Declining-Balance Depreciation
Learn the declining balance method of depreciation for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-03). With this method an asset loses the same percentage of its current value every year, so its value falls quickly at first and then more and more slowly.
You will use the formula \(S = V_0(1 - r)^n\) to find the salvage value, the amount of depreciation and the initial value of an asset, and see how this formula is like the compound interest formula and how it differs — useful for working out what a car or laptop will be worth in a few years.
Theory
Declining-balance depreciation is part of the Depreciation and loans topic in Year 12 Standard 1 (NSW). An asset loses the same percentage of its current value every year, so its salvage value after \(n\) years is \(S=V_0(1-r)^n\). This guide shows how to find the salvage value, the amount of depreciation and the purchase price, and why the yearly loss gets smaller.
Declining-balance depreciation takes the same percentage off an asset's current value each year. The value falls quickly at first and then more and more slowly.
The salvage value after \(n\) years is \(S=V_0(1-r)^n\), where \(V_0\) is the initial value, \(r\) is the depreciation rate per year written as a decimal \((20\%=0.20)\) and \(n\) is the number of years. Losing \(20\%\) means keeping \(80\%\), so each year the value is multiplied by \(0.80\). The amount of depreciation is \(V_0-S\).
It is compound interest in reverse: compound interest \(FV=PV(1+r)^n\) uses \((1+r)\) and grows, declining balance uses \((1-r)\) and shrinks. In this Year 12 Standard 1 (NSW) topic you find salvage values, amounts of depreciation and purchase prices.
With initial value \(V_0\), rate \(r\) per year (a decimal) and \(n\) years, the salvage value is:
The amount of depreciation is the purchase price minus the salvage value:
To find the purchase price, divide the salvage value by \((1-r)^n\):
Compare with the compound interest formula. Both multiply by the same factor each year and raise it to the power \(n\); compound interest uses \((1+r)\) and grows, declining balance uses \((1-r)\) and shrinks:
(The compound interest formula is sometimes written \(A = P(1+r)^n\).)
| Year \(n\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Value ($) | 24 000 | 18 000 | 13 500 | 10 125 | 7593.75 |
| Loss that year ($) | — | 6000 | 4500 | 3375 | 2531.25 |
How to solve a declining-balance depreciation problem
- Identify the initial value \(V_0\), the rate \(r\) (as a decimal) and the number of years \(n\).
- Find \(1-r\): losing \(r\) means keeping \(1-r\), for example \(1-0.20=0.80\).
- Substitute into \(S=V_0(1-r)^n\) and evaluate, or divide by \((1-r)^n\) to find \(V_0\).
- Answer the question asked: the value left (\(S\)) or the amount of depreciation (\(V_0-S\)).
Apply \(S=V_0(1-r)^n\) with \(V_0=9000\), \(r=0.20\), \(n=2\).
| \(S\) | \(=\) | \(9000(1-0.20)^2\) |
| \(\) | \(=\) | \(9000(0.80)^2\) |
| \(\) | \(=\) | \(9000 \times 0.64\) |
| \(\) | \(=\) | \(5760\) |
The salvage value after \(2\) years is \(\$5760\).
Find the salvage value, then subtract it from the purchase price.
| \(S\) | \(=\) | \(3600(1-0.25)^2\) |
| \(\) | \(=\) | \(3600(0.75)^2\) |
| \(\) | \(=\) | \(3600 \times 0.5625 = 2025\) |
| \(\text{Depreciation}\) | \(=\) | \(3600 - 2025\) |
| \(\) | \(=\) | \(1575\) |
The computer depreciates by \(\$1575\) in \(2\) years.
Losing \(30\%\) means keeping \(70\%\), so multiply by \(0.70\) each year.
| \(\text{Year 1: } S\) | \(=\) | \(40000(0.70)^1 = 28000\) |
| \(\text{Year 2: } S\) | \(=\) | \(40000(0.70)^2 = 19600\) |
| \(\text{Loss in year 1}\) | \(=\) | \(40000 - 28000 = 12000\) |
| \(\text{Loss in year 2}\) | \(=\) | \(28000 - 19600 = 8400\) |
The truck is worth \(\$28\,000\) then \(\$19\,600\); the loss is decreasing \((12000>8400)\).
Substitute into \(S=V_0(1-r)^n\), then divide to find \(V_0\).
| \(21870\) | \(=\) | \(V_0(0.90)^3\) |
| \(21870\) | \(=\) | \(V_0 \times 0.729\) |
| \(V_0\) | \(=\) | \(\dfrac{21870}{0.729}\) |
| \(\) | \(=\) | \(30000\) |
The forklift cost \(\$30\,000\).
Common pitfalls
Frequently asked questions
What is declining-balance depreciation?
Declining-balance depreciation takes the same percentage off an asset's current value every year. Because the percentage is taken from a smaller value each time, the dollar loss gets smaller each year.
What is the formula for declining-balance depreciation?
The salvage value after n years is S = V0 times (1 minus r) to the power n, where V0 is the initial value, r is the yearly rate as a decimal and n is the number of years. The amount of depreciation is V0 minus S.
Why do you multiply by 0.8 for 20 percent depreciation?
Losing 20 percent of the value means the asset keeps 80 percent of it, and 80 percent as a decimal is 0.8. So each year the value is multiplied by 1 minus 0.20, which is 0.80.
How do you find the original price from the salvage value?
Substitute into S = V0 times (1 minus r) to the power n and divide. A forklift worth $21870 after 3 years at 10 percent gives V0 = 21870 divided by 0.729, which is $30000.
Does declining-balance depreciation ever reach zero?
No. Each year the asset keeps a fraction of its current value, so the value gets closer and closer to zero but never reaches exactly zero.