You are trying to plan for retirement in 10 years, and currently you have $350,000 in savings...

Question:

You are trying to plan for retirement in 10 years, and currently you have $350,000 in savings account and $600,000 in stocks. In addition, you have plans on adding to your savings by depositing $10,000 per year in your savings account at the end of each of the next five years and then $15,000 per year at the end of each year for the final five years until retirement.

Assume your savings account returns 3.5 percent compounded annually while your investment in stock returns 10.5 percent compounded annually, how much will you have at the end of 10 years. If you expect to live for 20 years after you retire, and at retirement, you deposit all of your savings in a bank account paying 3.5 percent, how much can you withdraw each year after retirement to end up with a zero balance at the end of 20 years?

Payout Annuity:

Payout annuity refers to the fixed equal amounts withdrawn at the end of each uniform interval. It is dependent upon the amount in the account in the beginning, rate of interest on the savings and the number of years of withdrawal.

Answer and Explanation: 1

Step 1: Calculate the future value at the end of 10 years of the savings

There are 3 deposits:

1. $350,000 already present in the account

2. $10,000 per year for the first 5 years

3. $15,000 per year for the last 5 years

{eq}\begin{align*} {\rm\text{Future value of savings}}\left( {{F_S}} \right)& = \$ 350,000 \times FV{F_{\left( {0.035,10} \right)}} + \$ 10,000 \times FVA{F_{\left( {0.035,5} \right)}} \times FV{F_{\left( {0.035,5} \right)}} + \$ 15,000 \times FVA{F_{\left( {0.035,5} \right)}}\\ & = \$ 350,000 \times 1.4106 + \$ 10,000 \times 5.3625 \times 1.1877 + \$ 15,000 \times 5.3625\\ & = \$ 493,710 + \$ 63,690.41 + \$ 80,437.5\\ & = \$ 634,837.91 \end{align*} {/eq}

Step 2: Calculate the future value of investment in stocks

{eq}\begin{align*} {\rm\text{Future value of investment}}\left( {{F_i}} \right)& = \$ 600,000 \times FVA{F_{\left( {0.105,10} \right)}}\\ & = \$ 600,000 \times 16.3246\\ & = \$ 9,794,760 \end{align*} {/eq}

Step 3: Calculate the total future value at the end of 10 years.

{eq}\begin{align*} {\rm\text{Total FV}}& = {F_s} + {F_i}\\ & = \$ 634,837.91 + \$ 9,794,760\\ & = 10,429,597.91 \end{align*} {/eq}

Step 4: Calculate the amount of annual withdrawal (D)

This total value of $10,429,597.91 at the end of 10 years becomes the present value for annual withdrawals.

{eq}\begin{align*} \$ 10,429,597.91& = \frac{{D\left( {1 - {{\left( {1 + r} \right)}^{ - n}}} \right)}}{r}\\ \$ 10,429,597.91& = \frac{{D\left( {1 - {{\left( {1 + 0.035} \right)}^{ - 20}}} \right)}}{{0.035}}\\ \$ 10,429,597.91& = D \times 14.212\\ D& = \$ 733,858.56 \end{align*} {/eq}

Thus, he can withdraw $733,858.56 per year for 20 years from the savings account which leaves 0 amount at the end.


Learn more about this topic:

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Present and Future Value: Calculating the Time Value of Money

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Chapter 11 / Lesson 2
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Study the time value of money formula. Learn the time value of money definition and practice how to calculate time value of money to understand the relation to purchasing power.


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