Results 1 to 6 of 6

Thread: Accumulated Fund Value -

  1. #1
    Newbie
    Joined
    Mar 2008
    From
    America
    Posts
    8

    Accumulated Fund Value -

    An investor deposits $1000 at the beginning of each year for 10 years in a fund earning i effective p.a. The interest from this fund can be reinvested at only j effective p.a. Whereby $\displaystyle i=1.25j $

    Prove the total accumulated value at the end of 14 years is:

    $\displaystyle 1250(s_{\bar15|j}-s_{\bar5|j}-2) $

    If some one could help clarify, does the $1000 earn interest at i p.a and any extra earn interest at j p.a?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2008
    From
    Berkeley, Illinois
    Posts
    377
    Thanks
    1
    Quote Originally Posted by ninmaster View Post
    An investor deposits $1000 at the beginning of each year for 10 years in a fund earning i effective p.a. The interest from this fund can be reinvested at only j effective p.a. Whereby $\displaystyle i=1.25j $

    Prove the total accumulated value at the end of 14 years is:

    $\displaystyle 1250(s_{\bar15|j}-s_{\bar5|j}-2) $

    If some one could help clarify, does the $1000 earn interest at i p.a and any extra earn interest at j p.a?
    We note that the Accumulated value of an annuity due reinvestment fund is:

    $\displaystyle n + i(\frac {s_{\bar(n+1)|j}- (n + 1)}{j})$

    We also know that j = 0.8i.

    Plugging and chugging, our AV at end of year 10 is $\displaystyle 1250s_{\bar11|j} - 3750$

    Now, we need to roll that balance up 4 more years at a rate of j, so we multiply that by $\displaystyle (1 + j)^4$

    Try to multiply and rearrange terms from here and let me know how you do.
    Last edited by mathceleb; Apr 5th 2008 at 09:06 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Feb 2008
    From
    Berkeley, Illinois
    Posts
    377
    Thanks
    1
    Also, note one more property for this problem that will help you:

    $\displaystyle s_{\bar n|j} * (1 + j)^m = s_{\bar m+n|j} - s_{\bar m|j} $
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Mar 2008
    From
    America
    Posts
    8
    Im not too sure if your formula is correct. The investment is at the beginning of the year so the interest is earnt at the end of year right?

    Just putting in values they seem to be different, but it gives me ideas.

    $\displaystyle AV = 1000\left(10+i({(1+j)^{9} + 2(1+j)^8 + .... + 10}\right)) $

    $\displaystyle AV = 1000\left(10 + i\left((1+j)^{9} + 2(1+j)^8 + .... + 10\right)\right)
    $

    $\displaystyle AV = 1000\left(10 + i(Is_{\bar10|j})\right)
    $

    $\displaystyle AV = 1000\left(10 + i\left(\displaystyle\frac{\ddot{s}_{\bar10|j}-10}{j}\right)\right)$

    $\displaystyle AV = 1000\left(10 + 1.25\left(\displaystyle\frac{(1+j)^{11}-(1+j)^1}{j}-10\right)\right)$

    This would be for the end of year 10. Then we multiply by $\displaystyle (1+j)^4 $ as you suggested?

    $\displaystyle AV = 1000\left(10(1+j)^4 + 1.25\left(\displaystyle\frac{(1+j)^{15}-(1+j)^5}{j}-10(1+j)^4\right)\right)$

    $\displaystyle AV = 1000\left(10((1+j)^4-1.25(1+j)^4) + 1.25\left(\displaystyle\frac{(1+j)^{15}-(1+j)^5}{j}\right)\right)$

    $\displaystyle AV = 1000\left(10(1+j)^4(1-1.25) + 1.25\left(\displaystyle\frac{(1+j)^{15}-(1+j)^5}{j}\right)\right)$

    Factor out 1250

    $\displaystyle AV = 1250\left(\displaystyle\frac{(1+j)^{15} - (1+j)^5}{j} - 2\right)$

    $\displaystyle AV = 1250\left(\displaystyle\frac{(1+j)^{15} - 1 - ((1+j)^5 - 1)}{j} - 2\right)$

    Which simplifies to the answer, I hope my working seem correct?

    $\displaystyle AV = 1250\left(s_{\bar15|j}-s_{\bar5|j}-2\right)$
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Feb 2008
    From
    Berkeley, Illinois
    Posts
    377
    Thanks
    1
    Quote Originally Posted by ninmaster View Post
    Im not too sure if your formula is correct. The investment is at the beginning of the year so the interest is earnt at the end of year right?

    Just putting in values they seem to be different, but it gives me ideas.
    Your question states the payment is at the beginning of the year, so that is the annuity due reinvestment formula.

    The end of year formula as I understand it is similar, but it is as follows:

    $\displaystyle n + i(\frac {s_{\bar(n)|j}- n)}{j})$

    Looks like it is close to what you did.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member jonah's Avatar
    Joined
    Apr 2008
    Posts
    226
    Thanks
    27
    Quote Originally Posted by ninmaster View Post
    I hope my working seem correct?
    Almost.
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 10 years}}} = 1,000\left\{ {10 + i\left[ \begin{gathered}
    1\left( {1 + j} \right)^9 + 2\left( {1 + j} \right)^8 + \cdot \cdot \cdot \hfill \\
    + 8\left( {1 + j} \right)^2 + 9\left( {1 + j} \right)^1 + 10\left( {1 + j} \right)^0 \hfill \\
    \end{gathered} \right]} \right\} \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 10 years}}} = 1,000\left\{ {10 + i\left[ {1 \cdot \frac{{\left( {1 + j} \right) \cdot s_{\left. {\overline {\,
    {10} \,}}\! \right| j} - 10}}
    {j}} \right]} \right\} \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 10 years}}} = 1,000\left\{ {10 + \frac{i}
    {j}\left[ {\left( {1 + j} \right) \cdot \frac{{\left( {1 + j} \right)^{10} - 1}}
    {j} - 10} \right]} \right\} \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 10 years}}} = 1,000\left\{ {10 + \frac{{1.25j}}
    {j}\left[ {\frac{{\left( {1 + j} \right)\left( {1 + j} \right)^{10} - \left( {1 + j} \right)}}
    {j} - 10} \right]} \right\} \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 10 years}}} = 1,000\left\{ {10 + 1.25\left[ {\frac{{\left( {1 + j} \right)^{11} - \left( {1 + j} \right)}}
    {j} - 10} \right]} \right\} \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    AV_{{\text{at the end of 10 years}}} = 10,000 + 1,250\left[ {\frac{{\left( {1 + j} \right)^{11} - \left( {1 + j} \right)}}
    {j} - 10} \right]
    $

    Starting at the end of year 10, you have three accounts to consider:

    A $10,000, i % bond like fund that pays coupons annually for 4 years.

    The accumulated value of the reinvested interest at the end of 10 years from the annual beginning of year deposits as represented by $\displaystyle
    1,250\left[ {\frac{{\left( {1 + j} \right)^{11} - \left( {1 + j} \right)}}
    {j} - 10} \right]
    $. This amount is then to be accumulated for 4 more years.

    The annual end of year interest from the $10,000 bond like fund which will start at the end year 11 and end at the end of year 14. This is represented by $\displaystyle
    10,000i \cdot s_{\left. {\overline {\,
    4 \,}}\! \right| j}
    $
    We then have
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 14 years}}} = 10,000 \hfill \\
    + 1,250\left[ {\frac{{\left( {1 + j} \right)^{11} - \left( {1 + j} \right)}}
    {j} - 10} \right]\left( {1 + j} \right)^4 + 10,000i \cdot s_{\left. {\overline {\,
    4 \,}}\! \right| j} \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 14 years}}} = 10,000 \hfill \\
    + 1,250\left[ {\frac{{\left( {1 + j} \right)^{15} - \left( {1 + j} \right)^5 }}
    {j} - 10\left( {1 + j} \right)^4 } \right] \hfill \\
    + 10,000\left( {1.25j} \right) \cdot \frac{{\left( {1 + j} \right)^4 - 1}}
    {j} \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 14 years}}} = 10,000 + 1,250 \cdot \frac{{\left( {1 + j} \right)^{15} - \left( {1 + j} \right)^5 }}
    {j} \hfill \\
    - 12,500\left( {1 + j} \right)^4 + 12,500\left[ {\left( {1 + j} \right)^4 - 1} \right] \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 14 years}}} = 10,000 + \hfill \\
    1,250 \cdot \frac{{\left( {1 + j} \right)^{15} - \left( {1 + j} \right)^5 }}
    {j} \hfill \\
    - 12,500\left( {1 + j} \right)^4 + 12,500\left( {1 + j} \right)^4 - 12,500 \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 14 years}}} = 1,250 \cdot \frac{{\left( {1 + j} \right)^{15} - \left( {1 + j} \right)^5 }}
    {j} - 2,500 \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 14 years}}} = 1,250\left[ {\frac{{\left( {1 + j} \right)^{15} - \left( {1 + j} \right)^5 }}
    {j} - 2} \right] \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 14 years}}} = 1,250\left\{ {\frac{{\left[ {\left( {1 + j} \right)^{15} - 1} \right] - \left[ {\left( {1 + j} \right)^5 - 1} \right]}}
    {j} - 2} \right\} \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    \begin{gathered}
    AV_{{\text{at the end of 14 years}}} = 1,250\left[ {\frac{{\left( {1 + j} \right)^{15} - 1}}
    {j} - \frac{{\left( {1 + j} \right)^5 - 1}}
    {j} - 2} \right] \hfill \\
    \Leftrightarrow \hfill \\
    \end{gathered}
    $
    $\displaystyle
    AV_{{\text{at the end of 14 years}}} = 1,250\left( {s_{\left. {\overline {\,
    {15} \,}}\! \right| j} - s_{\left. {\overline {\,
    5 \,}}\! \right| j} - 2} \right)
    $

    Note: Methinks I detected two flaws in your work. One, you neglected/forgot/(or you were simply unaware) to factor the annual end of year interest from the $10,000 bond like fund which will start at the end year 11 and end at the end of year 14. Two, you added an inconsistency to an already flawed work when from
    $\displaystyle
    AV = 1000\left( {10\left( {1 + j} \right)^4 \left( {1 - 1.25} \right) + 1.25\left( {\frac{{\left( {1 + j} \right)^{15} - \left( {1 + j} \right)^5 }}
    {j}} \right)} \right)
    $
    there was sudden transformation into
    $\displaystyle
    AV = 1250\left( {\frac{{\left( {1 + j} \right)^{15} - \left( {1 + j} \right)^5 }}
    {j} - 2} \right)
    $
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: Dec 5th 2011, 05:26 PM
  2. Replies: 2
    Last Post: Oct 7th 2010, 10:12 AM
  3. Interest accumulated
    Posted in the Business Math Forum
    Replies: 4
    Last Post: Sep 15th 2009, 09:13 PM
  4. Accumulated value of an annuity
    Posted in the Business Math Forum
    Replies: 11
    Last Post: Sep 15th 2009, 01:23 PM
  5. Accumulated Amount of Money
    Posted in the Business Math Forum
    Replies: 1
    Last Post: Dec 3rd 2008, 06:41 PM

Search Tags


/mathhelpforum @mathhelpforum