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Teaching by Topic: Loans

Here are a bunch of tips, learning objectives, worksheets, and pre-built lesson plans to help you build your curriculum to teach students about loans!

Loans are a pivotal part of the overall financial picture. Whether you need to take out a loan for school, buy a house or car, or need extra money in a pinch, loans can be a great tool. Students should understand the power – and dangers – of loans. As a teacher or homeschooler, you are in a great position to inform your learners all about loan terminology and the lending process. Check out these great lesson plans, worksheets, games, and activities to help your students prepare for their financial futures!

National Standards for Personal Finance Education

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Our free cheat sheet covers every learning objective in the National Standards for Personal Finance Education and the corresponding Kids' Money Lesson Plans - we cover each and every standard!

Teaching Loans to K-5 th Graders

Teaching loans can start early in school, beginning with the basic concepts of borrowing, being trustworthy, and the idea of interest, which you can introduce as early as 1 st or 2 nd grade. However, upper elementary is the ideal time to teach explicit loan concepts. These activities can be used at specific levels to guide your students’ understanding of many concepts.

K-5 Loans Lesson Plans 

• 2 nd Grade KMLP: Opportunity Cost: This lesson introduces kids to the idea that if you buy one thing, you give up a chance to purchase another. They learn that major purchases often require loans, which fits into needs vs. wants and how we pay for them.
• 3 rd Grade KMLP: Interest: In this lesson , kids read a book, watch a video, and hold class discussions to get to the bottom of interest. They see how it works in their favor but can also be harmful in loans.
• 4 th Grade KMLP: Managing Credit: This lesson plan covers it all. Whether your students are learning about credit, interest rates, and why businesses lend money to one person over another. 
• 4 th Grade KMLP: Spending: In this lesson , students learn about car loans and how they factor into budgets. They see that loans can be a part of their spending plan and help them build economic success.
• Sharing a Story About Borrowing: In this lesson , students see the importance of borrowing responsibly, ensuring they take care of things and return them on time. You can reinforce that adults are rewarded with better loan terms when they borrow appropriately. ( 2 nd -5 th grade )
• 5 th Grade KMLP: Credit and Debit: This lesson dives into interest, loan vocabulary, and more. They also see that credit and loans can be valuable tools in their arsenal. They learn through a specific example of a car loan.
• Introduction to Borrowing: This lesson shows how important borrowing can be. It involves a lot of partner discussion, reasoning, and seeing both sides of loans and borrowing. ( 3 rd -5 th grade)
• Money Responsibility: In this lesson , students see how responsible financial decisions can work in their favor. They learn that loans can be a practical tool to reach goals. Kids learn money management techniques to keep their finances in order. ( 4 th -5 th grade)

K-5 Loans Worksheets

• Decision-Making Worksheets: These worksheets demonstrate how to separate choices into precise steps to learn how to improve their decision-making. ( 2 nd -5 th grade)
• Decision Tree: This worksheet has students list all the steps they took in a decision,
• increasing their financial know-how. ( 3 rd -5 th grade)
• Banking Terms Worksheet: This fun worksheet introduces essential banking terms, including several about loans. ( 4 th -5 th grade )
• Comparison Shopping: This worksheet shows students how to shop wisely, which you can transfer to home loans and big-ticket purchases. ( 4 th -5 th grade)
• Allowances and Spending Decisions: This worksheet shows students how to save for and afford large purchases and differentiate between wants and needs. ( 4 th -5 th grade)
• Responsibility Game: Pairs work on this worksheet and try to beat each other, solidifying their responsibility knowledge and awareness. ( 2 nd -4 th grade)
• Rights and Responsibilities: This worksheet shows kids the difference between rights and responsibilities. You can show them how this works in the loan realm. ( 2 nd -4 th grade)
• Budget Worksheet: This worksheet shows students how to budget and requires students to reflect, making this a good option for strengthening basic finances. ( 1 st -2 nd grade)
• Borrowing FAQs: This worksheet has students create replies to common questions about borrowing, helping them synthesize their understanding. ( 3 rd -5 th grade)
• Calculating Interest: This worksheet (scroll down to the last page) focuses on loans and interest, providing practice to students in real lending math examples. ( 3 rd grade)
• Elementary Entrepreneurs: This worksheet shows kids real-life applications, including a loan agreement activity. ( 1 st -4 th grade)
• Decision-Making Comic Strip: Students draw a comic to analyze their decisions, which educators can tailor to teaching loans. ( 3 rd -5 th grade)

K-5 Loans Games and Activities

• Vault Financial Literacy: This interactive learning activity provides students with instruction in multiple financial concepts, including loans and credit. 
• Cat Insanity: This fun game shows kids how important it is to be organized in finances, especially as they take out loans and need to repay them. They also learn about interest and compounding.
• Wise Pockets: This resource contains stories that teach various money lessons, including loans, credit, and borrowing. 

Tips for Teaching Loans to K-5 th Grade Students

Keep it “light.” Focus on games, songs, and videos that show the idea of borrowing. Connect to their lives by using examples of borrowing items from friends and the importance of returning them in good shape. You can introduce the idea of interest and how it affects total payments, especially for upper elementary classes.

Teacher Guides by Grade Level

• Money Lesson Plans for Kindergarten
• Money Lesson Plans for 1st Grade
• Money Lesson Plans for 2nd Grade
• Money Lesson Plans for 3rd Grade
• Money Lesson Plans for 4th Grade
• Money Lesson Plans for 5th Grade

Teaching Loans to 6 th -8 th Graders

Your middle schoolers are ready to dive into the concept of loans. They have a solid understanding of money, interest, and credit, and these activities can boost their loan understanding. 

6 th -8 th Loans Lesson Plans

• 7 th Grade KMLP: Personal Finance: This lesson shows students the ins and outs of multiple economic topics, including loans and borrowing. They learn about interest, budgeting for loan payments, and how credit scores can impact their futures.
• 8 th Grade KMLP: Managing Credit: In this lesson , kids see how people use loans for various purchases. They analyze credit reports, watch videos, and learn that not all lenders are fair and honest.
• Mortgages (BrainPOP): Kids will enjoy this lesson that walks them through the mortgage process. It has many supporting worksheets and interactive activities, including a quiz for you to check on their progress and comprehension.
• Life is About Choices: This lesson shows kids the impact of major purchases like cars and homes. It focuses on houses, covering topics like mortgages, types of home loans, and how to understand market trends. 
• Understanding Your Credit Score: This lesson highlights the significance of credit scores, showing learners how and why they can fluctuate. They also discover the power of good credit and how it can determine loan rates.
• Keeping Score: Why Credit Matters: People should know their credit, but many don’t have a good handle on their scores. This lesson walks students through maintaining good scores and knowing what kind of loan term to expect. 
• Establishing Credit: This lesson shows middle schoolers how to create a successful path regarding credit. They look at credit scores closely and see how to raise them and get the best loan possible.
• Purchasing a Vehicle: In this lesson , students see how buying a car really works. It shows them how to shop for vehicles and calculate loan totals.   
• Being a Responsible Borrower: In this lesson , kids learn the actual costs of borrowing and see the pros and cons involved. It focuses on auto loans, but the tasks can transfer to any loan type.
• Cars and Loans: This lesson is a good way to show kids how to get a loan and figure out the interest amounts. It stresses the idea of comparison shopping to ensure you get the best rates.

6 th -8 th Grade Loans Worksheets 

• Loans and Interest Activity: This worksheet has students determine simple interest on home loans, providing practical experience for their future.
• Middle School Money: Loan/Interest Calculations: This worksheet is a good exercise for students to figure how much interest can tack onto loans.
• Borrowing Money For a House: Students complete this worksheet to discover how loans work, calculate interest, and compare loan offers.
• Buying a House: Students fill out this worksheet set to see how to pay for a home and understand the costs involved.
• Mortgage Shopping: This worksheet has students compare loan offers from different lenders and find the best overall deal.
• Interactive: Shady Sam: This worksheet complements the game “Shady Sam” and serves to gauge student comprehension. 
• Using Your Credit – BizKids: This set of worksheets gives students practice finding how credit can directly affect their financial futures.
• Simple Interest Real-Life Activity: Kids complete this worksheet to see how various interest rates impact their loans.
• Beginner’s Budget and Money Management: This worksheet shows students the fundamentals, including writing checks and probing if loans are a good idea for everyone.
• Simple and Compound Interest: This worksheet has students compute simple and compound interest math problems, showing how each impacts total loan amounts.
• I’m Buying a Car : This worksheet shows students how to use online apps to determine the costs of purchasing a car. They learn about interest rates, vehicle financing, and how to shop for loans for the best duration and monthly payments for their incomes.
• Buying a Car: Cash Vs. Interest: This worksheet highlights the pros and cons of getting a car loan instead of using cash. They apply research skills and discover how to find interest rates quickly.
• Dream Car Loan: This worksheet has students fill in information about their dream car.
• They learn how much it would total with interest and see if they could realistically afford that car.
• Deciding Which Car and Car Loan You Can Afford: This worksheet shows students how to calculate and budget for monthly payments and more.
• Dream Car Math: Students complete calculations in this worksheet to find how interest impacts loan costs.

6 th -8 th Grade Loans Games and Activities

• Shady Sam: As students take the role of a predatory lender in this game , they see how easy it is to become prey to high-interest rates and unreasonable loans. 
• Buy A Car Project: This interactive activity lets students manipulate different car loan-related numbers to discover how the process works.
• Cartoons: Car Payments: In this fun activity , students investigate cartoons about car loans to expand their understanding of the concept.
• Compound Interest Game: Students play a game to compare compound vs. simple interest, learning the benefits of interest and how it can become a negative in loans.
• Car Payment Calculator: Students enter numbers in this calculator to learn how car loan payments vary based on down payments, interest rates, and durations.
• Misadventures in Money Management: This interactive game lets students see money dangers and how to escape them in an engaging platform. They learn that piling up too much debt can hurt their finances.
• Loan Myths and Realities: Students analyze statements about lending and decide if they are true or false in this activity .
• Break the Bank: In this game , kids pretend to be bankers giving loans to customers and then need to stop the evil payday lenders from tricking consumers.
• Financial Literacy Skills: This resource pack has loans and credit activities to allow students to develop a stronger understanding of loans.
• Financial Literacy: What Can Banks Offer You?: This activity gives students an extensive overview of banking, including the influence of interest and how the loan process works.
• Asking For A Loan: Students use their persuasive abilities to ask for a loan in this activity . They write a speech to help them reach their goal.
• Telling the Difference Between Loan Myths and Realities: Students analyze home loan sentences and decide whether they are true or false in this activity .
• Dream House Mortgage Calculations: Students research a job and see if their salaries would be enough to afford home loan payments in this activity .

Tips for Teaching Loans to 6 th -8 th Grade Students

Your middle schoolers are in a transition period. This time can be uncertain, so boost their confidence with team-building and partner work as they learn about loans. Use research-based activities often and keep exercises practical to their future lives.

• Money Lesson Plans for 6th Grade
• Money Lesson Plans for 7th Grade
• Money Lesson Plans for 8th Grade

Teaching Loans to 9 th -12 th Graders

Your high school students are closer than they realize to getting their first loan. The more exposure they have to how lending works, the better. In the lists below, you’ll find hands-on, in-depth activities to bolster student comprehension of various loan topics. 

9 th -12 th Loans Lesson Plans

• 9th Grade KMLP: Basic Economics: This lesson covers a broad array of topics, with a focus on personal loans. Kids decide the pros and cons of these loans to see if they are potentially a good idea for their situations. They dive into wants and needs and determine the role of loans.
• 12 th Grade KMLP: Managing Credit: In this lesson , students learn how to understand their credit history, boost credit scores, and deepen their comprehension of how credit works. They participate in varied activities to stay engaged and learn as much as possible.
• 12 th Grade KMLP: Spending : High schoolers see how loan payments work in this lesson , how interest impacts their loan amounts, and how to keep tabs on their spending. They look at mortgage examples to discover what home ownership and lending look like.
• Calculating Loan Payments: In this lesson , students look into a case study to find how amounts borrowed, down payments, and interest rates affect finances. They also see how to make informed choices about credit choices.  
• Qualifying For Loans: This lesson shows students how to distinguish between secured and unsecured loans and use that information to see how to get the best rates. They then decide if individuals will be eligible for specific loans.
• Determining How Down Payments Affect Loans: This lesson shows the significance of down payments on loan totals. It focuses on home loans and other types of loans, giving students a broad understanding of how lending works.
• Debt: The Good, The Bad, and The Ugly – BizKids: In this lesson , students find the good and bad of personal loans and other forms of credit. They discover budgeting and compound interest and see how to handle demanding financial situations.
• Personal Finance Unit – Credit and Interest: This lesson includes credit, credit reports and history, credit scores, and loans. It includes PowerPoints, worksheets, and other engaging activities.
• Describing Credit Scores: In this lesson , students learn how credit scores and how they can impact their loans. They also learn about the influences that change credit scores, helping them to get a loan with good terms.
• Determining How Down Payments Affect Loans: Students learn how varying down payment amounts impact monthly and total loan costs in this lesson . They also discover how installment loans can be helpful for affording larger purchases.

9 th -12 th Grade Loans Worksheets

• Creating a Buying Plan: This worksheet highlights the importance of planning, reflecting, and decision-making in the loan process.
• Decision Making: Your high schoolers should know how to self-analyze decisions, and this worksheet walks them through the steps to succeed. Educators can use this throughout the loans unit they teach. 
• Comparing Auto Loans: This worksheet has students compare loans to see the interest rates, loan duration, monthly payments, and the total cost over the life of the various offerings.
• Group Project Worksheet: Students partner up for this worksheet to see how to find and finance the car they can afford, showing them how loans operate as they work together.
• Cosigning Loans And Sharing Credit: In this worksheet , students find the benefits and drawbacks of taking out a loan with a cosigner.
• Qualifying For Loans: This worksheet shows students how to qualify for a loan, allowing them to see who would likely get a loan and what terms would go along with it.
• Down Payments and Loans: In this worksheet , students look at various loan terms to see how down payments can lower monthly and total costs.
• Loan Repayment: This worksheet lets kids compute simple interest on a loan, giving them a quick and valuable skill.
• Getting Banked: In this worksheet , students research a local bank to see their services, including loans and standard interest rates.
• Consumer Loan Checklist: This worksheet highlights the information students need to know when applying for a personal loan, and they can practice as they fill out the form.
• Reading About Credit Scores: Students read a handout and use the worksheet to take notes and think about credit scores, a crucial influence in home loans.
• Angela Builds Her Credit: This worksheet is a case study of Angela trying to build her credit, showing students how credit scores fluctuate for various reasons and how to qualify for the best loans.
• Reflecting on Needs and Wants: This worksheet shows students how to consider needs and wants, and teachers can connect this concept to home loans and buying a house. 
• Personal Credit Report: This worksheet asks students to find information in a credit report, identifying essential components that will boost or harm their credit standing.

9 th -12 th Grade Loans Games and Activities

• Differentiating Between Secured and Unsecured Loans: This activity highlights the differences between various loan types. 
• Identifying the Missing Credit Score Category: Students work together in this activity to discover what factors influence credit scores. 
• Role-Playing Borrowing and Lending: This partner activity provides kids practice being a lender and a borrower to discover what is important in the home loan process. 
• Mortgage Calculator: This tool is an excellent way for students to see how down payments and interest rates factor into their loan payments.
• Buying A Home: This activity lets students see the many factors that go into purchasing a home, including shopping for a mortgage and calculating mortgage costs.
• Shopping In Credit City: Students engage in a game that highlights the dangers of interest on personal loans and credit through real-world situations they may run into.
• Personal Loan Application: This activity shows how students would fill out a personal loan application, giving them practical experience. 
• Loans and Credit: This activity has students solve word problems by determining how the interest rate and loan length change total payments.
• Banking Basics Card Game: This hands-on activity covers the various services banks offer, including different loans and typical terms.
• Credits and Loans: This activity contains a PowerPoint and different handouts that students complete about loans, interest, payments, and debt.

Tips for Teaching Loans to 9 th -12 th Grade Students

Your high school students need to know the power and dangers of taking out loans. Make sure you give them time to explore and research, so they feel prepared for the real world. Differentiate instruction to individual students, providing student loan resources to college-bound kids and auto loan information to those looking to buy their first car. Let them synthesize their findings in PowerPoints or reports, so they feel confident in their learning.

• Money Lesson Plans for 9th Grade
• Money Lesson Plans for 10th Grade
• Money Lesson Plans for 11th Grade
• Money Lesson Plans for 12th Grade

More Teacher Resources

• Borrowing Money Lesson Plans
• Home Loans Lesson Plans
• Personal Loans Lesson Plans
• Auto Loans Lesson Plans
• All Money Management Lesson Plans

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Mortgage Formula Cheat Sheet: Home Loan Math Made Simple

Home loan math doesn’t have to be intimidating—all you need is a mortgage formula cheat sheet. Though a home loan does indeed involve a few equations, it’s fairly easy to break it all down into the kind of simple arithmetic every home buyer can understand and, more important, needs to know.

Mortgage formula cheat sheet

Take note: The latest figures available show the median home costs about $220,000 , so we’ll use that figure as a base for our calculations. Other figures we’ll use: an average family’s annual salary is about $54,000 and it carries  $7,630 in debt .

How much do you need for a down payment?

Though you can contribute as little as 3.5% of a home’s value for a down payment, lenders consider an ideal down payment to be 20%  of a home’s total price. So here’s the math on that for the average-priced home:

20% of $220,000 = $44,000 down payment

This would leave $176,000—the amount a home buyer will need for the mortgage.

Another reason to aim for 20% down: You’ll avoid paying private mortgage insurance, which is typically required under that threshold. And that will cost you about $1,000 per year, says  David Bakke of  Money Crashers .

(Still, if that hefty 20% is an unattainable goal, at least try to  put down 10% for a significantly better interest rate than you’d get with 3.5%.)

How much will a mortgage cost per month?

A mortgage can be paid off in numerous ways, but one of the most typical is to stretch those payments out over 30 years—that way, you break it down into bite-size pieces. Building off the numbers above, here’s how much your average mortgage would cost per month:

$176,000 at 4% interest rate = $840.25 monthly payment

Keep in mind, this monthly bill does not include property taxes, home insurance, HOA dues, or other home-related maintenance fees, which vary by area but are in the ballpark of a few hundred per year for a home at this price.

Also note that the longer you stretch out your mortgage payments, the more you’ll end up paying in interest. Over 30 years, the total you’ll fork over in interest amounts to $302,490.33!

But there are ways to lower the amount you pay in interest—like paying off your loan faster. Finish in 15 years, and you’ll end up paying only $234,333.13 in interest. Granted, for a 15-year loan you’ll have to cough up more per month—$1,301.85 instead of $840.25. But the upside is you’ll save a sizable chunk in interest over the life of your loan, and be mortgage-free in half the time. So if you can afford it, it’s an option worth considering.

How much mortgage can I afford?

Of course, you’ll want to buy a home that you can comfortably pay for. So, how do you know how much is too much, too little, or just right? The way they do this is by determining your debt-to-income ratio.

For most conventional loans, experts say you’ll want your DTI ratio lower than 36%. That means your debts don’t exceed more than about one-third of your income. But how does a mortgage fit into that?

To figure that out, start with your gross income (what you take home before taxes). Let’s say your family pulls in the U.S. average, which is $54,000 per year. Divide that over 12 months to get your monthly income.

$54,000 / 12 months = $4,500 income per month

Then total up your debts—including what you owe on credit cards, auto insurance, and college loans. Remember, debt includes only items that appear on a credit report, not recurring expenses like groceries or phone bills. Since the average American carries an average debt of $7,630 per year, we’ll use that number. Divide that by 12 to get your monthly debt:

$7,630 (average debt) / 12 months = $636 debt per month

Now, add that monthly debt to your average monthly mortgage payment of $840.25 to get your total debt owed per month:

$636 debt + $840.25 mortgage = $1,476.25 debt per month

Next, divide your monthly debts by your monthly income

$1,476.25 monthly debt /  $4,500 monthly income = 33% DTI

In this scenario, the debt-to-income ratio is 33%—just below the 36% cutoff. Which means this mortgage would most likely pass the bank’s muster with flying colors! Calculate your own DTI  here .

See? Not so hard. Granted, this is a simplified version of mortgage math; your own results will depend on your income, debts, and other circumstances. But if there’s one thing we hope you take away from this, it’s that mortgages are nothing to fear—a little knowledge goes a long way. And if you get stuck, there’s no need to copy from your neighbor’s paper, since we have this handy mortgage calculator to help you whiz through these permutations with ease.

Margaret Heidenry is a writer living in Brooklyn, NY. Her work has appeared in the New York Times Magazine, Vanity Fair, and Boston Magazine.

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6.8 The Basics of Loans

Learning objectives.

After completing this section, you should be able to:

• Describe various reasons for loans.
• Describe the terminology associated with loans.
• Understand how credit scoring works.
• Calculate the payment necessary to pay off a loan.
• Read an amortization table.
• Determine the cost to finance for a loan.

New car envy is real. Some people look at a new car and feel that they too should have a new car. The search begins. They find the model they want, in the color they want, with the features they want, and then they look at the price. That’s often the point where the new car fever breaks and the reality of borrowing money to purchase the car enters the picture. This borrowing takes the form of a loan.

In this section, we look at the basics of loans, including terminology, credit scores, payments, and the cost of borrowing money.

Reasons for Loans

Even if you want a new car because you need one, or if you need a new computer since your current one no longer runs as fast or smoothly as you would like, or you need a new chimney because the one on your house is crumbling, it’s likely you do not have that cost in cash. Those are very large purchases. How do you buy that if you don ’t have the cash? You borrow the money.

And for helping you with your purchase, the company or bank charges you interest.

Loans are taken out to pay for goods or services when a person does not have the cash to pay for the goods or services. We are most familiar with loans for the big purchases in our lives, such as cars, homes, and a college education. Loans are also taken out to pay for repairs, smaller purchases, and home goods like furniture and computers.

Loans can come from a bank, or from the company selling the goods or providing the service. The borrower agrees to pay back more than the amount borrowed. So there is a cost to borrowing that should be considered when deciding on a purchase bought with credit or a borrowed money. Even using a credit card is a form of a loan.

Essentially, a loan can be obtained for just about any purchase, large or small, that has a cost beyond a person’s cash on hand.

The Terminology of Loans

There are many words and acronyms that get used in relation to loans. A few are below.

APR is the annual percentage rate. It is the annual interest paid on the money that was borrowed. The principal is the total amount of the loan, or that has been financed. A fixed interest rate loan has an interest rate that does not change during the life of the loan. A variable interest rate loan has an interest rate that may change during the life of the loan. The term of the loan is how long the borrower has to pay the loan back. An installment loan is a loan with a fixed period, and the borrower pays a fixed amount per period until the loan is paid off. The periods are almost uniformly monthly. Loan amortization is the process used to calculate how much of each payment will be applied to principal and how much is applied to interest. Revolving credit , also known as open-end credit, is how most credit cards work but is also a kind of loan account. (We will learn about credit cards in Credit Cards ) You can use up to some specified value, called the limit, any way you want, and as long as you pay the issuer of the credit according to their terms, you can keep borrowing from this account.

These and other terminologies can be researched further at Forbes .

Credit Scores

Not everyone pays the same interest for the same loan. One person might get an APR of 2.9% while another pays 6.9%. These rates are based on your credit score.

Data about you and your credit is collected by three credit bureaus—Experian, Equifax, and TransUnion. They calculate your score using one of two main models: FICO and Vantage Score. The score they develop is based on the following categories:

• Payment History: Making your payments on time and not missing payments is by far the most important factor. All three credit types—revolving, installment, and open—contribute to this factor.
• Credit Utilization or Amount Owed: How much do you owe on your credit card accounts? This category is concerned with the ratio of how much you owe on revolving credit accounts relative to your available credit, also known as your credit utilization ratio. This is the only category that depends solely on your revolving credit accounts.
• Length of Credit History: This is the average age of your credit history, including the age of the oldest and newest accounts. All three types of credit accounts play a role in this category.
• Credit Mix: This number represents the different types of credit accounts you have, such as credit cards, car loan, mortgages, and whether you are successful managing both revolving and installment accounts.
• New Credit: Have you recently opened a new account or applied for new credit? Lenders want to know how much new credit you are taking on. So, if you are planning to buy a car and make another large purchase with a credit card, you may want to space these purchases out.

If you have done well in these categories, your credit score will be high, and you will qualify for lower interest rates because you are not perceived as being a risky investment. However, if you do poorly in these categories, your score will be low and you will pay higher interest rates since you present a greater risk.

Check out this nerdwallet article on credit scores to learn more!

Credit Scores Explained

Where Do Interest Rates Come From?

There are many factors that impact your interest rate beyond your credit score. Banks have the authority to set their own rates, so competition between banks impacts interest rates. Bank A doesn’t want to charge interest rates that are too high, since borrowers will find banks with better rates. Banks also don’t want to charge too little interest.

The too little interest is more involved than the too high. The bank needs to make a profit on its loans. Deposits at the bank are used by the bank to generate loans. The bank has to pay those depositors interest. The bank must charge more for loans they give than they pay to people with deposits in the bank.

Banks also borrow money from each other. These loans have an interest rate, and once more, the bank making a loan must make a profit. More directly, they must charge more for loans they give than they pay for loans they take. In the United States, banks may also borrow from the Federal Reserve, which also charges an interest rate, which is also called the discount rate.

This is where a bank’s prime rate comes from. A bank’s prime rate is the interest rate it will give to its very best customers, which means most customers will pay more than the bank’s prime rate. To confuse the issue, there is also the Wall Street Journal ’s prime rate. It is the average of the prime rates charged by individual banks. The Wall Street Journal surveys several banks to generate this value.

Banks also increase the interest rate charged to customers based on both the credit risk presented by the customer, and the risk associate with what the loan will be used for.

Calculating Loan Payments

Loan payments are made up of two components. One component is the interest that accrued during the payment period. The other component is part of the principal. This should remind you of partial payments from Simple Interest .

Over the course of the loan, the amount of principal remaining to be paid decreases. The interest you pay in a month is based on the remaining principal, just as in the partial payments of Simple Interest .

The amount of interest, I I , to be paid for one period of a loan with remaining principal P P is I = P × r n I = P × r n , where r r is the interest rate in decimal form and n n is he number of payments in a year (most often n n = 12). Since the interest is for the one period, the time is 1 and does not impact the calculation. Note, interest paid to lenders is always rounded up to the next penny.

Example 6.77

Interest for a monthly payment of a loan.

Find the interest to be paid for the period on loans with the following remaining principal and given annual interest rate. Each period is a month.

• Remaining principal is $13,450, interest rate is 6.75%
• Remaining principal is $8,460, interest rate is 5.99%
• Substituting $13,450 for the remaining principal P P , 0.0675 for r r , and n n = 12 since the period is a month into the formula, we find that the interest to be paid this period is I = P × r n = $ 13 , 450 × 0.0675 12 = $ 75.66 I = P × r n = $ 13 , 450 × 0.0675 12 = $ 75.66 .
• Substituting $8,460 for the remaining principal P P , 0.0599 for r r , and n n = 12 since the period is a month into the formula, we find that the interest to be paid this period is I = P × r n = $ 8 , 450 × 0.0599 12 = $ 42.18 I = P × r n = $ 8 , 450 × 0.0599 12 = $ 42.18 .

Your Turn 6.77

The payment of the loan has to be such that the principal of the loan is paid off with the last payment. In any period, the amount of interest is defined by the formula above, but changes from period to period since the principal is decreasing with each payment. The trick is knowing how much principal should be paid each payment so that the loan is paid off at the stated time. Fortunately, that is found using the following formula.

The payment, pmt pmt , per period to pay down a loan with beginning principal P P is p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 , where r r is the annual interest rate in decimal form, t t is the number of years of the loan, and n n is the number of payments per year (typically, loans are paid monthly making n n = 12).

Note, payment to lenders is always rounded up to the next penny.

Often, the formula takes the form p m t = P × ( r ) × ( 1 + r ) n ( 1 + r ) n − 1 p m t = P × ( r ) × ( 1 + r ) n ( 1 + r ) n − 1 , where r r is the interest rate per period (annual rate divided by the number of periods per year), and n n is the total number of payments to be made.

Example 6.78

Calculating the payment for a loan.

In the following, calculate the payment necessary to pay off the loan with the given details. The payments are monthly.

• A car loan taken out for $28,500 at an annual interest rate of 3.99% for 5 years.
• A home loan taken out for $136,700 and an annual interest rate of 5.75% for 15 years.
• The loan is for $28,500, which is the principal. The rate is 3.99%, so r r = 0.0399. The term of the loan is 5 years, so t t =5. Monthly payments means n n = 12. Substituting these values for P P , r r , n n , and t t into the formula p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 and calculating, we find the payment for the loan. p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 = $ 28 , 500 × ( 0.0399 / 12 ) × ( 1 + 0.399 / 12 ) 12 × 5 ( 1 + 0.0399 / 12 ) 12 × 5 − 1 = $ 28 , 500 × ( 0.003325 ) × ( 1.003325 ) 60 ( 1.003325 ) 60 − 1 = $ 115.6470437 0.220388273 = $ 524.75 p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 = $ 28 , 500 × ( 0.0399 / 12 ) × ( 1 + 0.399 / 12 ) 12 × 5 ( 1 + 0.0399 / 12 ) 12 × 5 − 1 = $ 28 , 500 × ( 0.003325 ) × ( 1.003325 ) 60 ( 1.003325 ) 60 − 1 = $ 115.6470437 0.220388273 = $ 524.75 The monthly payment needed is $524.75.
• The loan is for $136,000, which is the principal P P . The rate is 5.75% so r r = 0.0575. The term of the loan is 15 years, so n n = 15. Monthly payments mean n n =12. Substituting these values for P P , r r , n n , and t t into the formula p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 and calculating, we find the payment for the loan. p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 = $ 136 , 700 × ( 0.0575 / 12 ) × ( 1 + 0.0575 / 12 ) 12 × 15 ( 1 + 0.0575 / 12 ) 12 × 15 − 1 = $ 28 , 500 × ( 0.0047917 ) × ( 1.0047917 ) 180 ( 1.003325 ) 180 − 1 = $ 1 , 548.600986 1.364201118 = $ 1,135.18 p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 = $ 136 , 700 × ( 0.0575 / 12 ) × ( 1 + 0.0575 / 12 ) 12 × 15 ( 1 + 0.0575 / 12 ) 12 × 15 − 1 = $ 28 , 500 × ( 0.0047917 ) × ( 1.0047917 ) 180 ( 1.003325 ) 180 − 1 = $ 1 , 548.600986 1.364201118 = $ 1,135.18 The monthly payment needed is $1,135.18.

Your Turn 6.78

Using google sheets to calculate loan payments.

Google Sheets has a formula to calculate the monthly payment necessary to pay off a loan with a specified interest rate and term. The formula is the PMT formula. It uses the principal P P , interest rate r r , and t t years. Follow these steps

Open a Google worksheet and click on any cell.

Type =-PMT(r/12,12*t,P).

Hit the enter key.

The cell displays the payment.

Please note the negative sign. Since Google Sheets is a spreadsheet program, is sees the payment as funds leaving the account, and so they are, by default, negative. The negative sign in front of the formula makes the result positive.

For example, for a $50,000 loan at 10.9% interest for 7 years, you would type =-PMT(0.109/12,12*7,50000). The formula and the result are shown in Figure 6.18 .

Alternatively, you can also use an online calculator to find monthly payments for a loan, such as the one at Caluculator.net

Reading Amortization Tables

An amortization table or amortization schedule is a table that provides the details of the periodic payments for a loan where the payments are applied to both the principal and the interest. The principal of the loan is paid down over the life of the loan. Typically, the payments each period are equal. Importantly, one of the columns will show how much of each payment is used for interest, another column shows how much is applied to the outstanding principal, and another column shows the remaining principal or balance Figure 6.19 .

Example 6.79

Reading from an amortization table.

Using the partial amortization table (Figure 6.23) , answer the following questions.

• What is the loan amount (principal), the interest rate, and the term of the loan?
• How much is the monthly payment?
• How much remaining balance is there after the payment in month 15?
• How much was the interest in payment 10?
• What is the total of the interest paid after payment 18?
• What happens to the amount paid in interest each month?
• Reading the values at the top of the table, we see the principal is $10,000, the interest rate is 4.75%, and has a term of 20 years.
• The monthly payment is listed below the term of the loan, and is $64.62.
• The amount paid to interest decreases each month.

Your Turn 6.79

Reading an Amortization Table

Cost of Finance

There are often costs associated with a loan beyond the interest being paid. The cost of finance of a loan is the sum of all costs, fees, interest, and other charges paid over the life of the loan.

Example 6.80

Cost of financing a personal loan.

Irena signed for a loan of $15,000 at 6.33% for 5 years. When she took out the loan, Irena paid a $750 origination fee. Over the course of the loan, she pays $2,537.96 in interest. What was her cost to finance the loan?

The cost of finance is the sum is all interest and any fees paid for the loan. The fees paid were $750.00 and the interest was $2,537.96. Her cost of finance for this loan was $3,287.96.

Your Turn 6.80

Check your understanding, section 6.8 exercises.

• Did the payments increase by the same amount for each 1% jump in interest rate? Describe the pattern.
• Did the payments increase by the same amount for each 1% jump in interest rate?

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On this page

• 1. Interesting Interest
• 1a. Rule of 72
• 2. Credit Cards

3. Math of House Buying

• 4. Gold Rush?
• 5. Money Charts and Fibonacci
• 6. Superannuation

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by M. Bourne

This discussion is simplified so we don’t get lost in complications. Also, interest rates are changing all the time - check your local banks for latest rates.

There is no need to really use the formulas in this section. You can easily use Excel to calculate the values or you can use the many on-line calculators to find the values.

This site does not contain investment advice.

Don’t miss interactive mortgage calculators on this page.

Tom (see the chapter intro ) wants to buy a house sometime soon. He knows that real estate is a reasonably safe way to build wealth. It is slow, but generally low risk.

He will come across a bewildering choice of mortgages . A mortgage is just a fancy word for the process of borrowing money for a house.

The bank actually buys the house and keeps the title deed. You agree to pay off the borrowed amount each month, usually over a period of 20 to 30 years. When you have finally paid it off, you get the title deed, meaning you now fully own the house.

If you stop paying for any reason, the bank has the right to sell the house. After all, they own most of it.

Mortgage Example

Tom’s house will cost `$300,000`. He needs to pay a deposit of `10%` and will pay the remaining `90%` over `30` years at `8%` per annum.

So he will need to pay a deposit of `10%` of `300,000 = $30,000`.

The remaining amount he owes is `$270,000`.

Monthly Payments

The formula for the amount Tom has to pay each month is

`A=(Lxxr)/(1-(1+r)^-n)`

A = amount to pay each month L = loan amount (or principal ) r = interest rate (per year as a decimal - or divide by `12` to get the rate per month) n = number of payments

(This formula is based on the Sum of a Geometric Progression .)

So for our case, we have:

`L = 270,000` `r = 8% -: 12 = 0.0066667` `n = 30 xx 12 = 360`

So the amount is

`text(Payment)=(Lxxr)/(1-(1+r)^-n)`

`=(270000xx0.08/12)/(1-(1+0.08/12)^-360)`

`=$1981.16`

So Tom needs to find just under $2000 per month to buy his house. Of course, he won’t be paying rent so the extra amount is not too bad, he figures.

Back of the Envelope Calculation

A reader asked if there is a reasonable approximation that we can use for this formula when we want to do "back of the envelope" calculations. It turns out there is.

("Back of the envelope" is an expression meaning we can work it out quickly and easily on a scrap of paper.)

Most mortgages are for less than 30 years. The following chart illustrates the value of the denominator for the above expression, for interest rates of `6%` to `14%` up to 30 years.

For example, the graph for lowest curve, `6%`, is of the following form, where n is the number of years:

`1 - (1 + 0.06/12)^(12n)`

We can use this graph as follows.

If the mortgage is `25` years and the interest rate is `10%`, the graph tells us the denominator value is close to `0.91`.

So a `$100,000` loan would have a monthly payment of about

`(100,000xx0.1/12)/0.91=916`

A check in the actual formula yields a value of `$908.70`. So the approximation is not bad.

Interest amount in each payment

The formula for the amount of interest we pay in each installment is:

`"Interest"` `=(Lxxr(1-(1+r)^(p-n)))/(1-(1+r)^-n)`

Considering the example given above, in the first month of our mortgage repayments, we pay in interest:

`(170000xx0.0066667(1-(1+0.006667)^(1-360)))/(1-(1+0.006667)^-360)` `= 1798.79`

So early in the repayments, we pay a large amount off the interest (and very little off the principle: `{:1981.16 - 1798.79 = 182.37).`

Later in the process, say at the 25-year point (the 300th payment), the amount of interest we pay that month is:

`(170000xx0.0066667(1-(1+0.006667)^(300-360)))/(1-(1+0.006667)^-360)` `= 651.39`

Interactive Mortgage Payments Calculator

You can calculate any repayment here. Change any of the values for principal, interest or period and then click "Calculate". You can also see a graph of the equity (the amount you own) that has been built up in the property over time.

If you move your cursor over any of the graph bars that appear, a tooltip will tell you how much is still owing, and how much equity you have in the house.

The amounts show are for the beginning of the year in each case, so at the beginning of Year 1, you owe the full amount and have no equity, while at the beginning of Year 31 (in the example given), you have paid it all off and now you own the house outright.

NOTE: This interactive assumes zero house price inflation . If you are lucky, your house will increase in value and will be worth more than you paid for it. But that doesn't always happen.

Principal = $ (min 50000, max 10 million)

Interest p.a. = % (min 0.01, max 100)

Period = years (min 5, max 35)

Calculate Show equity

Copyright © 2013 www.intmath.com

Note: These calculations do not take into account the bank fees and we are assuming a fixed rate of interest for the whole period of the loan. Actual amounts charged by lenders are sure to be higher than this.

Six Months Later

Tom has been paying off his pride and joy for 6 months. He has paid a total of `$1981.16 xx 6` ` = $11886.96`.

He gets the first statement from the bank and expects to see a reasonable dent in the amount he owes. He is shocked to find that he still owes `$268,894.74`. How is this possible? He has paid `$12000` but only around `$1000` has come off the balance.

The formula for the balance is:

`text(Balance)=(L[1-(1+r)^(p-n)])/(1-(1+r)^-n)`

L = the loan amount r = interest rate per month as a decimal p = number of payments already made n = total number of payments to be made

In our example, the balance will be:

`=(270000[1-(1+0.08/12)^(6-360)])/(1-(1+0.08/12)^-360)`

`=$268\ 894.74`

With mortgages, the amount you are paying early in the loan period is mostly interest, and very little is coming off the principal. Towards the end of the loan, the interest amount is less and the principal starts to disappear more quickly.

Amount ($) still owing after p months.

The mistake a lot of people make is to sell the house quickly. The average mortgage is only 7 years. They own very little of the house by then because they have mostly been paying interest to the bank. If house prices have gone up a lot, they are ahead. But if house prices level off, or decline, then they lose a lot of money.

Reader Question

Reader David asked:

Is it possible to derive an equation for when the interest part of the payment for a loan exactly equals the principal part of the payment?

There is, David, and here's how to do it.

This question has two related parts — when the amount of interest in the payment is equal to the equity amount in the payment; and when the total amount of interest paid equals the total amount of principle paid. They are around the same time, but not exactly.

Interest payment equals principle payment

interest paid in month p = 0.5 × monthly payment

That is, when:

`(Lr(1-(1+r)^(p-n)))/(1-(1+r)^-n)` ` = 0.5xx(Lr)/(1-(1+r)^-n)`

Simplifying this (by multiplying throughout by `(1-(1+r)^-n)` and dividing by `Lr` gives:

`1-(1+r)^(p-n) = 0.5`

`(1+r)^(p-n) = 0.5`

L = 270000 r = 0.08/12 = 0.0066667 p = unknown n = 360

We only need `r` and `n`, so we substitute:

`(1+0.00666667)^(p-360) = 0.5`

Taking log of both sides:

`(p-360)log(1+0.00666667) ` `= log(0.5)`

Solving for `p`:

`p ` `= (log(0.5))/log(1+0.00666667) + 360` `~~256`

So the month where the amount paid in interest and the amount paid in principle is (very close to) the same, is the 256th month.

Total Interest paid equals Total Principle paid

The interest part of the total amount paid so far will equal the principal paid so far when the the balance owing equals the equity (the amount of the house we own).

We saw earlier that:

The amount of equity we have in the house is simply the house value minus what we still owe. So:

`text(Equity)=L - (L[1-(1+r)^(p-n)])/(1-(1+r)^-n)`

Setting these equal gives:

`(L[1-(1+r)^(p-n)])/(1-(1+r)^-n)` `=L - (L[1-(1+r)^(p-n)])/(1-(1+r)^-n)`

We need to solve for `p`, the number of payments already made.

Multiplying throughout by `1-(1+r)^-n` gives:

`L[1-(1+r)^(p-n)]` `=L[1-(1+r)^-n]` ` - L[1-(1+r)^(p-n)]`

Adding `L[1-(1+r)^(p-n)]` to both sides then cancelling the `L`'s gives:

`2[1-(1+r)^(p-n)]` `=[1-(1+r)^-n]`

`2[1 - ((1+r)^p)/(1+r)^n] =1-1/((1+r)^n)`

Multiplying throughout by `(1+r)^n` gives:

`2[(1+r)^n - (1+r)^p]` ` =(1+r)^n - 1`

`(1+r)^n - 2(1+r)^p = - 1`

Tidying this up and getting the `(1+r)^p` term on the left hand side:

`(1+r)^p = 0.5[(1+r)^n+1]`

`log(1+r)^p = log[0.5((1+r)^n+1)]`

This is equivalent to:

`p log(1+r) = log[0.5((1+r)^n+1)]`

`p = (log[0.5((1+r)^n+1)])/log(1+r)`

In the example given earlier on this page, we had:

Therefore, the required time is:

`p = (log[0.5((1+0.08/12)^360+1)])/log(1+0.08/12)` ` = 280.959 ~~ 281`

This means the principal equals the equity on the 281st payment, i.e. the 4th payment of the 23rd year.

Money Maths Lesson Plan Suggestion - House Buying

Simulate a house buying scenario in your district. Use actual advertisements for houses and for housing loans. Get students to find the best loan deals. Which is best - fixed or variable interest? What will they pay in total for the house? What will it be worth at the end of the loan period? How much will they have paid after 10 years and how much will they own by then?

It may not always be best to buy a house compared to renting. In Japan, house prices have still not recovered to where they were 16 years ago. Sure, interest rates are low there, but negative equity has been a huge problem for years. (This is when the amount that you owe on a house is more than what the house is worth.)

House prices, Japan. Index = 100 in 1975.

This means that house prices more than doubled at the peak in 1991 and have dropped back to 1975 levels since. In 2005/2006 there has been a modest increase.

Interestingly, house prices in US, Australia, UK, and most industrialised economies have shot up in recent years due to low interest rates. One wonders what the future holds for house prices in these countries...

If rents are low in your area, it may be better to rent a place and invest (not spend ) the difference.

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

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It is a good idea to try to save up money to buy large items or find 0% interest deals so you are not paying interest. However, this is not always possible, especially when buying a house or car. That is when it is important to understand how much interest you will be charged on your loan.

Example \(\PageIndex{1}\): Loan Payment Formula:

Ed buys an iPad from a rent-to-own business with a credit plan with payments of $30 a month for four years at 14.5% APR compounded monthly. If Ed had bought the iPad from Best Buy or Amazon it would have cost $500. What is the price that Ed paid for his iPad at the rent-to-own business? How much interest did he pay?

PMT = $30, r = 0.145, t = 4, n = 12

The price Ed paid for the iPad was $1,087.83. That’s a lot more that $500!

For some problems, you will have to find the payment instead of the present value. In that case, it is helpful to just solve the loan payment formula for PMT . Since PMT is multiplied by a fraction, to solve for PMT , you can just multiply both sides of the formula by that fraction. You should just think of the loan payment formula in two different forms, one solving for present value, P , and one solving for payment, PMT .

Example \(\PageIndex{2}\): Loan Formula—Finding Payment

Jack goes to a car dealer to buy a new car for $18,000 at 2% APR with a five-year loan. The dealer quotes him a monthly payment of $425. What should the monthly payment on this loan be?

P = $18,000, r = 0.02, n = 12, t = 5

Jack should have a monthly payment of $315.50, not $425.

Therefore, the dealer is trying to get Jack to pay $25,500 - $18,930 = $6,570 in additional principal and interest charges. This means that the quoted rate of 2% APR is not accurate, or the quoted price of $18,000 is not accurate, or both.

Example \(\PageIndex{3}\): Loan Formula—Mortgage

Morgan is going to buy a house for $290,000 with a 30-year mortgage at 5% APR. What is the monthly payment for this house?

P = $290,000, r = 0.05, n = 12, t = 30

Therefore, Morgan will pay $560,440.80 in principal and interest which means that Morgan will pay $290,000 for the principal of the loan and $560,440.80 - $290,000 = $270,440.80 in interest. This is enough to buy another comparable home. Interest charges add up quickly.

Example \(\PageIndex{4}\): Loan Formula—Refinance Mortgage

If Morgan refinanced the $290,000 at 3.25% APR what would her monthly payments be?

P = $290,000, r = 0.0325, n = 12, t = 30

If Morgan refinanced the mortgage at 3.25% APR, the monthly payment would now be $1262.10 instead of $1556.78.

How much money would Morgan save over the life of the loan at the new payment amount?

Then, subtract $560,440.80 - $454,356 = $106,084.80.

Morgan would save $106,084.80 in interest because of refinancing the loan at 3.25% APR.

Example \(\PageIndex{5}\): Loan Formula—Mortgage Comparison

With a fixed rate mortgage, you are guaranteed that the interest rate will not change over the life of the loan. Suppose you need $250,000 to buy a new home. The mortgage company offers you two choices: a 30-year loan with an APR of 6% or a 15-year loan with an APR of 5.5%. Compare your monthly payments and total loan cost to decide which loan you should take. Assume no difference in closing costs.

Option 1 : First calculate the monthly payment:

The monthly payment for a 30-year loan at 6% interest is $1498.88.

Now calculate the total cost of the loan over the 30 years:

The monthly payments are $1498.88 and the total cost of the loan is $539,596.80.

Option 2 : First calculate the monthly payment:

The monthly payment for a 15-year loan at 5.5% interest is $2042.71.

Now calculate the total cost of the loan over the 15 years:

The monthly payments are $2042.71 and the total cost of the loan is $367,687.80.

Therefore, the monthly payments are higher with the 15-year loan, but you spend a lot less money overall.