Present/future value | Year 11

01 / LEARN

Understand the idea.

Move along a time line

Present value is the amount at the starting date; future value is its value after a stated number of growth periods. For one deposit, $F V = P V {(1 + i)}^{n}$ . Draw a short timeline so the exponent counts interest periods between the two dates.

To work backwards from a target, divide by the whole growth factor: $P V = \frac{F V}{{(1 + i)}^{n}}$ . For a positive rate the required starting amount is smaller than the target, because interest supplies part of the final balance.

Interpret present value

Choose the rate per period and count periods consistently. A deposit at the end of Year 1 valued at the end of Year 4 earns three annual periods, not four. Quarterly compounding for two years similarly gives eight quarterly periods.

A minimum deposit must still meet the target after currency rounding. Round the calculated requirement upward to the next cent, then check its future value. Ordinary nearest-cent rounding can leave a deposit fractionally too small even though its displayed answer looks reasonable.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

Find the minimum deposit for AUD 5000 in 3 years at 4% annually.

Worked solution

1
Write the future-value target relationship.
$5000 = P V {(1.04)}^{3}$
The deposit earns three full annual periods at 4%.
2
Reverse the compound growth.
$P V = \frac{5000}{{(1.04)}^{3}} \approx 4444.981793$
Dividing by the three-year growth factor isolates the required starting deposit.
3
Round upward to a cent.
$P V_{minimum} = AUD 4444.99$
AUD 4444.98 is below the unrounded requirement, so the next cent is needed.
4
Verify that the rounded deposit meets the target.
$4444.99 {(1.04)}^{3} \approx 5000.009231 > 5000$
The selected cent amount produces at least AUD 5000 under the supplied model.

Answer

AUD 4,444.99

EXAMPLE 02

Worked example

Value AUD 3500 after 2 years at 5% nominal compounded quarterly.

Worked solution

1
Find the quarterly rate.
$i = \frac{0.05}{4} = 0.0125$
The nominal annual rate is split across four quarterly periods.
2
Count the interest periods.
$n = 4 (2) = 8$
Two years contain eight quarters.
3
Grow the starting amount.
$F V = 3500 {(1.0125)}^{8} \approx 3865.701354$
The balance receives the quarterly growth factor eight times.
4
State the future value to cents.
$F V \approx AUD 3865.70$
Round the final balance after retaining the full compounded value.

Answer

AUD 3,865.70

03 / YOUR TURN

Try it yourself.

Check your understanding

Present value of AUD 1210 in 2 years at 10% annually?

Show answer & reasoning

AUD 1,000.00

There are two annual growth periods, so $P V = \frac{1210}{{(1.1)}^{2}} = \frac{1210}{1.21} = AUD 1000.00$ . Growing this deposit by 10% twice returns AUD 1210.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Future value of AUD 2400 at 3% annually for 4 years?
2 marks
Present value of AUD 6000 in 2 years at 5% annually, nearest cent?
2 marks
Minimum deposit for AUD 8000 in 5 years at 6% annually?
2 marks
An investment reaches AUD 5500 after 8 quarters at 1.2% per quarter. Find the starting sum.
2 marks
For a fixed positive rate and future target, why does a longer term need less starting money?
2 marks
Count annual periods from the end of Year 1 to the end of Year 4.
2 marks

Read the worked solutions

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

Future value of AUD 2400 at 3% annually for 4 years?
2 marks
$2400 {(1.03)}^{4}$ .
AUD 2,701.22
Present value of AUD 6000 in 2 years at 5% annually, nearest cent?
2 marks
$\frac{6000}{{(1.05)}^{2}}$ .
AUD 5,442.18
Minimum deposit for AUD 8000 in 5 years at 6% annually?
2 marks
$\frac{8000}{{(1.06)}^{5}} = 5978.065383$ .
Round up to cents.
AUD 5,978.07
An investment reaches AUD 5500 after 8 quarters at 1.2% per quarter. Find the starting sum.
2 marks
$PV = \frac{5500}{{(1.012)}^{8}}$ .
AUD 4,999.41
For a fixed positive rate and future target, why does a longer term need less starting money?
2 marks
$PV = \frac{FV}{{(1 + i)}^{n}}$ ; increasing $n$ increases the denominator.
The larger growth factor produces a smaller present value.
Count annual periods from the end of Year 1 to the end of Year 4.
2 marks
The intervals are 1→2,2→3,3→4.
3

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: investing and borrowing. Supplied educational models cover interest, investment comparisons, shares and credit; they are not recommendations of real products or statements of personal tax liability.

Official curriculum source ↗

In this lesson

Understand the idea.

Move along a time line

Interpret present value

See the method in action.

Worked example

Write the future-value target relationship.

Reverse the compound growth.

Round upward to a cent.

Verify that the rounded deposit meets the target.

Worked example

Find the quarterly rate.

Count the interest periods.

Grow the starting amount.

State the future value to cents.

Try it yourself.

Check your understanding

Keep practising