Capital tax factor

The capital tax factor (CTF) is a single multiplier that captures the present-worth effect of all future CCA tax savings on an asset’s first cost. Instead of computing each year’s CCA, each year’s tax saving, and discounting them all back individually, you apply the CTF to the asset’s first cost and you’re done.

The formula:

$\text{CTF}=1-\frac{t\cdot d}{(i+d)}\cdot \frac{1+i/2}{1+i}$

where $t$ is the corporate tax rate, $d$ is the asset’s CCA rate, and $i$ is the (after-tax) MARR. The CTF is less than 1 — the present-worth-adjusted cost of the asset is less than its sticker price because of the tax shield. Multiply the first cost by CTF to get its after-tax PW.

The factor assumes:

• The asset is depreciated under standard CCA declining-balance.
• The Half-year rule applies in year 1 (this is what produces the $(1+i/2)/(1+i)$ correction).
• The asset is held forever (the geometric series of CCA tax shields runs to infinity).

The infinite-life assumption is the main approximation. In practice the asset is sold or scrapped at some point, which means some of the assumed tax shield doesn’t actually get realised. The CSF corrects for this by applying a comparable factor to the salvage value at the end of life. CTF and CSF together give the correct PW of the asset’s first cost and end-of-life salvage, including all the CCA tax effects in between.

How much does the infinite-life assumption over-count? At year $N$ the UCC remaining is approximately $P(1-d{)}^N$ (ignoring half-year), and the present value of all future tax shields from that remaining UCC is roughly $t\cdot d\cdot P(1-d{)}^N/(i+d)\cdot (1+i{)}^{-N}$. For a Class 8 asset ($d=0.20$) at $i=0.08$, $t=0.27$, $N=10$, that works out to about 1% of $P$; for a Class 1 building ($d=0.04$) at the same $i$ and $t$ over 10 years, it’s about 2.8% of $P$. So the slower the declining-balance rate relative to $i$, and the shorter the holding period, the more the CSF correction matters. For fast-depreciating assets held a long time, the correction is small and the CTF alone gets close to the right answer.

Derivation sketch. The CCA tax shield in year $n$ is approximately $t\cdot d\cdot UC{C}_{n-1}$, where $UC{C}_n$ decays geometrically at rate $(1-d)$. Discounting this infinite geometric stream at $i$ gives the closed form $\frac{td}{i+d}$. The half-year rule shifts the first-year shield by a half-year of growth/discount, which produces the $(1+i/2)/(1+i)$ correction. Subtracting this present-worth shield from 1 gives the CTF as the effective fraction of the first cost that’s not recovered through tax savings.

A worked example. Buy a $100,000 Class 8 asset ($d=20\%$) at after-tax MARR $i=8\%$, tax rate $t=27\%$:

$\text{CTF}=1-\frac{0.27\cdot 0.20}{0.08+0.20}\cdot \frac{1.04}{1.08}=1-0.1929\cdot 0.963=1-0.1858=\mathrm{0.8142.}$

The PW of the first cost, accounting for the infinite CCA tax shield, is 100{,}000 \cdot 0.8142 = \81{,}420. The other \18,580 is recovered through the present value of future tax savings.

For the salvage-side counterpart see Capital salvage factor. For the underlying framework see Capital cost allowance, Undepreciated capital cost, Half-year rule.