Depreciation refers to the situation where items or investments lose value over time. We will consider two types of depreciation in this course:
- Straight line depreciation - the value reduces by the same amount every time period.
- Reducing balance depreciation - the value reduces by a percentage of the previous value every time period.
Straight line depreciation
Straight line depreciation is a bit like simple interest in reverse because the principal is reduced by the same amount every time period. The graph showing the value at regular intervals will appear as a downward-sloping straight line. The slope of the line reflects the fixed quantity lost from the value in each period. Faster depreciation means a graph with a steeper slope.
The straight line method assumes the value of depreciation is constant per period.
The straight line graph above shows a $\$20000$$20000 initial value with an annual depreciation of $\$3000$$3000 p.a., the item reduces to a worth of $\$0$$0 after approximately $6.7$6.7 years.
Straight line depreciation can also be modelled using an arithmetic sequence. The recurrence relation for the graph above would be $A_{n+1}=A_n-3000$An+1=An−3000 where $A_0=20\ 000$A0=20 000 and $A_n$An represents the value at the end of the $n$nth year.
For an item with initial value $P$P, and depreciating by $d$d per period, the sequence which generates the value, $A_n$An, of the item at the end of each depreciation period is:
- Recursive form:
$A_{n+1}=A_n-d$An+1=An−d, where $A_0=P$A0=P
- Explicit form:
$A_n=P-nd$An=P−nd
Worked examples
Example 1
A car purchased for $\$20\ 000$$20 000 depreciates by $\$2500$$2500 per year.
(a) Write a recurrence relation for this situation.
Think: Use the form $A_{n+1}=A_n-d$An+1=An−d , where $A_0=$A0= the initial value and $d$d is the annual depreciation.
Do: $A_{n+1}=A_n-2500$An+1=An−2500, where $A_0=\$20\ 000$A0=$20 000
(b) How much is it worth after $6$6 years?
Think: We could use the answer key on the calculator to generate the sequence or simply use the explicit form of the sequence. After six years the depreciation amount will have been subtracted $6$6 times from the initial value.
Do:
| $A_6$A6 | $=$= | $P-nd$P−nd |
| $=$= | $\$20000-6\times\$2500$$20000−6×$2500 | |
| $=$= | $\$5000$$5000 |
Example 2
A car purchased for $\$15000$$15000 is worth $\$9000$$9000 after $8$8 years. By how much did it depreciate per year?
| $\text{Depreciation per year}$Depreciation per year | $=$= | $\frac{\text{Value lost}}{\text{Number of years}}$Value lostNumber of years |
| $=$= | $\frac{\$15000-\$9000}{8}$$15000−$90008 | |
| $=$= | $\$750$$750 |
Practice questions
question 1
The graph shows the depreciation of a car's value over 4 years.
What is the initial value of the car?
By how much did the car depreciate each year ?
After how many years will the car be worth $\$14400$$14400 ?
What is the value of the car after 4 years ?
Question 2
A car is initially purchased for $\$24000$$24000 depreciates by $\$1700$$1700 each year.
Write a recurrence relation for $V_n$Vn in terms of $V_{n-1}$Vn−1 that gives the book value of the car in dollars after $n$n years, and an initial condition for $V_0$V0.
Write both parts of the relation on the same line, separated by a comma.
Use your calculator to determine the book value of the car after $7$7 years.
After how many whole years will the book value of the car first fall below $\$11800$$11800?
Question 3
A boat, initially purchased for $\$46000$$46000, depreciates at a rate of $\$170$$170 for every $1000$1000 km of use.
Write a recurrence relation, $V_n$Vn, that gives the value of the boat, in dollars, after $n$n thousand kilometres.
Write both parts of the relation (including for $V_0$V0) on the same line, separated by a comma.
Use your calculator to determine the value of the boat after it has travelled $11000$11000 km.
Use your calculator to determine after how many kilometres the boat will first fall below $\$44808$$44808 in value.
Your answer should be a multiple of $1000$1000 km.
Reducing balance depreciation
In a similar way to how investments with compound interest increase by a percentage of the value at the start of a time period, assets that are subject to reducing-balance depreciation decrease in value by a percentage of the value at the start of each time period.
This is the more common form of depreciation. We will calculate this kind of depreciation using two methods:
- Using the reducing balance depreciation formula, similar to calculating compound interest.
- Using a geometric sequence to model the situation.
Reducing balance depreciation formula
The formula is just slightly different from the compound interest formula. The difference is that we are reducing the value so we must multiply the principal by a number less than $1$1 each time. For example, reducing by $5%$5% is the same as multiplying by $100%-5%$100%−5% or $95%$95% or $0.95$0.95. Therefore the formula has $1-i$1−i in the bracket instead of $1+i$1+i (or we could in fact consider it the same formula with a negative rate).
$A=P\left(1-i\right)^n$A=P(1−i)n
where: $P$P is the principal (or initial) amount
$i$i is the depreciation rate per time period
$n$n is the number of time periods
$A$A is the value of the item after being depreciated
Note: The amount an item is worth after depreciation is also called the expected value , book value or residual value.
Worked examples
Example 3
Kathleen deposited $\$6500$$6500 into a new superannuation account. This amount decreased by $2%$2% each year for $3$3 consecutive years.
What was the value of her superannuation after $3$3 years? Give your answer to $2$2 decimal places if necessary.
Think: How do we substitute these values into the depreciation formula?
Do:
| $A$A | $=$= | $P\left(1-i\right)^n$P(1−i)n | |
| $=$= | $6500\times\left(1-0.02\right)^3$6500×(1−0.02)3 | ||
| $=$= | $6500\times0.98^3$6500×0.983 |
Decreasing by $2%$2% is the same as finding $98%$98% |
|
| $=$= | $\$6117.75$$6117.75 |
Practice questions
Question 4
A pickup truck depreciated in value from $\$20000$$20000 to $\$15200$$15200 in $4$4 years.
At what rate per annum ($i$i) did it depreciate?
Write your answer as a percentage correct to 2 decimal places.
Question 5
Buzz's share portfolio of $\$91000$$91000 fell $5%$5% per month for the first $7$7 months of the Global Financial Crisis and then $2%$2% per month for the $8$8 months after that.
What was the value of his portfolio after $15$15 months?
Write your answer to the nearest cent.
Question 6
A microwave selling for $\$600$$600, depreciates at $15%$15% p.a. .
What is the percentage of the original value that will remain after 1 year?
What is the percentage of the original value that will remain after 2 years?
Write your answer as a percentage to 2 decimal places.
What is the percentage of the original value that will remain after 3 years?
Write your answer as a percentage to 2 decimal places.
How many full years will it take for the microwave to lose half its original value?
How many years will it take for the microwave to lose $90%$90% of its original value?
Depreciation modelled as a geometric sequence
Reducing balance depreciation is where the value at the start of each year is multiplied by a constant rate of depreciation. Therefore we can solve depreciation problems using the geometric sequence forms, where the common ratio will always be less than $1$1. Notice the explicit rule is the same as the depreciation formula.
For an item with initial value, $P$P, at a depreciation rate of $i$i per period, the sequence of the value of the item over time forms a geometric sequence with a starting value of $P$P and a common ratio of $r=\left(1-i\right)$r=(1−i).
The sequence which generates the value, $A_n$An, of the item at the end of each depreciation period is:
- Recursive form:
$A_{n+1}=rA_n$An+1=rAn, where $A_0=P$A0=P and $r=1-i$r=1−i
- Explicit form:
$A_n=P(1-i)^n$An=P(1−i)n
Worked example
Example 4
Danielle buys a car for $\$15\ 000$$15 000 and is told to expect it to depreciate at a rate of $x%$x% p.a. The progression of the depreciation is shown in the table below.
| Year | Car value at start of year ($) |
Depreciation ($) | Car value at end of year ($) |
|---|---|---|---|
| 1 | $15\ 000$15 000 | $2250$2250 | $12\ 750$12 750 |
| 2 | $12\ 750$12 750 | $1912.50$1912.50 | $10\ 837.50$10 837.50 |
| 3 | $10\ 837.50$10 837.50 | $1625.63$1625.63 | $9211.88$9211.88 |
(a) Determine $x%$x%, the depreciation rate of the car.
Think: We can use any of the table rows to do this. Use the formula
$\text{Depreciation rate}=\frac{\text{Depreciation in year n}}{\text{Value at start of year n}}\times100%$Depreciation rate=Depreciation in year nValue at start of year n×100%
Do: Using the values from Year 1 we find:
| Rate | $=$= | $\frac{2250}{15000}$225015000 |
| $=$= | $0.15$0.15 | |
| $=$= | $15%$15% |
Therefore, the depreciation rate is $15%$15% p.a.
Reflect: We can check we obtain the same result using the values from Year 2 or Year 3.
(b) Write a recursive rule for the investment.
Think: Each time we are decreasing by $15%$15%, which means multiplying by $0.85$0.85 as ($100%-15%=85%$100%−15%=85%). We want $n=1$n=1 to show the value at the end of year $1$1, therefore let $A_0=$A0= initial value.
Do: Write the rule $A_{n+1}=0.85A_n$An+1=0.85An, where $A_0=15\ 000$A0=15 000
(c) After how many whole years will the book value of the car first fall below $\$5000$$5000?
Think: We can use the answer key on the calculator to generate the sequence and count the years. However, if $n$n is large it may be best to use the formula and guess and check. Since $A=15\ 000\times0.85^n$A=15 000×0.85n we can try different values for $n$n to find when it first drops below $\$5000$$5000.
Do:
When $n=6$n=6, $A=\$5657.24$A=$5657.24 and when $n=7$n=7, $A=\$4808.66$A=$4808.66. Hence the value first drops below $\$5000$$5000 after approximately $7$7 years.
Practice questions
question 7
A brand new car depreciates in value each year and its value is modelled by
$V_n=0.88V_{n-1}$Vn=0.88Vn−1, $V_0=28000$V0=28000
where $V_n$Vn is the value, in dollars, of the car after $n$n years.
How much was the car purchased for?
As a percentage, what is the annual depreciation rate?
Use your calculator to determine the value of the car after $9$9 years.
Give your answer to the nearest cent.
When a car is worth less than $\$700$$700 it is deemed only useful for parts. At the end of which year is the car only useful for parts?
question 8
After one year, the value of a company’s machinery had decreased by $\$16720$$16720 from $\$88000$$88000. The value of the machinery depreciates by a constant percentage each year.
At what rate did the machinery depreciate in the first year?
Give your answer as a percentage.
What will the machinery be worth at the end of the second year?
Give your answer to the nearest cent.
Write a recurrence relation, $V_n$Vn, that gives the value of the machinery at the end of year $n$n.
Write both parts of the relation (including for $V_0$V0) on the same line, separated by a comma.
The company bought this machinery at the end of $2014$2014. When the value of the machinery falls below $\$5000$$5000, they will invest in new machinery. In which year will this occur?