16.06 Depreciation (Extended)
Depreciation is the decrease in value of an asset over time. Previously you may have dealt with depreciation using the straight line method, where the loss of value throughout each period is the same, and is based on the initial value.
For example, a business may depreciate a computer that is worth $\$1000$$1000 by $\$200$$200 (or $20%$20% of its original value) each year.
A more accurate method of depreciation is to calculate the loss of value at each period, based on its current value in that period. This is the declining-balance method where the rate of depreciation is constant.
For example, if the same company uses a declining-balance method, it would depreciate the value of the computer by $20%$20% of its current value each year.
As we will see it has a lot in common with how compound interest works; the major difference being that the value of the asset is decreasing over time. The salvage value (sometimes called resale value) is the value of the asset after a certain number of time periods.
Using repeated multiplication
When calculating the salvage value after a period of depreciation, we multiply the previous year's salvage value by the depreciation rate. For example, to find the value of a $\$3000$$3000 car in 3 years' time if it depreciates at a rate of $8%$8% per annum, we do the following:
- After the first year, the salvage value of the car would be $3000\times\left(1-0.08\right)$3000×(1−0.08) which simplifies to $3000\times0.92$3000×0.92.
- After the second year, the new salvage value of the car would decrease by the same factor. We multiply the new salvage value by $0.92$0.92 again to get $\left(3000\times0.92\right)\times0.92$(3000×0.92)×0.92, which we could also write as $3000\times0.92^2$3000×0.922.
- After the third year, the salvage value of the car would be $\left(3000\times0.92^2\right)\times0.92$(3000×0.922)×0.92, which we could also write as $3000\times0.92^3$3000×0.923.
Using a calculator and rounding to the nearest dollar, we can see that after $3$3 years the car would be worth $\$2336$$2336.
Using the declining-balance depreciation formula
Notice that we kept multiplying by the same number, once for each time period, to find its salvage value. Each year we wrote its value in the form $3000\times0.92^n$3000×0.92n, where $n$n is the number of time periods. From this we can develop the following formula.
The salvage value $S$S of an asset after $n$n time periods depends on the initial value $V_0$V0, and the rate of depreciation $r$r. It is given by the formula
$S=V_0\left(1-r\right)^n$S=V0(1−r)n
where $n$n is typically in years, and $r$r is the annual depreciation, as a decimal.
Notice the similarity between the formula for declining-balance depreciation and compound interest. In the depreciation formula the $r$r is subtracted, because the value is decreasing. Declining-balance depreciation and compound interest both calculate how much will be added or subtracted with each time period based on the current value, not the initial value, which is why their formulas are so similar.
Remember that this formula gives us the value at the end of $n$n terms. To find the amount the item has depreciated by, just find the difference between the initial value and the salvage value.
The depreciation, or the loss in value, of an asset with initial price $V_0$V0 and salvage value $S$S is given by
$\text{Loss }=V_0-S$Loss =V0−S
The declining-balance method doesn't ever reduce the value of the asset to zero, unlike straight line methods. This more accurately describes the real world; most of the time you can sell an asset for a small amount of money no matter how old it is.
Worked example
Example 1
A new car is valued at $\$20000$$20000 and has a depreciation rate of $10%$10% p.a. We can find the salvage value of the car over $3$3 years by calculating its value one year at a time.
| Year 1: | $S$S | $=$= | $\$20000\times\left(1-0.1\right)$$20000×(1−0.1) |
| $=$= | $\$20000\times0.9$$20000×0.9 | ||
| $=$= | $\$18000$$18000 |
| Year 2: | $S$S | $=$= | $\$18000\times0.9$$18000×0.9 |
| $=$= | $\$16200$$16200 |
| Year 3: | $S$S | $=$= | $\$16200\times0.9$$16200×0.9 |
| $=$= | $\$14580$$14580 |
Say we want to find the value of a second car that is currently worth $\$30000$$30000 and depreciating by $10%$10% p.a. after $12$12 years.
Calculating the depreciation for each year would require us to do $12$12 separate calculations! Instead, we are going to use the declining-balance formula to find the value straight away.
We know the initial value $V_0$V0, the rate of depreciation $r$r, and the number of periods $n$n, so we can substitute those in and solve for $S$S, the salvage value.
| $S$S | $=$= | $V_0\times\left(1-r\right)^n$V0×(1−r)n |
| $=$= | $\$30000\left(1-0.1\right)^{12}$$30000(1−0.1)12 | |
| $=$= | $\$8473$$8473 (nearest dollar) |
Finding the initial value of an asset
We can also use the formula $S=V_0\left(1-r\right)^n$S=V0(1−r)n to find the initial value of an asset given its salvage value. In this case, we would need to rearrange the formula to solve for $V_0$V0.
Take for example, a manufacturing plant that purchased machinery for its production line three years ago.
If the machinery has been depreciating by $20%$20% p.a. and the plant is able to sell it for $\$20000$$20000, we can work backwards to find the initial value of the machinery.
We know $S$S, the salvage value, and need to find $V_0$V0. We use the same equation and substitute the right values in before rearranging to find the initial value.
| $S$S | $=$= | $\$20000$$20000 | (Given) |
| $S$S | $=$= | $V_0\left(1-r\right)^n$V0(1−r)n | (Stating the formula) |
| $\$20000$$20000 | $=$= | $V_0\times$V0×$\left(1-0.2\right)^3$(1−0.2)3 | (Substituting the known values) |
| $V_0$V0 | $=$= | $\frac{\$20000}{\left(1-0.2\right)^3}$$20000(1−0.2)3 | (Rearranging the formula) |
| $=$= | $\$39063$$39063 | (Round to nearest dollar) |
Practice questions
Question 1
Brad purchased a drone for $\$700$$700, which depreciates at $11%$11% p.a.
What is its salvage value after $3$3 years?
Give your answer in dollars to the nearest cent.
Question 2
A laptop depreciated by $11%$11% p.a. and was valued at $\$668$$668 after $7$7 years. The owner of the laptop wants to calculate the original cost.
To find the initial value, $V_0$V0, the following equation needs to be rearranged:
$S=V_0\left(1-r\right)^n$S=V0(1−r)n
Which of the following is the rearranged formula?
$V_0=\frac{\left(1-r\right)^n}{S}$V0=(1−r)nS
A$V_0=S\left(1-r\right)^n$V0=S(1−r)n
B$V_0=\frac{S}{\left(1-r\right)^n}$V0=S(1−r)n
CWhat was the laptop's original price?
Give your answer in dollars to the nearest cent.
Question 3
A car salesman received a total commission of $\$3442$$3442 at the beginning of the month. He expects that since fewer cars are being purchased, his commission will decrease by $14%$14% each month over the next few months.
According to his prediction, what will be his commission in $6$6 months' time? Give your answer in dollars to the nearest cent.
Question 4
Marge purchased new office equipment for a total of $\$6560$$6560.
Calculate the value of the equipment after $3$3 years, to the nearest cent, if she uses straight line depreciation at $7%$7% per year.
Calculate the value of the equipment after $3$3 years, to the nearest cent, if she uses declining-balance depreciation at $7%$7% per year.
Assuming declining-balance depreciation, after how many full years will the equipment be worth $\$3000$$3000?