Answer :
In this case we are making deposits each year in out account. This will be compounded at a 6% interest.
After 20 years of working and making deposits, the principal accumulated has to be able to give $30,000 per year for 30 years.
We can start with the principal. We assume that during retirement, the account yield 6% interest from the capital still in the account.
Then, we can calculate this principal as the present value of an annuity with payments PMT = $30,000, rate of interest r = 0.06 and n = 30 years.
The calculation is:
[tex]\begin{gathered} PV=PMT\cdot\frac{1-(1+r)^{-n}}{r} \\ PV=30000\cdot\frac{1-(1.06)^{-30}}{0.06} \\ PV\approx30000\cdot\frac{1-0.17411}{0.06} \\ PV\approx30000\cdot\frac{0.82589}{0.06} \\ PV\approx30000\cdot13.7648 \\ PV\approx412944.93 \end{gathered}[/tex]Now we know the amount of capital that has to be accumulated with the deposits.
This value is the future value of an annuity of n = 20 years at a rate r = 0.06.
But now, we need to calculate the yearly deposit D.
We can calculate it as:
[tex]\begin{gathered} D=\frac{FV\cdot r}{(1+r)^n-1} \\ D=\frac{412944.93\cdot0.06}{(1.06)^{20}-1} \\ D\approx\frac{24776.70}{3.207-1} \\ D\approx\frac{24776.70}{2.207} \\ D\approx11225.73 \end{gathered}[/tex]Answer: the yearly deposit has to be $11,225.73.
