What You ll Learn. And Why. Key Words. interest simple interest principal amount compound interest compounding period present value future value

Transcription

1 What You ll Learn To solve problems involving compound interest and to research and compare various savings and investment options And Why Knowing how to save and invest the money you earn will help you manage your money more effectively and allow you to achieve your financial goals. Key Words interest simple interest principal amount compound interest compounding period present value future value

2 CHAPTER 5 Activate Prior Knowledge Determining the Percent of a Number Prior Knowledge for 5.1 To determine the percent of a number, write the percent as a decimal, then multiply. Example John earns $17 per hour. He will get an 8% raise. a) How much more will John earn per b) Determine John s new hourly hour? wage. Solution a) 8% Write 8% as a decimal $17 $1.36 John will earn $1.36 more per hour. b) Method 1: Add the increase to the hourly rate. Method 2: Add the percents. Original wage $17/h John s new wage is 100% 8%, Increase in wage $1.36/h or 108% of the original wage. New wage: $17/h $1.36/h 108% $18.36/h 1.08 $17 $18.36 John s new hourly wage is $18.36 per hour. Check 1. Selina earns $500 per week as a landscape gardener. She will get a 6% raise next week. a) How much more will Selina earn next week than this week? b) How much will she earn next week? 2. The labour rate at an auto shop is $75 per hour. Suppose the shop offers a discount of 25% every Tuesday. What will be the labour rate for 2 h work on Tuesday? 3. The student population of a high school is 865. The principal is planning for next year and predicts an increase in the student population of 4.5%. a) What student population is predicted in this high school for next year? b) Which method did you use to determine your answer? Explain your choice. 212 CHAPTER 5: Saving and Investing Money

3 5.1 Simple Interest When you deposit money in a bank account, you lend money to the bank. In return, the bank pays you interest for the use of the money. Investigate Calculating Simple Interest You will need grid paper. When Rachel was born, her grandparents deposited $1000 in an account that earns 6% interest per year for 5 years. Every year on her birthday, Rachel receives a cheque for the interest earned. Determine the interest earned after 1 year. Determine the total interest accumulated at the end of 5 years. Year Amount in Interest Total account earned (6%) interest accumulated 1 $ $ $ $ $1000 What patterns do you see in the table? Explain the patterns. Plot Total interest accumulated against Year. Describe how the total interest accumulated grows. Reflect Why is the interest earned the same each year? How does the total interest accumulated after 5 years compare with the interest earned after 1 year? If you know the number of years, how can you determine the total interest accumulated? Does the total interest accumulated grow linearly? Use the table and graph to justify your answer. Keep your graph for use in Section Simple Interest 213

4 Connect the Ideas Simple interest is calculated using this formula: I Prt I is the simple interest. P is the principal or the money invested. r is the annual, or yearly, interest rate as a decimal. t is the time in years. Short-term Simple interest is often earned on short-term investments of 1 year or less. When the interest rate is an annual rate, time in months or days must be written as a fraction of 1 year. Suppose you invest $4000 for 60 days at 6.5% per year. To determine the simple interest earned, use the formula: I Prt As a decimal, 6.5% As a fraction of 1 year, 60 days is. 365 Substitute: P 4000, r 0.065, and t I I The interest earned is $ One year is 365 days. Long-term Some long-term investments also earn simple interest. Each year, the interest earned is paid out to the investor. As a result, the principal each year is constant. So, the interest earned each year is constant. Suppose you invest $800 at 5% simple interest per year for 5 years. To determine the interest earned each year, find 5% of $800. This is: $ $40 $40 interest is earned each year for 5 years. You can also use the formula I Prt to find the interest earned each year. Substitute: P 800, r 0.05, and t 1 I Year Principal Interest Total earned (5%) interest 1 $800 $40 $40 2 $800 $40 $80 3 $800 $40 $120 4 $800 $40 $160 5 $800 $40 $200 Total interest ($) Total Interest Earned Year CHAPTER 5: Saving and Investing Money

5 In the table, the total interest increases by a constant amount, $40, each year. The points on the graph of Total interest against Year lie on a straight line. These are characteristics of linear growth. Practice 1. Determine the interest earned. Principal Rate Time a) b) c) d) e) $1200 4% 3 years $6000 2% 4 months $750 10% 15 months $2000 9% 120 days $500 6% 30 days When an investment matures, both the principal invested and the interest due are paid to the investor. This sum is called the amount of the investment. Amount Principal Interest, or A P I. Example Solution Peter invests $ for 5 months in an account that earns 3.4% interest per year. What amount will Peter receive when the investment matures? To determine the interest earned, use the formula: I Prt 5 Substitute: P , r 0.034, and t 5 12 I I The interest earned is $ The amount is the sum of the principal and interest. A P I Peter receives $ when the investment matures. 2. Determine the amount of each investment. a) $500 for 3 months at 2.25% per year b) $750 for 90 days at 5.2% per year c) $1000 for 1 year at 3.5% per year d) $300 for 6 months at 8% per year 5.1 Simple Interest 215

6 3. Denise deposits $5000 in a Guaranteed Investment Certificate (GIC). The GIC earns 3.5% simple interest per year for 7 years. a) How much interest does the GIC earn in 1 year? b) What is the total interest earned in 7 years? c) How are the answers in parts a and b related? Why are they related this way? Interest rates change frequently and interest is taxable income. To make the math easier, we assume fixed rates of interest and ignore taxes. 4. Governments and corporations sell bonds to raise money. Some bonds offer the option of receiving annual simple interest payments. Suppose a $500 bond earns 3.5% simple interest per year for 6 years. a) How much interest is earned each year? b) Determine the total interest earned at the end of each year. Record your results in a table. c) Plot Total interest against Year. d) What type of growth does the graph illustrate? Justify your answer. 5. Assessment Focus Josie holds the following bonds in her investment portfolio. Each bond earns simple interest at the given rate. A $1200 Ontario Savings Bond (OSB) earning 6.7% per year A $500 Canada Savings Bond (CSB) earning 5.9% per year A $5000 corporate bond earning 8.25% per year a) Each year, Josie must report the interest earned on these bonds on her tax return. What is the total interest she must report? b) Josie keeps the bonds for 4 years. What is the total interest earned? 6. Suppose a $100 investment earns 6% simple interest per year for 5 years. Does the interest earned double under each scenario? Justify your answers. a) The principal is doubled to $200. b) The interest rate is doubled to 12%. c) The time is doubled to 10 years. 7. Take It Further Brendan invested $1200 for 9 months. He received $1263 when the investment matured. What annual rate of interest did the investment earn? What is simple interest? How is simple interest calculated? Include examples in your explanation. 216 CHAPTER 5: Saving and Investing Money

7 5.2 Compound Interest Recall, from Section 5.1, that Rachel s grandparents deposited $1000 in an account that earns 6% simple interest per year. Each year, Rachel receives a cheque for $60. At the end of 5 years, Rachel has accumulated $300 in interest. Investigate Calculating Compound Interest Work with a partner. You will need a calculator and grid paper. Suppose Rachel s grandparents leave the interest in the account so that each year, the interest earned is added to the principal. Copy the table. Explain how the numbers in the second and third columns were calculated. Complete the table. Add a fourth column to the table labelled Total interest. Complete the column. What is the total interest at the end of 5 years? Plot Total interest against Year. Use the same grid you used in 5.1 Investigate. Compare the two graphs. What do you notice? Year Principal Interest for year earned (6%) 1 $ $ $ $ $ Reflect Why does the interest earned increase each year? At the end of 5 years, how much additional interest is earned by keeping the interest in the account instead of paying it out? Why might Rachel s grandparents choose to have the interest paid out each year? Why might they choose to keep it in the account? 5.2 Compound Interest 217

8 Connect the Ideas If you purchased a CSB in 2006, you earned 3.0% interest in the first year. Simple interest Interest rates in the 1970s were much higher than they are now. Suppose you bought a $500 Canada Savings Bond (CSB) at that time. If you kept the bond until it matured, you received 10.25% interest per year for 7 years. A regular interest CSB makes annual interest payments. So, it pays simple interest. Year Principal Interest Total earned (10.25%) interest 1 $ $51.25 $ $ $51.25 $ $ $51.25 $ $ $51.25 $ $ $51.25 $ $ $51.25 $ $ $51.25 $ The total interest is $ Compound interest In a compound interest CSB, the interest earned each year is added to the value of the bond. The following year, this interest also earns interest. Interest calculated this way is compound interest. Year Principal Interest Total earned (10.25%) interest The principal in the 2nd year includes the interest from the 1st year. The principal in the 3rd year includes the interest from the 2nd year, and so on. 1 $ $51.25 $ $ $56.50 $ $ $62.29 $ $ $68.68 $ $ $75.72 $ $ $83.48 $ $ $92.04 $ The total interest is $ CHAPTER 5: Saving and Investing Money

9 You redeem a bond if you cash it in before it matures. Compare simple and compound interest The CSB earning simple interest provides regular interest income, but less total interest. The CSB earning compound interest gives greater total interest, but the interest is not available until the bond matures or is redeemed. This is the trade-off between simple interest and compound interest. We can use a table and a graph to compare simple interest and compound interest. Simple interest CSB Compound interest CSB Year Total First interest differences 1 $ $ $ $ $ $ $ $51.25 $51.25 $51.25 $51.25 $51.25 $51.25 Year Total First interest differences 1 $ $ $ $ $ $ $ $56.50 $62.29 $68.68 $75.72 $83.48 $92.04 The first differences are equal. The total interest changes by a constant amount each year. The first differences are not equal. The total interest changes by an increasing amount each year. The points for simple interest lie on a straight line. The points for compound interest lie on a curve. We use a broken line and a broken curve to show the trend. From the table and the graph, simple interest illustrates linear growth. Compound interest illustrates non-linear growth. In the first year, the simple interest and the compound interest are the same. After the first year, the compound interest is greater than the simple interest. The difference between the compound interest and the simple interest increases each year. Money grows more rapidly when interest is compounded. Comparing Simple Interest and Compound Interest Total interest ($) Compound interest Simple interest Year 5.2 Compound Interest 219

10 Practice 1. Nadia receives a $300 compound interest CSB on her birthday. The bond earns 3% interest per year. Nadia keeps the bond for 4 years. a) Copy and complete this table. Year Principal Interest earned (3%) b) What is the total interest Nadia earns? 2. Suppose the bond in question 1 is a simple interest CSB. a) Copy and complete this table. Year Principal Interest earned (3%) b) What is the total interest Nadia earns? c) How much more interest is earned under compound interest? d) Where does the extra interest come from? 3. Refer to the tables you completed for the bonds in questions 1 and 2. On the same grid, plot Interest earned against Year for each bond. a) Compare the graphs. What do you notice? b) Why does money grow more rapidly under compound interest? 4. A principal of $100 earns 12.5% interest per year for 6 years. a) Determine the total interest at the end of each year under simple interest. Record your results in a table with these headings. Year Principal Interest earned (12.5%) Total interest b) Determine the total interest at the end of each year under compound interest. Record your results in a table. c) How much additional interest is earned under compound interest? d) On the same grid, plot Total interest against Year. Compare the two graphs. 220 CHAPTER 5: Saving and Investing Money

11 We can use a calculator or a spreadsheet to calculate the yearly growth in the amount of a compound interest investment. Example Solution Sadiki invests $250 in a compound interest GIC that earns 6.1% per year for 5 years. a) Determine the amount of the GIC at the end of each year. b) What is the amount of the GIC when it matures? The amount is the sum of the principal and interest. Use a calculator a) Use a table to organize the calculations. Year Principal Interest Amount at for year earned (6.1%) end of year 1 $ $ $15.25 $ $15.25 $ $ $ $16.18 $ $16.18 $ $ $ $17.17 $ $17.17 $ $ $ $18.21 $ $18.21 $ $ $ $19.33 $ $19.33 $ b) The amount of the GIC at maturity is $ Use a spreadsheet A 1 Year =A2+1 B C D Principal Interest Amount $ =B2*0.061 =B2+C2 =D2 =B3*0.061 =B3+C3 a) Open your spreadsheet program and start a new spreadsheet. Copy the headings, values, and formulas shown above. Select cells B2 to D3. Format the cells to show currency. Select cells A3 to D6. Fill Down to calculate values to Year 5. A B C D Year Principal Interest Amount $ $15.25 $ $ $16.18 $ $ $17.17 $ $ $18.21 $ $ $19.33 $ b) The amount of the GIC at maturity is $ Compound Interest 221

12 5. Nima inherits $ He invests the money in a compound interest GIC that earns 8.5% per year for 8 years. a) Determine the amount of the GIC at the end of each year. b) What is the amount of the GIC at maturity? c) How much interest did Nima earn? Which tools can help you solve this problem? 6. Suppose the GIC in question 5 earns simple interest. a) Determine the total interest earned. b) How much additional interest is earned under compound interest? c) Why might Nima choose the simple interest GIC instead of the compound interest GIC? 7. A principal of $5000 earns 12% compound interest per year for 7 years. a) Determine the amount at the end of each year. Organize your results in a table. b) Plot Amount against Year. c) Is the growth in the amount linear? Use the table and graph to justify your answer. 8. Assessment Focus The Canadian government also issues Canada Premium Bonds (CPBs). Jackson plans to invest $3000 for 4 years. Which bond would you recommend? A compound interest CSB earning 6.35% per year A simple interest CPB earning 7.3% per year Justify your recommendation. CPBs have a higher rate of interest than CSBs because CPBs can only be redeemed once a year. 9. Take It Further Which bond earns more interest? Justify your answer. A simple interest CPB that earns 3.0% per year in the first year, 3.25% per year in the second year, and 4.0% per year in the third year An Ontario Fixed Rate Bond that earns 3.6% compound interest per year for 3 years Explain the difference between simple interest and compound interest. Include tables and graphs in your explanation. 222 CHAPTER 5: Saving and Investing Money

13 5.3 The Amount of a Compound Interest Investment Interest rates change with economic conditions. Investigate The Amount of a Bond You will need grid paper. In the early 1980s, interest rates were much higher than they are now. Saroosh invested $1000 in a compound interest bond that earned 18% per year for 6 years. You will determine the amount at maturity in two ways. Copy and continue the table. What is the amount at maturity? Year Principal for year Interest earned (18%) Amount at end of year 1 $ $ $ $ $ $ Each year, the amount of the bond grows by 18%. Explain why increasing the amount by 18% is the same as multiplying it by Here is another way to calculate the amount. Explain the values in the Calculation column. Year Calculation Amount at end of year 1 $ $ $ $ Copy and complete the table until Year 6. Compare the two methods you used to complete the tables. Which involves less computation? Explain. Plot Amount at end of year against Year. Describe how the amount grows. Reflect What patterns do you see in the Calculation column in the second table? What type of growth is represented by the values in the Amount at end of year column? How do you know? 5.3 The Amount of a Compound Interest Investment 223

14 Connect the Ideas Pia invests $750 in a compound interest GIC that earns 6% per year for 10 years. To find the amount of the GIC at maturity, we reason as follows. 106% 1.06 Each year, the amount increases by 6%. So, each year, the amount is 100% 6%, or 106%, of its previous value. That is, each year, the amount is 1.06 times its previous value. To calculate the amounts, we repeatedly multiply by From Chapter 4, this is exponential growth. Year Calculation Amount $ $ $ $ There is a pattern in the calculations. We start with the principal, 750, and multiply by 1.06 each year. For example, the amount in 4 years is Similarly, the amount in 10 years is The amount at maturity is $ We can write an equation to give the amount of the GIC, A dollars, after n years. A n Principal This is , where 0.06 is the interest rate as a decimal. A similar pattern exists for other principals and interest rates. This suggests the following formula. Under annual compounding, interest is added to the principal at the end of every year. The amount of an investment under annual compounding is: A P(1 i) n A is the amount. P is the principal. i is the annual interest rate as a decimal. n is the number of years. 224 CHAPTER 5: Saving and Investing Money

15 Suppose you invest $3000 at 5.7% per year, compounded annually for 4 years. Calculate the amount Calculate the interest To determine the amount at maturity, use the formula: A P(1 i) n Substitute: P 3000, i 0.057, and n 4 A 3000( ) 4 Press: 3000 ( ) ^ 4 A The amount at maturity is $ Note: If these keystrokes do not work for your calculator, check your User s Manual. The interest is the difference between the amount and the principal, I A P,where I is the interest, A is the amount, and P is the principal. So, the interest earned is: $ $3000 $ Practice 1. Determine the amount of each investment. Each interest rate is an annual rate. a) A $2000 GIC earning 6.2% compounded annually for 4 years b) A $300 CSB earning 3.24% compounded annually for 5 years c) A $ corporate bond earning 7.4% compounded annually for 10 years d) A $2500 OSB earning 11.3% compounded annually for 8 years e) A $ GIC earning 9.4% compounded annually for 3 years 2. Determine the interest earned on each investment in question Juan invests $3500 at 4.2% per year, compounded annually. a) What is the amount after 5 years? b) What is the amount after 6 years? c) How much interest is earned in year 6? How do you know? 5.3 The Amount of a Compound Interest Investment 225

16 4. Gemma borrowed $2500 from her parents. She repaid the money at the end of 3 years. Suppose Gemma s parents had invested the money at 4.8% per year, compounded annually. a) What would have been the amount at the end of 3 years? b) How much interest would Gemma s parents have earned? The amount of a compound interest investment grows exponentially. Example Solution a) Make a table and a graph to illustrate the growth of $100 invested for 30 years at 6% per year, compounded annually. b) Use the table and graph to show that the growth in the amount is exponential. a) Make a table of values. Use the formula A P(1 i) n to calculate the amount every 5 years. Year Amount 0 $100 5 $100(1 0.06) 5 $ $100(1 0.06) 10 $ $100(1 0.06) 15 $ $100(1 0.06) 20 $ $100(1 0.06) 25 $ $100(1 0.06) 30 $ Use the table of values to draw a graph. Amount ($) $100 Invested for 30 Years at 6% per Year, Compounded Annually Year 226 CHAPTER 5: Saving and Investing Money

17 b) Divide each amount in the table by the preceding amount. Year Amount 0 $100 5 $100(1 0.06) 5 $ $100(1 0.06) 10 $ $100(1 0.06) 15 $ $100(1 0.06) 20 $ $100(1 0.06) 25 $ $100(1 0.06) 30 $ Growth factor In the table, the amounts grow by a constant factor, The graph is a curve that increases slowly at first and then more rapidly over time. These are characteristics of exponential growth. 5. Suppose you deposit $5000 in a retirement account that earns 10% per year, compounded annually. a) Draw a graph to show the growth in the amount over the next 25 years. b) Describe the growth. Justify your answer. 6. A principal of $100 is invested for 6 years at 8% per year, compounded annually. The graph shows the amount of the investment at the end of each year. $100 Invested for 6 Years at 8% per Year, Compounded Annually Amount ($) Year 6 Julian thinks that the amount is growing linearly since the points on the graph appear to lie on a straight line. Do you agree? Justify your answer. 5.3 The Amount of a Compound Interest Investment 227

18 7. The table shows the highest and lowest interest rates Year Annual interest rate ever offered on Canada Savings Bonds % Suppose you were able to purchase a $1000 CSB with % a 7-year term at each interest rate under annual compounding. a) Determine the amount of each CSB at maturity. b) How much additional interest is earned at the higher interest rate? 8. Assessment Focus Suppose a $100 investment earns 6% per year, compounded annually for 5 years. Does the amount double under each scenario? a) The principal is doubled. b) The interest rate is doubled. c) The time is doubled. Justify your answers. 9. Carrie saw a table like this in the business section of a newspaper. The table was in an article with the headline The Power of Time in Compounding. a) Copy and complete the table. b) What is the total interest earned in the first 5 years? The last 5 years? c) Using the results of parts a and b and your knowledge of exponential growth, explain what is meant by the power of time in compounding. Amount of a $ investment at 12% per year, compounded annually for 40 years Year Amount 0 $ Take It Further Sasha starts a new job. His salary is $ According to his contract, his salary will increase by 2.5% per year for the next 4 years. a) How does this situation relate to compound interest? b) What will Sasha s salary be in 4 years? How do you know? In Chapter 4, you learned that the equation y ab x describes exponential growth. Explain how the formula A P(1 i) n is related to y ab x. Include an example in your explanation. 228 CHAPTER 5: Saving and Investing Money

19 5.4 Compounding Periods Compound interest is often calculated more than once a year. For example, interest can be calculated and added to the principal semi-annually (twice a year) or quarterly (4 times a year). The semi-annual interest rate is one-half of the annual interest rate, while the quarterly interest rate is one-quarter of the annual interest rate. Investigate The Effect of Different Compounding Periods Work with a partner. Suppose you deposit $1000 for 1 year at an annual interest rate of 8%. Copy and complete each table. Find the amount at maturity under each compounding option. Option 1: Interest is compounded annually Year Compounding period Principal Interest earned (8%) Amount 1 1 $ Option 2: Interest is compounded semi-annually The annual interest rate is 8%, so the semiannual rate is 4%. Year Compounding period Principal Interest earned (4%) Amount 1 $ Option 3: Interest is compounded quarterly The annual interest rate is 8%, so the quarterly rate is 2%. Year Compounding period Principal Interest earned (2%) Amount 1 $ Reflect Why is the amount the greatest when the number of compounding periods is the greatest? 5.4 Compounding Periods 229

20 Connect the Ideas The time after which interest is calculated and added to the principal is called the compounding period. The table shows some common compounding periods. Frequency of compounding Annual Semi-annual Quarterly Monthly Daily Number of times interest is added during a year 1 (every year) 2 (every 6 months) 4 (every 3 months) 12 (every month) 365 (every day) Interest is usually quoted as an annual rate. If compounding occurs more than once a year, the annual interest rate is divided evenly among the compounding periods. Suppose a principal of $1500 is invested at an annual interest rate of 5% for 4 years. Annual compounding Semi-annual compounding If the interest is compounded annually, interest is calculated once a year. To find the amount, use: A P(1 i) n Substitute: P 1500, i 0.05, and n 4 A 1500(1 0.05) 4 A $ The amount after 4 years is $ If the interest is compounded semi-annually, interest is calculated twice a year. 1 The semi-annual rate is of 5% 2.5%. 2 In 4 years, there are 4 2, or 8 compounding periods. To find the amount, use: A P(1 i) n Substitute: P 1500, i 0.025, and n 8 A 1500( ) 8 A $ The amount after 4 years is $ CHAPTER 5: Saving and Investing Money

21 Monthly compounding 1 of 5% is a 12 repeating decimal that is tedious to write. We leave it as a fraction. If the interest is compounded monthly, interest is calculated 12 times a year. 1 5 The monthly rate is of 5% % In 4 years, there are 4 12, or 48, compounding periods. To find the amount, use: A P(1 i) n 0.05 Substitute: P 1500, i, and n A 12 $ The amount after 4 years is $ A 1500 ( 1 ) 48 Press: 1500 ( ) ^ 48 When the number of compounding periods increases, the number of times interest is earned on previous interest also increases. This results in a greater amount. The general formula for the amount of a compound interest investment is: A P(1 i) n A is the amount. P is the principal. i is the interest rate per compounding period as a decimal. n is the number of compounding periods. Compound Interest and Your Return Investment Amount Interest Rate Years Your investment options: $ % 4 $1850 $1800 $1750 $1700 $1650 $1600 $1550 $1500 $1450 $1400 $1350 Return in Dollars Yearly Total $1823 Quarterly Total $1830 Monthly Total $1831 Daily Total $ Compounding Periods 231

22 Practice 1. Use the formula A P(1 i) n to verify your answer for each option in Investigate. 2. An investment earns 12% interest per year for 6 years. Copy and complete the table. Determine i and n under each type of compounding. Type of Interest rate Number of compounding per period (i ) compounding periods (n) Annual Semi-annual Quarterly Monthly Daily 3. Suppose you deposit $750 in an investment account for 5 years at 6% per year. a) Determine the amount of the investment if interest is compounded: i) annually ii) quarterly iii) monthly b) Which type of compounding provides the greatest amount? Explain. 4. Determine the amount of each investment. The interest rate is an annual rate. a) A $2000 GIC that earns 8% compounded quarterly for 3 years b) A $1600 bond that earns 6% compounded monthly for 4.5 years c) A $750 savings account that earns 4% compounded daily for 1 year d) A $1500 bond earning 5% compounded semi-annually for 5.5 years 5. Determine the interest earned on each investment in question Assessment Focus Jamie thinks that you get the same amount of interest when interest is compounded semi-annually instead of annually. He reasons like this: With semi-annual compounding, the interest rate per period is halved, but the number of compounding periods is doubled. So, you should get the same amount as with annual compounding. Is Jamie correct? Explain your thinking. Include calculations in your explanation. 7. Determine the amount of each principal when invested at 4.8% per year, compounded monthly for 5 years. a) $250 b) $500 c) $1000 Does doubling the principal double the interest earned? Explain. 232 CHAPTER 5: Saving and Investing Money

23 We often have to choose between investments with different compounding periods. Example Solution Terence wins a contest. The prize is a $1000 bond that matures in 5 years. Terence has a choice between 2 bonds. Bond A: 8.5% per year, compounded quarterly Bond B: 8.0% per year, compounded monthly Which prize should Terence choose? Explain. Determine the amount of each bond. Use the formula: A P(1 i) n Bond A Substitute: P 1000, i, and n A 1000 ( ) 20 A The amount at maturity is $ Bond B 0.08 Substitute: P 1000, i and n A 1000 ( ) 60 A The amount at maturity is $ Terence should choose Bond A since it gives the greater amount. 8. Kiki has $ to invest for 7 years. Which is the better investment? Justify your answer. A corporate bond earning 7.75% per year, compounded semi-annually A GIC earning 7.25% per year, compounded quarterly 9. Take It Further Francis invests $2000 in a GIC that matures in 18 months. The GIC earns 5.6% per year, compounded quarterly. What is the amount at maturity? What is the difference between an annual interest rate of 5% compounded annually, 5% compounded semi-annually, and 5% compounded monthly? Include an example and calculations in your explanation. 5.4 Compounding Periods 233

24 Mid-Chapter Review Henri bought a $2000 simple interest CSB that earns 4.6% per year. He plans to keep the bond until it matures in 10 years. a) How much interest does he earn each year? b) What is the total interest earned over the term of the investment? 2. Adrian invests $4800 at simple interest for 45 days. The annual interest rate is 3.2%. What is the amount at maturity? 3. For a given principal, interest rate, and time, why is more interest earned under compound interest than simple interest? Include an example in your explanation. 4. Jeff plans to invest $2500 in a GIC for 5 years. He has two choices: Option 1: 6% simple interest per year with annual interest payments Option 2: 5.5% per year, compounded annually a) Which option earns more interest? How much additional interest does it earn? b) Why might Jeff choose the option that earns less interest? 5. Determine each amount. The interest is compounded annually. a) $775 at 9% for 5 years b) $ at 5.25% for 3 years c) $2000 at 11.4% for 20 years 6. Determine the interest earned on each investment in question When Lisa was born, her aunt deposited $5000 in an account that pays 8% per year compounded annually. Determine the amount in the account on Lisa s: a) 10th birthday b) 25th birthday 8. For a given principal, interest rate, and time, would you prefer: interest compounded annually? interest compounded monthly? Explain your thinking. Include an example and calculations in your explanation. 9. Suppose $750 is invested at 8.5% per year. What is the amount of the investment after 2 years under each type of compounding? a) annual b) semi-annual c) monthly d) daily 10. Determine each amount. a) $1850 invested for 7 years at 12% per year, compounded quarterly b) $2400 invested for 6 years at 9.6% per year, compounded monthly c) $900 invested for 15 years at 5.5% per year, compounded semi-annually 11. Determine which annual interest rate produces the greater amount for a principal of $675 invested for 1 year. 7.0% compounded annually 6.8% compounded monthly 234 CHAPTER 5: Saving and Investing Money

25 Research in Mathematics You will research financial information in Sections 5.5 and 5.8. Following a research plan can help make the process more effective and efficient. RESEARCH PLAN Step 1: Prepare to research State the information you are trying to find as a question. Identify key concepts, words, and phrases in your question. Use these as search words when you gather the information. What are some possible sources of information? How can you limit your search to Canadian sources? How will you record or organize your research? Step 2: Gather and evaluate information Who wrote the information? Who is the intended audience? Is the information current? When was it written or last updated? Is the information reliable? Is the information fact or opinion? Can you use this source to find other sources? Broaden or narrow your search words as needed. Step 3: Record information Record information in your own words. Document your sources. Work with a partner. Think about a topic you have researched in the past. Create a concept map of the strategies you used to research the topic. Literacy in Math: Research in Mathematics 235

26 5.5 Savings Alternatives Ima starts her first job. She needs to open a bank account so that she can cash her pay cheques and save the money she earns. The account representative at the bank tells Ima that the type of account she should open depends on her goals for the money and the length of time before the money is needed. Inquire Researching and Comparing Bank Accounts Three savings alternatives are commonly available at banks. Chequing accounts are best used to save money that will be spent within a 30-day period. Bills and everyday expenses are usually paid through a chequing account. Savings accounts are best used to set aside money that will be needed in the next 2 to 6 months, or to create an emergency fund. Term deposits are used to set aside a fixed sum of money (usually $500 or more) at a fixed interest rate for a specified term or length of time. Work in small groups. You will need either access to the Internet, print materials about the savings alternatives offered at a bank, or a representative from a bank. 1. Planning the research What research strategies might you use to gather information about the savings alternatives available at Canadian banks? How can you find the names of Canadian banks? Which banks would you use? Why? How might you record your research to prepare for comparing savings alternatives? Search words Financial Consumer Agency of Canada (FCAC) Canadian banks savings account chequing account term deposit 236 CHAPTER 5: Saving and Investing Money

27 2. Gathering information about savings and chequing accounts Research the interest rate. What is the interest rate? How is interest calculated? How often is interest paid? Is a minimum balance required to earn interest? Explain. Research costs and fees. Is there a charge for cheques? Are any transactions free? What costs are there for transactions? What fees are charged? How are these fees determined? How can you reduce the costs of the accounts? Does a minimum balance reduce costs? Describe any packages that reduce costs. Banking Investments Insurance Global Services Enter your search keyword(s) Search Savings Account Chequing Account Research plans that are available for students. How does the interest rate change as the length of the term increases? 3. Gathering information about term deposits What is the minimum deposit required? What terms are available? What is the interest rate? How does the interest rate change as the length of the term increases? What is the penalty for withdrawing money early? Are there any fees or charges? Explain. 4. Comparing savings alternatives Organize your research to show similarities and differences for the savings alternatives. 5. Compare with another group How is your research the same as the other group s research? What did you learn by comparing the research? 5.5 Savings Alternatives 237

28 Practice 1. What are two advantages and disadvantages of each savings alternative? a) savings account b) term deposit 2. What are some services for which banks charge fees? What can a customer do to reduce or eliminate these fees? 3. The table shows some of the costs associated with Mike s chequing account. One month, Mike s balance dropped to $750 and he made 15 transactions. How much will Mike pay in fees? Monthly fee Transactions $3.50, waived with a minimum monthly balance of $ free, $0.60 per additional transaction 4. A bank offers three different chequing accounts. Account Monthly Fee Transactions A $3.95, waived with minimum monthly balance of $ free, $0.65 per additional transaction B $8.95, waived with minimum monthly balance of $ free, $0.65 per additional transaction C $12.95, waived with minimum monthly balance of $3000 Unlimited a) Describe the type of customer for which each account is best suited. b) Why do you think banks charge higher fees for customers who have low account balances or who make a large number of monthly transactions? 5. Use your research to calculate and compare the interest earned on a principal of $3000 at the end of 1 year under each savings alternative. a) chequing account b) savings account c) term deposit 6. Jennie has accumulated $5000 in her chequing account. a) Why should Jennie consider moving some of the money to another savings alternative? b) What factors should Jennie consider in choosing another savings alternative? Reflect How did you decide to organize your research? Why did you make this decision? What else would you like to know about savings alternatives? Why is this important to you? How could you use your research in your own life? 238 CHAPTER 5: Saving and Investing Money

29 5.6 Present Value We invest money to plan for future expenses or to meet financial goals. The principal that must be invested today to obtain a given amount in the future is called the present value of the amount. Investigate Calculating Present Value Rajah is planning a trip to Europe in 4 years. He will need $3500 at that time. How much must Rajah invest today, at 7% per year, compounded annually, to have $3500 four years from now? Record your solution. How much interest does the investment earn? How do you know? Reflect Compare solutions with a classmate. If you used different strategies, explain your strategies. If you have different answers, try to find out why. Did you use the formula A P(1 i) n to solve the problem? If not, use the formula to solve the problem. 5.6 Present Value 239

30 Connect the Ideas When money is invested, the principal is also called the present value. The value of the investment over time is the amount or future value. Mi Young received $5000 on her 21st birthday. This is the amount of an investment her parents made on the day she was born. The investment earned 10.25% per year, compounded quarterly. Determine the principal invested Check the result Determine the interest earned We can determine the principal that Mi Young s parents invested. Use the formula: A P(1 i) n Substitute: A 5000, i, and n P( ) 84 To solve for P, divide each side by ( ) P Press: 5000 ( ) ^84 ( ) 84 P Mi Young s parents invested $ We can check that the answer is correct. Use the formula: A P(1 i) n Substitute: P , i, and n 84 A ( ) 84 A So, the answer is correct. The interest earned is the difference between the amount and the principal. That is, I A P I $5000 $ $ Why is the amount not exactly $5000? So, Mi Young s parents earned $ interest on their investment. 240 CHAPTER 5: Saving and Investing Money

31 Practice 1. Determine the present value, P, of each amount A. The interest rate is an annual rate. a) $1500 in 3 years at 6% compounded annually b) $2500 in 7 years at 4% compounded semi-annually c) $1300 in 5 years at 8.4% compounded monthly d) $ in 6 years at 8% compounded quarterly e) $5000 in 8.5 years at 10.5% compounded monthly f) $ in 10 years at 12.4% compounded annually 2. Determine the interest earned on each investment in question Irene starts college in 3 years. Her parents want to give Irene $5000 to help with her tuition. How much should they invest now in a bond earning 6.5% per year, compounded semi-annually? 1 4. HBD Corporation needs to replace its computers in 2 2 years at a cost of $ a) How much should it invest now at 4.35% per year, compounded monthly? b) How much interest will be earned? To find the present value, the amount A is divided by (1 i) n. A That is, P. This can be rewritten as P A(1 i) n. (1 i) n 1 Recall that x x 1. 1 So, (1 i ) n (1 i ) n. Example Solution Determine the present value of $7300 in 9 years at 8.4% per year, compounded semi-annually. Use the formula: P A(1 i) n i is 0.0 Press the negative key, 2 not the subtract key. n is Substitute: A 7300, i 0.042, and n 18 P 7300( ) 18 Press: 7300 ( ) ^ ( ) 18 P The present value is $ Choose any three parts in question 1. Use the formula: P A(1 i) n to find the present value. Which method do you prefer? Explain. 5.6 Present Value 241

32 6. Assessment Focus Yvette wants to take a trip in 4 years. She estimates that it will cost her $4500. Yvette has 2 options for saving this amount. Option A: An investment that earns 7.2% per year, compounded monthly Option B: An investment that earns 7.6% per year, compounded semi-annually Which option should Yvette choose? Justify your answer. 7. Determine the present value of each investment. a) $600 in 5 years at 8% per year, compounded monthly b) $1200 in 5 years at 8% per year, compounded monthly c) $600 in 10 years at 8% per year, compounded monthly d) $600 in 5 years at 16% per year, compounded monthly 8. Choose one of the examples from question 7. a) Does doubling the amount double the present value? b) Does doubling the time double the present value? c) Does doubling the interest rate double the present value? Justify your answers. 9. Spencer deposited money in a GIC that earned 4.8% per year, compounded monthly for a 2-year term. He receives $2500 at maturity. How much interest did Spencer earn? 10. Take It Further Janka is planning to attend college 4 years from now. Her grandparents have promised to pay her tuition for the first year. The current cost of tuition is $2100. It is expected to increase by an average of 3% per year for the next 4 years. How much should Janka s grandparents invest today, at 4.2% per year, compounded quarterly, to pay for Janka s tuition? What is the difference between the amount of an investment and the present value of an investment? Use an example in your explanation. 242 CHAPTER 5: Saving and Investing Money

33 5.7 Using the TVM Solver You have solved for A and P in the formula A P(1 i) n. However, you do not know enough mathematics to solve for i or n. To solve for these variables, you can use an application on your graphing calculator called the TVM (Time Value of Money) Solver. When interest is paid on a principal, the amount changes over time. This change is the time value of money. Inquire Solving Financial Problems with the TVM Solver You will need a TI-83 or TI-84 calculator. 1. Introducing the TVM Solver Set the calculator to 2 decimal places. Press: z ~~~ Open the TVM Solver. Press: O 11 You will use 5 of the 8 variables that appear on the screen. If necessary, enter the values shown at the right. N Number of years I% Annual interest rate as a percent PV Principal or present value FV Amount or future value C/Y Number of compounding periods per year For all problems in this section, set: PMT 0.00, P/Y 1.00, PMT: END When you use the TVM Solver: PV is negative because you pay money out when you invest a principal. FV is positive because you receive money when an investment matures. To exit the TVM Solver, press: yz 5.7 Using the TVM Solver 243

34 2. Solving for I (the annual interest rate) a) A principal of $400 amounted to $496 in 4 years. The interest was compounded quarterly. What was the annual interest rate? Enter the given values for N, PV, FV, and C/Y. Press Í(after you input each value. Why is PV 400? Why is C/Y 4? Move the cursor to the row for I%. Press ˆƒ Í( The annual interest rate is 5.41%, compounded quarterly. b) Suppose the interest was compounded annually instead of quarterly. Would a higher or lower annual interest rate be required for $400 to amount to $496 in 4 years? Make a prediction, then use the TVM Solver to find out. Were you correct? Explain. 3. Solving for N (the number of years) a) How long will it take for $1000 to double at 7% per year, compounded semi-annually? Enter the values for I%, PV, FV, and C/Y. Press Í(after you input each value. Why is FV 2000? Why is C/Y 2? Move the cursor to the row for N. Press ˆƒ Í( It takes a little more than 10 years for the money to double. The interest is compounded semi-annually, so round up to the nearest half year. It takes 10.5 years for the money to double. b) Suppose $5000 was invested instead of $1000. Would the doubling time be the same as in part a? Make a prediction, then use the TVM Solver to find out. Were you correct? Explain. 244 CHAPTER 5: Saving and Investing Money

35 4. Solving for FV (the amount) You can use the TVM Solver to investigate the effect of changing the interest rate on the amount of an investment. a) Copy this table. $5000 Invested for 10 Years At Each Rate, Compounded Annually Annual interest rate 6% 7% 8% 9% 10% 12% Amount b) Use the TVM Solver to complete the table. Complete the first column of the table. Enter the values for N, I%, PV, and C/Y. Press Í after you input each value. Move the cursor to the row for FV. Press ˆƒ Í( Why is C/Y 1? The amount is $ Record this value in the table. Use the formula A P(1 i) n to check your answer. Repeat these steps to complete the table. Why does it make sense that the formula A P(1 i) n can be written FV PV(1 i) n? c) Calculate the differences in the amounts in the table. Does an increase of 1% in the interest rate always have the same effect on the amount? Explain. Practice Record the calculator screen in each solution. 1. What interest rate, compounded semi-annually, is required for $500 to amount to $ in 8 years? 2. Amanda has $500 to invest for 7 years. What interest rate, compounded annually, does she require so that her principal doubles? 3. How long will it take $775 to amount to $1000 if it is invested at 6% per year, compounded semi-annually? 5.7 Using the TVM Solver 245

36 4. The screen at the right shows the time it takes for a $250 investment to double in value at 12% per year, compounded annually. Dena rounds N to 6 years. Peter rounds N to 7 years. Who is correct? Explain. 5. Jasmine invests $ that she won in a lottery. How long will it take for her winnings to amount to $ at each annual interest rate? a) 10% compounded quarterly b) 12% compounded annually 6. The table shows the highest and lowest interest rates ever offered on Canada Savings Bonds. How long would it take to double a $500 investment at each interest rate under annual compounding? Year Annual interest rate % % 7. A principal of $450 amounts to $600 in 5 years. a) Determine the annual interest rate under each type of compounding. i) annual ii) semi-annual iii) monthly iv) daily b) Compare the results in part a. What is the relationship between the frequency of compounding and the interest rate? Explain the relationship. 8. Suppose you invest $2500 at 6% per year, compounded annually. a) How long will it take for the investment to double? b) Will doubling the interest rate halve the time it takes for the investment to double? Justify your answer. 9. Investigate the effect on the amount when you change the length of time of an investment. a) Copy and complete the table. $5000 Invested at 8% Per Year, Compounded Semi-annually b) Calculate the differences in the amounts in the table. Time (years) Amount Does the amount increase by the same amount between the time periods? Explain. Reflect What is a goal that you have in the future? How much do you think it would cost to achieve that goal? Use the TVM solver to analyse different scenarios of how you might achieve your goal. Show what you did. 246 CHAPTER 5: Saving and Investing Money

37 Maximum Investor Materials TVM Solver on TI-83 or TI-84 graphing calculator 3 dice score pad timer Play in a group of 3. Each player starts with a principal of $100. Each player rolls one die. The player with the highest number starts. Play continues around to the left. Each player takes turns rolling the dice. The player chooses: one number to be the annual interest rate another number to be the number of compounding periods per year the third number to be the time in years The player has 1 min to use the numbers to calculate the amount. For example, the principal is $100 and suppose the dice show 5, 2, and 6. Here are two possible amounts. $100 invested at 6% per year, $100 invested at 5% per year, compounded semi-annually compounded semi-annually for 5 years. for 6 years. The second option is the better choice because it produces the greater amount. The amount calculated is the player s principal for the next round. If another player can determine a higher amount, it gets added to that player s principal. The first player to reach $1000 or more wins. GAME: Maximum Investor 247