# You plan to start saving for your retirement by depositing $10,000 exactly one year from now.... ## Question: You plan to start saving for your retirement by depositing$10,000 exactly one year from now. Each year you intend to increase your retirement deposit by 2%. You plan on retiring 30 years from now, and you will receive 4% interest compounded annually.

a) How much will have in your retirement account at the end of 30 years if you follow this strategy (i.e., you make 30 annual deposits)?

b) In year 10, you have a sudden expense, and you do not deposit any money at the end of year 10. All of your other deposits remain the same as in part a. In other words, you still deposit 12,189.94 at the end of year 11. How much money will you have in your retirement account at the end of 30 years? ## Skipped deposit: Skipped deposit can be explained as an amount that was required to be deposited by the depositor but could not be deposited due to certain reasons or conditions. Such deposits reduce the accumulated amount for the depositor. ## Answer and Explanation: 1 a. Calculation of amount in the retirement account: {eq}\begin{align*} {\rm\text{Amount}}& = \frac{{{\rm\text{Cash flow in year 1}}}}{{{\rm\text{Return rate}} - {\rm\text{Growth rate}}}} \times \left[ {1 - {{\left\{ {\frac{{(1 + {\rm\text{Growth rate}})}}{{(1 + {\rm\text{Return rate}})}}} \right\}}^{\rm\text{Number of periods}}}} \right]\\ & = \frac{{10,000}}{{0.04 - 0.02}} \times \left[ {1 - {{\left\{ {\frac{{(1 + 0.02)}}{{(1 + 0.04)}}} \right\}}^{30}}} \right]\\ & = 500,000 \times (1 - 0.5584765908)\\ & = \ 220,761.7046 \end{align*} {/eq}

b. Calculation of the amount in this case:

{eq}\begin{align*} {\rm\text{Amount}}& = \frac{{{\rm\text{Cash flow in year 1}}}}{{{\rm\text{Return rate}} - {\rm\text{Growth rate}}}} \times \left[ {1 - {{\left\{ {\frac{{(1 + {\rm\text{Growth rate}})}}{{(1 + {\rm\text{Return rate}})}}} \right\}}^{{\rm\text{Number of periods}}}}} \right] - {\rm\text{Value of the missed amount}}\\ &= \frac{{10,000}}{{0.04 - 0.02}} \times \left[ {1 - {{\left\{ {\frac{{(1 + 0.02)}}{{(1 + 0.04)}}} \right\}}^{30}}} \right] - 10,000 \times {(1 + 0.02)^{10}} \times {(1 + 0.04)^{20}}\\ & = 500,000 \times (1 - 0.5584765908) - (10,000 \times 2.6709668848)\\ & = 220,761.7046 - 26,709.668848\\ & = \\$ 194,052.035752 \end{align*} {/eq}