Use of Annuities (HSC SSCE Mathematics Standard): Revision Notes

Worked Example: Car Loan Repayment Table

Let's look at a practical example. Jessica takes out a car loan for $24,000 with an interest rate of $10\%$ per annum compound interest. She agrees to make regular yearly payments of $5,500 over six years.

Here's the annuity table for Jessica's loan:

Payment number	Payment paid	Interest paid	Principal reduction	Balance of annuity
0	0	0.00	0	24000.00
1	5500.00	2400.00	3100.00	20900.00
2	5500.00	2090.00	3410.00	17490.00
3	5500.00	1749.00	3751.00	13739.00
4	5500.00	1373.90	4126.10	9612.90
5	5500.00	961.29	4538.71	5074.19
6	5500.00	507.42	4992.58	81.61

Let's answer some questions about this table:

a) The interest paid when payment 1 is received

Looking at the table, find the row for payment number 1, then read across to the "Interest paid" column.

Interest paid = $2400.00

b) The principal reduction when payment 3 is received

Find payment number 3 and read the "Principal reduction" column.

Principal reduction = $3751.00

c) The balance of the annuity after payment 4 has been received

Find payment number 4 and read the "Balance of annuity" column.

Balance = $9612.90

d) The value of the last payment if the balance is to be zero after the 6th payment

After payment 6, there's still $81.61 remaining. To clear this, the final payment needs to be:

$$\text{Payment} = \text{Regular payment} + \text{Remaining balance}$$$$\text{Payment} = \$5500 + \$81.61$$$$\text{Payment} = \$5581.61$$

Final payment = $5581.61

e) The total amount of interest paid

Add up all the values in the "Interest paid" column:

$$\text{Total interest} = \$2400 + \$2090 + \$1749 + \$1373.90 + \$961.29 + \$507.42$$$$\text{Total interest} = \$9081.61$$

Total interest paid = $9081.61

Worked Example: Superannuation Growth Table

Michael is 60 years old and plans to retire at 65. His current superannuation fund has a balance of $500,000 and earns $7\%$ per annum compound interest. Michael contributes $20,000 each year to his super fund.

Here's the annuity table showing how Michael's super grows:

Payment number	Payment received	Interest earned	Principal increase	Balance of annuity
0	0	0.00	0	500000.00
1	20000.00	35000.00	55000.00	555000.00
2	20000.00	38850.00	58850.00	613850.00
3	20000.00	A	B	C
4	20000.00	47377.37	67377.37	744196.87
5	20000.00	52093.78	72093.78	816290.65

The table has three missing values: A, B, and C. Let's calculate them:

Finding A (Interest earned for payment 3):

We need to find the future value of the balance after one year of compound interest.

Use the formula: $FV = PV(1 + r)^n$

Where:

• $PV = 613850$ (the balance after payment 2)
• $r = 0.07$ (the interest rate)
• $n = 1$ (one year)

Calculating:

$$FV = 613850 \times (1.07)^1$$$$FV = 613850 \times 1.07$$$$FV = 656819.50$$

Now find the interest using: $I = FV - PV$

$$I = 656819.50 - 613850.00$$$$I = 42969.50$$

Therefore, A = $42,969.50

Finding B (Principal increase for payment 3):

The principal increase comes from two sources: the interest earned and the payment made.

$$\text{Principal increase} = \text{Interest earned} + \text{Payment received}$$$$\text{Principal increase} = \$42,969.50 + \$20,000$$$$\text{Principal increase} = \$62,969.50$$

Therefore, B = $62,969.50

Finding C (Balance of annuity after payment 3):

Add the principal increase to the previous balance.

$$\text{New balance} = \text{Previous balance} + \text{Principal increase}$$$$\text{New balance} = \$613,850 + \$62,969.50$$$$\text{New balance} = \$676,819.50$$

Therefore, C = $676,819.50