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Study Guides > Mathematics for the Liberal Arts

Which Formula to Use?

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
Now that we have surveyed the basic kinds of finance calculations that are used, it may not always be obvious which one to use when you are given a problem to solve. Here are some hints on deciding which equation to use, based on the wording of the problem.

Loans

The easiest types of problems to identify are loans.  Loan problems almost always include words like loan, amortize (the fancy word for loans), finance (i.e. a car), or mortgage (a home loan). Look for words like monthly or annual payment. The loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying interest on the loan.

Loans 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 (the principal, or amount of the loan).
  • d is your loan payment (your monthly payment, annual payment, etc)
  • r is the annual interest rate in decimal form.
  • k is the number of compounding periods in one year.
  • N is the length of the loan, in years.
 

Interest-Bearing Accounts

Accounts that gain interest fall into two main categories.  The first is on where you put money in an account once and let it sit, the other is where you make regular payments or withdrawals from the account as in a retirement account. Interest
  • If you're letting the money sit in the account with nothing but interest changing the balance, then you're looking at a compound interest problem. Look for words like compounded, or APY. Compound interest assumes that you put money in the account once and let it sit there earning interest.

COMPOUND INTEREST

[latex-display]P_{N}=P_{0}\left(1+\frac{r}{k}\right)^{Nk}[/latex-display]
  • PN is the balance in the account after N years.
  • P0 is the starting balance of the account (also called initial deposit, or principal)
  • r is the annual interest rate in decimal form
  • k is the number of compounding periods in one year
    • If the compounding is done annually (once a year), k = 1.
    • If the compounding is done quarterly, k = 4.
    • If the compounding is done monthly, k = 12.
    • If the compounding is done daily, k = 365.
  • The exception would be bonds and other investments where the interest is not reinvested; in those cases you’re looking at simple interest.

SIMPLE INTEREST OVER TIME

[latex-display]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex-display]
  • I is the interest
  • A is the end amount: principal plus interest
  • [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
  • r is the interest rate in decimal form
  • t is time
The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.
  Annuities
  • If you're putting money into the account on a regular basis (monthly/annually/quarterly) then you're looking at a basic annuity problem.  Basic annuities are when you are saving money.  Usually in an annuity problem, your account starts empty, and has money in the future. 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.

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.
 
  • If you're pulling money out of the account on a regular basis, then you're looking at a payout annuity problem.  Payout annuities are used for things like retirement income, where you start with money in your account, pull money out on a regular basis, and your account ends up empty in the future. 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.

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
Remember, the most important part of answering any kind of question, money or otherwise, is first to correctly identify what the question is really asking, and then determine what approach will best allow you to solve the problem.

Try It

For each of the following scenarios, determine if it is a compound interest problem, a savings annuity problem, a payout annuity problem, or a loans problem. Then solve each problem.
  1. Marcy received an inheritance of $20,000, and invested it at 6% interest. She is going to use it for college, withdrawing money for tuition and expenses each quarter. How much can she take out each quarter if she has 3 years of school left?

    Answer: This is a payout annuity problem. She can pull out $1833.60 a quarter.

  2. Paul wants to buy a new car. Rather than take out a loan, he decides to save $200 a month in an account earning 3% interest compounded monthly. How much will he have saved up after 3 years?

    Answer: This is a savings annuity problem. He will have saved up $7,524.11

  3. Keisha is managing investments for a non-profit company.       They want to invest some money in an account earning 5% interest compounded annually with the goal to have $30,000 in the account in 6 years. How much should Keisha deposit into the account? 

    Answer: This is compound interest problem. She would need to deposit $22,386.46.

  4. Miao is going to finance new office equipment at a 2% rate over a 4 year term. If she can afford monthly payments of $100, how much new equipment can she buy? 

    Answer: This is a loans problem. She can buy $4,609.33 of new equipment

  5. How much would you need to save every month in an account earning 4% interest to have $5,000 saved up in two years?

    Answer: This is a savings annuity problem. You would need to save $200.46 each month

In the following video, we present more examples of how to use the language in the question to determine which type of equation to use to solve a finance problem. https://youtu.be/V5oG7lLTECs In the next video example, we show how to solve a finance problem that has two stages, the first stage is a savings problem, and the second stage is a withdrawal problem. https://youtu.be/CNkvwMuLuis

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