# You plan to retire 40 years from now. After retirement you want to be able to withdraw $20,000...

## Question:

You plan to retire 40 years from now. After retirement you want to be able to withdraw $20,000 annually at the end of each year from retirement account for 20 years. You plan to save a given amount of money at the end of each of the next 40 years in your retirement account. Assuming that funds invested will earn a 10% annual rate, how much do you need to invest each year?

## Saving for Retirement:

401(K) is a type of retirement saving account that individuals contribute to on a regular basis. The future value of the account determines the amount of money available fore retirement spending.

## Answer and Explanation: 1

If you will be able to earn 10% per year, the discounted value of the annuity of spending , at the beginning of retirement, is:

- {eq}\dfrac{20,000*(1 - (1 + 10\%)^{-20})}{10\%} = 170,271.27 {/eq}

Given a 10% annual interest rate, denote the annual deposit by *M*. For the deposits to be able to fund future expenditures, we must have:

- {eq}\dfrac{M*((1 + 10\%)^{40} - 1)}{10\%} = 170,271.27 {/eq}

- {eq}M * 442.5925557 = 170,271.27 {/eq}

- {eq}M = 384.71 {/eq}

That is, the amount you need to invest each year is $384.71.

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Chapter 21 / Lesson 15An 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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