# Your Best Interest: Finding the Right Place to Save Money

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1 Chapter 1 Your Best Interest: Finding the Right Place to Save Money The Bottom Line What is the effect on your future savings when you change values such as principal, interest rate, and time in simple and compound interest formulas? Using linear, step, and exponential functions, students will see how the money in savings accounts and other saving options (e.g., money market accounts) can change over time. They will explore how banks set up savings accounts, and they will discover that some types of accounts may not be good for making money, and in fact may result in losing money. Standards Met in This Chapter CCSSM Standards and Practices F.BF.1: Write a function that describes a relationship between two quantities. F.IF.7: Graph functions expressed symbolically and show key features of the graph by hand in simple cases or using technology for more complicated cases; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. (National Governors Association Center for Best Practices and Council of Chief State School Officers [NGA Center and CCSSO] 2010) 11

2 On the Money: Math Activities to Build Financial Literacy, High School Jump\$tart Standards Spending and Saving: Apply strategies to monitor income and expenses, plan for spending and save for future goals. Standard 1: Develop a plan for spending and saving. Financial Decision Making: Apply reliable information and systematic decision making to personal financial decisions. Standard 1: Recognize the responsibilities associated with personal financial decisions. (Jump\$tart Coalition for Personal Financial Literacy 2015) Balance Sheet This chapter is about helping students to understand savings options. The first of its two activities consider interest as it pertains to saving, with a focus on the impact of associated fees and penalties. The second activity turns to different ways to save money at various interest rates and to the benefits of making regular deposits to accounts. Students calculate compound interest, graph exponential functions that are nearly linear functions, graph step functions, apply the compound interest formula to financial situations, write exponential functions, solve exponential equations, and analyze functions using tables and graphs. The following three sections describe the mathematical and financial literacy concepts addressed by the activities in this chapter, as well as the knowledge and experiences that high school students may bring to the lesson and take from it upon its completion. Mathematics This lesson includes various kinds of functions: linear, step, and exponential. Savings accounts in banks are based on compound interest, typically compounded daily, and therefore grow exponentially. The formulas to calculate types of interest are provided in figure 1.1; they are also included on the Savings Accounts Reference Page at the end of this chapter. Formulas for Calculating A =P(1+r) Interest Simple Interest: A =P(1+r) Compound Interest: Variables: A =P ( 1+ r n) nt A =P ( 1+ r n) nt A represents amount of money, based on the other variables P represents principal amount (initial investment) r represents annual nominal interest rate (not reflecting the compounding) n represents number of periods the interest is compounded per year t represents number of years the money is invested Fig Formulas for calculating simple and compound interest 12

3 Your Best Interest: Finding the Right Place to Save Money The first activity in this chapter (accompanied by Activity Sheet 1.1) uses the compound interest formula for students to explore the value of money placed in a savings account across several scenarios and asks students to look for patterns in resulting amounts (MP.8; NGA Center and CCSSO 2010). When interest rates are low, such as 0.01% (0.0001), the amount in an account looks flat over time. For example, the graph for f(x) = x looks nearly equivalent to the graph for g(x) = 250 in the short term (F.IF.7). This provides an opportunity for students to look at situations that essentially can be represented by constant functions (linear functions with a slope of 0). Additionally, students will find that account holders who do not maintain minimum balances often are charged monthly fees. If a bank deducts a fee at the end of every month, the situation might be represented by a step function. For example, in an account with no growth for 30 days and a fee deduction of \$5, followed by no change for 30 days and a fee deduction of \$5, and so on, the resulting amounts can be represented by a step function. Students will compare these situations to the way in which these functions are represented to notice common features within a family of functions (MP.7). In the second activity (accompanied by Activity Sheet 1.2), students explore and interpret compound interest, apply the compound interest formula to financial situations, write expressions, solve exponential equations, and analyze functions using tables and graphs (F.BF.1 and F.IF.7). Specifically, students will write expressions for functions that describe the relationship between two quantities when considering the amount and variables such as the annual interest rate, number of years invested, fees, and regular deposits (MP.7). They will write functions, including piecewise functions (specifically step functions), to describe interest rate scenarios (F.BF.1). Financial Literacy Teenagers tend to have little income, so it is often difficult for them to think about saving. Yet the impact of saving over time is an important concept to understand when managing one s finances. Saving is different from investing. A savings plan (savings account) involves little or no risk, but it also provides little or no opportunity to make money (through interest); an investing plan involves more risk but can provide a higher rate of return (Faulkenberry n.d.). Savings accounts, checking accounts, and certificates of deposit (CDs) are all safe ways to save because their assets are insured by the Federal Deposit Insurance Corporation (FDIC). Savings accounts are liquid assets, which means that savings can be turned into cash quickly if that cash is needed to fix a flat tire or to buy concert tickets. On the other hand, investing focuses on long-term goals, with a main goal of making money from the money that is invested. Investors often do not know how much money they will earn from an investment, and they could even lose money! Investments are therefore riskier than savings options but offer more potential to make money. Additionally, interest rates and bank fees have changed dramatically in the past decade, thereby changing the landscape of saving and investing. In recent years, the annual percentage yield (APY) for savings accounts has been so low that when considered in combination with fees, it is often not a good idea to open a savings account. Table 1.1 illustrates some alarming numbers from savings accounts at large national banks. 13

4 On the Money: Math Activities to Build Financial Literacy, High School Table 1.1. Basic savings accounts facts and figures (March 2016) Basic Savings APY Minimum Opening Balance Fees Fees Waived When Minimum Daily Balance Is... JPMorgan Chase 0.01% \$25 \$5/month \$300, or recurring deposit of \$25 or account holder is under 18 Bank of America 0.01% \$25 \$5/month \$300 Citibank 0.01% \$500 \$4.50/month \$500 or more Wells Fargo 0.01% \$25 \$5/month \$300, or recurring deposit of \$25 or account holder is under 18 HSBC North America Holdings 0.01% (compounded quarterly) \$0.01 or \$1.00 to earn APY \$0 N/A Did you know? Businessman and investor Warren Buffett made his first investment by age eleven and by age thirteen was running a small business. At age seventy-six, he announced that he would give an estimated \$62 billion away to charity (Biography.com n.d.). Savings accounts are no longer a reasonable option for growing money, and they can be a way to lose money. Note that if you maintained only the minimum balance for most of the banks listed in the table, your money would be gone within a year (an important discovery included in the first activity of this chapter). Why do savings accounts still exist? They are liquid assets, and the money in them is perhaps less likely to be spent than if it were in a checking account or a piggy bank. So savings accounts can be effective for budgeting or planning for future expenses but not for accumulating more money. Some other ways of saving yield slightly higher interest (e.g., CDs and money market accounts), which enable savings to grow slightly. However, these options often mean the money is more difficult to access at a moment s notice (meaning it is less liquid). In the second activity in this chapter, students explore these different savings options. Together, the explorations in this chapter lead students to consider the importance of long-term goals and of saving early. Student Knowledge and Experience High school students may have varying knowledge of the advantages and disadvantages of different saving strategies. Specifically, investigating these options can help students to recognize that (1) there are significant differences with different savings options and (2) there are major advantages to saving at a younger age. Teaching students about the difference between needs and wants as well as the mathematics and the financial benefits of saving and minimizing debt can help plant a seed for good practices in the future practices that help students of all backgrounds. Mathematically, high school students should be familiar from earlier grades with the idea that rewriting an expression can provide insight into its meaning. Students in middle school generally spend a great deal of time on linear equations and functions, including writing equivalent expressions. Lesson Plan Learning Targets Analyze and compare savings options and the functions that can be used to best model them. Determine the effects of changing initial investments, term lengths, and interest rates on savings accounts. 14

5 Your Best Interest: Finding the Right Place to Save Money Resources and Tools Activity Sheet 1.1: Savings Account or Piggy Bank? (pp ) Activity Sheet 1.2: More Saving Options (pp ) Savings Accounts Reference Page (p. 32) (Available for download and printing at Key Language Calculators and/or computers for spreadsheet functionality annual nominal interest rate: Annual rate of interest that does not take into account the compounding of interest within the year. annual percentage yield (APY): Annual rate of interest that takes into account the compounding of interest within the year. compound interest: Interest that is earned on the starting balance (principal) and on interest previously earned. growth factor: The factor by which a quantity multiplies itself over time. interest rate: The percentage of the principal that is charged to a borrower (as interest) by a lender. investing: Expending money with an expectation of future financial gain. principal: Original amount of money invested (or loaned). saving: Holding onto money, rather then spending it. simple interest: Interest based on the starting balance (principal) only. Sales Pitch Ask students to each record on sticky notes some answers to this question (with each idea recorded on a separate sticky note): Why might a young person want to start saving money NOW? Have students bring their sticky notes to the front of the room and place them on the board in one of two categories: Needs and Wants. Point out that people save money for both wants and needs, and that sometimes needs must be prioritized over wants. Tell the students that saving money requires knowing both how to save and how to invest. Explain that knowing how to save and invest money is important in being able to have the things we really want to have, even if it means giving up things in the short term. Bring or show a piggy bank and a sock with some cash in each. Explain to students that you are wondering if you want to save for a want or need, how might you best save your money? Should you keep your money in the piggy bank or a sock, or are you better off putting your money in a savings account at the bank? Invite students to share arguments on both sides of this decision. Opening Explain to students that the series of activities in this lesson will explore savings, including how savings accounts work, and other ways to save money. Ask students to think about how much money they think they would have in 6 months if they deposited \$200 in a savings account today. 15

8 On the Money: Math Activities to Build Financial Literacy, High School 3. Celebrate America Saves Week. Founded in 2007, this is a program coordinated by American Saves and American Savings Education Council that promotes good savings habits. Educators and youth organizations can sign up to participate in America Saves Week using this website: 4. Read an article about the Top Seven Reasons to Save Money Now. This article (Caldwell 2015) can support the classroom discussion with additional background on the major reasons why people save their money. 5. Read 3 Reasons Why You Should Save More Money (McWhinnie 2014). In contrast to the purely practical concerns of the Caldwell article above, this article talks about how saving money can help us to increase our happiness, reach our dreams, and cope with emergencies. References and Resources Biography.com [website]. Warren Buffett Biography (no date). warren-buffett Caldwell, Miriam. Learn the Most Important Reasons to Save Your Money. About.com [website] (2014). Money.htm. Chen, Tim. Maximize Your Opportunities To Earn The Highest Interest Rates On Savings Accounts. Forbes magazine, January 13, /01/13/maximize-your-opportunities-to-earn-the-highest-interest-rates-on-savingsaccounts/#2715e4857a0b2b9e4d56157b. Faulkenberry, Ken. What Is the Difference Between Saving and Investing? Arbor Asset Allocation Model Portfolio AAAMP Blog RSS (no date). Hamm, Trent. Personal Finance 101: Money Market Accounts Versus Normal Savings Accounts. The Simple Dollar.com [website] (2014). Investopia.com. Certificate of Deposit (CD) [web video] (2014). video/play/certificate-deposit-cd/. Jump\$tart Coalition for Personal Financial Literacy. National Standards in K 12 Personal Finance Education. 4th ed. Washington, D.C.: Jump\$tart Coalition for Personal Financial Literacy, McWhinnie, Eric. 3 Reasons Why You Should Save More Money. The Cheat Sheet, December 30, Money Market Account Basics [web video]. YouTube.com, October 31, My Money Coach [website]. 10 Reasons Why You Should Save Money (Even When Borrowing Is Cheap & Easy) (no date). National Governors Association Center for Best Practices and Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, D.C.: NGA Center and CCSSO,

9 Your Best Interest: Finding the Right Place to Save Money Ross, Allison. CD Early Withdrawal Can Come at a Hefty Cost. Bankrate.com [website] (no date). Saving vs. Investing [web video]. YouTube.com, June 7, watch?v=sohgdxlj9hy. Wesley, Daniel. Choosing a Certificate of Deposit (CD) vs. a Savings Account. CreditLoan.com [website] (no date). 19

10 On the Money: Math Activities to Build Financial Literacy, High School Activity Sheet 1.1 Savings Account or Piggy Bank? Most banks and credit unions offer savings accounts as a simple way for consumers to keep the money they save safe. But is a savings account the best way to keep your money safe, or are you better off putting cash in a piggy bank? This is the question you will answer at the end of this activity. 1. On May 1st, Harry, Larry, Sherri, and Mary each decide to put aside some money, and they are wondering if they should put their money in a savings account. For the questions in this activity, you can look up and use current interest rates and fees or use the common 2015 interest rate and fee for the most basic savings account listed below: Minimum to open an account: \$25 Annual nominal interest rate: 0.01% (interest is compounded daily) Monthly fee: \$5 fee per month, waived if balance is over \$300 a. Complete the table for each student s balance with your chosen interest rate. Note that May, July, August, and October each have 31 days, and June and September have 30 days. Person Starting Balance May June July August September October Harry \$150 Larry \$500 Sherri \$325 Mary \$275 b. Graph each person s balance on the graph on the following page with time in months plotted on the horizontal axis and balance plotted on the vertical axis. 20

11 Your Best Interest: Finding the Right Place to Save Money 600 Balance Months Describe and write a function that best models the following: a. Balance as a function of time when not subjected to monthly fees b. Balance as a function of time when subjected to monthly fees 3. Based on your mathematical models, for whom do you suggest depositing money in a bank savings account (and why)? For whom do you suggest putting money in a piggy bank (and why)? SAVINGS ACCOUNT: PIGGY BANK: 4. Youth Accounts. Some banks offer better rates for young people! You can start a savings account with no minimums, no fees, and higher interest rates. Explore how these differences affect the money saved and earned in savings accounts. a. At Best Bank USA, a Youth Savings Account earns a 0.75% annual nominal interest rate compounded daily. Write a function to model the balance from Best Bank savings accounts for young people. 21

12 On the Money: Math Activities to Build Financial Literacy, High School b. Complete the table using this information from Best Bank USA. Person Starting Balance May June July August September October Harry \$150 Larry \$500 Sherri \$325 Mary \$275 c. Compare the results in this table to the table in question 1a. What do you notice? d. Even Better Bank offers a Youth Savings Account for youth up to age 21. Youth earn a 4.00% annual nominal rate compounded daily on the first \$1000 and 0.75% afterwards. Write a piecewise function that models the amount of money earned from Even Better Bank Youth Savings Accounts. e. Complete the table using Even Better Bank terms. Person Starting Balance May June July August September October Harry \$150 Larry \$500 Sherri \$325 Mary \$275 f. Compare the results from this table with the table in question 1a and with the table in question 4b. What do you notice? 22

13 Your Best Interest: Finding the Right Place to Save Money 5. Savings vs. Investing. Harry, Larry, Sherri, and Mary want to know how long it will take them to make \$20 if they have their savings in one of these Youth Accounts. Use your work from question 4 to determine the time needed for each person to earn \$20 at Best Bank USA and to earn \$20 at Even Better Bank. Enter your answers in the table below. Person Harry Larry Sherri Mary Time (in days) to earn \$20 at BEST BANK Time (in days) to earn \$20 at EVEN BETTER BANK 6. Is putting money in a general savings account a good way to make more money? Is putting money in a youth savings account a good way to make more money? Explain. 7. Name two reasons to start saving money early (as a child or young teenager). 23

14 On the Money: Math Activities to Build Financial Literacy, High School Activity Sheet 1.2 More Saving Options Option #1: Online Savings Accounts Online Savings Accounts Fast Facts: Online savings accounts often have better rates than savings accounts offered by physical banks because online banks do not have the same overhead (operating expenses) that physical banks have. As with savings accounts at physical banks, you can withdraw your money from online savings accounts as you need it. You want to buy a car when you graduate from high school, which is two years away. After working three jobs in the summer and adding to the money you already saved, you have \$6000. You have to decide how much money to put into savings and how much money to use for needs or fun over the next two years. 1. Without doing any calculations, respond to the following questions. a. How much would you like to spend on a car in two years? b. How much of the \$6000 would you need to put into a savings account in order to afford your car? c. Where is the best place to save money if you would like it to earn more money? Suppose that you deposit \$3000 in an online savings bank that offers 0.85% annual nominal interest rate compounded daily with no fees. Explore how your money would accrue over a two-year period. 2. Write a function to model the growth of the savings option. 3. What will be the balance in your account at the end of two years? 24

15 Your Best Interest: Finding the Right Place to Save Money 4. What is the growth factor for this savings option? 5. If you have to withdraw money from the account at the end of one year, what effect would that have on the amount you saved at the end of that year? 6. What are the pros and cons of using an online savings account to save money? ********************************************************************************* 7. How is an online savings account similar to and different from a CD? 8. How is an online savings account similar to and different from a money market deposit account? 9. Revisit your responses to question 1, and update your responses as needed. a. How much would you like to spend on a car in two years? b. How much of the \$6000 would you need to put into a savings account in order to afford your car? c. Where is the best place to save money if you would like it to earn more money? 10. What changes did you make and why? 25

16 On the Money: Math Activities to Build Financial Literacy, High School Option #2: Certificates of Deposit (CDs) Certificates of Deposit (CDs) Fast Facts: CDs are certificates you purchase for a period of time (the longer the time, the higher the rate). If you buy a one-year CD at a 1.35% annual percentage yield (APY), at the end of the year, you get your principal back plus 1.35% interest. If you need your money before the year is up, you have to pay a penalty. For a one-year loan at \$3000, that might be \$10 (depending on when you withdraw the money). You want to buy a car when you graduate from high school, which is two years away. After working three jobs in the summer and adding to the money you already saved, you have \$6000. You have to decide how much money to put into CDs and how much money to use for needs or fun over the next two years. 1. Without doing any calculations, respond to the following questions. a. How much would you like to spend on a car in two years? b. How much of the \$6000 would you need to put into a CD in order to afford your car? c. Where is the best place to save money if you would like it to earn more money? Suppose that you deposit \$3000 in a CD that offers 1.25% annual nominal interest rate compounded monthly. Explore how your money would accrue over a two-year period. 2. Write a function to model the growth of the CD option. 3. How much money will you receive when you cash in your CD after two years? 26

17 Your Best Interest: Finding the Right Place to Save Money 4. What is the growth factor for the CD option? 5. If you have to withdraw money from the CD at the end of one year, what impact would that have on the amount you saved at the end of that year? 6. What are the pros and cons of using a CD to save your money? ********************************************************************************* 7. How is a CD similar to and different from an online savings account? 8. How is a CD similar to and different from a money market deposit account? 9. Revisit your responses to question 1, and update your responses as needed. a. How much would you like to spend on a car in two years? b. How much of the \$6000 would you need to put into a CD in order to afford your car? c. Where is the best place to save money if you would like it to earn more money? 10. What changes did you make and why? 27

18 On the Money: Math Activities to Build Financial Literacy, High School Option #3: Money Market Deposit Accounts Money Market Deposit Accounts Fast Facts: Money market deposit accounts pay a higher interest rate than savings accounts, and the rate varies based on how much money you deposit (the more you deposit, the higher the rate). They require a higher deposit than other options. You can access your money at any time, but limits are set (such as 3 6 withdrawals a month). You want to buy a car when you graduate from high school, which is two years away. After working three jobs in the summer and adding to the money you already saved, you have \$6000. You have to decide how much money to put into a money market deposit account and how much money to use for needs or fun over the next two years. 1. Without doing any calculations, respond to the following questions. a. How much would you like to spend on a car in two years? b. How much of the \$6000 would you need to put into a money market deposit account in order to afford your car? c. Where is the best place to save money if you would like it to earn more money? Suppose that you deposit \$3000 in a money market account that offers 1% annual nominal interest rate compounded daily. Explore how your money would accrue over a two-year period. 2. Write a function to model the growth of the money market option. 3. How much money will you have in your account at the end of two years? 28

19 Your Best Interest: Finding the Right Place to Save Money 4. What is the growth factor for this money market option? 5. If you have to withdraw money from the account at the end of one year, what impact would that have on the amount you saved at the end of that year? 6. What are the pros and cons of using a money market deposit account to save money? ********************************************************************************* 7. How is a money market deposit account similar to and different from a CD? 8. How is a money market deposit account similar to and different from an online savings account? 9. Revisit your responses to question 1, and update your responses as needed. a. How much would you like to spend on a car in two years? b. How much of the \$6000 would you need to put into a money market deposit account in order to afford your car? c. Where is the best place to save money if you would like it to earn more money? 10. What changes did you make and why? 29

20 On the Money: Math Activities to Build Financial Literacy, High School Analysis of Options 1. Complete the table below, comparing the advantages and disadvantages of each type of saving option. Summary of Findings Saving Option Advantages Disadvantages Online savings account Certificate of deposit Money market deposit account 30

21 Your Best Interest: Finding the Right Place to Save Money Paying for the Car One thing that is true for online savings accounts and money market accounts but is not true for CDs is that you can continue to add to your principal at any time. For example, you can add \$25 every month to your account. Use the table below to see how your money accrues in a savings account if you deposit \$25 a month for 6 months. Use an annual nominal interest rate of 0.9% compounded daily and \$3000 as a starting principal. Beginning balance Interest earned May June July August September October + Deposit New balance 2. Write a function to model the balance in the account that includes the additional \$25 per month. 3. How much money would you have in your account at the end of two years? 4. If you want to purchase a car that costs \$5000, how would you save your money to reach that goal? Include the starting principal amount you would need, the savings option(s) you would use (online savings account, CD, money market deposit account), the additional amount of money you would deposit each month, and why. 5. If you want to purchase a car that costs \$10,000, how would you save your money to reach that goal? Include the starting principal amount you would need, the savings option(s) you would use (online savings account, CD, money market deposit account), the additional amount of money you would deposit each month, and why. 6. For the car you are hoping to purchase, how much money will you need to deposit and which option(s) would you use to reach your goal? 31

22 On the Money: Math Activities to Build Financial Literacy, High School Savings Accounts Reference Page Transaction limits By law, you are permitted 6 withdrawals or outgoing transfers per month from a savings account. If you use it more than this, you are subject to fees. (If you use your debit card more than 6 times, you will pay a fee.) Compounding interest Savings account interest rates are compounded daily (usually, though this is not guaranteed). Annual percentage yield (APY) This is the amount you would earn if you invested a principal value for a full year. This rates takes compounding of interest within the year into account. Variable rates Savings account rates are subject to change without notice and can shift daily at the discretion of the bank or credit union. Fees Banks charge fees for your savings accounts if you fall below a certain amount of money. The table below shows the fee structure for 5 of the largest banks in the United States. Table 1.2. Basic savings accounts figures from five big banks in the United States (February 2015) Basic Savings APY Minimum Balance Fees Fees Waived When... JPMorgan Chase 0.03% \$25 \$4/month \$300 or add \$25 or under 18 Bank of America 0.01% \$25 \$5/month \$300 Citibank 0.01% \$100 \$10/month \$500 or more Wells Fargo 0.01% \$25 \$5/month \$300 or recurring deposit of \$25 or more HSBC North America Holdings 0.01% (compounded quarterly) A =P(1+r) \$1 or \$100 to earn APY Not posted \$500 Compound Interest: A =P ( 1+ r n) nt Simple Interest: A =P(1+r) Variables: A represents amount of money, based on the other variables. A =P ( 1+ r n) nt P represents principal amount (initial investment). r represents annual interest rate. n represents number of periods the interest is compounded per year. t represents number of years the money is invested. (Sometimes the variable i is substituted into the formula, where i = r n ). 32

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