of $500 is made into an annuity account at the end of every six-month period for three years. The account earns 6% compounded semi-annually. a. Fill in the table below to model the growth of the annuity account over the three years. Number of Compounding Periods Deposited Amount Beginning Balance Amount Earning Interest Interest Earned in Compounding Period Ending Balance 1 (6 months) $500 $0 $0 $0 $500 2 (12 months) $500 $500 $500 3 (18 months) $500 $ (24 months) 5 (30 months) 6 (36 months) b. How did changing from adding the deposit at the beginning to the end affect the ending balance? 2012, TESCCC 04/25/13 page 1 of 4

17 Annuities, Retirement Plans, and Life Insurance Mathematical Models with Applications An is one in which the payment or deposit is made at the end of each period. An is one in which the payment or deposit is made at the beginning of each period. Formulas can simplify calculations and be used to find the future value of either type of annuity. Future Value of an Ordinary Annuity Future Value of an Annuity Due FV n 1r 1 P r FV P 1r n 1r 1 r P = annuity payment or deposit r = interest rate per period in decimal form n = number of compounding periods Sample 3 Annuity payments of $200 are made monthly for ten years at 8% compounded monthly. What are the predicted values at the end of ten years if using an ordinary annuity? What are the predicted values at the end of ten years if using an annuity due? How do they compare? Sample 4 How could the TVM Solver be used to calculate future values of annuities? Check your answers to Sample 3 and see if your conclusions are accurate. 2012, TESCCC 04/25/13 page 2 of 4

18 Annuities, Retirement Plans, and Life Insurance Mathematical Models with Applications Up to this point the annuities investigated have started with a zero balance and increased in value. In some cases annuities will begin with a and will decrease due to fixed payments that are made at specified time intervals while the account still collects a set interest. Sample 5 Pat will be retiring from his job at the end of the year. He estimates that in addition to his social security and 401K at work, he will need $15,000 per year. He received an inheritance from his grandfather and would like to determine how much money he would have to put in an ordinary annuity at 5% compounded annually to receive payments of $15,000 per year for the next 20 years. Calculate the present value that he would have to deposit. How does it compare to how much money he will receive over the 20 years? In order to provide for the family in the event of death (terminal or critical illness), many people purchase life insurance. Life insurance is a contract between the insurer and the policy owner whereby a benefit is paid to the designated beneficiary of the insured. Different types of life insurance are available including term, whole life, universal life, and variable life. Sample 6 A 35-year old woman wants to purchase a 20-year term life insurance policy with a face value of $150,000. The annual rate is $7.75 per $1,000 of face value. What would be the annual premium? If premiums are paid in intervals less than one year, a percentage of the annual premium is paid for each specified interval. Semiannually 54% Quarterly 28% Monthly 10% What would be the premium for each semiannual payment? Quarterly payment? Monthly payment? How do these compare to the cost of one annual payment? 2012, TESCCC 04/25/13 page 3 of 4

19 Practice Problems Annuities, Retirement Plans, and Life Insurance Mathematical Models with Applications 1. Paul s company will match him dollar-for-dollar up to 6% of his monthly salary to invest into an annuity. Paul makes $4,000 per month and invests the full 6% of his salary plus the company s matching fund. In 8 years, what will be the future value of an annuity due, if the annuity is at 4.5% annual interest compounded monthly? 2. When Jackson was born, his parents established a college fund annuity in which they deposit $150 at the end of each month. The annual interest is 5.5% compounded monthly. How much will be in his college account when he turns eighteen? 3. Marcia invests $2,500 into an IRA at the beginning of each year as part of her retirement plan. If she plans to retire in 20 years, how much will be in the IRA, if it earns 7.5% compounded annually? 4. The state must keep sufficient funds to cover all lottery winners. If a winner is to receive $50,000 per year for 20 years from the beginning day they win, how much money must the state have on deposit at 6% compounded annually in order to make the payments? 5. Tomas purchased a $500,000, 20-year life insurance policy. The rate is $0.825 per $100 of face value. Tomas has decided to pay monthly at 8.75% of the annual rate. What will be his monthly payments? How much could he have saved by paying one annual premium? 2012, TESCCC 04/25/13 page 4 of 4

20 Stocks, Bonds, and Mutual Funds KEY Mathematical Models w ith Applications Although savings and annuities are considered safe investments, some of the riskier investments can offer better returns over a longer period of time. These investments include stocks, bonds, and mutual funds. Certain factors can impact the value of these investments such as the success of companies, the economic and political climate, interest rates, taxes, and unemployment. Stocks Companies that go public can sell ownership in their company in the form of stock. A share of stock is a unit of ownership in a particular company. Dividends are amounts sometimes paid on a quarterly basis to shareholders from profits made by the company over the previous period. Most stocks are traded on the New York Stock Exchange (NYSE) or the National Association of Securities Dealer Automated Quotations (NASDAQ). Stock transactions can be done by an individual, especially now with online access; these transactions many times have a flat fee for each transaction. Stock transactions can also be carried out by a stockbroker who usually charges a commission fee. Sample 1 52-WEEK HIGH LOW STOCK DIV YLD% PE VOL 100s CLOSE NET CHG T , ETE , DIS , Use the stock listings in the table above to answer the following questions: a. How many shares of Disney (DIS) stock were traded on this day? 15,226,100 shares were traded. b. What was the closing price of a share of Disney stock? The closing price on this day was $32.49 per share. c. Is the closing price of Disney stock higher or lower than the previous day? The closing price for this day is 17 higher per share. d. What was the price of a share of Disney stock on the previous day? = $32.32 was the price of a share on the previous day. e. What was the percent decrease between the high and low of Disney stock over the past 52-week period? ( )/36.79 x 100 = 28.51% decrease f. If you owned 25 shares of Disney stock, how much will you receive in dividends for the quarter? (.35)(25) = $8.75 in dividends for the quarter. 2012, TESCCC 12/06/12 page 1 of 3

21 Stocks, Bonds, and Mutual Funds KEY Mathematical Models w ith Applications g. PE represents the price to earnings ratio. What would be the earnings per share for Disney stock? = 32.49/earnings Earnings = $2.07 h. How can the PE values of the three stock options be used to recommend stock options? The lower the PE value, the better the investment option. Stock T: the PE value means an investment of $19.56 earns the company $1 or for every investment of $1 the company earns $ Stock ETE: the PE value means an investment of $20.97 earns the company $1 or for every investment of $1 the company earns $ Stock DIS: the PE value means and investment of $15.67 earns the company $1 or for every investment of $1, the company earns $ i. Suppose you purchased 50 shares of Energy Transfer Equity (ETE) at its lowest price the last 52-Week period and sold it on this day. You paid $9.95 for an online fee for each transaction. Determine if you made or lost money. Show calculations to back up your conclusion. Purchase price = 50(26.99) = $1, Total fees = 2(9.95) = $19.90 Sale price = 50(32.00) = $ = $ profit was realized Mutual Funds For some, an easier way to invest in the stock market is to invest in a group of stocks managed by a professional who develops a portfolio of various stocks and bonds called a mutual fund. A fee is paid to the manager of the portfolio. Invested funds from several investors are pooled and the fund manager buys and sells stock to try and achieve the best return for the investors. Bonds Three types of bonds are available to the public: treasury bonds (issued by federal government), municipal bonds (issued by state and local governments), and corporate bonds (issued by corporations). All bonds are a type of loan to governments or corporations that they agree to pay back to investors with interest. Bonds earn simple interest which is paid in addition to the face value of the bond to investors. Bonds are considered to be safer investments than the stock market, but they usually have a lower return. Sample 2 Why would a person want to include several investment options in their financial planning for the future? Answers will vary. The answers should address safe investments versus those that are riskier. They should also address higher and lower returns. 2012, TESCCC 12/06/12 page 2 of 3

22 Practice Problems Stocks, Bonds, and Mutual Funds KEY Mathematical Models w ith Applications 52-WEEK STOCK DIV YLD% PE VOL 100s CLOSE NET CHG HIGH LOW T , Use the table above to answer the following questions. 1. How many shares of AT&T (T on NYSE) stock were traded on this day? 22,894,400 shares were traded on this day. 2. What was the closing price of a share of AT&T stock? The closing price of a share of stock was $ Is the closing price of AT&T stock higher or lower than the previous day? The closing price on this day is lower by 4 than it was on the previous day. 4. What was the price of a share of AT&T stock on the previous day? The price on the previous day was $ What was the percent increase between the high and low of AT&T stock over the past 52- week period? ( )/33.32 x 100 = 28.96% increase 6. If you owned 50 shares of AT&T stock, how much will you receive in dividends for the quarter? (1.60)(50) = $80 7. Explain the meaning of the PE ratio for this company. For every investment of $1 the company earns $ Suppose you purchased 150 shares of AT&T at its highest price in the last 52-week period and sold it on this day. You paid $10.50 for an online fee for each transaction. Determine if you made or lost money. Show calculations to back up your conclusion. Purchase price = 150(42.97) = $6, Total fees = 2(10.50) = $21.00 Sale price = 150(37.88) = $ = -$ loss was realized Solve the following problem using a system of equations. 9. Mike received a bonus of $6,000. He wants to invest the money in stocks that have historically had an annual yield of 10% and in a mutual fund that had a 5% annual yield. Mike does not want to earn more than $500 or it will impact his taxes adversely. What amount should he invest in stocks and what amount should he invest in the mutual fund? s = amount in stock m = amount in mutual fund s + m = s m = 500 Solve by any method (graphing, algebraic, matrix) s = $4,000, m = $2, , TESCCC 12/06/12 page 3 of 3

23 Stocks, Bonds, and Mutual Funds Mathematical Models with Applications Although savings and annuities are considered safe investments, some of the riskier investments can offer better returns over a longer period of time. These investments include,, and. Certain factors can impact the value of these investments such as the success of companies, the economic and political climate, interest rates, taxes, and unemployment. Stocks Companies that go public can sell ownership in their company in the form of. A is a unit of ownership in a particular company. are amounts sometimes paid on a quarterly basis to shareholders from profits made by the company over the previous period. Most stocks are traded on the New York Stock Exchange (NYSE) or the National Association of Securities Dealer Automated Quotations (NASDAQ). Stock transactions can be done by an individual, especially now with online access; these transactions many times have a flat fee for each transaction. Stock transactions can also be carried out by a stockbroker who usually charges a commission fee. Sample 1 52-WEEK HIGH LOW STOCK DIV YLD% PE VOL 100s CLOSE NET CHG T , ETE , DIS , Use the stock listings in the table above to answer the following questions: a. How many shares of Disney (DIS) stock were traded on this day? b. What was the closing price of a share of Disney stock? c. Is the closing price of Disney stock higher or lower than the previous day? d. What was the price of a share of Disney stock on the previous day? e. What was the percent decrease between the high and low of Disney stock over the past 52-week period? f. If you owned 25 shares of Disney stock, how much will you receive in dividends for the quarter? 2012, TESCCC 04/25/13 page 1 of 3

24 Stocks, Bonds, and Mutual Funds Mathematical Models with Applications g. PE represents the price to earnings ratio. What would be the earnings per share for Disney stock? h. How can the PE values of the three stock options be used to recommend stock options? i. Suppose you purchased 50 shares of Energy Transfer Equity (ETE) at its lowest price in the last 52-Week period and sold it on this day. You paid $9.95 for an online fee for each transaction. Determine if you made or lost money. Show calculations to back up your conclusion. Mutual Funds For some, an easier way to invest in the stock market is to invest in a group of stocks managed by a professional who develops a portfolio of various stocks and bonds called a. A fee is paid to the manager of the portfolio. Invested funds from several investors are pooled and the fund manager buys and sells stock to try and achieve the best return for the investors. Bonds Three types of are available to the public: treasury bonds (issued by federal government), municipal bonds (issued by state and local governments), and corporate bonds (issued by corporations). All bonds are a type of loan to governments or corporations that they agree to pay back to investors with interest. Bonds earn simple interest which is paid in addition to the face value of the bond to investors. Bonds are considered to be safer investments than the stock market, but they usually have a lower return. Sample 2 Why would a person want to include several investment options in their financial planning for the future? 2012, TESCCC 04/25/13 page 2 of 3

25 Stocks, Bonds, and Mutual Funds Mathematical Models with Applications Practice Problems 52-WEEK HIGH LOW STOCK DIV YLD% PE VOL 100s CLOSE NET CHG T , Use the table above to answer the following questions. 1. How many shares of AT&T (T on NYSE) stock were traded on this day? 2. What was the closing price of a share of AT&T stock? 3. Is the closing price of AT&T stock higher or lower than the previous day? 4. What was the price of a share of AT&T stock on the previous day? 5. What was the percent increase between the high and low of AT&T stock over the past 52- week period? 6. If you owned 50 shares of AT&T stock, how much will you receive in dividends for the quarter? 7. Explain the meaning of the PE ratio for this company. 8. Suppose you purchased 150 shares of AT&T at its highest price in the last 52-week period and sold it on this day. You paid $10.50 for an online fee for each transaction. Determine if you made or lost money. Show calculations to back up your conclusion. Solve the following problem using a system of equations. 9. Mike received a bonus of $6,000. He wants to invest the money in stocks that have historically had an annual yield of 10% and in a mutual fund that had a 5% annual yield. Mike does not want to earn more than $500 or it will impact his taxes adversely. What amount should he invest in stocks and what amount should he invest in the mutual fund? 2012, TESCCC 04/25/13 page 3 of 3

26 Feeling Financially Secure Mathematical Models w ith Applications Joanna Bellows received a $50,000 inheritance that she wants to invest to supplement her retirement plan. Joanna already pays into an annuity through her employer that will make up part of her retirement portfolio. She estimates that she will retire in 25 years. Select three investment options from the list below that Joanna can use to invest her money. Savings: 4.5% compounded quarterly Savings: 4.0% compounded daily Stocks: 8.5% annual rate historically Mutual fund: 6.5% annual rate Bond: 5.0% annual rate Possible Points Requirements Awarded Points 10 Divide the $50,000 between the investment options you have selected. 30 Using the investment options you selected, calculate the projected value after 25 years. 60 Written report: Show calculations, tables, graphs, and formulas. (40 points) Write a summary of the value of the investments after 25 years. (10 points) Write a summary of your reasons for selecting the options and the amount of money invested. (10 points) Final Grade: 2012, TESCCC 12/06/12 page 1 of 1

27 Evaluating Financial Planning KEY Mathematical Models w ith Applications Elvira is planning on putting $5,000 into a savings investment for five years. The bank has offered her three different options for savings investment. Simple interest at 6% Compounded quarterly at 6% Compounded monthly at 6% 1. Fill in the table below to model the relationships between time in years and the amount using each option. Time in Years Simple Compounded Quarterly Compounded Monthly 0 $5,000 $5,000 $5,000 1 $5,300 $5, $5, $5,600 $5, $5, $5,900 $ $5, $6,200 $6, $6, $6,500 $6, $6, x (0.06)x 5000( x) 5000( )4x 5000( )12x 2. Graph the function of each option on the coordinate plane below. Label the axes. 3. Which model is the best for comparing the investment options? Explain your reasoning. With the graph you can very easily see that the simple interest is always less than the other options, and the difference increases over time. However, with the graph it is difficult to compare the two compounded options because the yields are too close. The table is a better method to compare the compounded options. 4. What option would give Elvira a higher yield? Explain your reasoning. Investing the $5,000 in a savings account at 6% annual interest compounded monthly would give Elvira the maximum yield. Reasoning will vary. 2012, TESCCC 12/06/12 page 1 of 3

28 Evaluating Financial Planning KEY Mathematical Models w ith Applications 5. Annuity payments of $25 are made monthly for twenty years at 7.75% compounded monthly. Use the formula below to predict the value at the end of twenty years if using an ordinary annuity. Use the formula below to predict the value at the end of twenty years if using an annuity due. Show all your calculations. Check your answer using the TVM Solver. How do the values of the two different methods compare after twenty years? Future Value of an Ordinary Annuity Future Value of an Annuity Due FV n 1 r 1 P r FV P r 1 r n 1 1 r Future Value of an Ordinary Annuity FV FV = $14, FV Future Value of an Annuity Due FV = $14, The annuity due method would result in a higher yield, because it would have one additional interest payment. 6. Christine will be retiring in one year and is preparing by analyzing her retirement plan. Although her pension and Social Security will be enough for her basic needs, Christine would like an additional $10,000 per year with which she can travel. She wants to invest an amount into an annuity now that will pay her $10,000 per year for the next 15 years. If the annuity is compounded annually at an annual percent rate of 6.5%, how much will Christine need to invest up-front? How does this compare to the amount she will receive after 15 years? Christine will need to invest $100, After 15 years she will receive $150,000. This is $49, more than she invested. 2012, TESCCC 12/06/12 page 2 of 3

29 Evaluating Financial Planning KEY Mathematical Models w ith Applications 52-WEEK STOCK DIV YLD% PE VOL 100s CLOSE NET CHG HIGH LOW DOW , Use the table above to answer the following questions. 7. What was the price of a share of DOW stock on the previous day? The previous day the price was $ What was the percent increase between the high and low of DOW stock over the past 52-week period? ( )/33.01 x 100 = 45.29% increase 9. If you owned 50 shares of DOW stock, how much will you receive in dividends for the quarter? (1.68)(50) = $84.00 will be received in dividends over the quarter. 10. Suppose you purchased 150 shares of DOW at its lowest price in the last 52-week period and sold it on this day. You paid $12.00 for an online fee for each transaction. Determine if you made or lost money. Show calculations to back up your conclusion. Purchase price: 150(33.01) = $4, Fees: 2(12) = $24.00 Selling price: 150(38.50) = $5, ( ) = $ in profit on the stock Solve the following question using a system of equations. 11. Julie has $20,000 she wants to invest in savings and mutual funds. The savings account will pay an APR of 4%. The mutual fund will pay an APR of 6%. To remain in the same tax bracket, Julie does not want to earn over $950 in interest for the year. What amount should she invest in savings and what amount should she invest in the mutual fund? x = savings amount y = mutual fund amount x + y = x y = 950 Solving by a method of choice: x = $12,500 y = $7,500 Julie should invest $12,500 in the savings account and invest $7,500 in the mutual fund. 12. Why is it important to have a diversity of investment options in a retirement plan? Answers will vary. Sample: Safe investments like savings and annuities usually have lower return on the investment, but are more secure. Riskier investments like stocks, mutual funds, and bonds usually have higher returns, but also have a greater chance of loss on the investment. A stronger financial plan will have some of each in order to balance the risk and higher yield. 2012, TESCCC 12/06/12 page 3 of 3