Sinking fund

**yellowtree** · April 9th 2011, 06:12 AM

Hi,
I’m 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

**emakarov** · April 10th 2011, 04:51 AM

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}= \begin{cases} b_n(1+r_n)+4,000, & n< 25\\ b_{25}(1+r_{25}), & n=25 \end{cases} $ and $r_{n}= \begin{cases} 8\%, & b_n\ge 20,000\\ 7\%, & 5,000\le b_n<20,000\\ 6\%, & \mbox{otherwise} \end{cases} $

**yellowtree** · April 11th 2011, 04:45 AM

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

**emakarov** · April 11th 2011, 07:42 AM

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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