Santana is planning for his pension savings:
- He plans to work for $20$ years and then retire.
- He expects to receive an annuity income of $\$\, 20000$ at the start of each year of his retirement for $30$ years.
- If the interest rate remains unchanged at $3\,\%$, how much should Santana put into his pension account at the start of each working year $?$.
*I tried to use the PV due annuity formula $$ {A\left(1 + i\right)\left[1 - \left(1 + i\right)^{-n}\right] \over i} $$ to solve this question but couldn't get the answer. It might be something to do with the $20$ and $30$ years but I don't know what is wrong. The answer is: $\$\,14589$. Please help $!$. Thank you $!$.
Indeed it is not a good idea to use a formula for such a question, math from high school may help in understanding it better. Consider the following equations:
After 20-year work, the pension account, $$ T_{t=20} = \sum_{i=1}^{20}S_{aving}\times(1+r_{ate})^i $$ For the next 30 years, total saving decreases. At the last year, $$ T_{t=50} = T_{t=20}\times(1+r_{ate})^{30-1} -\sum_{i=1}^{30}a_{nnuity}\times(1+r_{ate})^{30-i} $$ We expect $T_{t=50} = 0$. Then solve the equation, we get $$ \begin{aligned} \left(\sum_{i=1}^{20}S_{aving}\times(1+r_{ate})^i\right)\times(1+r_{ate})^{30-1} &= \sum_{i=1}^{30}a_{nnuity}\times(1+r_{ate})^{30-i}\\ \left(\sum_{i=1}^{20}S_{aving}\times(1+0.03)^i\right)\times(1+0.03)^{30-1} &= \sum_{i=1}^{30}20000\times(1+0.03)^{30-i}\\ 27.67648 \times S_{aving}\times 2.35656 &\approx 951508.31412 \\ S_{aving} &\approx 14589 \end{aligned} $$