Declining-Balance Depreciation
Learn declining-balance depreciation for NSW Year 12 Mathematics Standard 2. Also called the diminishing-value method, it reduces an asset's value by a fixed percentage of its current value each year, so the salvage value after \(n\) years is \(S=V_0(1-r)^n\).
You will use the formula to find the salvage (book) value, the amount of depreciation, the rate, the original value or the number of years, and compare declining balance with straight-line depreciation numerically and graphically — a core Standard 2 financial mathematics skill for vehicles, machinery and equipment.
Theory
Declining-balance depreciation is part of the depreciation topic in Year 12 Standard 2 (NSW). An asset loses a fixed percentage of its current value each 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 rate, and how to compare declining balance with straight-line depreciation.
Declining-balance depreciation (also called the diminishing-value method) reduces the value of an asset by a fixed percentage of its current value each year. Because the percentage is taken from a smaller value every year, the dollar amount lost gets smaller over time.
The salvage value (or book value) after \(n\) years is given by \(S=V_0(1-r)^n\), where \(V_0\) is the original value, \(r\) is the annual rate written as a decimal, and \(n\) is the number of years. The total amount of depreciation is \(V_0-S\).
In this Year 12 Standard 2 (NSW) topic you use the formula (or a spreadsheet) to find the salvage value, the amount of depreciation, the rate, the original value or the number of years, and you compare declining balance with straight-line depreciation (\(S=V_0-Dn\)) both numerically and graphically.
With original value \(V_0\), annual rate \(r\) (a decimal) and \(n\) years, the salvage (book) value is:
The amount the asset has depreciated is the fall from its original value:
Rearranging the formula lets you find the rate from a known original and salvage value:
How to work with declining-balance depreciation
- Identify the original value \(V_0\), the rate \(r\) (write the percentage as a decimal) and the number of years \(n\).
- Salvage value: substitute into \(S=V_0(1-r)^n\).
- Amount of depreciation: subtract, \(V_0-S\).
- Find a missing quantity: rearrange for the rate \(r\), the original value \(V_0\) or the number of years \(n\) (using logarithms for \(n\)).
- Compare: against straight-line \(S=V_0-Dn\), numerically or on a graph.
Apply \(S=V_0(1-r)^n\) with \(V_0=18000\), \(r=0.15\), \(n=3\).
| \(S\) | \(=\) | \(18000(1-0.15)^3\) |
| \(\) | \(=\) | \(18000(0.85)^3\) |
| \(\) | \(=\) | \(18000 \times 0.614125\) |
| \(\) | \(=\) | \(11054.25\) |
The salvage value after \(3\) years is \(\$11054.25\).
Find the salvage value, then subtract it from the purchase price.
| \(S\) | \(=\) | \(54000(0.80)^4 = 22118.40\) |
| \(\text{depreciation}\) | \(=\) | \(54000 - 22118.40\) |
| \(\) | \(=\) | \(31881.60\) |
The ute depreciates by \(\$31881.60\) over \(4\) years.
Substitute into \(S=V_0(1-r)^n\) and solve for \(r\).
| \(25600\) | \(=\) | \(40000(1-r)^2\) |
| \((1-r)^2\) | \(=\) | \(\dfrac{25600}{40000} = 0.64\) |
| \(1-r\) | \(=\) | \(\sqrt{0.64} = 0.80\) |
| \(r\) | \(=\) | \(0.20 = 20\%\) |
The boat depreciates at \(20\%\) per annum.
Work out each value after \(3\) years and compare.
| \(\text{straight-line } S\) | \(=\) | \(16000 - 3200\times3 = 6400\) |
| \(\text{declining } S\) | \(=\) | \(16000(0.80)^3 = 8192\) |
| \(8192\) | \(>\) | \(6400\) |
Declining balance leaves the higher value, \(\$8192\) versus \(\$6400\).
Common pitfalls
Frequently asked questions
What is declining-balance depreciation?
Declining-balance (or diminishing-value) depreciation reduces an asset's value by a fixed percentage of its current value each year. Because the percentage is taken from a smaller amount every year, the dollar value lost gets smaller over time and the value falls along a decay curve.
What is the formula for declining-balance depreciation?
The salvage or book value after n years is S = V0 times (1 minus r) to the power n, where V0 is the original value, r is the annual rate written as a decimal, and n is the number of years. The total depreciation is V0 minus S.
How is declining balance different from straight-line depreciation?
Straight-line depreciation subtracts the same dollar amount each year using S = V0 minus Dn, giving a straight line. Declining balance subtracts the same percentage each year using S = V0 times (1 minus r) to the power n, giving a curve that falls quickly at first and then levels off.
How do you find the rate of depreciation?
Rearrange the formula to r = 1 minus (S divided by V0) to the power one over n. For example, a boat that falls from $40000 to $25600 in 2 years gives (1 minus r) squared equal to 0.64, so 1 minus r is 0.8 and the rate is 20 percent per annum.
Why does the value never reach zero?
Each year the asset keeps the fraction (1 minus r) of its current value, so it always retains some value no matter how many years pass. The salvage value gets closer and closer to zero but never becomes exactly zero, unlike straight-line depreciation which can reach zero.
What does salvage value or book value mean?
The salvage value, also called the book value, is what the asset is worth after depreciating for n years. It is the value S you get from the formula S = V0 times (1 minus r) to the power n.