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6.4 Savings & Retirement Plans

6.4 Savings & Retirement Plans
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  • 21. Savings Goals

    For an increasing annuity, the monthly deposit M that you must make for n months, at an annual percentage rate of r (in decimal form), to achieve a balance of A is

    You start saving for retirement at age 25. You want to have $1 million when you retire in 42 years. You invest in a savings plan that earns 6%, compounded monthly.

    1. How much should you deposit each month?
    2. Suppose you wait until you are 30 to start saving. How much more do you have to deposit each month compared to the amount in part (a)?
    • Worked-Out Solution
      1. Use the formula given above with A = $1,000,000, r = 6%, and n = 12(42) = 504.

      2. Use the formula given above with A = $1,000,000, r = 6%, and n = 12(37) = 444.

        By waiting 5 more years to start your retirement plan, you would need to deposit

        more each month to end up with the same balance.

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  • 22. Savings Goals

    For an increasing annuity, the monthly deposit M that you must make for n months, at an annual percentage rate of r (in decimal form), to achieve a balance of A is

    You want to have $20,000 to help pay for your child's college education in 18 years. You invest in a savings plan that earns 4.8%, compounded monthly.

    1. How much should you deposit each month?
    2. Suppose you want to have the money in 10 years. How much more do you have to deposit each month compared to the amount in part (a)?
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     __   _      ___      ______    ______    ______  
    | || | ||   / _ \\   /_   _//  /_   _//  /_____// 
    | '--' ||  / //\ \\  `-| |,-    -| ||-   `____ `  
    | .--. || |  ___  ||   | ||     _| ||_   /___//   
    |_|| |_|| |_||  |_||   |_||    /_____//  `__ `    
    `-`  `-`  `-`   `-`    `-`'    `-----`   /_//     
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  • 23. Annual Percentage Yield

    The annual percentage yield (APY) is the rate at which an investment increases each year. The formula for the APY of an investment with an annual percentage rate of r that is compounded n times a year is

    Find the APY for an investment that earns 6% for each compounding period.

    1. a.  Daily
    2. b.  Monthly
    3. c.  Quarterly
    1. d.  Semiannually
    2. e.  Annually
    • Worked-Out Solution
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     _    _      ___     ______      ___     _    _   
    | || | ||   / _ \\  |      \\   / _ \\  | |  | || 
    | || | ||  / //\ \\ |  --  //  / //\ \\ | |/\| || 
    | \\_/ || |  ___  |||  --  \\ |  ___  |||  /\  || 
     \____//  |_||  |_|||______// |_||  |_|||_// \_|| 
      `---`   `-`   `-` `------`  `-`   `-` `-`   `-` 
                                                      
    
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  • 24. Annual Percentage Yield

    The annual percentage yield (APY) is the rate at which an investment increases each year. The formula for the APY of an investment with an annual percentage rate of r that is compounded n times a year is

    Find the APY for an investment that earns 7% for each compounding period.

    1. a.  Daily
    2. b.  Monthly
    3. c.  Quarterly
    1. d.  Semiannually
    2. e.  Annually
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      ______     ___     _    _     _____      _____  
     /_   _//   / _ \\  | || | ||  |  ___||   / ___// 
       | ||    | / \ || | || | ||  | ||__     \___ \\ 
      _| ||    | \_/ || | \\_/ ||  | ||__     /    // 
     /__//      \___//   \____//   |_____||  /____//  
     `--`       `---`     `---`    `-----`  `-----`   
                                                      
    
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  • 25. Annual Percentage Yield

    The annual percentage yield (APY) is the rate at which an investment increases each year. The formula for the APY of an investment with an annual percentage rate of r that is compounded n times a year is

    For what compounding period is the APY the same as the APR? Explain your reasoning.

    • Worked-Out Solution

      For annual compounding, the annual percentage yield is the same as the annual percentage rate.

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      _____     _____     ____       ___     ______   
     / ____||  |  ___||  |  _ \\    / _ \\  |      \\ 
    / //---`'  | ||__    | |_| ||  | / \ || |  --  // 
    \ \\___    | ||__    | .  //   | \_/ || |  --  \\ 
     \_____||  |_____||  |_|\_\\    \___//  |______// 
      `----`   `-----`   `-` --`    `---`   `------`  
                                                      
    
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  • 26. Adjustable Percentage Yield

    The annual percentage yield (APY) is the rate at which an investment increases each year. The formula for the APY of an investment with an annual percentage rate of r that is compounded n times a year is

    Which of the following earns more interest annually?

    1. An investment with an APY of 6%
    2. An investment with an APR of 5.9%, compounded monthly
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     _____      ______    _____    __   __    ______  
    |  __ \\   /_   _//  / ____||  \ \\/ //  /_____// 
    | |  \ ||   -| ||-  / //---`'   \ ` //   `____ `  
    | |__/ ||   _| ||_  \ \\___      | ||    /___//   
    |_____//   /_____//  \_____||    |_||    `__ `    
     -----`    `-----`    `----`     `-`'    /_//     
                                             `-`      
    
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