Inflation/tax/fees | Year 12

01 / LEARN

Understand the idea.

Apply deductions to the right quantity

When tax applies only to interest, calculate the interest first. If the interest rate is $r$ and the stated tax fraction is $t$ , the retained interest is $P r (1 - t)$ . The original principal is not reduced by that interest tax.

Fees need both a base and a date. A fixed dollar fee is subtracted as a cash flow; a percentage fee multiplies its stated base. Applying a fee before rather than after interest can change the result, so follow the given order.

Purchasing power compares two growth factors

Real return measures purchasing power. If money grows by $1 + r_{net}$ while prices grow by $1 + f$ , divide these factors. The quotient tells how much more or less the investment can buy; subtract $1$ to express that change as a rate.

The exact formula is $r_{real} = \frac{1 + r_{net}}{1 + f} - 1$ . Subtracting inflation is only an approximation. For constant annual inflation over $n$ years, convert a future balance to today’s purchasing power by dividing by ${(1 + f)}^{n}$ .

r_{r e a l} = \frac{1 + r_{n e t}}{1 + f} - 1

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build the model

AUD 5,000 earns 6% for one year. A stated 20% tax applies only to interest. Find the net interest and final balance.

Worked solution

1
Calculate interest before tax.
$I_{gross} = 5000 \times 0.06 = 300$
The 6% return is earned on the AUD 5,000 principal.
2
Apply the stated tax to interest only.
$T = 300 \times 0.20 = 60$
The taxable base is AUD 300 of interest, not the full account balance.
3
Subtract tax to find retained interest.
$I_{net} = 300 - 60 = AUD 240.00$
The remaining 80% of the interest belongs to the account.
4
Add retained interest to the principal.
$B = 5000 + 240 = AUD 5240.00$
Tax has reduced the earnings, while the original principal remains in the balance.

Answer

Net interest AUD 240.00; balance AUD 5,240.00.

EXAMPLE 02

Use the model to make a decision

An investment has a 5% net nominal annual return while prices rise 3%. Find the exact real annual return.

Worked solution

1
Write the money and price growth factors.
$1 + r_{net} = 1.05, 1 + f = 1.03$
Returns and price inflation must be compared over the same one-year interval.
2
Find the purchasing-power multiplier.
$g_{real} = \frac{1.05}{1.03} \approx 1.0194174757$
The amount the investment can buy depends on money growth relative to price growth.
3
Express the real increase as a percentage.
$\begin{aligned} r_{real} & = \frac{1.05}{1.03} - 1 \\ \approx 0.0194174757 = 1.9417 % \end{aligned}$
Removing one converts the purchasing-power multiplier into a return rate.

Answer

1.9417%

03 / YOUR TURN

Try it yourself.

Check your understanding

A 7% annual return is taxed at a stated 25% of interest. Find the after-tax interest rate.

Show answer & reasoning

5.2500%

A 25% tax leaves 75% of the interest. The retained rate is $0.07 (1 - 0.25) = 0.0525 = 5.2500 %$ under the stated annual deduction model.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

AUD 8,000 earns 4% interest, then a fixed AUD 50 fee is charged. Find the final balance.
3 marks
A 7% annual return is taxed at a stated 25% of interest. Find the after-tax interest rate.
3 marks
Prices rise 2.5% annually for 4 years. What future amount has the purchasing power of AUD 10,000 today?
3 marks
A balance of AUD 15,000 in 5 years is measured when inflation has been 3% annually. Express it in today’s dollars.
3 marks
An account grows 2% while prices grow 4%. Calculate its real return and interpret the sign.
3 marks
AUD 3,000 earns 5%, then a fee of 1% of the post-interest balance is charged. Find the final balance.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

AUD 8,000 earns 4% interest, then a fixed AUD 50 fee is charged. Find the final balance.
3 marks
Interest is $8000 \times 0.04 = 320$ .
Balance is $8000 + 320 - 50 = 8270$ .
AUD 8,270.00
A 7% annual return is taxed at a stated 25% of interest. Find the after-tax interest rate.
3 marks
The retained fraction of interest is 0.75.
Net rate is $0.07 \times 0.75 = 0.0525$ .
5.2500%
Prices rise 2.5% annually for 4 years. What future amount has the purchasing power of AUD 10,000 today?
3 marks
Increase the target by the price growth factor.
Future target is $10000 {(1.025)}^{4}$ .
AUD 11,038.13
A balance of AUD 15,000 in 5 years is measured when inflation has been 3% annually. Express it in today’s dollars.
3 marks
Discount the future dollars by cumulative inflation.
Real amount is $\frac{15000}{{(1.03)}^{5}}$ .
AUD 12,939.13
An account grows 2% while prices grow 4%. Calculate its real return and interpret the sign.
3 marks
Real factor is $\frac{1.02}{1.04}$ .
The result is below 1, so the real rate is negative.
$- 1.9231 %$ ; purchasing power falls.
AUD 3,000 earns 5%, then a fee of 1% of the post-interest balance is charged. Find the final balance.
3 marks
Post-interest balance is $3000 \times 1.05 = 3150$ .
Retain 99% after the stated fee: $3150 \times 0.99 = 3118.50$ .
AUD 3,118.50

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics financial models: investments and annuities, superannuation, income streams and borrowing. Original practice uses supplied hypothetical rates, taxes and fees; statutory or commercial conditions are not inferred.

Official curriculum source ↗

In this lesson

Understand the idea.

Apply deductions to the right quantity

Purchasing power compares two growth factors

See the method in action.

Build the model

Calculate interest before tax.

Apply the stated tax to interest only.

Subtract tax to find retained interest.

Add retained interest to the principal.

Use the model to make a decision

Write the money and price growth factors.

Find the purchasing-power multiplier.

Express the real increase as a percentage.

Try it yourself.

Check your understanding

Keep practising