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Again, software includes a built-in function for sum-of-the-years'-digits (SYD) method. Following is the function that returns the \$18,000 annual depreciation value for Year 3.

# Fractional Period Depreciation

With the sum-of-the-years'-digits method, fractional years require fairly intensive layering for every year (e.g., if a ten-year asset is acquired on July 1, 20X1, depreciation for 20X1 is the depreciable base times 10/55 times 6/12 (relating to six months of use); depreciation for 20X2 is the depreciable base times 10/55 times 6/12 (reflecting the last six months of the first layer), plus the depreciable base times 9/55 times 6/12 (reflecting the first six months of the next layer)). Returning to our \$100,000, four-year lived asset; if the asset was acquired on April 1, Year 1, the resulting depreciation amounts are calculated as:

Admittedly, the above table is a bit "busy," but if you take time to trace each of the amounts, it will be a good key to your understanding.

Before moving away from the sum-of-the-years'-digits, you may find it tedious to be adding numbers like 10 + 9 + 8 + . . . + 1 = 55. But, mathematicians long ago figured out a short cut for this calculation: (n (n + 1))/2, where n is the number of items in the sequence. Thus, for an asset with a ten year life: (10 (10 + 1)/2 = 10(11)/2 = 110/2 = 55. Try this on your own for the four-year life, and make sure your result is "10." Try again for a 15 year life asset, and make sure you get "120." Do you see that the sum-of the- years'-digit's fraction for the 4th year of use would be 12/120? Remember, you count backwards - Year one is 15/120, Year two is 14/120, Year three is 13/120, and Year four would be 12/120.

# Changes in Estimates

Obviously, the initial assumption about useful life and residual value is only an estimate. Time and new information may suggest that the initial assumptions need to be revised, especially if the initial estimates prove to be materially off course. It is well accepted that changes in estimates do not require re-doing the prior period financial statements; after all, an estimate is just that, and the financial statements of prior periods were presumably based on the best information available at the time. Therefore, rather than correcting prior periods' financial statements, such revisions are made prospectively (over the future) so that the remaining depreciable base is spread over the remaining life.

To illustrate, let's return to the straight-line method. Assume that two years have passed for our \$100,000 asset that was initially believed to have a four-year life and \$10,000 salvage value; as of the beginning of Year 3, new information suggests that the asset will have a total life of seven years (three more than originally thought), and have a \$5,000 salvage value. As a result, the revised remaining depreciable base (as of January 1, 20X3) will be spread over the remaining five years, as follows:

 Depreciation Expense Accumulated Depreciation at End of Year Annual Expense Calculation Year 1 \$22,500 \$22,500 (\$100,000 - \$10,000)/4 Year 2 \$22,500 \$45,000 (\$100,000 - \$10,000)/4 Year 3 \$10,000 \$55,000 (\$100,000 - \$45,000 - \$5,000)/5 Year 4 \$10,000 \$65,000 (\$100,000 - \$45,000 - \$5,000)/5 Year 5 \$10,000 \$75,000 (\$100,000 - \$45,000 - \$5,000)/5 Year 6 \$10,000 \$85,000 (\$100,000 - \$45,000 - \$5,000)/5 Year 7 \$10,000 \$95,000 (\$100,000 - \$45,000 - \$5,000)/5

The depreciation amounts for Years 3 through 7 are based on spreading the "revised" depreciable base over the last five years of remaining life. The "revised" depreciable base is \$50,000, and is calculated as the original cost (\$100,000) minus the depreciation already taken (\$45,000), and minus the revised salvage value (\$5,000).

## Tax Laws

Although this book is about financial and managerial accounting, it is certainly necessary to call your attention to the unique features of depreciation under the tax code. First, it is important to note that tax methods and financial accounting methods are not always the same; that is certainly true when it comes to the subject of depreciation. For example, when the economy "slows down" governments will often try to stimulate economic investment activity by providing special incentives that are realized through rapid depreciation for tax purposes (even immediate write-off in some cases). Now, you may wonder how this is supposed to help the economy. Well, suppose you were thinking of buying a new truck for use in your trade or business. If the government said you could reduce your taxable income by the amount of the purchase price immediately (rather than depreciating the asset over a much longer period of time), you see how this might prompt you to buy and bring about an incremental improvement in the economy.

The history of the tax laws is marked by many changes to the rates and methods that are permitted in any given year. As a result, it is difficult to generalize about the operation of the tax code as it relates to depreciation. But, in general, the USA tax rules provide for a depreciation technique known as the Modified Accelerated Cost Recovery System (MACRS - called "makers"). MACRS provides a general depreciation system and an alternative system - and within those systems are generally provisions relating to the 200% declining balance, 150% declining balance, and straight-line techniques.

Further, the tax system will generally stipulate the useful life of an asset rather than leaving it to the imagination of the taxpayer. For example, a race horse over two-years old when placed in service is assumed to have a three-year life; obviously very few stones are left unturned. The tax code tends to be very complete in identifying assets and their lives. As a general rule, the tax code lives tend to be "favorable" to taxpayers, and generally result in depreciation occurring at a faster rate than under generally accepted accounting principles.

It is noteworthy that the government has reduced the depreciation calculations down to percentage values that are reproduced in numerous reference tables. This reduces the possibility of error and makes it easy for someone who never studied depreciation methods to still come up with the right amount of depreciation in any given year.

You may be bothered to consider that a company would use one accounting method for financial reporting and another for tax. But, this is often the case, and there is nothing devious involved. Accounting rules are about measuring economic activity of a business and require a proper scheme of matching revenues and cost to achieve this objective. Meanwhile, the tax code must be followed, and it often changes to meet the revenue or social objectives of the government. As a result, temporary (and sometimes not so temporary) differences will arise between accounting and tax measurements. Records of these differences must be maintained, making the accounting task all the more challenging for a complex business organization.

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