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Math Help - Need help with interest (Nominal)

  1. #1
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    Exclamation Need help with interest (Nominal)

    I need help with how to calculate nominal annual rate.

    1) A sum of $850 is invested for 10 years and the interest is compounded quarterly. There is $1000 in the account at the end of 10 years. What is the nominal annual rate?

    So to set this up is the equation:
    1000 = 850(1+r/4)^(4*10) and we solve for r? I don't know if that equation is correct. But when I solve for it the answer comes out to be 0.0163 or 1.63%. Please confirm with me if that answer is correct.


    2) An investment grows by 3% per year for 10 years. By what percent does it increase over the 10-year period?

    Is the formula: P(1 + 0.03/1)^(1*10) and we are trying to find the effective rate. So should the answer be 34.39%?


    3) If the balance, M, at time t in years, of a bank account that compounds its interest payments monthly is given by
    M = M0(1.07763)^t
    Note: In the M0, the 0 is a subscript.

    What is the nominal annual rate?
    Again I need help with nominal annual rate. In the back of the book it says the nominal annual rate is 7.5%. But I do not get how to find that answer. Please shed some light on this issue and help me with finding out how to solve this problem.

    Thank you so much.
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  2. #2
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    Quote Originally Posted by krzyrice View Post
    I need help with how to calculate nominal annual rate.

    1) A sum of $850 is invested for 10 years and the interest is compounded quarterly. There is $1000 in the account at the end of 10 years. What is the nominal annual rate?

    So to set this up is the equation:
    1000 = 850(1+r/4)^(4*10) and we solve for r? I don't know if that equation is correct. But when I solve for it the answer comes out to be 0.0163 or 1.63%. Please confirm with me if that answer is correct.


    2) An investment grows by 3% per year for 10 years. By what percent does it increase over the 10-year period?

    Is the formula: P(1 + 0.03/1)^(1*10) and we are trying to find the effective rate. So should the answer be 34.39%?


    3) If the balance, M, at time t in years, of a bank account that compounds its interest payments monthly is given by
    M = M0(1.07763)^t
    Note: In the M0, the 0 is a subscript.

    What is the nominal annual rate?
    Again I need help with nominal annual rate. In the back of the book it says the nominal annual rate is 7.5%. But I do not get how to find that answer. Please shed some light on this issue and help me with finding out how to solve this problem.

    Thank you so much.
    1. A = P(1 + r)^t.

    Since this is being compounded quarterly for 10 years, there will be 40 time periods and the rate will be the rate every quarter.

    So 1000 = 850(1 + r)^{40}

    \frac{20}{17} = (1 + r)^{40}

    \left(\frac{20}{17}\right)^{\frac{1}{40}} = 1 + r

    r = \left(\frac{20}{17}\right)^{\frac{1}{40}} - 1.

    This is the quarterly interest rate. So multiply by 4 to get the yearly interest rate.
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  3. #3
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    Quote Originally Posted by krzyrice View Post
    2) An investment grows by 3% per year for 10 years. By what percent does it increase over the 10-year period?

    Is the formula: P(1 + 0.03/1)^(1*10) and we are trying to find the effective rate. So should the answer be 34.39%?
    Year 0: A = P

    Year 1: A = 1.03 \cdot P

    Year 2: A = 1.03\cdot 1.03 \cdot P

    Year 3: A = 1.03 \cdot 1.03 \cdot 1.03 \cdot P.


    Can you see that for year n the amount will be

    A = 1.03^n\cdot P.


    So in 10 years: A = 1.03^{10}\cdot P

    \approx 1.3439 \cdot P.

    So it has increased by just over 34.39\%.
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    Thank you so much. I got the same answers but wasn't sure if I did them right.
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