Over the years I have been amazed by the amount of math used in tax laws, and some of them are complicated math. We have been talking about depreciation for years, and we know there are percentage applied each corresponding to the life of the property, and other variables, but have you seen a literature where the actual percentages are derived? I have not, and the purpose of this article is to derive math formulas used in IRS tables.

First a little background on the tax rules.

When you purchase a major item for your business such as a printer that can last for multiple years, you cannot deduct the full amount of cost in the first year, but instead you should deduct a portion of the cost for multiple years, this process is called depreciation.

The current tax depreciation system used in the United States is Modified Accelerated Cost Recovery System, or MACRS. Almost everyone uses the extensive tables in IRS Publication 946, a 120-page document, or computer software to to compute the depreciation amount, but we will see how these numbers come out.

The percentage you can use to depreciate depends on:

• Life: the number of years (Y) the property can last. It does not matter how long it actually lasts, IRS says it last N years, then you need to depreciate for N years.

• Declining balance rate: the multiplying factor (D) on the straight line rate (1/Y). Therefore the depreciation amount is D * 1/Y over the remaining cost basis (the total cost basis less the accumulated depreciation in previous years).

• Convention: the discount for the first year. With half year (HY) convention, the first year depreciation rate is reduced by 50%, as well as the last year as if you only used the business property for only half a year. With mid-quarter convention and the property placed in service in the first quarter, the depreciate rate is reduced as 3.5/4 (87.5%) for the first year, and 12.5% for the last year.

If we use above method alone, then the cost basis can never be fully deprecated, so there is an additional rule that we switch to straight line rate [over the remaining years] when the straight line rate is equal or higher.

The following is reproduced from Table A-1 of IRS Publication 946 for MACRS applicable percentage for property class 3-, 5-, 7-, 10-, 15-, and 20-Year Property with Half-Year Convention.

Recovery Years 3-Year 5-Year 7-Year 10-Year 15-Year 20-Year
1 33.33 20.00 14.29 10.00 5.00 3.750
2 44.45 32.00 24.49 18.00 9.50 7.219
3 *14.81 19.20 17.49 14.40 8.55 6.677
4 7.41 *11.52 12.49 11.52 7.70 6.177
5   11.52 *8.93 9.22 6.93 5.713
6   5.76 8.92 7.37 6.23 5.285
7     8.93 *6.55 *5.90 4.888
8     4.46 6.55 5.90 4.522
9       6.56 5.91 *4.462
10       6.55 5.90 4.461
11       3.28 5.91 4.462
12         5.90 4.461
13         5.91 4.462
14         5.90 4.461
15         5.91 4.462
16         2.95 4.461
17           4.462
18           4.461
19           4.462
20           4.461
21           2.231

Let us derive the formula, for half-year convention as an example.

For the first year, the percentage is $\small&space;\frac{1}{Y}*D*50% = \frac{D}{2Y}$, According to IRS Publication 946, the 3-, 5-, 7-, and 10-year classes use 200% and the 15- and 20-year classes use 150% declining balance depreciation. So for 20-year class property, the depreciation percentage is $\small&space;\frac{1.5}{2\times30} = 3.75%$, which is consistent with what in the table, similarly we can verify all other classes in the table.

For the second year, the remaining balance is $\small&space;1-\frac{D}{2Y}$, and the rate is: $\small&space;\frac{D}{Y}$, so the depreciation percentage is $\small&space;(1-\frac{D}{2Y})\frac{D}{Y}$.

For the third year, the remaining balance is $\small&space;(1-\frac{D}{2Y})(1-\frac{D}{Y})$, and the rate is still $\small&space;\frac{D}{Y}$, so the depreciation percentage is $\small&space;(1-\frac{D}{2Y})(1-\frac{D}{Y})\frac{D}{Y}$.

Generally, for the Nth year, the depreciation percentage is: $\small&space;(1-\frac{D}{2Y})(1-\frac{D}{Y})^{N-2}\frac{D}{Y}$

Let us verify for the 5th year 15-Year class property, we have D=1.5, Y=15, N=5, so the percentage should be: $\small&space;(1-\frac{1.5}{2*15})(1-\frac{1.5}{15})^{5-2}\frac{1.5}{15} = 7.695%$.

and it is.

Now we need to find the year X in which we should switch to straight line depreciation. At year X, we have $\small&space;Y-(X-1)+\frac{1}{2} = Y-X+\frac{3}{2}$ years remaining, we added 1/2 because the first year is treated as half year, so we need another 1/2 to make up the full Y years. The straight line rate is $\small&space;\1/(Y-X+\frac{3}{2})$. over the remaining balance, we equalize this to D/Y to find X:

$\small&space;\frac{1}{Y-X+\frac{3}{2}} = \frac{D}{Y}$.

Solving the equation we have:

$\small&space;X=Y(1-\frac{1}{D}) + \frac{3}{2}$.

The depreciation rate is:

$\small&space;(1-\frac{D}{2Y})(1-\frac{D}{Y})^{N-2}/(Y-X+\frac{3}{2})$

for all years except the last year which is the half of the above percentage.

Let us verify for 7-Year class property, Y=7, D=200%, so $\small&space;X=7(1-\frac{1}{2})+\frac{3}{2}=5$, so from 5th year, it is switches to straight line depreciation, and the depreciation rate for 5th, 6th, and 7th years is:

$\small&space;(1-\frac{2}{2*7})(1-\frac{2}{7})^{5-2}/(7-5+\frac{3}{2}) = 8.925%$

Please note also how the percentage alternates between 8.92 and 8.93 over the years. IRS tries to be as precise as possible beyond the second decimal point.

To summarize, for half year convention, the depreciation percentage is:

$\large&space; P = \left\{ \begin{array}{ll} 1-\frac{D}{2Y} & N = 1 \\ (1-\frac{D}{2Y})(1-\frac{D}{Y})^{N-2}\frac{D}{Y} & 2 \leq N < \lceil{Y(1-\frac{1}{D})+\frac{3}{2}}\rceil \\ (1-\frac{D}{2Y})(1-\frac{D}{Y})^{N-2}\frac{1}{(Y-N+\frac{3}{2})} & \lceil{Y(1-\frac{1}{D})+\frac{3}{2}}\rceil \leq N \leq Y \\ (1-\frac{D}{2Y})(1-\frac{D}{Y})^{N-2}\frac{1}{2(Y-N+\frac{3}{2})} & N = Y + 1 \\ \end{array} \right.$

where $\large&space;\lceil{x}\rceil$ is the least integer greater than or equal to x.

From this derivation and verification we can see the tax code is not a mess of rules, in it there is strict logic and mathematical precision. We will see other examples. By the way, the commonwealth of Virginia does not honor the federal accelerated depreciation, so every year we need add a portion of the extra depreciation as compared to linear depreciation to the Virginia income in early years, and subtract deficiency in depreciation in later years, and this appears as mysteriously as “fixed date conformity”.

The equation can be extended to any convention. Let p denotes the portion for the first year the asset is in service, for example, if an asset is placed in service in December with Mid-Quarter convention, then d=1/8. The general equation is:

$\large&space; P = \left\{ \begin{array}{ll} 1-\frac{pD}{Y} & N = 1 \\ (1-\frac{pD}{Y})(1-\frac{D}{Y})^{N-2}\frac{D}{Y} & 2 \leq N < \lceil{Y(1-\frac{1}{D})+2-d}\rceil \\ (1-\frac{pD}{Y})(1-\frac{D}{Y})^{N-2}\frac{1}{(Y-N+2-p)} & \lceil{Y(1-\frac{1}{D})+2-d}\rceil \leq N \leq Y \\ (1-\frac{pD}{Y})(1-\frac{D}{Y})^{N-2}\frac{1}{2(Y-N+2-p)} & N = Y + 1 \\ \end{array} \right.$