Investigating Investment Formulas Using Recursion Grade 11


 Sandra Davidson
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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 multistep 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 onehalf 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. PreAssessment: Distribute Attachment A, PreAssessment. 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. PostAssessment: Distribute Attachment C, Investigating Investment Formulas PostAssessment. 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 prerankings 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 timeconsuming 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 payoff or final amount to be paid in month 1. Ask, What is the payoff 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 closedform 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 highlevel 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, PreAssessment Attachment B, PreAssessment Answer Key Attachment C, Investigating Investment Formulas PostAssessment Attachment D, Investigating Investment Formulas PostAssessment Answer Key Attachment E, Defining Variables Attachment F, Deriving the Formula Attachment G, Deriving the Formula Answer Key 6
7 Attachment A PreAssessment 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 PreAssessment 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 PostAssessment 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 PostAssessment 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 fiveyear 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
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