Analysing and Modelling Reducing Balance Loans and Annuities (VCE SSCE General Mathematics): Revision Notes

Worked Example: Flora's Loan (Annual Compounding)

Problem: Flora borrows $8000 at an interest rate of 13% per annum, compounding annually. She makes yearly payments of $2100. Construct a recurrence relation to model this loan.

Solution:

Step 1: Identify the principal and payment amount

• $V_0 = 8000$ (the amount borrowed)
• $D = 2100$ (the yearly payment)

Step 2: Calculate the growth multiplier $R$

Using the formula $R = 1 + \frac{r}{100 \times p}$ where $r = 13$ and $p = 1$ (annual compounding):

$R = 1 + \frac{13}{100 \times 1} = 1 + 0.13 = 1.13$

Step 3: Write the recurrence relation

$V_0 = 8000, \quad V_{n+1} = 1.13V_n - 2100$

This formula tells us that each year, the balance is multiplied by 1.13 (adding 13% interest), then $2100 is subtracted (the payment).

Worked Example: Alyssa's Loan (Monthly Compounding)

Problem: Alyssa borrows $1000 at an interest rate of 15% per annum, compounding monthly. She makes monthly payments of $250. Construct a recurrence relation to model this loan.

Solution:

Step 1: Identify the principal and payment amount

• $V_0 = 1000$
• $D = 250$

Step 2: Calculate the growth multiplier $R$

Using the formula $R = 1 + \frac{r}{100 \times p}$ where $r = 15$ and $p = 12$ (monthly compounding):

$R = 1 + \frac{15}{100 \times 12} = 1 + \frac{15}{1200} = 1 + 0.0125 = 1.0125$

Step 3: Write the recurrence relation

$V_0 = 1000, \quad V_{n+1} = 1.0125V_n - 250$

Worked Example: Analysing Alyssa's Loan Balance

Problem: Alyssa's loan can be modelled by the recurrence relation:

$V_0 = 1000, \quad V_{n+1} = 1.0125V_n - 257.85$

a) Use your calculator to find the balance of the loan after four payments.

b) Find the balance of the loan after two payments have been made. Round your answer to the nearest cent.

Solution:

Part a: Finding the balance after four payments

Step 1: Enter the initial value

• Type 1000 and press ENTER or EXE

Step 2: Set up the iteration

• Type × 1.0125 - 257.85 and press ENTER (or EXE)

Step 3: Continue pressing ENTER four times

Each press calculates the next value in the sequence:

Answer: The balance after four payments is $0.04 (to the nearest cent).

This shows that four payments are almost enough to pay off the loan completely!

Part b: Finding the balance after two payments

Look at the third line of the calculator output (remember: the first line shows $V_0$, the second shows $V_1$, and the third shows $V_2$).

Answer: $506.23 (to the nearest cent)

Worked Example: Reza's Annuity Investment

Problem: Reza invests $12,000 in an annuity that earns interest at the rate of 6% per annum, compounding monthly, providing him with a monthly income of $2035.

a) Model this annuity using a recurrence relation.

b) Use your calculator to find the value of the annuity after the first four months. Round your answer to the nearest cent.

Solution:

Part a: Constructing the recurrence relation

Step 1: Identify the principal and payment amount

• $V_0 = 12000$ (the initial investment)
• $D = 2035$ (the monthly payment received)

Step 2: Calculate the growth multiplier $R$

Using the formula $R = 1 + \frac{r}{100 \times p}$ where $r = 6$ and $p = 12$:

$R = 1 + \frac{6}{100 \times 12} = 1 + \frac{6}{1200} = 1 + 0.005 = 1.005$

Step 3: Write the recurrence relation

$V_0 = 12000, \quad V_{n+1} = 1.005V_n - 2035$

Part b: Finding the value after four months

Step 1: Enter the initial value

• Type 12000 and press ENTER or EXE

Step 2: Set up and perform the iteration

• Type × 1.005 - 2035 and press ENTER (or EXE) four times

Calculator output:

Answer: The value of the annuity after four months is $4040.55 (to the nearest cent).

Interpretation: Even though Reza has withdrawn $8140 (4 × $2035) over four months, he still has $4040.55 remaining in his annuity because his investment has been earning interest.