Annuities and Loans

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

 

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 the math behind 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.

 

Savings Annuities

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.

Glass jar labeled "Retirement." Inside are crumpled $100 bills

An annuity can be described recursively in a fairly simple way. Recall that basic compound interest follows from the relationship

[latex]{{P}_{m}}=\left(1+\frac{r}{k}\right){{P}_{m-1}}[/latex]

For a savings annuity, we simply need to add a deposit, d, to the account with each compounding period:

[latex]{{P}_{m}}=\left(1+\frac{r}{k}\right){{P}_{m-1}}+d[/latex]

Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general.

Example

Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. Write an explicit formula that represents this scenario.

Generalizing this result, we get the savings annuity formula.

Annuity Formula

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

  • 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 this?

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 assumes that you put money in the account once and let it sit there earning interest.

  • Compound interest: One deposit
  • Annuity: Many deposits.

Examples

A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?

This example is explained in detail here.

 

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A conservative investment account pays 3% interest. If you deposit $5 a day into this account, how much will you have after 10 years? How much is from interest?

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 Financial planners typically recommend that you have a certain amount of savings upon retirement.  If you know the future value of the account, you can solve for the monthly contribution amount that will give you the desired result. In the next example, we will show you how this works.

Example

You want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?

View the solving of this problem in the following video.

 

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Solving For Time

We can solve the annuities formula for time, like we did the compounding interest formula, by using logarithms. In the next example we will work through how this is done.

Example

If you invest $100 each month into an account earning 3% compounded monthly, how long will it take the account to grow to $10,000?

This example is demonstrated here:

Payout Annuities

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.

Black and white aerial shot of hands exchanging money

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]

  • 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

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.

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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.

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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?

Loans

Conventional Loans

In the last section, you learned about payout annuities. In this section, you will learn about conventional loans (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.

Hand holding green pen, which has just written "Approved!" in a circle

One great thing about loans is that they use exactly the same formula as a payout annuity. To see why, imagine that you had $10,000 invested at a bank, and started taking out payments while earning interest as part of a payout annuity, and after 5 years your balance was zero. Flip that around, and imagine that you are acting as the bank, and a car lender is acting as you. The car lender invests $10,000 in you. Since you’re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.

Loans Formula

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

  • 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.

Like before, the compounding frequency is not always explicitly given, but is determined by how often you make payments.

When do you use this?

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.

  • Compound interest: One deposit
  • Annuity: Many deposits
  • Payout Annuity: Many withdrawals
  • Loans: Many payments

Example

You can afford $200 per month as a car payment. If you can get an auto loan at 3% interest for 60 months (5 years), how expensive of a car can you afford? In other words, what amount loan can you pay off with $200 per month?

Details of this example are examined in this video.

 

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Example

You want to take out a $140,000 mortgage (home loan). The interest rate on the loan is 6%, and the loan is for 30 years. How much will your monthly payments be?

View more about this example here.

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Janine bought $3,000 of new furniture on credit. Because her credit score isn’t very good, the store is charging her a fairly high interest rate on the loan: 16%. If she agreed to pay off the furniture over 2 years, how much will she have to pay each month?

Calculating the Balance

With loans, it is often desirable to determine what the remaining loan balance will be after some number of years. For example, if you purchase a home and plan to sell it in five years, you might want to know how much of the loan balance you will have paid off and how much you have to pay from the sale.

Pair of glasses resting on a Mortgage Loan Statement

To determine the remaining loan balance after some number of years, we first need to know the loan payments, if we don’t already know them. Remember that only a portion of your loan payments go towards the loan balance; a portion is going to go towards interest. For example, if your payments were $1,000 a month, after a year you will not have paid off $12,000 of the loan balance.

To determine the remaining loan balance, we can think “how much loan will these loan payments be able to pay off in the remaining time on the loan?”

Example

If a mortgage at a 6% interest rate has payments of $1,000 a month, how much will the loan balance be 10 years from the end the loan?

This example is explained in the following video:

 

Oftentimes answering remaining balance questions requires two steps:

  1. Calculating the monthly payments on the loan
  2. Calculating the remaining loan balance based on the remaining time on the loan

Example

A couple purchases a home with a $180,000 mortgage at 4% for 30 years with monthly payments. What will the remaining balance on their mortgage be after 5 years?

More explanation of this example is available here:

Solving for Time

Recall that we have used logarithms to solve for time, since it is an exponent in interest calculations. We can apply the same idea to finding how long it will take to pay off a loan.

Try It

Joel is considering putting a $1,000 laptop purchase on his credit card, which has an interest rate of 12% compounded monthly. How long will it take him to pay off the purchase if he makes payments of $30 a month?

FYI

Home loans are typically paid off through an amortization process, amortization refers to paying off a debt (often from a loan or mortgage) over time through regular payments. An amortization schedule is a table detailing each periodic payment on an amortizing loan as generated by an amortization calculator.

If you want to know more, click on the link below to view the website “How is an Amortization Schedule Calculated?” by MyAmortizationChart.com. This website provides a brief overlook of Amortization Schedules.

Which Formula to Use?

 

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

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

  • 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]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex]

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

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

  • 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?

  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?

  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? 

  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? 

  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?

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.

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.

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