5/5 FA 7.2

Last updated over 2 years ago

40 questions

Note from the author:

OBJECTIVES & STANDARDS
Math Objectives
Use the compound interest formula given a financial situation and interpret its results
Evaluate expressions given variable definitions
Common Core Math Standards
Link to all CCSS Math
CCSS.HSF.LE.A.1.C
CCSS.HSF.LE.A.2
CCSS.HSF.BF.A.1
Personal Finance Objectives
Demonstrate how compound interest can affect future loan balances
Explain the difference between APR and APY
Calculate the future value of periodic investments
National Standards for Personal Financial Education
Managing Credit
1a: Describe how credit card grace periods, methods of interest calculation, and fees affect borrowing costs
3b: Compare the cost of borrowing $1,000 using consumer credit options that differ in rates and fees

Instructions

OBJECTIVES & STANDARDS
Math Objectives
Use the compound interest formula given a financial situation and interpret its results
Evaluate expressions given variable definitions
Common Core Math Standards
Link to all CCSS Math
CCSS.HSF.LE.A.1.C
CCSS.HSF.LE.A.2
CCSS.HSF.BF.A.1
Personal Finance Objectives
Demonstrate how compound interest can affect future loan balances
Explain the difference between APR and APY
Calculate the future value of periodic investments
National Standards for Personal Financial Education
Managing Credit
1a: Describe how credit card grace periods, methods of interest calculation, and fees affect borrowing costs
3b: Compare the cost of borrowing $1,000 using consumer credit options that differ in rates and fees

Intro

Explore It

Learn It

Practice It

Required

Learn It

Application: Loans and the Compound Interest Formula

5/5 FA 7.2

Last updated over 2 years ago

40 questions

Note from the author:

OBJECTIVES & STANDARDS
Math Objectives
Use the compound interest formula given a financial situation and interpret its results
Evaluate expressions given variable definitions
Common Core Math Standards
Link to all CCSS Math
CCSS.HSF.LE.A.1.C
CCSS.HSF.LE.A.2
CCSS.HSF.BF.A.1
Personal Finance Objectives
Demonstrate how compound interest can affect future loan balances
Explain the difference between APR and APY
Calculate the future value of periodic investments
National Standards for Personal Financial Education
Managing Credit
1a: Describe how credit card grace periods, methods of interest calculation, and fees affect borrowing costs
3b: Compare the cost of borrowing $1,000 using consumer credit options that differ in rates and fees

Instructions

OBJECTIVES & STANDARDS
Math Objectives
Use the compound interest formula given a financial situation and interpret its results
Evaluate expressions given variable definitions
Common Core Math Standards
Link to all CCSS Math
CCSS.HSF.LE.A.1.C
CCSS.HSF.LE.A.2
CCSS.HSF.BF.A.1
Personal Finance Objectives
Demonstrate how compound interest can affect future loan balances
Explain the difference between APR and APY
Calculate the future value of periodic investments
National Standards for Personal Financial Education
Managing Credit
1a: Describe how credit card grace periods, methods of interest calculation, and fees affect borrowing costs
3b: Compare the cost of borrowing $1,000 using consumer credit options that differ in rates and fees

Intro

CALCULATE: Calculating Interest
Interest on a loan can be calculated in a variety of ways.  Examine the two tables and use them to answer the questions below.

Required

How is Table 2’s calculation different?

Required

Explore It

Compound Interest
The process of including interest in the balance when calculating new interest, like we did in Table 2 above, is called Compounding Interest. In that table, interest was compounded one time each year.  What happens if we decide to compound the accumulated interest more frequently, like monthly or daily?  Is there an easy way to calculate your balance when compounding?  Let’s explore these questions.
Let’s start with the exponential growth equation that you’ve used many times before:


Where:
A = Total Balance after interest has accrued
P = Principal (This may be referred to by other terms such as starting balance or loan amount)
r = the annual interest rate
t = the amount of time in years

Required

If the annual interest rate was 8%,
a. How would you calculate the monthly interest rate?
b. How would you calculate the daily interest rate?

Required

If we update the formula above to include this time interval, it looks like this now:

We have one more problem.  The variable t represents years, but if we’re compounding monthly or daily, we need the exponent to reflect the number compounds, NOT the number of years.

Required

If you are compounding monthly over 3 years, how many times do you compound?

Required

If you are compounding daily for 3 years, how many times do you compound?

Required

Learn It

Ok, we’re almost there.  Let’s update our formula one more time.

The equation above is called the compound interest formula and it can be used to calculate the account balance when interest is compounded n times per year.

Using the Compound Interest Formula
Now that we have a formula that can be used to calculate balances that use compound interest, let’s put it into practice.
If Alana has a loan balance of $12,000 that is compounded monthly at a rate of 4.5%.  Assuming that she makes no payments on the account, what will her balance be in 4 years?
Review the completed example below.

Required

Example 1
P = $5000
r = 6.25%
Compounded daily for 5 years

Required

Practice It

Required

P = $8,500
r = 3.175%
Compounded monthly for 10 years

Required

P = $28,000
r = 9%
Compounded annually for 5 years

Required

P = $275
r = 6.25%
Compounded daily for 20 years

Required

Marko has a credit card balance of $875 that is compounded daily at an annual percentage rate (APR) of 19.62%. If he makes no payments, what will his balance be in 2 years?

Required

Bethany has $37,525 in student loans that accumulate interest at 3.73% annualized interest that is compounded monthly. If she starts making payments in 4 years, what will her loan balance be?

Required

Camilla took a $23,500 secured auto loan that allows her to make no payments for 18 months. If interest is still compounded monthly at a rate of 6.3%, what will her balance be in 18 months when she starts making payments?

Required

Shawn’s mortgage balance of $263,000 is compounded daily at a rate of 4.075% annualized interest. If no payments are made for 6 months, what will the balance be after this time has passed?

Learn It

VIDEO: APR vs APY
When shopping for a loan, you may see the terms APR and APY.  What are these terms and how do they apply to compound interest?  Watch the video at the top of the linked article, then use it to answer the questions.

Example:  
If a loan charges 1% interest per month, what is its APR and APY?

Required

Application: Loans and the Compound Interest Formula

Level 1
Frequency of Compounding and APY
We’ve seen that the frequency of compounding can have a significant effect on the overall interest accumulated.  Let’s take a deeper dive by calculating the APY of the same situation using different compounding periods.
Scenario: Alonso bought a $3,000 gaming computer using his credit card.  A special financing offer allows him to make no payments for 2 years but interest will still accrue at a rate of 16.92% APR and be added to the bill if he doesn't pay full balance in 2 years.

Required

Level 2:  
Compound Interest Over Irregular Periods
Many scenarios given to you when learning about the compound interest formula include whole years.  Life doesn’t always work in nice whole numbers! How would we handle a situation where we wanted to calculate a balance after 19 months or 27 months?

Scenario: Kira takes a $12,000 loan that has a 5 year term at 4.25% compounded daily.  The terms of the loan state that she can make no payments for 6 months.  After reading the fine print, Kira notices that interest will continue to accumulate during this 6 month grace period that will be rolled over into her loan balance.

Required

If the compound interest formula is written in years, what value should you use for t to represent 6 months?

Required

Scenario: Lindsay makes an investment of $5,000 in an account that yields 2.25% annual rate of return compounded monthly.

Required

Assuming that she makes no additional deposits or withdrawals, what is Lindsay’s investment balance after 18 months?

Required

After 18 months, Lindsay decides to deposit an additional $6,500 into this account. What will her balance be if she makes no changes for 28 additional months?

Required

Level 3:
Compound Interest and Student Loans
One form of student loan, called an unsubsidized loan, allows you to defer payments until after you graduate from college.  The catch is that you will still accumulate interest during this time.  Let’s look at a scenario where we will need to use a variation of the compound interest formula.
Scenario
Maya has been accepted to her dream school.  She received federal aid, scholarships and other forms of financial aid.  After applying these to her tuition, she is still responsible for $11,500 per semester.  She is approved for an unsubsidized loan at a rate of 3.73% compounded semiannually to cover the remaining costs.  Based on her course catalog, Maya plans to attend school two semesters per year for four years.

Required

The End!  Make good Choices this weekend :) 

5/5 FA 7.2

5/5 FA 7.2

How is Table 2’s calculation different?

If the annual interest rate was 8%,a. How would you calculate the monthly interest rate?b. How would you calculate the daily interest rate?

If you are compounding monthly over 3 years, how many times do you compound?

If you are compounding daily for 3 years, how many times do you compound?

Example 1P = $5000r = 6.25%Compounded daily for 5 years

P = $8,500r = 3.175%Compounded monthly for 10 years

P = $28,000r = 9%Compounded annually for 5 years

P = $275r = 6.25%Compounded daily for 20 years

Marko has a credit card balance of $875 that is compounded daily at an annual percentage rate (APR) of 19.62%. If he makes no payments, what will his balance be in 2 years?

Bethany has $37,525 in student loans that accumulate interest at 3.73% annualized interest that is compounded monthly. If she starts making payments in 4 years, what will her loan balance be?

Camilla took a $23,500 secured auto loan that allows her to make no payments for 18 months. If interest is still compounded monthly at a rate of 6.3%, what will her balance be in 18 months when she starts making payments?

Shawn’s mortgage balance of $263,000 is compounded daily at a rate of 4.075% annualized interest. If no payments are made for 6 months, what will the balance be after this time has passed?

If the compound interest formula is written in years, what value should you use for t to represent 6 months?

Assuming that she makes no additional deposits or withdrawals, what is Lindsay’s investment balance after 18 months?

After 18 months, Lindsay decides to deposit an additional $6,500 into this account. What will her balance be if she makes no changes for 28 additional months?

Briefly describe how each month’s interest is calculated in Table 1.

How is Table 2’s calculation different?

Would you prefer Table 1 or Table 2 if you were earning interest on an investment? Explain your answer.

Would you prefer Table 1 or Table 2 if you were being charged interest on a credit card or loan? Explain your answer.

If the annual interest rate was 8%,a. How would you calculate the monthly interest rate?b. How would you calculate the daily interest rate?

Write a general expression for converting an annual interest rate into other time intervals where n is used to represent the time interval.

If you are compounding monthly over 3 years, how many times do you compound?

If you are compounding daily for 3 years, how many times do you compound?

If n represents the time interval and t represents time in years, write an expression that calculates the number of times you compound in t years.

Example 1P = $5000r = 6.25%Compounded daily for 5 years

Example 2Carmine takes a loan for $11,500 at a rate of 8% that is compounded quarterly. Assuming she makes no payments for the first 2 years, what is her loan balance?

P = $8,500r = 3.175%Compounded monthly for 10 years

P = $28,000r = 9%Compounded annually for 5 years

P = $275r = 6.25%Compounded daily for 20 years

Marko has a credit card balance of $875 that is compounded daily at an annual percentage rate (APR) of 19.62%. If he makes no payments, what will his balance be in 2 years?

Bethany has $37,525 in student loans that accumulate interest at 3.73% annualized interest that is compounded monthly. If she starts making payments in 4 years, what will her loan balance be?

Camilla took a $23,500 secured auto loan that allows her to make no payments for 18 months. If interest is still compounded monthly at a rate of 6.3%, what will her balance be in 18 months when she starts making payments?

Shawn’s mortgage balance of $263,000 is compounded daily at a rate of 4.075% annualized interest. If no payments are made for 6 months, what will the balance be after this time has passed?

What do APR and APY stand for?

What is the key difference between APR and APY?

Shawn knows that his student loan interest is calculated at a rate of 0.35% per month.a. What is Shawn's APR?b. What is Shawn's APY?c. Which of the percentages represents the actual cost that Shawn will pay for his student loans if he makes no payments?

Why does the savings account advertise using APY while the credit card company uses APR?

Use the compound interest formula to calculate Alonso’s balance after 2 years using each compounding period, assuming he makes no payments. Write your answer to fill in the Account Balance column, make sure to show your work.

How much more is Alonso paying by having his interest compounded daily versus yearly?

How does the frequency of compounding impact the overall cost of Alonso’s total interest?

Now let’s calculate the APY of each compounding period to see what percentage rate Alonso is actually paying annually. Recall the formula:Where:r = Annual Percentage Raten = number of times the account is compoundedCalculate the APY for each compounding period.

Most credit cards are compounded monthly or daily, which means that Alonso is paying about 1.5% more annually than the advertised APR. What effect could this knowledge have had on Alonso’s purchasing decisions?

If the compound interest formula is written in years, what value should you use for t to represent 6 months?

What will Kira’s loan balance be after 6 months?

How much interest will be accumulated during this 6 month grace period?

Assuming that she makes no additional deposits or withdrawals, what is Lindsay’s investment balance after 18 months?

After 18 months, Lindsay decides to deposit an additional $6,500 into this account. What will her balance be if she makes no changes for 28 additional months?

What is Maya’s student loan balance after one semester?

Why can’t we use the compound interest formula to calculate Maya’s balance after graduation?

Use this formula to calculate Maya’s student loan balance after 4 years of undergraduate education.

How much interest has accumulated in 4 years?

Use the Student Loan Calculator to calculate how much Maya’s monthly payment will be after school is done, assuming she has a 10 year term to pay it off.a. How much is Maya’s student loan payment each month?b. What is the total cost of her initial $92,000 in loans?

What conclusions could Maya draw from comparing her student loan options at her top two schools?

Briefly describe how each month’s interest is calculated in Table 1.

Would you prefer Table 1 or Table 2 if you were earning interest on an investment? Explain your answer.

Would you prefer Table 1 or Table 2 if you were being charged interest on a credit card or loan? Explain your answer.

Write a general expression for converting an annual interest rate into other time intervals where n is used to represent the time interval.

If n represents the time interval and t represents time in years, write an expression that calculates the number of times you compound in t years.

Example 2Carmine takes a loan for $11,500 at a rate of 8% that is compounded quarterly. Assuming she makes no payments for the first 2 years, what is her loan balance?

What do APR and APY stand for?

What is the key difference between APR and APY?

Shawn knows that his student loan interest is calculated at a rate of 0.35% per month.a. What is Shawn's APR?b. What is Shawn's APY?c. Which of the percentages represents the actual cost that Shawn will pay for his student loans if he makes no payments?

Why does the savings account advertise using APY while the credit card company uses APR?

Use the compound interest formula to calculate Alonso’s balance after 2 years using each compounding period, assuming he makes no payments. Write your answer to fill in the Account Balance column, make sure to show your work.

How much more is Alonso paying by having his interest compounded daily versus yearly?

How does the frequency of compounding impact the overall cost of Alonso’s total interest?

Now let’s calculate the APY of each compounding period to see what percentage rate Alonso is actually paying annually. Recall the formula:Where:r = Annual Percentage Raten = number of times the account is compoundedCalculate the APY for each compounding period.

Most credit cards are compounded monthly or daily, which means that Alonso is paying about 1.5% more annually than the advertised APR. What effect could this knowledge have had on Alonso’s purchasing decisions?

What will Kira’s loan balance be after 6 months?

How much interest will be accumulated during this 6 month grace period?

What is Maya’s student loan balance after one semester?

Why can’t we use the compound interest formula to calculate Maya’s balance after graduation?

Use this formula to calculate Maya’s student loan balance after 4 years of undergraduate education.

How much interest has accumulated in 4 years?

Use the Student Loan Calculator to calculate how much Maya’s monthly payment will be after school is done, assuming she has a 10 year term to pay it off.a. How much is Maya’s student loan payment each month?b. What is the total cost of her initial $92,000 in loans?

What conclusions could Maya draw from comparing her student loan options at her top two schools?

If the annual interest rate was 8%,
a. How would you calculate the monthly interest rate?
b. How would you calculate the daily interest rate?

Example 1
P = $5000
r = 6.25%
Compounded daily for 5 years

P = $8,500
r = 3.175%
Compounded monthly for 10 years

P = $28,000
r = 9%
Compounded annually for 5 years

P = $275
r = 6.25%
Compounded daily for 20 years

If the annual interest rate was 8%,
a. How would you calculate the monthly interest rate?
b. How would you calculate the daily interest rate?

Example 1
P = $5000
r = 6.25%
Compounded daily for 5 years

Example 2
Carmine takes a loan for $11,500 at a rate of 8% that is compounded quarterly. Assuming she makes no payments for the first 2 years, what is her loan balance?

P = $8,500
r = 3.175%
Compounded monthly for 10 years

P = $28,000
r = 9%
Compounded annually for 5 years

P = $275
r = 6.25%
Compounded daily for 20 years

Shawn knows that his student loan interest is calculated at a rate of 0.35% per month.
a. What is Shawn's APR?
b. What is Shawn's APY?
c. Which of the percentages represents the actual cost that Shawn will pay for his student loans if he makes no payments?

Now let’s calculate the APY of each compounding period to see what percentage rate Alonso is actually paying annually. Recall the formula:

Where:
r = Annual Percentage Rate
n = number of times the account is compounded

Calculate the APY for each compounding period.

Use the Student Loan Calculator to calculate how much Maya’s monthly payment will be after school is done, assuming she has a 10 year term to pay it off.
a. How much is Maya’s student loan payment each month?
b. What is the total cost of her initial $92,000 in loans?

Example 2
Carmine takes a loan for $11,500 at a rate of 8% that is compounded quarterly. Assuming she makes no payments for the first 2 years, what is her loan balance?

Shawn knows that his student loan interest is calculated at a rate of 0.35% per month.
a. What is Shawn's APR?
b. What is Shawn's APY?
c. Which of the percentages represents the actual cost that Shawn will pay for his student loans if he makes no payments?

Now let’s calculate the APY of each compounding period to see what percentage rate Alonso is actually paying annually. Recall the formula:

Where:
r = Annual Percentage Rate
n = number of times the account is compounded

Calculate the APY for each compounding period.

Use the Student Loan Calculator to calculate how much Maya’s monthly payment will be after school is done, assuming she has a 10 year term to pay it off.
a. How much is Maya’s student loan payment each month?
b. What is the total cost of her initial $92,000 in loans?