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10.E: Compound Interest- Applications Involving Single Payments (Exercises)

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    34967
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    10.1: Application - Long-Term GICs

    Mechanics

    Calculate the amount of each interest payment as well as the total interest paid over the entire term for each of the following interest payout GICs.

    Principal (PV) Nominal Interest Rate (IY) Term
    1. $43,000 6% quarterly 4 years
    2. $38,100 6.6% monthly 2 years

    Calculate the missing pieces of the puzzle (as indicated by ?) for the following compound interest GICs. Cells with N/A are not required.

    Principal Nominal Interest Rate and Term Compounding Frequency Maturity Value Interest Amount Earned Equivalent Fixed Interest Rate
    3. $17,400 4.3% for 4 years semi-annually ? ? N/A
    4. $25,000 2.9% for first year; 3.9% for second year quarterly ? ? ?%quarterly
    5. $51,000 3.9% for 5 years monthly ? ? N/A
    6. $72,375 1.85% for 6 months; then 2.1% for 18 months; then 2.35% for 2 years Semi-annually ? ? ?%annually

    Calculate the missing pieces of the puzzle (as indicated by ?) for the following escalator rate GICs. Each interest rate listed is for a one-year term.

    Principal Nominal Interest Rate and Term Compounding Frequency Maturity Value Interest Amount Earned Equivalent Fixed Interest Rate
    7. $2,680 0.75%; 1.25%; 1.85% Annually ? ? ?% annually
    8. $134,000 1.15%; 1.95%; 2.5%; 4%; 7% Semi-annually ? ? ?% semi-annually
    9. $84,630 0.15%; 1.1%; 3.9%; 6.8% Monthly ? ? ?% annually
    10. $15,985 0.25%; 0.75%; 1.9%; 2.75%; 7.1% Quarterly ? ? ?% annually

    Applications

    1. Sanchez placed $11,930 into a five-year interest payout GIC at 4.2% compounded monthly. Calculate the interest payout amount every month.
    2. RBC is offering you a four-year fixed rate compound interest GIC at 3.8% compounded quarterly. TD Canada Trust is offering you a variable rate compound interest GIC at 3.4% for the first two years, followed by 4.25% for the remaining two years. If you have $10,000 to invest, which GIC should you select? How much more interest will you earn from the selected GIC than from the alternative?
    3. Your five-year variable rate GIC is due to mature. Your monthly compounded interest rate was 1.5% for 22 months, 1.75% for 9 months, 1.6% for 13 months, and 1.95% for the remainder. If you originally invested $7,165, what is the maturity value? What equivalent monthly compounded fixed rate of interest did the GIC earn?
    4. For three different five-year compound interest GICs, you can choose between fixed rates of 4.8% compounded monthly, 4.83% compounded quarterly, or 4.845% compounded semi-annually. On a $25,000 investment, how much more interest will the best option earn compared to the worst option?
    5. The RBC RateAdvantage escalator GIC offers annually compounded interest rates on a five-year nonredeemable GIC of 0.75%, 2.25%, 2.5%, 3%, and 5%. Calculate the maturity value of a $40,000 investment. Determine the annually compounded equivalent fixed rate earned on the investment.
    6. TD Canada Trust is offering its five-year Stepper GIC at annually escalating rates of 1.15%, 2%, 2.75%, 3.5%, and 4.5%. All rates are compounded semi-annually. Alternatively, it is offering a five-year fixed rate GIC at 2.7% compounded monthly. What total interest amount does an $18,000 investment earn under each option?
    7. Your escalator rate GIC has annually compounded interest rates of 1.15%, 1.95%, 3.2%, 4.7%, and 6.6% in subsequent years. On a $15,140 investment, calculate the following:
      1. The amount of interest earned in each year.
      2. The annually compounded equivalent fixed rate.

    Challenge, Critical Thinking, & Other Applications

    1. You have invested $6,365 into a three-year interest payout GIC earning 3.75% compounded annually. If you invest each of your interest payout amounts into a Builder GIC (which allows you to contribute regularly to the GIC) earning 2.4% compounded monthly, how much interest in total do you earn over the three-year term?
    2. Three years ago you placed $50,000 into an RBC five-year fixed rate, nonredeemable, compound interest GIC earning 3.9% compounded quarterly. Interest rates have risen since that initial investment, and banks today are advertising rates of 4.5% compounded semi-annually on two-year compound interest GICs. Your RBC GIC allows you to redeem your investment early, but you would incur a 0.75% penalty per year on the maturity value of the account. Should you keep your current RBC GIC, or should you take the early withdrawal penalty and invest in a new two-year GIC? Show calculations to support your recommendation.
    3. Calculate the interest earned on each of the following five-year GICs. Rank the GICs from best to worst based on the amount of interest earned on a $15,000 investment.
      1. An interest payout GIC earning 4.5% compounded quarterly.
      2. A fixed rate compound interest GIC earning 4.2% compounded monthly.
      3. A variable rate quarterly compound interest GIC earning consecutively 3.9% for 1.5 years, 4.25% for 1.75 years, 4.15% for 0.75 years, and 4.7% for 1 year.
      4. An escalator rate GIC earning semi-annually compounded rates of 1.25%, 2%, 3.5%, 5.1%, and 7.75% in successive years.

    10.2: Application - Long-Term Promissory Notes

    Mechanics

    Calculate the unknown variable (indicated with a ?) for the following noninterest-bearing promissory notes.

    Principal Issue Date Due Date Sale Date Discount Rate Sale Proceeds
    1. $10,000 August 14, 2010 November 14, 2015 February 14, 2012 5.95% compounded quarterly $?
    2. $19,000 May 29, 2010 May 29, 2015 September 29, 2012 8.7% compounded monthly $?
    3. $? September 30, 2009 September 30, 2014 March 30, 2012 6.8% compounded semi-annually $21,574.34
    4. $31,300 June 3, 2009 June 3, 2015 November 3, 2012 ?% compounded monthly $27,268.08

    Calculate the unknown variable(s) (indicated with a ?) for the following interest-bearing promissory notes.

    Issue Amount Term of Note Interest Rate on Note Date of Sale (before maturity) Discount Rate Sale Proceeds
    5. $51,000 6 years 7.75% compounded quarterly 2½ years 12% compounded semi-annually $?
    6. $18,200 4½ years 9.45% compounded monthly 1 year, 6 months 15% compounded quarterly $?
    7. $5,350 3¼ years 6.95% compounded monthly 1½ years ?% compounded semi-annually $5,587.02
    8. $2,900 4 years 8.8% compounded semi-annually 9 months ?% compounded monthly $3,570.13
    9. $? 5 years 7.5% compounded quarterly 2 years, 6 months 13.25% compounded semi-annually $11,705.14
    10. $26,945.75 6 years 10.3% compounded annually ? years, ? months 19.3% compounded monthly $32,046.52

    Applications

    1. Determine the proceeds of the sale on a seven-year noninterest-bearing promissory note for $1,600, discounted 45 months before its due date at a discount rate of 8.2% compounded quarterly.
    2. If a noninterest-bearing four-year promissory note is discounted 1½ years before maturity at a discount rate of 6.6% compounded semi-annually to have proceeds of $14,333.63, what is the principal amount of the note?
    3. Determine the proceeds of the sale on a six-year interest-bearing promissory note for $5,750 at 6.9% compounded monthly, discounted two years and three months before its due date at a discount rate of 9.9% compounded quarterly.
    4. A three-year interest-bearing promissory note for $8,900 at 3.8% compounded annually is sold one year and two months before its due date at a discount rate of 7.1% compounded monthly. What is the amount of the discount on the sale?
    5. Two years and 10 months before its due date, an eight-year interest-bearing promissory note for $3,875 at 2.9% compounded semi-annually is discounted to have sale proceeds of $4,182.10. What monthly compounded discount rate was used?
    6. On May 1, 2012, a six-year interest-bearing note for $8,800 at 4.99% compounded monthly dated August 1, 2009, is discounted at 8% compounded semi-annually. Determine the proceeds on the note.
    7. A seven-year interest-bearing note for $19,950 at 8.1% compounded quarterly is issued on January 19, 2006. Four years and 11 months later, the note is discounted at 14.55% compounded monthly. Determine the proceeds on the note and how much interest the original owner of the note realized.

    Challenge, Critical Thinking, & Other Applications

    1. A $36,555 interest-bearing note at 5% compounded monthly is issued on October 15, 2011, for a term of 87 months. Fifty-seven months later, the note is sold to yield a discount amount of $11,733.41. What quarterly compounded discount rate is being used?
    2. On December 12, 2012, an eight-year promissory note at 6.2% compounded semi-annually with three years and three months remaining until its due date is sold. The discount rate is 10.9% compounded monthly, resulting in a discount of $49,349.87. Calculate the original principal of the note.
    3. A 10-year, $100,000 note at 7.5% compounded quarterly is to be sold. The prevailing discount rate for promissory notes of this type today, six years before maturity, is 9% compounded annually. Prevailing discount rates are forecast to rise by 2% every year. The seller of the note will sell the note today, one year from today, or two years from today, depending on which sale date produces the highest nominal proceeds. Rank the various alternatives and recommend a sale date.

    10.3: Application - Savings Bonds

    For all questions in this section, refer to the two tables for savings and premium bonds for series numbers and interest rates on various bonds.

    Mechanics

    For each of the following R-bonds, calculate the total interest earned as of the date of redemption.

    Series Principal Redemption Date
    1. P64 $33,000 November 1, 2011
    2. S111 $17,800 February 1, 2012
    3. P72 $52,300 January 30, 2013
    4. S104 $18,600 January 2, 2012

    For each of the following C-bonds, calculate the maturity value and the total interest earned as of the date of redemption.

    Series Principal Redemption Date
    5. P47 $18,900 December 15, 2011
    6. S87 $76,400 September 18, 2011
    7. S103 $29,900 November 12, 2011
    8. P55 $2,500 February 15, 2013
    9. P50 $6,200 March 21, 2012
    10. S110 $45,700 October 15, 2011

    Applications

    1. A $24,900 Series S91 R-bond CSB is redeemed on December 1, 2011. Calculate the total interest earned.
    2. A $65,500 Series P57 C-bond CPB is redeemed on April 1, 2013. Calculate the maturity value and the total interest earned.
    3. A $103,100 Series S117 C-bond CSB is redeemed on September 3, 2011. Calculate the maturity value and the total interest earned.
    4. Calculate the total interest earned on a $57,600 Series P59 R-bond CPB if it is redeemed in 2013.
    5. Calculate the total interest earned on a $38,200 Series S98 R-bond CSB if it is redeemed on August 15, 2011.
    6. What amount would the owner of a $22,200 Series S93 C-bond CSB receive if the bond is redeemed on January 31, 2012? How much of that amount is interest?
    7. How much more would the owner of a Series P43 C-bond CPB have than the owner of a Series P45 C-bond CPB if both invested $10,000 and both were redeemed in 2014?
    8. How much more total interest would the owner of a Series P36 R-bond CPB have earned than the owner of a Series P38 R-bond CPB if both invested $100,000 and both were redeemed in 2012?

    Challenge, Critical Thinking, & Other Applications

    1. Investing is always a bit of a guessing game as to when to invest. Looking at 2005 Series S92 to S97 C-bond CSBs, rank the 2005 series from best to worst if an equal amount of money had been invested into each and all were matured on their anniversary dates in 2011. Show calculations to support your rankings.
    2. Most people do not have one single investment at any one time. Rather, they have multiple investments started at different points as money becomes available to them to invest. Let's say an investor is able to invest $3,000 each November 1 and April 1 starting with November 1, 2004, and he purchases C-bond CSBs with the money each time. Calculate the total maturity value of his investments on November 1, 2011 (not including the $3,000 he would invest on that day).

    10.4: Application - Strip Bonds

    Mechanics

    For each of the following, calculate the purchase price of the strip bond and the gain realized over the term.

    Face Value Years Remaining Until Maturity Yield
    1. $157,500 15 4.1965%
    2. $250,000 9.5 3.6926%
    3. $12,050 30 4.0432%
    4. $81,735 17.5 4.2561%

    For each of the following, calculate the semi-annual yield on the strip bond and the gain realized over the term.

    Purchase Price Years Remaining Until Maturity Face Value
    5. $9,087.13 26 $100,000
    6. $17,537.39 12 $50,000
    7. $24,947.97 21.5 $75,000

    For each of the following, determine the actual yield realized by the investor and the gain realized over the time the strip bond was held.

    Purchase Sale Face Value
    8. 22 years before maturity; Market yield 6.3815% 14.5 years before maturity; Market yield 7.8643% $80,000
    9. 14 years before maturity; Market yield 10.1831% 5 years before maturity; Market yield 9.1777% $5,000
    10. 28 years before maturity; Market yield 6.8205% 11.5 years before maturity; Market yield 7.2173% $500,000

    Applications

    1. A $15,000 face value Government of Manitoba strip bond has 19.5 years left until maturity. If the current market rate is posted at 6.7322%, what is the purchase price for the bond?
    2. A $300,000 face value Government of Canada strip bond will mature on June 1, 2041. If the yield on such strip bonds is 4.6849%, what was the purchase price on December 1, 2012?
    3. An $87,000 face value strip bond from the Government of Alberta matures on February 14, 2033. If an investor paid $23,644.78 on February 14, 2011, what was the semi-annually compounded yield on the strip bond?
    4. A $150,000 face value bond was issued on August 8, 2008, with a 30-year maturity. If an investor paid $40,865.24 on February 8, 2013, what was the market rate of return on the strip bond at the time of the purchase?
    5. An investor purchased a $7,500 face value strip bond for $2,686.01 on May 29, 2006. The strip bond had been issued on May 29, 2002, with a 25 -year maturity. The investor sold the strip bond on November 29, 2012, for $3,925.28.
      1. What was the market yield when the investor purchased the strip bond?
      2. What was the market yield when the investor sold the strip bond?
      3. What actual yield did the investor realize on the strip bond?
    6. A $70,000 face value strip bond was issued on October 6, 1998, with 30 years remaining until maturity. An investor purchased the bond on April 6, 2005, when market yields were 4.9529%. The investor sold the bond on October 6, 2012, when market yields were 5.2185%.
      1. What was the purchase price of the strip bond?
      2. What was the sale price of the strip bond?
      3. What was the actual yield the investor realized on the strip bond?
      4. What was the total gain realized by the investor?

    Challenge, Critical Thinking, & Other Applications

    1. On August 1, 2005, a $100,000 Government of Canada strip bond was issued with a 25-year maturity. An investor paid $30,597.14 for the strip bond. On February 1, 2012, the investor sold the bond for $44,604.78. What was the difference in yields in the market between the issue date and the selling date?
    2. Sebastien is the finance manager for a large corporation that needs to raise $100 million for a project today. Rounded to the nearest million, what face value of strip bonds would need to be issued if they are to mature in five years and the current market yield on five-year strip bonds is posted at 5.7391%?
    3. Antoine just inherited $50,000 from his grandfather's estate. If he is considering investing in 20-year maturity strip bonds posted at 5.8663%, how many $1,000 denomination strip bonds could he purchase with his inheritance?
    4. Consider the following three situations. For each, use a $100,000 strip bond issued with eight years until maturity for your calculations.
      • Assume the market yield is a constant 4.5%. Calculate the purchase price each year until maturity. Plot the eight purchase prices onto a line chart.
      • Assume the market yield initially is set at 3% when issued and rises by 0.375% each year until maturity. Plot the eight purchase prices onto the same line chart.
      • Assume the market yield initially is set at 6% when issued and decreases by 0.375% each year until maturity. Plot the eight purchase prices onto the same line chart. Looking at the line charts for all three situations, what do you see? Explain the results.

    10.5: Application - Inflation, Purchasing Power, and Rates Of Change

    Mechanics

    Solve the following inflation questions for the unknown variable (indicated with a ?). Note that the inflation values in each question represent an average for an actual time period in Canadian history.

    Beginning Value Ending Value Annual Inflation and Term
    1. $30,000.00 ? 2.68% for 20 years
    2. $50,288.00 ? 2.58% for 3 years; then 0.35% for 2 years
    3. ? $40,000.00 5.77% for 25 years
    4. $15,000.00 $48,905.33 ?% for 17 years
    5. $21,000.00 $78,969.53 7.22% for ? years

    For questions 6 and 7, use the information provided to calculate the purchasing power of a dollar from the base year to the end of the specified term. Note that the inflation values in each question are the actual averages for the specified time period in Canadian history.

    Base Year Annual Inflation and Term
    6. 1978 9.68% for 5 years
    7. 1989 5.33% for 2 years; then 0.91% for 3 years

    Solve the following rates of change questions for the unknown variable (indicated with a ?). Round all answers to two decimals.

    Beginning Value Ending Value Percent Change and # Times in a Row
    8. 98,000 ? 3.5% for 10 periods
    9. 235,000 ? 2.3% for 4 periods, then −1.6% for 2 periods
    10. ? 47,500 −5% for 5 periods

    Applications

    1. In 1982, the average price of a new car sold in Canada was $10,668. By 2009, the average price of a new car had increased to $25,683. What average annual rate of change in car prices has been experienced during this time frame?
    2. From September 8, 2007, to November 7, 2007, the Canadian dollar experienced one of its fastest periods of appreciation against the US dollar. It started at $0.9482 and rose 0.2514% per day on average. What was the final value of the Canadian dollar rounded to four decimals on November 7, 2007?
    3. The average price of a detached home in Calgary in 2005 was $246,308. Prices rose by 10.4834% on average per year for four years, and then changed by −2.7248%. What is the new price of a home in Calgary in 2010 rounded to the nearest thousand dollar?
    4. Example 10.5.1 used the average inflation rate of Canada for the past 97 years as an indicator of future inflation. Some critics argue that using “outdated” inflation rates is not a good representation of the future. Recall that you are 20 years old in 2012 and the comfortable retirement income is $40,000. The historical inflation rate over the past 40 years for Canada has averaged 4.5% from 1971 to 2011.
      1. Recalculate your required retirement income at age 65.
      2. What will be the purchasing power of a 2012 dollar in 45 years?
    5. In the late 1970s and early 1980s, Canada went through a period of high inflation. How did the purchasing power of a 1975 dollar decrease if the average inflation rate from May 1975 to May 1985 was 8.2% per year?
    6. The world population in 2011 was approximately 6.9 billion people. At the current annual population growth rate of 1.1%, how long will it take the world's population to double (round to the nearest year)?

    Challenge, Critical Thinking, & Other Applications

    1. Cadbury Canada is thinking about launching a new chocolate bar. In the first year, sales are forecast to hit 750,000 units. The marketing department has then forecast sales increases of 50% in each of the next five years, followed by annual increases of 15% in the years following. Cadbury will launch a new chocolate bar only if it forecasts annual sales of 22 million units by the end of the first 15 years. Should Cadbury Canada launch the new chocolate bar?
    2. How much money would have been required 20 years ago to have the same purchasing power of $10,000 today if the average rate of inflation per year for the period has been 2%?
    3. Measures of life expectancy account for various factors, including smoking and body mass index. Since 1975, when 33.3% of the population smoked, smoking has been declining at a rate of 1.71% per year on average. In the same time frame, the average body mass index (BMI) started at 25.2 and increased by 0.42% per year on average. While reduced smoking prolongs life, the increasing BMI shortens life. The average life expectancy of a Canadian is 80.7 years old. It is forecasted, though, to decrease by 0.0756% per year on average based on the trends in smoking and BMI. Assume these trends continue and calculating the following (rounded to the nearest year):
      1. In what year will half of the population be considered obese (a BMI average of 30)?
      2. In what year will less than 10% of the population be smokers?
      3. In what year will life expectancy fall below 75 years of age?
    4. Future inflation rates can only be speculated about. Using the recommended $40,000 of retirement income in 2012 when you are 20 years old, create a range of possible retirement incomes needed when you retire at age 65. The average Canadian lives to 80.7 years old, so your retirement income needs to be regularly updated after your retirement to reflect the increased costs of living.
      1. Start with a low inflation rate of 2%. Forecast the needed retirement income at age 65, 70, and 75 years.
      2. Repeat the process using inflation rates of 3%, 4%, and 5%.
      3. Graph the results on a line chart. For all three retirement ages, specify the range of income required. Calculate the average income needed at each age of retirement.
      4. Why do you think it is important to do these types of calculations?

    Review Exercises

    For all questions that involve Canada Savings Bonds, refer to the two tables on savings and premium bonds as needed.

    Mechanics

    1. Guido placed $28,300 into a five-year regular interest GIC with interest of 6.3% compounded semi-annually. Determine the total interest Guido will earn over the term.
    2. An eight-year, $35,000 noninterest-bearing promissory note is discounted 6% compounded quarterly and sold to a finance company three years and nine months after issue. What are the proceeds of the sale?
    3. A $9,800 Series S100 R-bond CSB was redeemed on March 1, 2012. Calculate the total interest earned on the investment.
    4. On April 27, 1990, Graham purchased a $100,000 face value 25-year Government of Quebec strip bond. The market yield on such bonds was 13.3177%. What was the purchase price for the strip bond?
    5. In retirement, Bill determined that he could safely invest $50,000 into a nonredeemable five-year GIC with a posted rate of 5.35% compounded semi-annually. Calculate the maturity value and the amount of interest earned on the investment.
    6. In 2012, a $10,000 Series P52 C-bond CPB was redeemed. Calculate the total interest earned on the bond.
    7. In 1979, Shania earned $10 per hour at her place of employment. From 1979 to 1982, the average annual inflation in Canada was 11.36%. Determine what Shania's hourly wage needed to be in 1982 to keep up with inflation.
    8. In September 2004, Google employed 2,688 workers. Over the next two years, the number of employees grew at an average of 86.7843% per year. How many employees did Google have in September 2006?

    Applications

    1. Entegra Credit Union offers a five-year escalator GIC with annual rates of 1.65%, 2.4%, 2.65%, 2.95%, and 3.3%. Determine the maturity value and total interest earned on an investment of $6,000 along with the equivalent five-year fixed rate.
    2. A 21-month $6,779.99 promissory note bearing interest of 7.5% compounded monthly was sold on its date of issue to a finance company at a discount rate of 9.9% compounded monthly. Determine the proceeds of the sale.
    3. A $20,500 Series S104 C-bond CSB was redeemed on its maturity date in 2011. Calculate the maturity value and the total interest earned on the bond.
    4. On January 19, 2012, a Series P36 C-bond CPB was redeemed for $19,549.90. How much interest is included in the maturity value?
    5. On September 30, 2011, a $34,000 Series S93 C-bond CSB was redeemed.
      1. Calculate the maturity value and total interest earned on the bond.
      2. How much more interest could have been earned if the bond was not redeemed until October 1, 2011?
    6. On February 7, 1990, a $100,000 face value 25-year Government of Saskatchewan strip bond was purchased for $9,365.85. On February 7, 2005, the investor sold the strip bond for $65,900.
      1. What was the posted yield on strip bonds on the date of issue?
      2. What was the posted yield on strip bonds when it was sold?
      3. What actual semi-annual rate of return did the original investor realize on the strip bond investment?
    7. General Motors has watched its market share decline over the past several decades. In 1980, GM held a 46% share of the market. From 1980 to 2003, it declined on average by 2.09% annually. From 2003 to 2007, it declined on average by 9.36% annually. Calculate GM's 2007 market share rounded to one decimal in percentage format.

    Challenge, Critical Thinking, & Other Applications

    1. Canada suffered a prolonged period of deflation in the early 1930s. From 1930 to 1933, the rate of deflation averaged 8.27% per year. Calculate the purchasing power of a 1930 dollar in 1933.
    2. On October 10, 1995, a $100,000 Government of Canada 30-year strip bond was purchased at a posted yield of 8.1258%. The investor sold the strip bond on October 10, 2005, and realized an actual yield of 15.9017% on the investment. What was the prevailing yield on strip bonds when the strip bond was sold on October 10, 2005?
    3. From 2004 to 2008, the average family after-tax annual earnings increased from $68,200 by 2.2677% per year. The inflation rate during that time period was 1.71%, 2.43%, 2.19%, and 3.13% in successive years, respectively. Determine the amount that Canadian family after-tax annual earnings have increased or decreased in 2008. Show calculations to support your answers.
    4. The research and development department forecasts that it will require $100 million in funding for a project scheduled for implementation four years from today. If the company wants to place $80 million into a $500 million 25-year strip bond, what is the minimum by which the yield in the market needs to change for the department to have sufficient funds when the project is started?
    5. On November 1, 2004, Andy invested $50,000 into a Series P40 R-bond CPB. On the 2005, 2007, and 2009 anniversary dates, he took his regular interest payment and acquired a CSB Series C-bond on the same date. On the 2006 and 2008 anniversary dates, he took his regular interest payment and acquired a CPB Series C-bond on the same date. On November 1, 2010, calculate the maturity value of all of his investments. Determine the total interest earned from November 1, 2004, to November 1, 2010.

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