Applied Math - Finances - Debt

Back to home

Introduction

Taking out a loan can improve our position in life. If we do not want to pay rent but want to instead own a home, we can take out a loan. If we need a new reliable vehicle for work, we can take out a loan.

Lenders will not give us that money for free. We pay the lenders interest. We often have to make choices based on that interest.

Example:

1.

A family has saved $10,000. They want to put the money toward their loans to save interest. They have a $100,000 mortgage loan at 3% interest and a $49,000 loan for land they purchased next to their home at 7% interest. They do not plan on paying off either loan, so they want to know which loan to put the money towards in order to save the most interest. Which loan should they put the money towards?

Some might think they should try to put money toward the $100,000 loan first as it is larger, but the money should actually go toward the $49,000 in order to save interest. The interest rate is higher for this loan. If the money is put toward the $100,000 loan, then a simple interest calculation of the interest owed for one year is

\[\$90,000 \cdot 0.03 + \$49,000 \cdot 0.07 =\] \[\$2,700 + \$3,430 =\] \[\$6,130\]

If the money is put toward the smaller loan with higher interest, the interest owed is

\[\$100,000 \cdot 0.03 + \$39,000 \cdot 0.07 =\] \[\$3,000 + \$2,730 =\] \[\$5,730\]

They saved $400 in interest for the first year by putting their money down on the higher interest loan!



2.

A family’s monthly budget is very tight. They have the following monthly bills:

One spouse completes a project for work they do as an independent contractor. They receive $3,500 for the work. In order to reduce their monthly bills, which loan(s) should they pay down with the $3,500? Why?

The family should pay off the credit card first. They have enough money to pay the entire principal (the amount they still owe). That means they will have $200 extra dollars a month in their budget. The remaining $1,000 could go toward the $350 a month car loan. Although this will not save them as much interest, it will get them closer to having another loan paid off and having another extra $350 a month in their budget.

If the family’s primary goal is to have more money in their account next month, they should pay off the smallest loan first. If the family’s goal is to save the most money in the long run, they should pay off the highest interest loan first. In this case, both were the same.

Credit card debt is almost always the first thing that should be paid off in a budget as it usually has the highest interest rate and a low principal balance.



3.

A family has a comfortable amount of money left over after bills are paid each month. They have the following monthly payments:

The family has $10,000 on hand and would like to save money in the long term. Which of these three loans should they pay down with $10,000?

The family should pay $10,000 to the mortgage, as it has the highest interest rate. They will save more money in the long term as they will save interest. They will not be freeing up any money in their monthly budget, but the problem states that long-term savings are the priority.



4.

Imagine there are two families with the same income and same debt (below),

but different strategies for debt. Family A makes early payments toward the smallest debt first, then uses any extra money to pay off the next smallest debt. Family B always puts their money toward the debt with the highest interest rate. Assume each family has $10,000 to pay toward debt each year. Which family will have paid less money in interest in four years?

For simplification purposes of this problem, we assume simple interest is calculated once per year for each debt, we assume that their monthly payments are not reducing the principal, and we assume that the $10,000 is paying down a principal with no loss due to interest at the beginning of each year.

If a family pays off a loan, they will have that monthly payment now available to pay off other loans for that year. Also assume that interest is only compounded once per year for simplification.

Family A pays $10,000 toward the smaller car loan, according to their strategy. Family B pays $10,000 to the larger car loan as it has more interest.

First year:

\[\text{family A pays \$10,000 on the \$20,000 loan}\] \[\text{family A's interest for year 1}\] \[\$140,000 \cdot 0.05 + \$10,000 \cdot 0.03 + \$30,000 \cdot 0.07 = \$9,400\]

\[\text{family B pays \$10,000 on the \$30,000 loan}\] \[\text{family B's interest for year 1}\] \[\$140,000 \cdot 0.05 + \$20,000 \cdot 0.03 + \$20,000 \cdot 0.07 = \$9,000\]

Year two: Family A finishes off the smallest loan in the second year. Family B continues to pay on the highest interest loan.

\[\text{family A pays another \$10,000 on the \$20,000 loan}\] \[\text{family A's interest for year 2}\] \[\$140,000 \cdot 0.05 + \$0 \cdot 0.03 + \$30,000 \cdot 0.07 = \$9,100\]

\[\text{family B pays another \$10,000 on the \$30,000 loan}\] \[\text{family B's interest for year 2}\] \[\$140,000 \cdot 0.05 + \$20,000 \cdot 0.03 + \$10,000 \cdot 0.07 = \$8,300\]

Year three: Family A now starts paying down the 7% car loan. Family A now has \(\$350 \cdot 12 = \$4,200\) more to pay down loans, as they paid off a loan early. Family B finishes off the 7% loan.

\[\text{family A pays \$14,200 on the \$30,000 loan}\] \[\text{family A's interest for year 3}\] \[\$140,000 \cdot 0.05 + \$15,800 \cdot 0.07 = \$8,106\]

\[\text{family B pays another \$10,000 on the \$30,000 loan}\] \[\text{family B's interest for year 3}\] \[\$140,000 \cdot 0.05 + \$20,000 \cdot 0.03 + \$0 \cdot 0.07 = \$7,600\]

Year four: Family A has \(\$14,200\) to pay down loans, as they paid off a loan early. Family B has paid off a loan and now has \(\$450 \cdot 12 = \$5,400\) extra to pay down loans. They pay $15,400 on the mortgage loan.

\[\text{family A pays \$14,200 on the \$30,000 loan}\] \[\text{family A's interest for year 4}\] \[\$140,000 \cdot 0.05 + \$1,600 \cdot 0.07 = \$7,122\]

\[\text{family B pays another \$10,000 on the \$30,000 loan}\] \[\text{family B's interest for year 4}\] \[\$114,600 \cdot 0.05 + \$20,000 \cdot 0.03 = \$6,330\]

Which family paid less in interest? Family A paid \[\$9,400 + \$9,100 + \$8,106 + \$7,122 = \$33,728\]. Family B paid \[\$9,000 + \$8,300 + \$7,600 + \$6,330 = \$31,230\]. Family A freed up more money in their budget to pay down loans, but family B saved more money by paying down high interest loans first.

Practice Problems

1. A family has saved $10,000. They want to put the money toward their loans to save interest. They have a $100,000 mortgage loan at 3% interest and a $49,000 loan for land they purchased next to their home at 4% interest. They do not plan on paying off either loan, so they want to know which loan to put the money towards in order to save the most interest. Which loan should they put the money towards?



2. A family has a comfortable amount of money left over after bills are paid each month. They have the following monthly payments:

The family has $10,000 on hand and would like to save money in the long term. Which of these three loans should they pay down with $10,000?



3. A family’s monthly budget is very tight. They have the following monthly bills:

One spouse completes a project for work they do as an independent contractor. They receive $4,500 for the work. In order to reduce their monthly bills, which loan(s) should they pay down with the $4,500? Why?



4. Imagine there are two families with the same income and same debt (below),

but different strategies for debt. Family A makes early payments toward the smallest debt first, then uses any extra money to pay off the next smallest debt. Family B always puts their money toward the debt with the highest interest rate. Assume each family has $10,000 to pay toward debt each year. Which family will have paid less money in interest in four years?

If a family pays off a loan, they will have that monthly payment now available to pay off other loans for that year. Also assume that interest is only compounded once per year for simplification.

Theory Questions

1. In question 4, one family tackled the smallest loans first. Paying off one loan gave them more money each year to pay down other loans. What would happen to these two families if we continued after four years? Do you think a family should always pay off smaller loans first, or always pay off high interest loans first?