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Math Help - Sinking fund

  1. #1
    Newbie
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    Sinking fund

    Hi,
    Iím trying to solve this problem;

    A bank has three types of account in which the interest rate depends on the amount invested. The ordinary account offers a return of 6% and is available to any customer. Extra account offers 7%, only available to customers with 5,000$ or more to invest, and super extra account offers 8% and is available only to customers with 20,000$ or more to invest. In each case interest is compounded annually and is added to the investment at the end of the year. A person saves 4,000$ at the beginning of the year for 25 years. Calculate the total amount saved assuming the money is transferred to a higher-interest account at the earliest opportunity.

    Any help would be much appreciated.
    Thanks
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  2. #2
    MHF Contributor
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    718
    Here is an Excel table that I made: fund.xls.
    Code:
    Year	Balance         Interest
    1	$4,000.00	0.06
    2	$8,240.00	0.07
    3	$12,816.80	0.07
    4	$17,713.98	0.07
    5	$22,953.95	0.08
    6	$28,790.27	0.08
    7	$35,093.49	0.08
    8	$41,900.97	0.08
    9	$49,253.05	0.08
    10	$57,193.29	0.08
    11	$65,768.76	0.08
    12	$75,030.26	0.08
    13	$85,032.68	0.08
    14	$95,835.29	0.08
    15	$107,502.12	0.08
    16	$120,102.28	0.08
    17	$133,710.47	0.08
    18	$148,407.31	0.08
    19	$164,279.89	0.08
    20	$181,422.28	0.08
    21	$199,936.06	0.08
    22	$219,930.95	0.08
    23	$241,525.42	0.08
    24	$264,847.46	0.08
    25	$290,035.25	0.08
    26	$313,238.07	0.08
    The second column contains the balance b_n at the beginning of year n. The third column contains the interest rate r_n used at the end of year n. Then b_1=4,000 and
    b_{n+1}=<br />
\begin{cases}<br />
b_n(1+r_n)+4,000, & n< 25\\<br />
b_{25}(1+r_{25}), & n=25<br />
\end{cases}<br />
and r_{n}=<br />
\begin{cases}<br />
8\%, & b_n\ge 20,000\\<br />
7\%, & 5,000\le b_n<20,000\\<br />
6\%, & \mbox{otherwise}<br />
\end{cases}<br />
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  3. #3
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    Thanks for the reply. The answer you got is the same as in the textbook. However this question is in preparation for an exam and I will not be able to use EXCEL. If anyone is able to produce the an answer without using excel I would be very grateful.
    Thanks again for your answer.
    Cheers
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  4. #4
    MHF Contributor
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    718
    Well, you probably need a calculator unless you are amazing at calculations, rounding and estimates. In fact, you can do all 25 iterations in reasonable time. It is also possible to calculate b_5, after which the interest rate stabilizes. Let r=1+r_5=\dots=1+r_{25}=1.08. Then

    b_6=r b_5+4000
    b_7=r(r b_5+4000)+4000=r^2b_5+4000(r+1)
    b_8=r(r(r b_5+4000)+4000)+4000=r^3b_5+4000(r^2+r+1)

    Following the pattern and using the formula for the sum of geometric progression,

    b_{5+k}=r^kb_5+4000(r^{k-1}+\dots+r+1)=r^k+4000(r^k-1)/(r-1)

    Thus,

    b_{25}=b_{5+20}=r^{20}b_5+4000(r^{20}-1)/(r-1)

    and b_{26}=rb_{25}.
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