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In our case, the savings in costs are just like increases in revenues. So, each ICF = $110,000. The entire second item can be written as 110,000(1-0.4)/(1+0.1) ...
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Explaining the Capital Budgeting Formula
Let’s use the following example to illustrate the formulas. Suppose you have a project which requires an initial investment (in a piece of equipment) of $200,000. The equipment has a CCA rate of 0.3. The project will last 3 years. At the end of year 3, the equipment will be sold for $35,000. The project will reduce production cost by $110,000 per year. The initial working capital requirement is $25,000. An additional amount of $8,000 is required for year 1. All will be recovered at the end of year 3. The tax rate is 40% and the discount rate is 10%. What is the NPV?
Let’s first get the CCA schedule using the half-year rule:
Since the salvage value S = $35,000 is less than UCC 3 = $83,300, we use the first formula:
Let’s examine one item at a time.
The first item is initial investment. C = $200,000. Since it is an investment or cash outflow, we have a negative sign in front it.
The second item is the total PV of after tax, incremental cash flows (ICF). It is the net benefit of having a project. In our case, the savings in costs are just like increases in revenues. So, each ICF = $110,000. The entire second item can be written as 110,000(1-0.4)/(1+0.1) + 110,000(1-0.4)/(1+0.1) 2 + 110,000(1-0.4)/(1+0.1) 3 =$109,421. Since this is a benefit, we have a positive sign in front it.
The third item is the net working capital. The summation goes from time zero to year n. Notice that we only keep track of the changes (hence the greek letter ∆). In other words, we only worry about the net addition and reductions. In our case, the initial requirement is $25,000, so ∆(NWC) 1 = $25,000. The second year, we need another $8,000, so ∆(NWC) 1 = $8,000. No new requirement or recovery in the second year, so ∆(NWC) 2 = 0. By year 3, we will recover everything, which means ∆(NWC) 3 = - $33,000. (This is $25,000 + $8,000). Why the negative sign in front of $33,000? Because this is not additional requirement, rather, it is the recovery, the opposite of
0 1 2 3 4
Machine cost ($120,000)
NWC ($10,000)
CIF(1-T) $30,000 $30,000 $30,000 $30,
CCA*T 4,800 8,640 6,912 5,
Operating CF $34,800 $38,640 $36,912 $35,
Salvage value $30,
CCA tax shields *^ 6,
NWC recovery $10,
Net CF ($130,000) $34,800 $38,640 $36,912 $82,
PV of CF ($130,000) $31,636 $31,934 $27,733 $56,
NPV $17,
CCA tax shields:
UCC 4 = $55,296, S = $30,000, Remaining capital continuing to generate tax shield: 55,296 - 30,000 = 25,
Then, dCT / (k + d) = 0.2(25,296)(0.4)/(0.1 + 0.2) = $6,746.
What if salvage value is $60,000? Everything else remains the same.
The Time Line Method
Here, we only need to make two changes. First, change the salvage value from $30,000 to $60,000; second, delete the line headed with “CCA tax shield”. Why delete this item? In the previous case when S < UCC4, we lose S = $30,000 from the capital base, since this is how much you have recovered from the sale of the old machine. In other words, the book value of capital investment at year 4 after selling the old machine is $55,296 - $30,000 = $25,296. So, from year 5 and on, you only continue to enjoy tax shields from this amount, which is the remaining book value of the capital cost. This is the base we used to calculate the PV of all future CCA tax shields of $6,746.
Now, if the salvage value is $60,000, which is bigger than UCC 4 = $55,296, you will recover all the book value of capital investment. So you will not enjoy any future tax shields from CCA, since there is no CCA to deduct anymore! That is why we don’t have that line “CCA tax shield” anymore. All you get from CCA tax shields are the four numbers for each year under CCA*T: $4,800, $8,640, $6,912, and $5,530. In any event, in this case, the time line method will lead to the following results:
0 1 2 3 4 Machine cost ($120,000) NWC ($10,000) CIF(1-T) $30,000 $30,000 $30,000 $30, CCA*T 4,800 8,640 6,912 5, Operating CF $34,800 $38,640 $36,912 $35, Salvage value $60, NWC recovery $10,
Net CF ($130,000) $34,800 $38,640 $36,912 $105, PV of CF ($130,000) $31,636 $31,934 $27,733 $72,
NPV $33,
The formula Method
Everything is the same as before, except that S = $60,000 and UCC 4 = $55,296, then you can double- check that you get NPV = $33,381.
