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Math Help - Annual effective rate of interest

  1. #1
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    Annual effective rate of interest

    A bank offers the following certificates of deposit:
    Term in years / Nominal annual interest rate compounded semi-annually
    1 - 5%
    2 - 6%
    3 - 7%
    4 - 8%
    The bank does not permit early withdrawals. The certificates mature at the end of the term. During the next 6 years the bank will continue to offer these certificates of deposit. An investor initially deposits $10,000 in the bank and withdraws both principle and interest at the end of 6 years. Calculate the maximum annual effective rate of interest the investor can earn over the 6-year period. (The answer is given as 7.466%)

    So I found the accumulated values for each to be, respectively: 10506.3 , 11225.1 , 12292.6 , 13685.7 ... but I'm not sure how to set it up to get the annual effective rates of interest.

    Thank you!
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  2. #2
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    Quote Originally Posted by tbl9301 View Post
    A bank offers the following certificates of deposit:
    Term in years / Nominal annual interest rate compounded semi-annually
    1 - 5%
    2 - 6%
    3 - 7%
    4 - 8%
    The bank does not permit early withdrawals. The certificates mature at the end of the term. During the next 6 years the bank will continue to offer these certificates of deposit. An investor initially deposits $10,000 in the bank and withdraws both principle and interest at the end of 6 years. Calculate the maximum annual effective rate of interest the investor can earn over the 6-year period. (The answer is given as 7.466%)

    So I found the accumulated values for each to be, respectively: 10506.3 , 11225.1 , 12292.6 , 13685.7 ... but I'm not sure how to set it up to get the annual effective rates of interest.

    Thank you!
    You must also determine which Option is best ?:
    4 years @8% + 2 yrs @ 6%
    or
    2 years @6% + 4 years @8%
    or
    3 years @7% + 3 yrs at 7%
    or
    6 years @5%

    I found the accumulated values for each to be, respectively: 10506.3 , 11225.1 , 12292.6 , 13685.7
    How did you get those numbers?
    I ask because:
    (1.05)^{6} \times 10000 = 13400.96
    3400.96/6 = 566.83 average per year
    &
    5% compounded semiannually is 2.5% per 6 months.
    (1.025)^{12} \times 10000 = 13448.89
    Total interest earned is $3448.89 or $574.81 average per year.

    I do not understand your 10506.3.
    Could you explain what the 10506.3 value actually means?

    .
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  3. #3
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    Oh I was setting it up like, for the 5% for 1 year:
    10000(1 + .05/2)2*1 = 10000(1.025)2 = 10506.30

    Is that the wrong formula to use for this case?
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  4. #4
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    Quote Originally Posted by tbl9301 View Post
    Oh I was setting it up like, for the 5% for 1 year:
    10000(1 + .05/2)2*1 = 10000(1.025)2 = 10506.30

    Is that the wrong formula to use for this case?
    That helps me understand the question.

    5%::  \left ( \left ( 1 + \dfrac{0 .05}{2} \right ) ^2 - 1 \right ) 100 :: yields 5.0625 %

    6%:: \left( \left ( 1+\dfrac{0.06}{2} \right)^2 - 1 \right ) 100 :: yields 6.0900%

    7%:: \left( \left ( 1+\dfrac{0.07}{2} \right)^2 - 1 \right ) 100 :: yields 7.1225%

    8%:: \left ( \left ( 1+\dfrac{0.04}{2} \right)^2 - 1 \right ) 100 :: yields 8.1600%

     \dfrac {4 ( 8.1600 ) + 2 ( 6.0900 )}{6} = \dfrac{44.82}{6} = 7.4700
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  5. #5
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    or (1.04^8 1.03^4)^(1/6) - 1 = .0746555.... ; ~7.47%

    Amount invested does not affect calculation.
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