Declining-balance depreciation

The declining-balance formula S = V0(1 - r)^n, depreciation tables and graphs, finding the rate or the time, and comparing declining-balance with straight-line depreciation.

A new car loses value the moment you drive it out of the dealership. Phones, laptops and work tools do the same. Businesses track this loss, called depreciation, because the tax office lets them claim it as a cost. The usual method takes the same percentage off the value every year, which is compound interest running backwards.

The declining-balance formula

With declining-balance depreciation, the value falls by a fixed percentage of the current value each year. Losing $r$ of the value leaves $(1 - r)$ of it, so after $n$ years:

Declining-balance depreciation

$$S = V_0(1 - r)^n$$

$S$ is the salvage value (the value after $n$ years).
$V_0$ is the original value.
$r$ is the depreciation rate per year, as a decimal.
$n$ is the number of years.

The total depreciation is $V_0 - S$.

Example 1●

A tradie buys a new ute for $\$42\,000$. It depreciates at $18\%$ p.a. using the declining-balance method. Find its value after $3$ years and the total depreciation.

$S = 42\,000(1 - 0.18)^3 = 42\,000(0.82)^3$

Losing 18% each year leaves 82%, so the multiplier is 0.82.

$S \approx \$23\,157.46$

Evaluate.

Depreciation $= 42\,000 - 23\,157.46 = \$18\,842.54$

Depreciation is the value lost.

Declining-balance vs straight-line

In Year 11 you used straight-line depreciation, where the same dollar amount is lost each year: $S = V_0 - Dn$. The two methods behave very differently.

Year	Declining balance, $18\%$ ($\$$)	Straight line, $\$5000$ a year ($\$$)
$0$	$42\,000$	$42\,000$
$1$	$34\,440.00$	$37\,000$
$2$	$28\,240.80$	$32\,000$
$3$	$23\,157.46$	$27\,000$
$4$	$18\,989.11$	$22\,000$
$5$	$15\,571.07$	$17\,000$
$6$	$12\,768.28$	$12\,000$
$7$	$10\,469.99$	$7000$
$8$	$8585.39$	$2000$

Declining balance loses the most in the first year, when the value is biggest, then loses less and less. Its graph is a curve that flattens out and never reaches zero.
Straight line loses the same amount every year. Its graph is a straight line that reaches zero.
Here the ute is worth less under declining balance for the first $5$ years, and more from year $6$ on. The graphs cross between years $5$ and $6$.

InteractiveDepreciation duel

Try this

Check the table above, then drag the declining-balance rate up. Does the crossover move earlier or later? Which method suits a laptop that is almost worthless after $3$ years?

Finding the rate or the time

Example 2●●●

A ute bought for $\$42\,000$ is worth $\$18\,000$ after $4$ years. Find the declining-balance rate, to $2$ decimal places.

$18\,000 = 42\,000(1 - r)^4$

Substitute what is known.

$(1 - r)^4 = \dfrac{18\,000}{42\,000} \approx 0.428\,57$

Divide by the original value.

$1 - r = \sqrt[4]{0.428\,57} \approx 0.8091$

Undo the power of 4 with a fourth root.

$r \approx 0.1909$, so the rate is about $19.09\%$ p.a.

Rearrange and convert to a percentage.

To find the number of years, try whole values of $n$ until the value first falls below the target, or use the table or graph.

Your turn

A laptop costs $\$2400$ and depreciates at $30\%$ p.a. using the declining-balance method. Find its value after $2$ years.

Show the working

$S = 2400(0.7)^2 = 2400 \times 0.49 = \$1176$.

Flashcards

Declining-balance formula

$S = V_0(1 - r)^n$

Total depreciation

$V_0 - S$: the original value minus the salvage value.

Shape of a declining-balance graph

A decreasing curve that flattens out and never reaches zero.

Practice

A laptop costs $\$2400$ and depreciates at $30\%$ p.a. (declining balance). Find its value after $2$ years.
A tractor bought for $\$30\,000$ depreciates at $15\%$ p.a. (declining balance). Find the total depreciation over $3$ years, to the nearest cent.
A machine bought for $\$30\,000$ depreciates by $\$4500$ each year using the straight-line method. Find its value after $4$ years.
Which best describes the graph of value against time for declining-balance depreciation?
Equipment worth $\$50\,000$ depreciates at $20\%$ p.a. (declining balance). After how many whole years is it first worth less than $\$20\,000$?
An e-bike is worth $\$12\,750$ after losing $15\%$ of its value in one year. What was it worth a year ago?
Which expression gives the value after $6$ years of a $\$9000$ camera that depreciates at $12\%$ p.a.?
A ute bought for $\$42\,000$ is worth $\$18\,000$ after $4$ years of declining-balance depreciation. Find the rate as a percentage, to $2$ decimal places.
Exam style: A delivery van costs $\$58\,000$ and depreciates at $22.5\%$ p.a. using the declining-balance method. Find the total depreciation over $5$ years, to the nearest cent.
Exam style: Equipment costing $\$20\,000$ can be depreciated straight-line by $\$2500$ a year, or by declining balance at $25\%$ p.a. After $3$ years, how much lower is the declining-balance value than the straight-line value?