# Payout Annuities

In the last section you learned about annuities. In an annuity, you start with nothing, put money into an account on a regular basis, and end up with money in your account.
In this section, we will learn about a variation called a **Payout Annuity**. With a payout annuity, you start with money in the account, and pull money out of the account on a regular basis. Any remaining money in the account earns interest. After a fixed amount of time, the account will end up empty.
Payout annuities are typically used after retirement. Perhaps you have saved $500,000 for retirement, and want to take money out of the account each month to live on. You want the money to last you 20 years. This is a payout annuity. The formula is derived in a similar way as we did for savings annuities. The details are omitted here.

*P0*is the balance in the account at the beginning (starting amount, or principal).

*d*is the regular withdrawal (the amount you take out each year, each month, etc.)

*r*is the annual interest rate (in decimal form. Example: 5% = 0.05)

*k*is the number of compounding periods in one year.

*N*is the number of years we plan to take withdrawals

### Example 9

After retiring, you want to be able to take $1000 every month for a total of 20 years from your retirement account. The account earns 6% interest. How much will you need in your account when you retire? In this example,*d*= $1000 the monthly withdrawal

*r*= 0.06 6% annual rate

*k*= 12 since we’re doing monthly withdrawals, we’ll compound monthly

*N*= 20 since were taking withdrawals for 20 years We’re looking for

*P0*; how much money needs to be in the account at the beginning. Putting this into the equation: [latex]\begin{align}&{{P}_{0}}=\frac{1000\left(1-{{\left(1+\frac{0.06}{12}\right)}^{-20(12)}}\right)}{\left(\frac{0.06}{12}\right)}\\&{{P}_{0}}=\frac{1000\times\left(1-{{\left(1.005\right)}^{-240}}\right)}{\left(0.005\right)}\\&{{P}_{0}}=\frac{1000\times\left(1-0.302\right)}{\left(0.005\right)}=\$139,600 \\ \end{align}[/latex] You will need to have $139,600 in your account when you retire. Notice that you withdrew a total of $240,000 ($1000 a month for 240 months). The difference between what you pulled out and what you started with is the interest earned. In this case it is $240,000 - $139,600 = $100,400 in interest.

### Example 10

You know you will have $500,000 in your account when you retire. You want to be able to take monthly withdrawals from the account for a total of 30 years. Your retirement account earns 8% interest. How much will you be able to withdraw each month? In this example, We’re looking for*d*.

*r*= 0.08 8% annual rate

*k*= 12 since we’re withdrawing monthly

*N*= 30 30 years

*P0*= $500,000 we are beginning with $500,000 In this case, we’re going to have to set up the equation, and solve for

*d*. [latex]\begin{align}&500,000=\frac{d\left(1-{{\left(1+\frac{0.08}{12}\right)}^{-30(12)}}\right)}{\left(\frac{0.08}{12}\right)}\\&500,000=\frac{d\left(1-{{\left(1.00667\right)}^{-360}}\right)}{\left(0.00667\right)}\\&500,000=d(136.232)\\&d=\frac{500,000}{136.232}=\$3670.21 \\ \end{align}[/latex] You would be able to withdraw $3,670.21 each month for 30 years.

### Try it Now 3

A donor gives $100,000 to a university, and specifies that it is to be used to give annual scholarships for the next 20 years. If the university can earn 4% interest, how much can they give in scholarships each year?## Licenses & Attributions

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