For retirement planning, you decided to deposit $1,000 per month and increase your deposit by $100 per month. How much will you have at the end of 10 years if the bank pays 3% annually, compounded monthly?
The given problem can be solved easily by using the compound interest table. In this case, it is given that the amount of monthly savings in first month is \(\$ 1000\) and it increases by \(\$ 100\) every month. The bank pays \(3 \%\) interest rate, compounded monthly. Therefore, the rate of interest is \(3 / 12\), that is, \(0.25 \%\).
Formula used to calculate the value of \(P\) is as follows:
$$ P=A(P / A, i, n)+G(P / G, i, n) $$
Here,
\(P=\) initial investment=?
\(A=\) amnual amount that is uniform over the time period- \(\$ 1,000\)
\(G=g r a d i e n t\) value- \(\$ 100\)
\(\mathrm{l}=\) rate of return \(=0.25 \%\)
\(n=\) time period \(=120\) months
Substituting the values into the above equation and solve for \(P_{2}\) we get:
$$ \begin{aligned} P &=\$ 1,000(P / A, 0.25 \%, 120)+\$ 100(P / G, 0.25 \%, 120) \\ &=\$ 1,000 \times 103.563+\$ 100 \times 5852.112 \\ &=\$ 103,563+\$ 585,211.2 \\ &=\$ 688,7742 \end{aligned} $$
Therefore, the initial investment is \(\$ 688.774 .2\)
Now, compute the future value of this investment
$$ F=P(F / P, l, n) $$
Here,
$$ \begin{aligned} &P=\text { initial investment- } \$ 688,774.2 \\ &F=\text { future value }=? \\ &I=\text { rate of retum }=0.25 \% \\ &n=\text { time period }=120 \end{aligned} $$
Substituting the values into the above equation and solve for \(\mathrm{F}_{2}\) we get:
$$ \begin{aligned} F &=\$ 688,7742(P / F, 0.25 \%, 120) \\ &=\$ 688,7742 \times 1.349 \\ &=\$ 929,156.395 \end{aligned} $$
Therefore, the depositor can get \(\$ 929,156\) (approx.) at the time of retirement.
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