
Sinking fund
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 higherinterest account at the earliest opportunity.
Any help would be much appreciated.
Thanks

1 Attachment(s)
Here is an Excel table that I made: Attachment 21414.
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 $\displaystyle b_n$ at the beginning of year $\displaystyle n$. The third column contains the interest rate $\displaystyle r_n$ used at the end of year $\displaystyle n$. Then $\displaystyle b_1=4,000$ and
$\displaystyle b_{n+1}=
\begin{cases}
b_n(1+r_n)+4,000, & n< 25\\
b_{25}(1+r_{25}), & n=25
\end{cases}
$ and $\displaystyle r_{n}=
\begin{cases}
8\%, & b_n\ge 20,000\\
7\%, & 5,000\le b_n<20,000\\
6\%, & \mbox{otherwise}
\end{cases}
$

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

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 $\displaystyle b_5$, after which the interest rate stabilizes. Let $\displaystyle r=1+r_5=\dots=1+r_{25}=1.08$. Then
$\displaystyle b_6=r b_5+4000$
$\displaystyle b_7=r(r b_5+4000)+4000=r^2b_5+4000(r+1)$
$\displaystyle b_8=r(r(r b_5+4000)+4000)+4000=r^3b_5+4000(r^2+r+1)$
Following the pattern and using the formula for the sum of geometric progression,
$\displaystyle b_{5+k}=r^kb_5+4000(r^{k1}+\dots+r+1)=r^k+4000(r^k1)/(r1)$
Thus,
$\displaystyle b_{25}=b_{5+20}=r^{20}b_5+4000(r^{20}1)/(r1)$
and $\displaystyle b_{26}=rb_{25}$.