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You plan to work for the next 39 years and earn 6.5% on your investments during your working...

Question:

You plan to work for the next 39 years and earn 6.5% on your investments during your working years. In retirement, you expect to need $82,000 per year during each of your 25 years of retirement and (during this time) you expect to earn 4.6% on investments.

How much do you need to invest at the end of each year (while working) to allow this to happen?

Annuity:

Annuity refers to the series of an equal sum of money deposited or paid at an interval period. Some example of an annuity is the insurance premiums, monthly mortgage payments, and monthly pension.

Answer and Explanation: 1

Step 1 of 2:

Calculate the required future value of the retirement fund using the following formula:

{eq}PV_{oa}=A*\frac{1-(1+r)^{-n}}{r}\\ whereas:\\ A=annuity\\ r=interest~rate\\ n=number~of~periods\\ {/eq}

{eq}\begin{align*} PV_{oa}&=82,000*\frac{1-(1+.04)^{25}}{.04}\\ &=82,000*\frac{1-.3751}{.04}\\ &=82,000*\frac{.6249}{.04}\\ &=82,000*15.6221\\ &=1,281,010.56 \end{align*} {/eq}

The required amount of the retirement fund to have an annual withdrawal of $82,000 for 25 years, which is put in an account earnings 4% is $1,281,010.56


Step 2 of 2:

Calculate the required annual investment for 39 years using the following formula:

{eq}Annuity=\displaystyle \frac{LV}{\frac{(1+r)^{n}-1}{r}}\\ whereas:\\ LV=loan~value\\ r=interest~rate\\ n=number~of~periods\\ {/eq}


{eq}\begin{align*} Annuity&=\frac{1,281,010.56}{\frac{(1+.065)^{39}}{.065}}\\ &=\frac{1,281,010.56}{\frac{(1.065)^{39}-1}{.065}}\\ &=\frac{1,281,010.56}{\frac{11.658286-1}{.065}}\\ &=\frac{1,281,010.56}{\frac{10.658286}{.065}}\\ &=\frac{1,281,010.56}{163.97363}\\ &=7,812.30\\ \end{align*} {/eq}

The required annual investment at the end of each year for 39 years is $7,812.30


Learn more about this topic:

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How to Find the Value of an Annuity

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Chapter 21 / Lesson 15
9.4K

An annuity is a type of savings account that pays back the investor in the future. Learn the formula used to calculate an annuity's value, and understand the importance of labeling specific numbers to calculate an output over time.


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