Borrowing money from the bank to buy a car or a house usually means getting a reducing balance loan. The most common reducible balance loan type is a mortgage. This is when we borrow money to buy a property.

This is the sort of loan where we are charged compound interest as the fee for borrowing money, and at regular time periods, usually months, we are required to make repayments to the financial institution to slowly pay off the loan. The repayments are usually all the same amount, except for the last repayment which may be less than the full amount - since payments are generally rounded up we see the cumulative affect of small additional payments.

When we make each repayment the amount owed (the balance) reduces, hence the name for this type of loan.

As well as calculating the length of time it takes to pay off a loan, we are often interested in the total amount paid on the loan and the total amount of interest paid to the bank.

\text{Total amount paid = Total of full repayments + Final adjusted repayment}

\text{Total interest paid = Total amount paid - Initial amount borrowed}

Examples

Example 1

Ivan takes out a car loan for \$24\,000. He is charged 8.1\% per annum interest, compounded monthly. Ivan makes repayments of \$450 at the end of each month.

Complete the values in the empty cells in the table below. Give your answers correct to the nearest cent.

Month	Opening Balance	Interest	Repayment	Closing Balance
1	24\,000	162	450	23\,712
2
3

Worked Solution

Create a strategy

We can complete the table by first finding the opening balance, then the interest, then the repayment, and then the closing balance for each month.

Apply the idea

To find the opening balance, take the closing balance from the previous month:

\text{Opening balance} = \$23\,712

To find the interest, multiply the opening balance by the interest rate per month. To find the interest rate per month we can divide the annual interest rate of 8.1\%=0.081 by 12.

\displaystyle \text{Interest}	\displaystyle =	\displaystyle 23\,712 \times \left(\dfrac{0.081}{12}\right)
	\displaystyle =	\displaystyle \$160.06

The repayment is the same every month.

\text{Repayment}=\$450

To find the closing balance, add the opening balance and the interest, then subtract the repayment.

\displaystyle \text{Closing Balance}	\displaystyle =	\displaystyle 23\,712+160.06-450
	\displaystyle =	\displaystyle \$23\,422.06

Follow the same process for month 3 to get the following table:

Month	Opening Balance	Interest	Repayment	Closing Balance
1	\$24\,000	\$162	\$450	\$23\,712
2	\$23\,712	\$160.06	\$450	\$23\,422.06
3	\$23\,422.06	\$158.10	\$450	\$23\,130.15

Idea summary

\text{Total amount paid = Total of full repayments + Final adjusted repayment}

\text{Total interest paid = Total amount paid - Initial amount borrowed}

Model a reducing balance loan with recurrence relations

Since a reducing balance loan uses compound interest as well as making regular repayments, this type of recurrence relation involves a first order linear recurrence relation similar to an investment with regular withdrawals such as an annuity .

For a principal loan, P, at the compound interest rate of r per period and a payment of d per period, the sequence of the value of the loan over time forms a first order linear recurrence. The recursive sequence which generates the value, V_{n}, of the loan at the end of each instalment period is: V_n=V_{n-1} \times (1 + r)- d, where V_0=P.

The recursive sequence which generates the value, V_n of the investment/loan at the beginning of each instalment period is: V_n=V_{n-1} \times (1 + r)- d, where V_1=P.

Examples

Example 2

Bart borrows \$61\,000 from a banking institution. He is charged 6.6\% per annum interest, compounded monthly. At the beginning of each month, before interest is charged, he makes a repayment of \$400.

Complete the table below:

Month	Opening Balance	Repayment	Interest	Closing Balance
1	61\,000	400	333.30	60\,933.30
2
3
4

Worked Solution

Create a strategy

We can complete the table by first finding the opening balance, the repayment, then the interest, and lastly the closing balance.

Apply the idea

To find the opening balance, take the closing balance from the previous month:

\text{Opening balance} = \$60\,933.30

The repayment is still the same every month.

\text{Repayment}=\$400

To find the interest, subtract the repayment from the opening balance then multiply the result by the monthly interest rate. To find the rate per month we divide the annual rate of 6.6\%=0.066 by 12.

\displaystyle \text{Interest}	\displaystyle =	\displaystyle (60\,933.30 - 400) \times \left(\dfrac{0.066}{12}\right)
	\displaystyle =	\displaystyle \$332.93

To find the closing balance, subtract the repayment from the opening balance, then add the interest.

\displaystyle \text{Closing Balance}	\displaystyle =	\displaystyle 60\,933.30-400+333.30
	\displaystyle =	\displaystyle \$60\,866.23

Follow the same process for months 3 and 4 to get the following table:

Month	Opening Balance	Repayment	Interest	Closing Balance
1	\$61\,000	\$400	\$333.30	\$60\,933.30
2	\$60\,933.30	\$400	\$332.93	\$60\,866.23
3	\$60\,866.23	\$400	\$332.56	\$60\,798.79
4	\$60\,798.79	\$400	\$332.19	\$60\,730.98

Write a recursive rule that gives the closing balance, B_n, at the end of month n.

Write both parts of the rule including B_0.

Worked Solution

Create a strategy

Modify the recursive rule V_{n}=V_{n-1}\times (1+r)-d, where V_{0} = P, and find r,\,d,\, and P.

Apply the idea

Each month the repayment is d=\$400. The original investment was P=\$61\,000.

The balance is increased by 6.6\% =0.066 each year. To find the rate per month we divide by 12 to get: r=\dfrac{0.066}{12}=0.0055.

Since we need to subtract the repayment before calculating the interest we need to multiply (1+r), by the previous closing balance minus the repayment, V_{n-1}-d. So the recursive rule becomes: V_{n}=(V_{n-1}-d)\times (1+r).

\displaystyle B_{n}	\displaystyle =	\displaystyle (1+r)( B_{n-1} - d),\,B_{0}=P	Write the recursive rule
	\displaystyle =	\displaystyle (1+0.0055)(B_{n-1} - 400),\,B_{0}=61\,000	Substitute r, \,d, \,P
\displaystyle B_{n}	\displaystyle =	\displaystyle 1.0055(B_{n-1} - 400),\,B_{0}=61\,000	Simplify

Use your calculator to determine how much is owing on the loan after 4 years. Give your answer to the nearest cent.

Worked Solution

Create a strategy

Enter the recursive rule and closing balance we found in part (b) into your calculator and look for the loan balance owed after 4 years.

Apply the idea

4 years is equal to 48 months. So when generating the values for the recursive rule in your calculator, look for the value when N=48.

Using your calculator you should get that the value for the 48th month is \$57\,347.43.

At the end of which year and month will the loan have been repaid?

Worked Solution

Create a strategy

Enter the recursive rule and closing balance we found in part (b) into your calculator and find when the balance of the loan first drops below \$0.

Apply the idea

Using your calculator you should get that the value drops below 0 at N=328. This means after 328 months. To find the number of years, we divide by 12.

\displaystyle 328\div 12	\displaystyle \approx	\displaystyle 27.333\ldots	Divide the number of months by 12
	\displaystyle =	\displaystyle 12\dfrac{4}{12}	Write as a mixed numeral

This tells us that 328 months is equal to 27 years and 4 months.

So at the end of 27 years and 4 months the loan will be repaid.

Idea summary

For a principal loan, P, at the compound interest rate of r per period and a payment of d per period, the sequence of the value of the loan over time forms a first order linear recurrence. The sequence which generates the value, V_{n}, of the loan at the end of each instalment period is:

Recursive sequence: V_n=V_{n-1} \times (1 + r)- d,\, \text{where } V_0=P

The sequence which generates the value, V_n of the investment/loan at the beginning of each instalment period is:

Recursive sequence:V_n=V_{n-1} \times (1 + r)- d,\, \text{where } V_1=P

Reducing balance loans with the financial application

Often it's more convenient to analyse various situations for a reducible balance loan using the financial facility of our calculator. Remember that the present value will be the value of the loan and is entered as a positive value, as the lender is receiving money from the bank. The payments will be negative as these are paid to the bank.

In real life, banks usually calculate interest on loan accounts monthly but people can choose to make fortnightly or even weekly repayments.

When the number of payments is not equal to the number of compounding periods the financial application of the calculator is a great tool.

N is the total number of payments, so: N=\text{Payments per year} \times \text{Number of years} and P/Y is number of payments per year, and C/Y is the number of times interest is calculated per year.

Examples

Example 3

Mr. and Mrs. Gwen held a mortgage for 25 years. Over that time they made monthly repayments of \$4500 and were charge a fixed interest rate of 4.4\% per annum, compounded monthly.

We will use the financial solver on your CAS calculator to determine how much they initially borrowed.

Which variable on the CAS calculator do we want to solve for?

I\%

Pmt

Worked Solution

Create a strategy

Consider which variable is our unknown.

Apply the idea

We want to find the amount they initially borrowed, which is the present value or PV.

The correct answer is option A.

Fill in the table for each of the following:

	Value
N
I\%
Pmt
FV
PpY
CpY

Worked Solution

Create a strategy

Use the given information to find each value, keeping in mind that the Pmt should be opposite in sign to FV.

Apply the idea

N is the number of monthly payments made during the loan which is 25\times 12 =300 months.

I\% is the annual interest rate which is 4.4\%.

Pmt is the amount Mr. and Mrs. Gwen repay each month which is \$4500.

FV is the amount of money owed at the end of the loan. This should be zero.

PpY is the number of repayments made each year which is 12.

CpY is the number of compounding periods each year which is also 12.

	Value
N	300
I\%	4.4\%
Pmt	4500
FV	0
PpY	12
CpY	12

Hence, state how much Mr. and Mrs. Gwen initially borrowed, correct to the nearest dollar.

Worked Solution

Create a strategy

Using technology, enter the values from part (b) into a financial solver function, and solve for the initial value of the loan PV.

Apply the idea

Using your calculator and pressing Enter in the PV box you should get PV=817\,926.

So the initial value of the loan is \$817\,926.

Example 4

Valerie borrows \$345\,000 to buy an apartment. The bank offers a reducing balance loan with an interest rate of 2.35\% p.a. compounded monthly. Valerie chooses to make fortnightly payments of \$1250 in order to pay off the loan. Use the financial application on your calculator to answer the following questions. Assume there are 26 fortnights in a year.

What is the balance, in dollars, after 100 weeks? Round your answer to the nearest cent.

Worked Solution

Create a strategy

Use the formula N=\text{Payments per year} \times \text{Number of years}.

Apply the idea

Since the interest is compounded monthly but the repayments are fortnightly, we need to use find the value of N. 100 weeks is \dfrac{100}{52} years.

\displaystyle N	\displaystyle =	\displaystyle \text{Payments per year} \times \text{Number of years}	Write the formula
	\displaystyle =	\displaystyle 26\times \dfrac{100}{52}	Multiply the number of payments by the years
	\displaystyle =	\displaystyle 50	Evaluate

From the information given, the values we need to enter into our financial application are:

	Value
N	50
I\%	2.35\%
Pmt	-1250
PV	345\,000
FV
PpY	26
CpY	12

Using your calculator and pressing Enter in the FV box you should get FV=-297\,029.57.

The balance after 100 weeks is \$297\,029.57.

Approximate how long it takes her to pay off the loan in years. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the financial application to find the value of N for FV=0.

Apply the idea

Enter the following values into your calculator:

	Value
N
I\%	2.35\%
Pmt	-1250
PV	345\,000
FV	0
PpY	26
CpY	12

Using your calculator and pressing Enter in the N box you should get N \approx 317.609. This is the number of total repayments. To find the number of years, we need to divide by the number of payments per year which is 26.

\displaystyle \text{Years}	\displaystyle =	\displaystyle 317.609 \div 26	Divide the number of payments by 26
	\displaystyle \approx	\displaystyle 12.22	Evaluate and round

So the number of years to pay off her loan is 12.22.

Example 5

An entrepreneur borrows \$1\,200\,000 from a bank at an interest of 1.85\% p.a. compounded weekly and makes \$5000 per week payments into the loan account.

Assume there are 52 weeks in a year.

If N is the number of payments, complete the table of values showing the variables required to use the financial application of your calculator to determine how long it takes to pay off the loan.

Variable	Value
N	-
I(\%)	\%
PV
Pmt
FV
P/Y
C/Y

Worked Solution

Create a strategy

Use the given information.

Apply the idea

\displaystyle I(\%)	\displaystyle =	\displaystyle 1.85\%	The annual interest rate
\displaystyle PV	\displaystyle =	\displaystyle 1\,200\,000	The amount borrowed
\displaystyle Pmt	\displaystyle =	\displaystyle -5000	The payment, which is negative since they are giving the money back
\displaystyle FV	\displaystyle =	\displaystyle 0	At value of the loan when it is repayed
\displaystyle P/Y	\displaystyle =	\displaystyle 52	Number of weekly payments per year
\displaystyle C/Y	\displaystyle =	\displaystyle 52	Number of compounding periods per year

Variable	Value
N	-
I(\%)	1.85\%
PV	1\,200\,000
Pmt	-5000
FV	0
P/Y	52
C/Y	52

Determine the whole number of weeks it will take until the entrepreneur pays off their loan.

Worked Solution

Create a strategy

Enter the values from part (a) into your calculator's financial application to solve for N.

Apply the idea

By pressing Enter in the box for N we should get N=250.914. This is the number of weekly payments, so we should round up to make sure we pay the loan back.

So it will take 251 weeks to pay off the loan.

Calculate the amount of the final payment of the loan, in dollars.

Worked Solution

Create a strategy

Find the future value for 251 payments.

Apply the idea

Enter the following values in your financial application to find the future value if 251 whole payments are made:

Variable	Value
N	251
I(\%)	1.85\%
PV	1\,200\,000
Pmt	-5000
FV
P/Y	52
C/Y	52

You should get FV=429.346. Since FV is positive this means that we have paid \$429.346 extra, so we need to subtract this from the usual payment amount of \$5000.

\displaystyle \text{Final payment}	\displaystyle =	\displaystyle 5000-429.346	Subtract the extra amount
	\displaystyle =	\displaystyle \$4570.65	Evaluate and round

Hence determine the total amount the entrepreneur pays over the duration of the loan, in dollars.

Worked Solution

Create a strategy

Add all the payments.

Apply the idea

So 250 payments were of \$5000 and the last payment was \$4570.65.

\displaystyle \text{Total paid}	\displaystyle =	\displaystyle 250\times 5000+4570.65	Add the payments
	\displaystyle =	\displaystyle \$1\,254\,570.65	Evaluate and round

Idea summary

When using a financial application for a reducing balance loan:

N is the number of years of the loan.

I\% is the annual interest rate.

PV is the initial value of the loan.

Pmt is the amount of repayment each time period.

FV is the amount of money owed at the end of the loan.

PpY s the number of repayments made each year.

CpY is the number of compounding periods each year.

If the number of payments and compounding periods are not equal:

\displaystyle N=\text{Payments per year} \times \text{Number of years}

\bm{N}

is the total number of payments

Outcomes

ACMGM097

use a recurrence relation to model a reducing balance loan and investigate (numerically or graphically) the effect of the interest rate and repayment amount on the time taken to repay the loan

ACMGM098

with the aid of a financial calculator or computer-based financial software, solve problems involving reducing balance loans; for example, determining the monthly repayments required to pay off a housing loan

6.08 Reducing balance loans

Ideas

Reducing balance loan tables

Examples

Example 1

Model a reducing balance loan with recurrence relations

Examples

Example 2

Reducing balance loans with the financial application

Examples

Example 3

Example 4

Example 5

Outcomes

ACMGM097

ACMGM098

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