Solving Harder Financial Problems Revision Notes for VCE SSCE General Mathematics

Solving Harder Financial Problems

Introduction

Financial situations don't always remain constant. Sometimes the conditions of a reducing balance loan need to change, requiring adjustments to regular repayments so the loan can be fully repaid. Similarly, changes in interest rates can affect:

Payments received from an annuity
The balance of an investment with compound interest

When these changes occur, a financial solver on your CAS calculator becomes essential for finding new payment amounts or updated balances.

infoNote

Key principle: When conditions change partway through a financial arrangement, you must analyse the problem in two separate parts:

The period before the change
The period after the change occurs

Important rounding rule

chatImportant

Critical Rule: Never Round Too Early

Banks and financial institutions never round the value of an investment or loan until:

The investment is withdrawn, or
The loan is fully repaid

This means you must use unrounded values for all intermediate calculations. Rounding too early will lead to incorrect final answers.

Changing the regular payment to an investment

Understanding the concept

Investors sometimes want to change how much they regularly contribute to an investment. When this happens, you need to:

First calculate the investment value up to the point of change
Then use that value as the starting point for the second period
Apply the new regular payment amount for the remaining time

lightbulbExample

Worked Example: Investment with changing regular payments

Problem: Derek invests $50,000 into a compound interest investment with an annual interest rate of 6.1%, compounding annually. He adds $8,000 per year immediately after interest is calculated.

After five years, Derek increases his additional investment to $10,000 per year.

Find the value of Derek's investment after twelve years in total.

Solution:

Step 1: Calculate the investment value after the first five years

Open the finance solver and enter:

$N = 5$ (five years before the change)
$I\% = 6.1$ (annual interest rate)
$PV = -50000$ (initial investment amount)
$PMT = -8000$ (additional amount added each year)
$Pp/Y = 1$ (annual payments)
$Cp/Y = 1$ (interest compounds annually)

Solve for $FV$ .

Result: $FV = 112414.364...$

Step 2: Calculate the investment value for the remaining seven years

Now change the finance solver settings:

$N = 7$ (seven years after the change)
$I\% = 6.1$ (same interest rate)
$PV = -112414.364...$ (use the full unrounded value from Step 1)
$PMT = -10000$ (new additional amount added)
$Pp/Y = 1$ (annual payments)
$Cp/Y = 1$ (interest compounds annually)

Solve for $FV$ .

Result: $FV = 254343.79745...$

Step 3: Write the final answer

The value of Derek's investment after twelve years is $254,343.80 (rounded to the nearest cent).

infoNote

Exam tip: Notice how the future value from the first period becomes the present value for the second period. This is the key to solving two-part financial problems.

Changing the interest rate

Understanding interest rate changes

When interest rates change during the term of a loan, the regular repayment amount often needs to be adjusted to ensure:

The loan is still fully repaid by the original end date
No money is owed at the end of the term

This same approach applies to annuities and investments.

lightbulbExample

Worked Example: Loan with changing interest rate

Problem: Adrian borrows $150,000 for 25 years at an interest rate of 6.8% per annum, compounding monthly.

For the first three years, Adrian repays $1,041.11 each month.

After 3 years, the interest rate rises to 7.2% per annum. Adrian still wants to pay off the loan in 25 years, so he makes 263 monthly payments of $1,076.18 followed by a final payment.

Calculate the final payment needed to ensure the loan is fully repaid at the end of 25 years. Round to the nearest cent.

Solution:

Step 1: Find the loan balance after three years

Open the finance solver and enter:

$N = 36$ (number of monthly payments in 3 years)
$I\% = 6.8$ (annual interest rate)
$PV = 150000$ (initial loan amount)
$PMT = -1041.11$ (monthly repayments)
$Pp/Y = 12$ (monthly payments)
$Cp/Y = 12$ (interest compounds monthly)

Note: You can enter $N$ as $3 \times 12$ and the calculator will compute 36 for you.

Solve for $FV$ .

Result: $FV = -142391.8359...$

The negative sign indicates this is money still owed on the loan.

Step 2: Calculate the remaining loan balance after 263 payments at the new rate

Since the loan must still be repaid in 25 years total, there are 22 years remaining.

Change the finance solver settings:

$N = 264$ (which is $22 \times 12$ monthly payments)
$I\% = 7.2$ (the new interest rate)
$PV = 142391.8359...$ (the full unrounded balance after 3 years)
$PMT = -1076.18$ (new monthly payment amount)
$Pp/Y = 12$ (monthly payments)
$Cp/Y = 12$ (interest compounds monthly)

Solve for $FV$ .

Result: $FV = 0.1173...$

Step 3: Calculate the final payment

Since the future value is $0.12 (when rounded), the final payment decreases by this amount:

$\text{Final payment} = \$1076.18 - \$0.12 = \$1076.06$

Answer: The final payment will be $1076.06

chatImportant

Critical warning: If you round prematurely during this calculation, you will get an incorrect answer of $1076.08. Always use the full unrounded values from your calculator until the final step.

Key finance solver variables

When using the finance solver on your CAS calculator, you'll work with these variables:

Variable	Meaning
$N$	Number of payment periods
$I\%$	Annual interest rate (as a percentage)
$PV$	Present Value (initial amount)
$PMT$	Payment amount per period
$FV$	Future Value (final amount)
$Pp/Y$	Payments per year
$Cp/Y$	Compounds per year

infoNote

Sign conventions:

Money you invest or pay out is entered as negative
Money you receive or owe is positive

Exam tips

bookmarkSummary

Key Points to Remember:

Split changed problems into two parts: Always identify the point where conditions change and solve for each period separately
Never round until the end: Use the full calculator value (all decimal places) when transferring the future value from one period to become the present value for the next period
Check your answer makes sense:
- For investments, the final value should be larger than what was put in
- For loans, the final payment should be similar to the regular payment amount
Time conversions: Remember to convert years to the correct number of periods:
- Monthly: multiply years by 12
- Quarterly: multiply years by 4
- Annual: use years as-is

Remember!

bookmarkSummary

Essential Takeaways:

When financial conditions change mid-term, always solve the problem in two separate parts
Use unrounded values for all intermediate steps to avoid incorrect final answers
The future value from the first period becomes the present value for the second period
Changes in interest rates or regular payments require recalculating the remaining balance
Always check that your final answer is rounded only at the very end, to the nearest cent for money values

Solving Harder Financial Problems (VCE SSCE General Mathematics): Revision Notes