Accumulated Fund Value -

**ninmaster** · April 5th 2008, 03:33 AM

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 $i=1.25j$

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

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

If some one could help clarify, does the $1000 earn interest at i p.a and any extra earn interest at j p.a?

**mathceleb** · April 5th 2008, 08:51 AM

Originally Posted by ninmaster

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 $i=1.25j$

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

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

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:

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

We also know that j = 0.8i.

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

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

Try to multiply and rearrange terms from here and let me know how you do.

**mathceleb** · April 5th 2008, 03:54 PM

Also, note one more property for this problem that will help you:

$s_{\bar n|j} * (1 + j)^m = s_{\bar m+n|j} - s_{\bar m|j}$

**ninmaster** · April 5th 2008, 03:54 PM

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.

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

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

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

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

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

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

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

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

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

Factor out 1250

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

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

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

$AV = 1250\left(s_{\bar15|j}-s_{\bar5|j}-2\right)$

**mathceleb** · April 5th 2008, 04:47 PM

Originally Posted by ninmaster

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:

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

Looks like it is close to what you did.

**jonah** · September 19th 2008, 10:42 AM

Originally Posted by ninmaster

I hope my working seem correct?

Almost.
$ \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} $
$ \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} $
$ \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} $
$ \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} $
$ \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} $
$ 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 $ 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 $ 10,000i \cdot s_{\left. {\overline {\, 4 \,}}\! \right| j} $
We then have
$ \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} $
$ \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} $
$ \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} $
$ \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} $
$ \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} $
$ \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} $
$ \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} $
$ \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} $
$ 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
$ 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
$ AV = 1250\left( {\frac{{\left( {1 + j} \right)^{15} - \left( {1 + j} \right)^5 }} {j} - 2} \right) $

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