### Learning Outcomes

- Calculate the balance on an annuity after a specific amount of time
- Discern between compound interest, annuity, and payout annuity given a finance scenario
- Use the loan formula to calculate loan payments, loan balance, or interest accrued on a loan
- Determine which equation to use for a given scenario
- Solve a financial application for time

## Removing Money from 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.

### Payout Annuity Formula

[latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex]

*P*is the balance in the account at the beginning (starting amount, or principal)._{0}*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

Like with annuities, the compounding frequency is not always explicitly given, but is determined by how often you take the withdrawals.

### When do you use this?

Payout annuities assume that you take money from the account on a regular schedule (every month, year, quarter, etc.) and let the rest sit there earning interest.

- Compound interest: One deposit
- Annuity: Many deposits.
- Payout Annuity: Many withdrawals

### Example

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?

View more about this problem in this video.

### Try It

### Evaluating negative exponents on your calculator

With these problems, you need to raise numbers to negative powers. Most calculators have a separate button for negating a number that is different than the subtraction button. Some calculators label this (-) , some with +/- . The button is often near the = key or the decimal point.

If your calculator displays operations on it (typically a calculator with multiline display), to calculate 1.005-240 you’d type something like: 1.005 ^ (-) 240

If your calculator only shows one value at a time, then usually you hit the (-) key after a number to negate it, so you’d hit: 1.005 yx 240 (-) =

Give it a try – you should get 1.005-240 = 0.302096

### Example

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?

A detailed walkthrough of this example can be viewed here.

### Try It

### Try It

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?