Inflation & Appreciation
Apply compound interest to inflation and appreciation for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-02). Inflation is the rise in everyday prices each year, and appreciation is an increase in the value of an asset such as a house, land or a painting — both grow by the same percentage each year.
You will use the compound interest formula to find a future price or value, build the value year by year, and check whether a pay rise or a savings account keeps up with inflation — so you can see what groceries might cost later or what a property could be worth.
Theory
Appreciation (an item gaining value) and inflation (prices rising) grow by the same percentage each year, so they work exactly like compound interest: \(FV=PV(1+r)^n\). This NSW Year 12 Mathematics Standard 1 guide shows how to find a future price or value, build the value year by year, and check whether a pay rise keeps up with inflation.
Inflation is the rise in the prices of everyday goods each year, so the same items cost more in future. Appreciation is an increase in the value of an asset, such as a house, land, a painting or a collectible, each year.
Both grow by the same percentage each year, so they are compound growth: \(FV=PV(1+r)^n\), where \(PV\) is the value or price now, \(FV\) is the value or price after \(n\) years and \(r\) is the yearly rate as a decimal. Because the percentage is taken of a bigger amount each year, the dollar increase grows every year.
If savings or a pay rise grow by less than inflation, the money buys less — its real value falls. Compare the multipliers, e.g. \(1.025\) against \(1.035\).
Cash kept for \(n\) years has real value \(\dfrac{\text{amount}}{(1+r)^n}\): \(\$1000\) at \(5\%\) inflation is worth \(\$952.38\), \(\$907.03\), \(\$863.84\), … — a falling curve that flattens out towards zero.
Inflation and appreciation use the compound growth formula:
The increase in price or value is:
How to solve an inflation or appreciation problem
- Identify the value now \(PV\), the yearly rate \(r\) as a decimal and the number of years \(n\).
- Calculate \(FV=PV(1+r)^n\), working out the power first, and give money to the nearest cent.
- Answer the question asked: the new value \(FV\), or the increase \(FV-PV\), or a comparison of multipliers.
Inflation is compound growth, with \(r=0.025\).
| \(FV\) | \(=\) | \(PV(1+r)^n\) |
| \(\) | \(=\) | \(120(1.025)^4\) |
| \(\) | \(=\) | \(120\times1.103813\) |
| \(\) | \(=\) | \(132.46\) |
The shoes will cost \(\$132.46\).
Appreciation is compound growth, with \(r=0.035\).
| \(FV\) | \(=\) | \(650{,}000(1.035)^5\) |
| \(\) | \(=\) | \(650{,}000\times1.18768631\) |
| \(\) | \(=\) | \(771{,}996.10\) |
The unit is worth \(\$771{,}996.10\).
Each year's increase is \(7\%\) of that year's opening value.
| Year | Opening | Increase | Closing |
|---|---|---|---|
| 1 | \(\$4500.00\) | \(\$315.00\) | \(\$4815.00\) |
| 2 | \(\$4815.00\) | \(\$337.05\) | \(\$5152.05\) |
| 3 | \(\$5152.05\) | \(\$360.64\) | \(\$5512.69\) |
| \(\text{Check: } 4500(1.07)^3\) | \(=\) | \(5512.69\) |
| \(\text{Increase}\) | \(=\) | \(5512.69-4500\) |
| \(\) | \(=\) | \(1012.69\) |
(i) \(\$4815.00\), \(\$5152.05\), \(\$5512.69\). (ii) \(\$1012.69\).
Compare the pay-rise multiplier with the inflation multiplier.
| \(\text{New salary}\) | \(=\) | \(62{,}000\times1.025 = 63{,}550.00\) |
| \(\text{To keep pace}\) | \(=\) | \(62{,}000\times1.035 = 64{,}170.00\) |
\(63{,}550 < 64{,}170\), so the salary rises more slowly than prices. The nurse is worse off in real terms: the new salary is \(\$620\) short of keeping pace with prices.
Common pitfalls
Frequently asked questions
What is inflation?
Inflation is the rise in the prices of everyday goods and services over time, so the same items cost more in future. In Australia it is measured by the Consumer Price Index (CPI).
What is appreciation?
Appreciation is an increase in the value of an asset, such as a house, land, a painting or a collectible, over time. It is the opposite of depreciation.
How do you calculate a future price with inflation?
Use FV equals PV times (1 plus r) to the power n, with the inflation rate as a decimal. For example, $120 shoes with 2.5% inflation for 4 years cost 120 times 1.025 to the power 4, which is $132.46.
Is inflation simple or compound growth?
Inflation is compound growth. Each year the percentage is taken of the new, higher price, so the dollar increase grows every year. That is why you use the power (1 plus r) to the power n.
How do you know if a pay rise keeps up with inflation?
Compare the multipliers. If the pay rise multiplier is smaller than the inflation multiplier, prices are rising faster than pay. For example, a 2.5% rise (1.025) against 3.5% inflation (1.035) leaves the worker worse off in real terms.