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Math Help - Annual Percentage Growth Rate

  1. #1
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    Annual Percentage Growth Rate

    The value of an asset rose from 110,000 to 694,694 in 26 years.

    Calculate the average annual percentage growth rate of the value of the asset assuming:

    (a) growth in annual discrete 'jumps'

    (b) continuous growth


    Any help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by Archimedes View Post
    The value of an asset rose from 110,000 to 694,694 in 26 years.

    Calculate the average annual percentage growth rate of the value of the asset assuming:

    (a) growth in annual discrete 'jumps'
    The value grows from v to (1+r/100)v in 1 year, so after 26 years will have grown from v to (1+r/100)^{26} v, where r is the percentage annual growth rate. So using the numbers in the question we have:

    694694 = (1+r/100)^{26} 110000

    So after some algebra we have:

    r=\left(\left(\frac{694694}{110000}\right)^{1/26}-1\right)\times 100

    RonL
    Last edited by CaptainBlack; August 1st 2008 at 07:09 AM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Archimedes View Post
    The value of an asset rose from 110,000 to 694,694 in 26 years.

    Calculate the average annual percentage growth rate of the value of the asset assuming:

    (a) growth in annual discrete 'jumps'

    (b) continuous growth
    With continuous growth at r\% annual rate we have:

     <br />
\frac{dv}{dt}=\frac{r}{100} \ v<br />

    where time is measured in years. So:

    v=e^{(r/100)t}v_0

    or in our case:

     <br />
694694=e^{(r/100)26}\ 110000<br />
,

    and after some algebra:

     <br />
r=\ln\left( \frac{694694}{110000} \right) \times \frac{100}{26}<br />

    RonL
    Last edited by CaptainBlack; August 1st 2008 at 07:09 AM.
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  4. #4
    Member jonah's Avatar
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    So after some algebra we have:

    ,

    and after some algebra:


    There's a slight typo on CaptainBlack's part. 100000 in both cases should have been 110,000.
    Equivalently,
    (a)
    P = 110,000
    F = 694,694
    j = nominal rate compounded annually [average annual percentage growth rate of the value of the asset assuming: (a) growth in annual discrete 'jumps']
    m = 1 (as in annually)
    t = 26 years
    <br />
F = P\left( {1 + \frac{j}<br />
{m}} \right)^{tm}  \Leftrightarrow j = m\left[ {\left( {\frac{F}<br />
{P}} \right)^{\frac{1}<br />
{{tm}}}  - 1} \right] \approx .0734569906<br /> <br /> <br />
    (b)
    P = 110,000
    F = 694,694
    j = nominal rate compounded continuously [average annual percentage growth rate of the value of the asset assuming: (b) continuous growth]
    t = 26 years
    <br />
F = Pe^{jt}  \Leftrightarrow j = \frac{{\ln \left( {\frac{F}<br />
{P}} \right)}}<br />
{t} \approx .07088427289<br /> <br /> <br />
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by jonah View Post
    There's a slight typo on CaptainBlack's part. 100000 in both cases should have been 110,000.
    Equivalently,
    (a)
    P = 110,000
    F = 694,694
    Deliberate error to see if everyone was awake

    RonL
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