Depreciation existed in the US Tax Code as early as 1909^{1} , however before 1954 all regulation regarding depreciation was regarding asset lives, not how to calculate depreciation over those lives. Various methods for calculating depreciation were used, however, Straight-Line depreciation was the most common. The most common complaint with this method was that its assumption of uniform depreciation over the life of the asset was unrealistic, arguing that an asset more likely depreciates on an accelerated scale. This leads to an overpayment of taxes in the first part of an asset’s life, and an underpayment in the later part.

In 1954, the Internal Revenue Code of 1954 allowed the use of the Declining-Balance method, or the SOYD (Sum Of Years Digits) method to calculate depreciation.^{2} It also permitted the taxpayer to switch between Declining-Balance and Straight-Line depreciation at any point in the asset’s life. These methods of depreciation operate using the assumption that the asset depreciates on a curve. The Declining-Balance method has the asset depreciate by a fixed percentage of the book value every year, while the SOYD method has the value decrease based on what year of its life it is in related to the total lifespan.

In 1981, the Economic Recovery Tax Act of 1981 replaced the various methods of depreciation in use with the ACRS(Accelerated Cost Recovery System). The ACRS used a classification system that sorted each asset based on its life. These ranged from 3 to 19 years long, and were not an estimate of its usable life so much as an assigned period over which the cost of the asset would be recovered. The ACRS was created for the purpose of decreasing taxation in the early part of an asset’s life, thus increasing investment.^{3} The controversy surrounding this method claimed that the ACRS led to exaggeration of corporate income due to rapid depreciation, which decreased taxes but led to inaccurate earning reports.

In 1986, the Tax Reform Act of 1986 introduced the Modified Accelerated Cost Recovery System, which is still in use today. The MACRS increases the lifespan of assets, which lowers the yearly depreciation of an asset.

Below is a chart comparing how depreciation would be handled under the different methods used over the years. All numbers assume a $10,000 Purchase of an asset with a 5-year life and a salvage value (where salvage is not assumed by the standards of the method) of $2000.

Year | Straight Line | SOYD | Declining Balance (20%) | MACRS |

0 | $10,000.00 | $10,000.00 | $10,000.00 | $10,000.00 |

1 | $8,400.00 | $7,333.33 | $8,000.00 | $6,000.00 |

2 | $6,800.00 | $5,200.00 | $6,400.00 | $3,600.00 |

3 | $5,200.00 | $3,600.00 | $5,120.00 | $2,160.00 |

4 | $3,600.00 | $2,533.33 | $4,096.00 | $2,000.00 |

5 | $2,000.00 | $2,000.00 | $3,276.80 | $2,000.00 |

# Straight Line Depreciation

$Dn = (I-S)/N$^{4}

* D _{n} = Depreciation charge during year n

* I = Cost of the asset, including installation expenses

* S = Salvage value at the end of the asset's useful life

* N = Useful life.

The shorthand description is that you are dividing the reduction in the value of the asset into equal size chunks and taking one chunk per year as depreciation. The straight line is both the depreciation, a horizontal straight line over time, and the book value, a straight line from cost basis to salvage value.

- Ex: If you have a $10,000 car (a 5 year asset) and it's value at the end of 5 years is $1,000 to find how much it depreciation is per year you take the initial value minus the final value and divide by the lifetime of the asset.

Year | Depreciation | Book Value |

1 | $1,800 | $8,200 |

2 | $1,800 | $6,400 |

3 | $1,800 | $4,600 |

4 | $1,800 | $2,800 |

5 | $1,800 | $1,000 |

# Sum of Years Digits (SOYD)

The sum of years digits method is an old accelerated depreciation method. There is a larger drop in the value of the asset in the early years rather than the later years.

The sum of years (SY) for an asset n years is:

$SY = n(n+1)/2$

This is just the sum of the integers from one to n. If n=5 this is 15.

In SOYD, the annual depreciation in year m is given by:

$SYDm= (n+1-m)/SY*adjusted cost$

This is a fancy way of saying that you take more chunks of depreciation early and fewer chunks later. In the case of n=5, you take 5 chunks in year 1, 4 chunks in year two and so on.

Ex:

A new machine costs $160,000, has a useful life of 10 years,

and can be sold for $15,000 at the end of its useful life. It is expected that

$5000 will be spent to dismantle and remove the machine at the end of

its useful life. Determine the sum-of-year depreciation schedule for this machine.

$SY = n(n+1)/2$

SY = 10(10+1)/2 = $55

adjusted cost = $160000- ($15000-$5000) =$150,000

Year | Depreciation, SYDm | Accumulated Depreciation, ASYDm | Book Value, SYB |

1 | $27,272 | $27,272 | $132,727 |

2 | $24,545 | $51,818 | $108,181 |

3 | $21,818 | $73,636 | $86,363 |

4 | $19,090 | $92,727 | $67,272 |

5 | $16,363 | $109,090 | $50,909 |

6 | $13,636 | $122,727 | $37,272 |

7 | $10,909 | $133,636 | $26,363 |

8 | $8,181 | $141,818 | $18,181 |

9 | $5,454 | $147,272 | $12,727 |

10 | $2,727 | $150,000 | $10,000 |

# Declining Balance Methods

$Dn = a*I(1-a)^n$^{5}

- Dn = Depreciation charge during year n
- I = Cost of the asset, including installation expenses
- a = multiplier

- Single Declining Balance Method

* For single declining balance method $a = 1/N$.

* N is the useful life of the asset.

*Ex: For a $10,000 car (a 5 year asset).

Year | Depreciation | Book Value |

1 | $10,000*1/5 = $2,000 | $8,000 |

2 | $8,000 * 1/5 = $1,600 | $6,400 |

3 | $6,400 * 1/5 = $1,280 | $5,120 |

4 | $5,120 * 1/5 = $1,024 | $4,096 |

5 | $4,096 * 1/5 = $819.20 | $3,276.80 |

- Double Declining Balance Method

* For double declining balance method $a = 2/N$.

* N is the useful life of the asset.

*Ex: For a $10,000 car (a 5 year asset).

Year | Depreciation | Book Value |

1 | $10,000*2/5 = $4,000 | $6,000 |

2 | $8,000 * 2/5 = $2,400 | $3,600 |

3 | $6,400 * 2/5 = $1,440 | $2,160 |

4 | $5,120 * 2/5 = $864 | $1,296 |

5 | $4,096 * 2/5 = $518.40 | $777.60 |

## Depreciation of Declining Balance Methods

- Total declining balance depreciation (TDB).

$TDB = I[1-(1-a)^n]$^{6}

- Book value at the end of n years (B
_{n}).

$Bn = I - TDB = I(1-a)^n$^{7}

- When the book value (B
_{n}) is greater than the salvage value: if depreciation by declining balance in any year is less than (or equal to) what it would be by straight line depreciation, switch to and use the straight line depreciation method for the rest of the project's depreciable life.^{8}

- If the book value is (B
_{n}) is less than the salvage value (S), then stop depreciating the asset when B_{n}= S. Adjust depreciation whenever B_{n}is less than S to match S.

# Sample Questions

**Question** (Sample Question)

Use the straight line depreciation model to find annual depreciation of your company's lime green and tangerine TIG welders to lose their value as the welders you employ are less than excited to be seen using them. Also find the book value after year five. The welders were bought on sale, due to the color scheme, for $1032.22 each; they have an estimated salvage value of $400.00; and a useful life of 15 years.

**Answer**

First let's find the useful equations.

Straight Line Method of Depreciation:

D_{n} = (I - S) / N

,where

D_{n} = Annual Depreciation

I = Cost of the asset

S = Salvage value at the end of the asset's useful life

N = Useful life

Book Value:

B_{n} = I - (D_{1} + D_{2} + D_{3} + … + D_{n})

,where

B_{n} = Book value for year n

D_{n} = Depreciation for year n

Determine the useful information in the problem statement

I = $1032.22

S = $400.00

N = 15 years

Solve

D_{n} = ($1032.22 - $400.00) / 15 year

D_{n} = $42.15 / year

B_{5} = $1032.22 - 5($42.15)

B_{5} = $821.47