Creating and analysing a table can be time-consuming, and it is often more convenient to analyse various situations for a reducible balance loan using the finance solver of a calculator. To use the financial solver you will need to know:

N - the total number of payments.
I\% - the annual interest rate.
PV - the present value of the loan or investment.
PMT or Pmt - the amount paid at each payment.
FV - the future value of the loan or investment.
PpY or P/Y - the number of payments per year.
CpY or C/Y - the number of times interest is compounded per year.

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 1

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 2

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.

Idea summary

When using a financial application for a reducing balance loan:

N - the total number of payments.
I\% - the annual interest rate.
PV - the present value of the loan or investment.
PMT or Pmt - the amount paid at each payment.
FV - the future value of the loan or investment.
PpY or P/Y - the number of payments per year.
CpY or C/Y - the number of times interest is compounded per 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

U3.AoS2.3

the concepts of financial mathematics including simple and compound interest, nominal and effective interest rates, the present and future value of an investment, loan or asset, amortisation of a reducing balance loan or annuity and amortisation tables

U3.AoS2.8

use a table to investigate and analyse on a step–by-step basis the amortisation of a reducing balance loan or an annuity, and interpret amortisation tables

U3.AoS2.4

the use of first-order linear recurrence relations to model compound interest investments and loans, and the flat rate, unit cost and reducing balance methods for depreciating assets, reducing balance loans, annuities, perpetuities and annuity investments

U3.AoS2.9

use technology with financial mathematics capabilities, to solve practical problems associated with compound interest investments and loans, reducing balance loans, annuities and perpetuities, and annuity investments

6.02 Reducing balance loans with technology

Ideas

Reducing balance loans with the financial application

Examples

Example 1

Example 2

Outcomes

U3.AoS2.3

U3.AoS2.8

U3.AoS2.4

U3.AoS2.9

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