Question

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?

Answer 1

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.