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32-456: Capital property eligible for CCA excludes: CCA is calculated on undepreciated capital cost ("UCC"), which is generally defined as: Where the UCC for a class is negative, a recapture of depreciation is deemed to take place, thus adding to taxable income and bringing the balance of UCC back to zero. Where UCC for a class is positive, but all assets with respect to that class have been disposed of,

64-400: A 2 r n ) n + 1  for  a ≥ 0 , r ≥ 0. {\displaystyle \prod _{k=0}^{n}ar^{k}=a^{n+1}r^{n(n+1)/2}=({\sqrt {a^{2}r^{n}}})^{n+1}{\text{ for }}a\geq 0,r\geq 0.} This corresponds to a similar property of sums of terms of a finite arithmetic sequence : the sum of an arithmetic sequence is

96-408: A n − 1 2 / a n − 2 {\displaystyle a_{n}=a_{n-1}^{2}/a_{n-2}} for every integer n > 2. {\displaystyle n>2.} This is a second order nonlinear recurrence with constant coefficients. When the common ratio of a geometric sequence is positive, the sequence's terms will all share the sign of

128-569: A − b − c ) {\displaystyle CCA=tdUCC-{\frac {1}{2}}td\left(a-b-c\right)} Under the Income Tax Act : Part XI of the Income Tax Regulations provides for the calculation rules for CCA, and Schedule II outlines the various classes of capital property that are eligible for it. Special rules are in place to deem certain assets to be in separate classes, thus not becoming part of

160-398: A geometric series . The n th term of a geometric sequence with initial value a = a 1 and common ratio r is given by and in general Geometric sequences satisfy the linear recurrence relation This is a first order, homogeneous linear recurrence with constant coefficients . Geometric sequences also satisfy the nonlinear recurrence relation a n =

192-510: A geometric series is a series summing the terms of an infinite geometric sequence , in which the ratio of consecutive terms is constant. For example, the series 1 2 + 1 4 + 1 8 + ⋯ {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{8}}+\cdots } is a geometric series with common ratio ⁠ 1 2 {\displaystyle {\tfrac {1}{2}}} ⁠ , which converges to

224-474: A terminal loss is deemed to take place, thus deducting from taxable income and bringing the balance of UCC back to zero. CCA itself is generally calculated using the following items: For assets subject to the full-year rule: C C A = t d U C C {\displaystyle CCA=tdUCC} For assets subject to the half-year rule: C C A = t d U C C − 1 2 t d (

256-410: A century or two later by Greek mathematicians , for example used by Archimedes to calculate the area inside a parabola (3rd century BCE). Today, geometric series are used in mathematical finance , calculating areas of fractals, and various computer science topics. The infinite product of a geometric progression is the product of all of its terms. The partial product of a geometric progression up to

288-405: A tax shield. Since a tax shield is a way to save cash flows , it increases the value of the business, and it is an important aspect of business valuation . The concept was originally added to the methodology proposed by Franco Modigliani and Merton Miller for the calculation of the weighted average cost of capital of a corporation . The reason that he was able to earn additional income

320-404: Is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2. Examples of a geometric sequence are powers r of a fixed non-zero number r , such as 2 and 3 . The general form of a geometric sequence is where r is the common ratio and a is the initial value. The sum of a geometric progression's terms is called

352-498: Is an example of a converging series for a geometric progression , this can be simplified further to become: P V = I t d i + d {\displaystyle PV={\frac {Itd}{i+d}}} The net present after-tax value of a capital investment then becomes: I ( 1 − t d i + d ) {\displaystyle I\left(1-{\frac {td}{i+d}}\right)} For capital investments where CCA

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384-419: Is because the cost of debt (i.e. 8% interest rate) is less than the return earned on the investment (i.e. 10%). The 2% difference makes income of $ 80 and another $ 100 is made by the return on equity capital. Total income becomes $ 180 which becomes taxable at 20%, leading to the net income of $ 144. In most business valuation scenarios, it is assumed that the business will continue forever . Under this assumption,

416-627: Is calculated under the half-year rule, except where otherwise specified, with respect to the following classes. (Minimum 5 years and Maximum 40 years) In contrast to the practice followed in the United States for depreciation there is no penalty for failing to claim Capital Cost Allowance. Where a taxpayer claims less than the amount of CCA to which he is entitled the pool remains intact, and available for claims in future years. Unclaimed amounts are not subject to recapture. Because assets subject to CCA are generally pooled by class, and CCA

448-1356: Is calculated under the half-year rule, the CCA tax shield calculation is modified as follows: P V = 1 2 ( I t d i + d ) + 1 2 ( I t d i + d ) ( 1 1 + i ) = I t d i + d [ 1 2 + 1 2 1 + i ] = I t d i + d [ 1 2 ( 1 + i ) + 1 2 1 + i ] = ( I t d i + d ) ( 1 + 1 2 i 1 + i ) {\displaystyle {\begin{aligned}PV&={\frac {1}{2}}\left({\frac {Itd}{i+d}}\right)+{\frac {1}{2}}\left({\frac {Itd}{i+d}}\right)\left({\frac {1}{1+i}}\right)\\&={\frac {Itd}{i+d}}\left[{\frac {1}{2}}+{\frac {\frac {1}{2}}{1+i}}\right]\\&={\frac {Itd}{i+d}}\left[{\frac {{\frac {1}{2}}\left(1+i\right)+{\frac {1}{2}}}{1+i}}\right]\\&=\left({\frac {Itd}{i+d}}\right)\left({\frac {1+{\frac {1}{2}}i}{1+i}}\right)\\\end{aligned}}} Therefore,

480-412: Is equivalent to taking the geometric mean of the partial progression's first and last individual terms and then raising that mean to the power given by the number of terms n + 1. {\displaystyle n+1.} ∏ k = 0 n a r k = a n + 1 r n ( n + 1 ) / 2 = (

512-411: Is generally calculated on a declining-balance basis, specific techniques have been developed to determine the net present after-tax value of such capital investments. For standard scenarios under the full-year rule and half-year rule models, the following standard items are employed: More specialized analysis would need to be applied to: Capital cost allowance will be calculated as follows: Therefore,

544-561: Is not valid for a < 0 {\displaystyle a<0} or r < 0 , {\displaystyle r<0,} which is the formula in terms of the geometric mean. A clay tablet from the Early Dynastic Period in Mesopotamia (c. 2900 – c. 2350 BC), identified as MS 3047, contains a geometric progression with base 3 and multiplier 1/2. It has been suggested to be Sumerian , from

576-567: The Tax shield in year n = I t d ( 1 − d ) n − 1 {\displaystyle Itd(1-d)^{n-1}} , and the present value of the taxation credits will be equal to I t d ∑ n = 1 ∞ ( 1 − d ) n − 1 ( 1 + i ) n {\displaystyle Itd\sum _{n=1}^{\infty }{\frac {(1-d)^{n-1}}{(1+i)^{n}}}} As this

608-615: The absolute value of the common ratio is smaller than 1, the terms will decrease in magnitude and approach zero via an exponential decay . If the absolute value of the common ratio is greater than 1, the terms will increase in magnitude and approach infinity via an exponential growth . If the absolute value of the common ratio equals 1, the terms will stay the same size indefinitely, though their signs or complex arguments may change. Geometric progressions show exponential growth or exponential decline, as opposed to arithmetic progressions showing linear growth or linear decline. This comparison

640-625: The Canada Revenue Agency, the Supreme Court of Canada has interpreted the Capital Cost Allowance in a fairly broad manner, allowing deductions on property which was owned for a very brief period of time, and property which is leased back to the vendor from which it originated. These decisions demonstrate the flexibility of the Capital Cost Allowance as a legal tax reduction strategy. A notable example of how

672-574: The Capital Cost Allowance can impact business activity was seen in the Canadian film industry in the 1970s, when the government of Pierre Trudeau introduced new regulations to facilitate the production of Canadian films by increasing the Capital Cost Allowance for film production to 100 per cent in 1974. While some important and noteworthy films were made under the program, and some film directors who released their first films in this era emerged as among Canada's most important and influential filmmakers of

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704-432: The Capital Cost Allowance for film production was reduced to 50 per cent, although films that had entered production under the program continued to be released for another few years afterward. Tax shield A tax shield is the reduction in income taxes that results from taking an allowable deduction from taxable income . For example, because interest on debt is a tax-deductible expense, taking on debt creates

736-474: The era, the new regulations also had an entirely unforeseen side effect: a sudden rush of low-budget horror and genre films , intended as pure tax shelters since they were designed not to turn a conventional profit. Many of the films, in fact, were made by American filmmakers, whose projects had been rejected by the Hollywood studio system as not commercially viable. The period officially ended in 1982, when

768-454: The first term. When the common ratio of a geometric sequence is negative, the sequence's terms alternate between positive and negative; this is called an alternating sequence. For instance the sequence 1, −3, 9, −27, 81, −243, ... is an alternating geometric sequence with an initial value of 1 and a common ratio of −3. When the initial term and common ratio are complex numbers, the terms' complex arguments follow an arithmetic progression . If

800-478: The general pool for the class. Certain elections are available to taxpayers to transfer or reclassify assets from one class to another. Additional allowances are prescribed with respect to specified circumstances. Specialized calculations for certain classes are also outlined in: Part XVII of the Income Tax Regulations provides for specialized calculation rules for CCA with respect to capital property acquired for use in earning income from farming and fishing. CCA

832-444: The net present after-tax value of a capital investment is determined to be: I [ 1 − ( t d i + d ) ( 1 + 1 2 i 1 + i ) ] {\displaystyle I\left[1-\left({\frac {td}{i+d}}\right)\left({\frac {1+{\frac {1}{2}}i}{1+i}}\right)\right]} In cases where claims have been contested or disallowed by

864-465: The number of terms times the arithmetic mean of the first and last individual terms. This correspondence follows the usual pattern that any arithmetic sequence is a sequence of logarithms of terms of a geometric sequence and any geometric sequence is a sequence of exponentiations of terms of an arithmetic sequence. Sums of logarithms correspond to products of exponentiated values. Let P n {\displaystyle P_{n}} represent

896-554: The product up to power n {\displaystyle n} . Written out in full, Carrying out the multiplications and gathering like terms, The exponent of r is the sum of an arithmetic sequence. Substituting the formula for that sum, which concludes the proof. One can rearrange this expression to Rewriting a as a 2 {\displaystyle \textstyle {\sqrt {a^{2}}}} and r as r 2 {\displaystyle \textstyle {\sqrt {r^{2}}}} though this

928-460: The sum of ⁠ 1 {\displaystyle 1} ⁠ . Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors. While Greek philosopher Zeno's paradoxes about time and motion (5th century BCE) have been interpreted as involving geometric series, such series were formally studied and applied

960-463: The term with power n {\displaystyle n} is ∏ k = 0 n a r ( k ) = a n + 1 r n ( n + 1 ) / 2 . {\displaystyle \prod _{k=0}^{n}ar^{(k)}=a^{n+1}r^{n(n+1)/2}.} When a {\displaystyle a} and r {\displaystyle r} are positive real numbers, this

992-410: The value of the tax shield is: (interest bearing debt) x (tax rate). Using the above examples: Geometric progression A geometric progression , also known as a geometric sequence , is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio . For example, the sequence 2, 6, 18, 54, ...

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1024-471: Was taken by T.R. Malthus as the mathematical foundation of his An Essay on the Principle of Population . The two kinds of progression are related through the exponential function and the logarithm : exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression yields an arithmetic progression. In mathematics ,

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