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# Introduction

## What you’ll learn to do: Determine how to choose the correct formula to use for a given scenario and utilize a spreadsheet to analyze a specific real-world problem

For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. In this section, we will explore specific kinds of accounts that gain interest over time, like retirement accounts. We will also explore how mortgages and car loans, called installment loans, are calculated.

### Learning Outcomes

• Explore annuity and loan formulas, identifying which formula applies in a given scenario
• Use online calculators and spreadsheets to perform calculations with annuity and loan formulas

# Annuities

## Savings Annuity

For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.   For a savings annuity, we add a deposit, d, to the account with each compounding period. The formula is a bit more involved than our previous formulas. As such, we will focus on understanding what the formula calculates and when to use it. Then we rely on an annuity calculator or spreadsheet to do our calculations for us.

### Annuity Formula

[latex-display]P_{N}=\frac{d\left(\left(1+\frac{r}{k}\right)^{Nk}-1\right)}{\left(\frac{r}{k}\right)}[/latex-display]
• PN is the balance in the account after N years.
• d is the regular deposit (the amount you deposit each year, each month, etc.)
• r is the annual interest rate in decimal form.
• k is the number of compounding periods in one year.
If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year. For example, if the compounding frequency isn’t stated:
• If you make your deposits every month, use monthly compounding, k = 12.
• If you make your deposits every year, use yearly compounding, k = 1.
• If you make your deposits every quarter, use quarterly compounding, k = 4.
• Etc.

### When do you use the Annuity Formula vs the Compound Interest Formula?

Recall that compound interest assumes that you put money in the account once and let it sit there earning interest. Annuities assume that you put money in the account on a regular schedule (every month, year, quarter, etc.) and let it sit there earning interest.
• Compound interest: One-time deposit that sits and earns interest for a certain number of compounding periods.
• Annuity: Fixed amount deposits made each compounding period.

### Examples

Let's consider some scenarios and discuss whether to use the Compound Interest formula or the Annuity formula.  Note that you are not being asked to solve the computational questions for these examples.   Example 1: Suppose that you plan to have $100 from your paycheck automatically deposited into a savings account each month. If the savings account has an annual interest rate of 5.5%, compounded monthly, how much will you have saved after 10 years? Answer: The interest in being compounded, which might at first glance suggest the Compound Interest formula, but there is a fixed monthly deposit made into the account every compounding period. Thus, this scenario would require using the Annuity Formula. Example 2: Suppose that you just won a small lottery and now have$5000 that you plan to use to open a new savings account.  If you do not plan any additional deposits into the account and if the account has an annual interest rate of 57%, compounded monthly, how much will you have saved after 15 years?

Answer: Since interest is compounding and there is only a one-time deposit being made into the account, this scenario would require using the Compound Interest formula.

## Removing Money from Annuities

So far, we have discussed a type of annuity called a Savings Annuity in which you start with nothing, put money into an account on a regular basis, and end up with (more) money in your account. Next, we consider a type of annuity 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 takes a good bit of work to derive and the details are omitted here. ### Payout Annuity Formula [latex-display]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex-display] • 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 ### note the similarities and differences When using formulas in applications, it is helpful to note the similarities and differences in the formulas so you don't mix them up. Compare the formulas for savings annuities vs payout annuities. Savings Annuity Payout Annuity [latex-display]P_{N}=\frac{d\left(\left(1+\frac{r}{k}\right)^{Nk}-1\right)}{\left(\frac{r}{k}\right)}[/latex] [latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex-display] 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-time deposit that sits and earns interest for a certain number of compounding periods. • Annuity (or Savings Annuity): Fixed amount deposits made each compounding period. • Payout Annuity: Fixed amount withdrawals taken out each compounding period. ### Example Consider the following problem: 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? Which formula would you use for this problem: Compound Interest, Savings Annuity, or Payout Annuity?

Answer: Since you would be withdrawing a fixed amount every compounding period, the Payout Annuity formula is the one we would apply here.

Here is another example of the type of scenario that would require the Payout Annuity formula: 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?