Depreciation
By:
John Powell and Alan Sun Mathematics of Finance Dr. Janusz Kawczak
Abstract Depreciation is the decline in value of an object over time and can be calculated in many different methods. However, these methods are all estimates of an objects value or “Book Value”, and can differ widely. Due to the importance of Depreciation in today’s financial world, we decided to utilize each method of Depreciation and find the most effective method. To achieve this, we used a test case and used its variable to find the annual depreciations that would occur for each method. Once the results were found, the comparison showed that the Straight Line Method of Depreciation, where there is an uniform annual decline of value over a set period of time, is the most efficient and beneficial method. This conclusion reinforces the current beliefs in the financial world. Background Depreciation is the term used to describe the decline in value of a physical asset over a certain span of time. Most objects lose value as they are used due to damage from use, aging, obsolescence, and impairment. However the exact amount of value that is lost and the current “book value” of an asset cannot be easily known. Depreciation is thus used as an estimate of this decline and is used for several important purposes in today’s business world. Depreciation has been a factor in human economics ever since the advent of trade. At its most basic, depreciation states that an older object is usually not as valuable as a newer one. However, with the formation of large corporations and the creation of our modern economic structure, depreciation has taken on a greater importance with a genuine need for accurate calculations of depreciation. Today most companies need a precise estimate of their physical assets’ depreciation for two main reasons. First, companies must compare the expenses in using and buying equipment and materials to the revenue generated in that year. Without an accurate estimate for depreciation, a company’s profit and current value could not be known. Secondly, companies must ascertain the value of their assets and determine that objects are not appraised as being more costly than their actual worth. In fact, the value of such assets directly affects the taxes that must be paid each year and depreciation is used to find the exact amount that is deductible. The exact amount depreciated annually is based on several factors. The main criteria is the method used in assessing an object's depreciation. The main methods range from deducting a set amount for a certain period of time to evaluating the exact hours that and object is used to generate revenue. Also, different physical assets can be evaluated in drastically dissimilar manners. In the United States, objects can have a depreciable life span ranging from 5 years to 50 years or more depending on several factors. For example, computers and electronics that become
obsolete quickly will undergo faster depreciation than trucks or factories that have much longer life-spans. Most countries today have set laws and regulations that determine the calculation of depreciation. In America, the law uses Straight Line along with Declining Balanace as an orthodox measure to make sure that all the companies and corporations in the United States conform with each other when reporting their company’s data and filing their tax reports. Research Problem Depreciation has become a complex operation in today’s business world. With our current economical situation, the precise estimate of capital and assets becomes crucial. Still, there are many differing ways of calculating depreciation, ranging from the simplistic straight line, to amortization that includes monetary inflation. Using a specific method can result in drastically different results, and there can be methods that are more beneficial to a company. In fact, the proper method of depreciation can lower taxable income and portray larger assets. We thus decided to research the methods of depreciation and determine which one creates the best results. Through setting up a test case and calculating depreciation for each different type we can discover the method that ameliorates a company’s financial situation Methods for Finding Depreciation 1.1- Straight-Line Method The Straight-Line Method is the most popular and simplistic way to find depreciation. In this method, it is assumed that depreciation occurs at a constant rate per unit of time. The constant rate yields a linear equation appearing as straight line when graphed. In Eq. 1 R is the depreciation allowance, W is the wearing value, which is original value minus scrap value, and n is the number of useful years that the item can be used. Below is a Basic Notation for Depreciation table. (Guthrie, 2004)
R
W n
1
Basic Notation for Depreciation
C Original cost of the capital asset S Scrap Value
W Wearing Value, W C S
Rt Depreciation allowance for year t Bt Book Value at the end of year t
r Depreciation rate per year, per service hour, or production unit.
n Useful life of capital asset in years, service hours, or production units
Example 1.1: Your family buys a car for $20,000 and it has an estimated life of 8 years with a trade in value of $3,500. Find the straight-line depreciation allowances and make a schedule.
R
End of Year 0 1 2 3 4 5 6 7 8
W 20, 000 3,500 16500 $2062.50 n 8 8
Accumulated Depreciation … $2,062.50 $4,125.00 $6,187.50 $8,250.00 $10,312.50 $12,375.00 $14,437.00 $16,500.00 Book Value of Car $20,000.00 $17,937.50 $15,875.00 $13,812.50 $11,750.00 $9,687.50 $7,625.00 $5,562.50 $3500.00
Annual Depreciation Allowance … $2,062.50 $2,062.50 $2,062.50 $2,062.50 $2,062.50 $2,062.50 $2,062.50 $2,062.50
In the example above the car depreciates at a rate of $2062.50 a year for 8 years. Each year the car depreciates by another $2,062.50 until it is traded in at the end of the eight year for $3,500. The car is not always traded in at the end of its “estimated” life but can be used for longer periods. However its “book value” remains at $3,500. 1.2- Service Hours Method The Service Hours Method is similar to the Straight-Line Method except it calculates “book value” based on the hours an object has been used instead of set amount of depreciation for each year. When using this system of depreciation a company has to keep up with all of the hours of usage so that it can calculate the correct depreciation amount. This makes it much more difficult to obtain than the straight-line method and makes it a lot less practical in real life. (Guthrie 2004).
Rt rH t
In Eq. 2 Rt is the depreciation allowance for year t, r is the rate in dollars per service hour,
2
H t represents the service hours expended in year t. (Guthrie 2004).
1.3-Production Units Method The Production Units Method of depreciation uses the number of actual units produced to determine the depreciation allowance. To get the depreciation allowance the depreciation rate is multiplied by the number of items produced. This method works well when factory production is up because when a company produces more they can write off larger amounts as depreciation. This method is also very similar to the straight-line method except that n is the estimated life in units produced and the rate of depreciation is expressed in dollars per unit. (Book Depreciation 2004-2005)
Rt rU t
In Eq. 3 r is the rate in dollars per unit, U t are the number of output units. (Guthrie 2004). 2.1-Sum of the Year’s Digits The Sum of the Year’s Digits method is a way to calculate depreciation where the depreciation rate diminishes as it gets closer to the end of the estimated life of the item. To calculate the rate of depreciation the number of years of the estimated life of the item are added up and placed in
3
the denominator and the numerator is the number of years that are left in the items estimated life. Then the rate of depreciation is multiplied by the wearing value to get the depreciation value. Below is Eq. 4 which shows the depreciation formula. (Depreciation Schedule 2004-2005)
r1
n n 1 1 , … , rn , r2 1 2 ... n 1 2 ... n 1 2 ... n
Rt rW t
4
In Eq. 4 rt is the depreciation rate per year and W is the wearing value. Example 2.1: A clothing company buys several cash registers for $5,000 each. Each cash register has an expected life of 6 years with a trade-in value of $900. Find the sum of the year’s depreciation and make a schedule. Wearing Value: W C S $5,000 $900 $4,100 Denominator of the Depreciation Rate: 1 2 ... 6 21
Rt rW R1 t
End of Year 0 1 2 3 4 5 6
6 5 $4,100 $1171.43 R2 $4,100 $976.19 , and so on. 21 21
Annual Depreciation Allowance … $1,171.43 $976.19 $780.95 $585.71 $390.48 $195.24 Accumulated Depreciation … $1,171.43 $2,147.62 $2,928.57 $3,514.28 $3,904.76 $4,100 Book Value of Car $5,000.00 $3,828.57 $2,852.38 $2,071.43 $1,485.72 $1,095.24 $900
2.2 Declining Balance Method The Declining Balance Method of depreciation uses a fixed rate of depreciation that is multiplied by the book value to get the depreciation allowance. In this formula the depreciation rate stays constant but since the depreciation allowance depends on the book value as well, the amount of money that is depreciated changes at every step. The larger amounts of depreciation will take
place in the first few years and gradually the amount of depreciation will get smaller and smaller. The rate will be a percentage of the straight-line rate so r is equal to 1/n. Typical rates of depreciation are 150% and 200%. 200% depreciation is called the double declining balance because the rate is r = 2(1/n). Double declining balance gives an accelerated write-off but doesn’t take the book value to scrap value. For this reason the last two or three years of depreciation are calculated with the straight-line method. Below is Eq. 5 which calculates the depreciation allowance at the end of the year t. (Declining Balance 2005)
Rt 2 / n Bt 1
Book Value Perspective: Rt Bt 1 Bt 3.1- Amortization of Total Depreciation All of the methods of depreciation that we have seen so far haven’t taken into account the time value of money. This method of depreciation assumes that the amount you paid for the item grows over time and each successive book value estimate will gain interest. In this method the depreciation allowances are like rent and are subtracted from the present value like an ordinary annuity. Because this method is like an ordinary annuity the depreciation allowances will all be constant. Below is Eq. 6 which is used to calculate the depreciation allowances. (Guthrie 2004)
5
C S 1 i R an i
n
6
Ex. 3.1: The School of Fine Arts is buying new drawing tables for the students at a cost of $10,000 and they have a life expectancy of 5 years with a trade in value of $2,000.Find the annual depreciation allowances using the amortization of total depreciation method and assuming an investment rate of 6%(1).
C S (1 i)n Ran i
$10, 000 $2, 000(1.06)5 Ra5 6%
$8505.48 Ra5 6% R $2355.70
3.2- Sinking Fund Method In the Sinking Fund Method of depreciation it is assumed that there is a savings account set up to replace an asset at the end of its life. The target value is equal to the wearing value and the depreciation allowances consist of the payment for the fund and the interest that it collects. Gradually as this method moves on the depreciation allowances will get marginally larger. The formula for the deposit into the fund is calculated by using the future value of an ordinary
annuity. Below is Eq. 7 which is the formula for calculating depreciation through the sinking fund method. (Guthrie 2004)
CS t 1 Rt 1 i S ni
Results The problem below will be calculated using several different methods to compare the depreciation allowances between the different methods. When we compare the depreciation allowances and the difficulty of solving the depreciation allowances we will determine which methods work best and why companies use these methods. A paper mill that operates 24 hours everyday, 365 days a year, buys a new tree grinder for $1,000 with an estimated life of 5 years (8,760 hours a year) with a trade-in value of $200. Calculate the depreciation using different methods. (For the Service Hours Method it has a useful life of 54,750 hours and for Amortization of Total Depreciation the interest rate is 5%) Straight-Line Method
7
R
End of Year 0 1 2 3 4 5 Service Hours Method
W $1, 000 $200 $160 per year depreciation n 5
Annual Depreciation Allowance … $160 $160 $160 $160 $160 Accumulated Depreciation … $160 $320 $480 $640 $800 Book Value $1,000 $840 $680 $520 $360 $200
r
W $1, 000 $200 1.83 ¢ per hour n 43,800
$0.0183*8, 760 $160 per year depreciation
End of Year 0 1 2 3 4 5 Annual Depreciation Allowance … $160 $160 $160 $160 $160 Accumulated Depreciation … $160 $320 $480 $640 $800 Book Value $1,000 $840 $680 $520 $360 $200
Sum of the Year’s Digits Wearing Value: W C S $1,000 $200 $800 Depreciation rate denominator: 1 2 ... 5 15
R1
End of Year
5 4 $800 $266.67, , R2 $800 $213.33 , and so on. 15 15
Accumulated Depreciation … $266.67 $480.00 $640.00 $746.67 $800.00 Book Value $1,000 $733.33 $520.00 $360.00 $253.33 $200.00
Annual Depreciation Allowance 0 … 1 $266.67 2 $213.33 3 $160.00 4 $106.67 5 $53.33 Amortization of Total Depreciation
C S 1 i
n
Ran i $1,000 $200(1.05)5 Ra5 5%
$1, 000 $156.71 Ra5 5% $843.29 Ra5 5% R $194.78
End of Year Annual Depreciation Allowances … $194.78 $194.78 $194.78 $194.78 $20.88 Accumulated Depreciation … $194.78 $389.56 $584.34 $779.12 $800.00 Book Value
0 1 2 3 4 5
$1,000 $805.22 $610.44 $415.66 $220.88 $200.00
Depreciation Method Comparison
1000 900 800 700 600
Book Value
500 400 300 200 100 0 1 2 3 4 5 6
Straight Line Method Service Hours Method Sum of the Year’s Digits Amortization
Years
Sum of the Year’s Digits
1200
Book Value
1000 800 600 400 200 0 1 2 3 4 5 6
Years
Straight Line Method/ Service Hours Method
1200 1000
Book Value
800 600 400 200 0 1 2 3 4 5 6
Years
Amortization
1200 1000 800 600 400 200 0 1 2 3 4 5 6
Conclusion After our example analysis and research of contemporary practices we concluded that the Straight Line Depreciation method is the best method of depreciating assets for multiple reasons. When used in conjunction with the other methods, Straight Line Depreciation gives consistent results that also are much easier to calculate and obtain in the real world. However, the different methods can not always be directly compared as their yearly depreciation has inherent differences based on the method they’re calculated. For example, when using Amortization to depreciate an object’s value, there is the additional factor of interest that is not present in Straight Line Depreciation or other similar methods. However, it is still possible to compare some of the data, and through the summation of the Book Values, we can find the total amount an asset contributes to a company’s worth over the span of five years. To find this you just use
BVT BV1 BV2 BV3 BV4 BV5
ex :1000 840 680 520 360 200 $3,600
Using this you obtain: Straight Line Method: $3,600.00 Service Hours Method: $3,600.00 Sum of the Year’s Digits: $3066.66 Amortization: $3252.20 Thus as seen from these calculations it is evident that the total summation of this specific asset’s book value for five years can change greatly when differing methods are used. For example, when comparing Amortization to Straight Line Method the difference can be seen as:
BVStraightLine BVAmortization $3600 $3252.20 $347.80
This means that the company is valued lower by $347.80 over the span of five years when Amortization is used to depreciate the asset instead of straight line method. Similarly, using the Sum of the Year’s Digits method also results in a much smaller asset value for five years with an even larger difference:
BVStraightLine BVSumYearsDigits $3600 $3066.66 $533.34
With these two comparisons, it is seen that the more exponential decline in value that occurs with Amortization and Sum of the Year’s Digits results in much lower asset value for the first few years. Usually towards the end of an objects estimated life the book values for each estimation even out towards the salvage value. However, the gap is so large for the first year or so of depreciation, with differences in the annual depreciation (AD) of up to:
AD1straightline AD1 Amortization AD1straightline
100 % Difference
$160 $266.67 100 66.67% $160
With such a large drop in value, it can be seen that straight line is preferable. Still, there is the case of Service Hours Method which produced the exact annual depreciation. This is due to the machine’s consistent usage without major variations on a day to day basis. Often though, Service Hours Method usually is much more costly to estimate as it involves logging every hour that an asset is used to generate revenue. The same problem occurs with Production Units Method. Finally, with Straight Line Depreciation’s ease of use, ability to generate consistent and congruent results for many different objects, and its favorable depreciation, it is the best depreciation estimate out of the many methods. This conclusion is supported by the actions of not only this analysis, but by the legislation the United State Government implemented that required the use of Straight Line Depreciation in combination with accelerated recovery by declining balance.
References Book Depreciation Retrieved July 08, 2005 on the World Wide Web: http://coen.boisestate.edu/mkhanal/book.htm Declining Balance, 2005, Wikipedia, Retrieved July 09, 2005 on the World Wide Web: http://en.wikipedia.org/wiki/Depreciation#Declining-balance Depreciation Schedule Solution Matrix Ltd. 2004-2005 Retrieved July 08, 2005 on the World Wide Web: http://www.solutionmatrix.com/depreciation-schedule.html Guthrie, G. L., Lemon, L.D. (2004). Mathematics of Interest Rates and Finance. Upper Saddle River, NJ: Prentice Hall