apital Budgeting Formula Not in the book. Wei s summary If salvage value S is less than U n : If salvage value S is greater than U n : Note: IF t : incremental cash flows (could be negative) )(NW): change in net working capital. Positive if additional W required for a year; negative if W is released for a year.
Explaining the apital 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 A 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 A schedule using the half-year rule: Year A U 0 $200,000 1 $30,000 $170,000 2 $51,000 $119,000 3 $35,700 $83,300 Since the salvage value S = $35,000 is less than U 3 = $83,300, we use the first formula: Let s examine one item at a time. 1) The first item is initial investment. = $200,000. Since it is an investment or cash outflow, we have a negative sign in front it. 2) The second item is the total PV of after tax, incremental cash flows (IF). It is the net benefit of having a project. In our case, the savings in costs are just like increases in revenues. So, each IF = $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. 3) 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 (NW) 1 = $25,000. The second year, we need another $8,000, so (NW) 1 = $8,000. No new requirement or recovery in the second year, so (NW) 2 = 0. By year 3, we will recover everything, which means (NW) 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
investment. Now, there is a negative sign in front of the who third item. This is to reflect the fact that a positive NW represents an investment or a cash outflow. So, the entire third item is 25,000 + 8,000/(1+0.1) - 33,000/(1+0.1) 3 = $7,479 In other words, in PV terms we need to invest $7,470 in NW (this is the financing cost). Since NW is cash outflow, we have a negative sign in front it. 4) The fourth item is simply the PV of salvage value, which is 35,000/(1+0.1) 3 = $26,296 We have a positive sign in front of it, since this is cash inflow. 5) The fifth item is the PV of all the future tax shields from A assuming the equipment will last forever, under the half-year rule. It is 0.3*(200,000)(0.4)/(0.1+0.4)*[(1+0.5*0.1)/(1+0.1)] = $57,273. We have a positive sign in front of it, since this is tax savings. 6) The last item is the tax shield adjustment. When we calculate item 5, we assume that the equipment would last forever. But we know that we will lose it after three years. Therefore, we will not be able to enjoy A tax shields after year three. We must subtract an amount from item 5 to reflect this lost. At the end of year 3, the U is $83,300 and the salvage value is $35,000. If the equipment is not sold and is held forever, then you would continue to depreciate $83,300 from year 3 on. No adjustment is needed in this case, since item 5 is the case of equipment lasting forever. But when you sell it for $35,000, you have recovered $35,000, only the remaining amount (which is $83,300 - $35,000 = $48,300) will continue to generate tax shields forever. In other words, you will lose $35,000 (which is salvage value, S) from the capital base. Since this is lost forever, so the present value of tax shields from this amount is dst/(k+d) (here we don t use the half-year formula anymore since it is year 3 already). What about 1/(1+k) n? This is simply to bring the PV of tax shield loss to today. In our case, the entire sixth item is 1/(1+0.1) 3 [0.3*35,000*0.4/(0.1+0.3)] = $7,889. We have a negative sign in front of it, since we lose it. Put all items together: NPV = -$200,000 + $109,421 - $7,479 + $26,296 + $57,273 - $7,889 = - $22,378 **************** If we change the example just by one number: make the salvage value to be $90,000. In this case, since S > U 3, we will use the second formula. Why? Since by selling the equipment, you will have recovered all the remaining book value of the equipment ($83,300). You should not enjoy any A tax shields from that point on. That is why we put the entire U in the formula. Anyway, in this case, we only need to adjust the fourth item (which is now $90000/(1.1) 3 = $67,618) and the last item (which is now 1/(1+0.1) 3 [0.3*83,300*0.4/(0.1+0.3)] = $18,775). So the new NPV is NPV = -$200,000 + $109,421 - $7,479 + $67,618 + $57,273 - $18,775 = $8,058
Example A new machine: A class: 8 (d=20%) Life 4 years Price $100,000 Freight & installation: $20,000 Salvage value: $30,000 Additional NW: $10,000 Effect on costs: decrease costs by $50,000 per year Tax rate: 40% ost of capital (k): 10% Depreciation base: = $100,000 + $20,000 = $120,000 (No additional NW required during life. But in general, NW changes every year)
Time Line Method Depreciation schedule (half year rule) Year A U 0 $120,000 1 $12,000 108,000 2 21,600 86,400 3 17,280 69,120 4 13,824 55,296 ash flow analysis 0 1 2 3 4 Machine cost ($120,000) NW ($10,000) IF(1-T) $30,000 $30,000 $30,000 $30,000 A*T 4,800 8,640 6,912 5,530 Operating F $34,800 $38,640 $36,912 $35,530 Salvage value $30,000 A tax shields * 6,746 NW recovery $10,000 Net F ($130,000) $34,800 $38,640 $36,912 $82,276 PV of F ($130,000) $31,636 $31,934 $27,733 $56,196 NPV $17,499 A tax shields: U 4 = $55,296, S = $30,000, Remaining capital continuing to generate tax shield: 55,296-30,000 = 25,296 Then, dt / (k + d) = 0.2(25,296)(0.4)/(0.1 + 0.2) = $6,746.
Formula Method Depreciation schedule (half year rule) Year A U 0 $120,000 1 $12,000 108,000 2 21,600 86,400 3 17,280 69,120 4 13,824 55,296 U 4 = $55,296 > S =$30,000, Therefore use = $120,000 )(NW) 0 = $10,000 )(NW) t = 0 (T = 1, 2, 3) )(NW) 4 = - $10,000 IF t = 50,000 (t = 1,4) S = $30,000 T = 0.4 d = 0.2 k = 0.1 n = 4 NPV = $17,499
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 A tax shield. Why delete this item? In the previous case when S < U 4, 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 A tax shields of $6,746. Now, if the salvage value is $60,000, which is bigger than U 4 = $55,296, you will recover all the book value of capital investment. So you will not enjoy any future tax shields from A, since there is no A to deduct anymore! That is why we don t have that line A tax shield anymore. All you get from A tax shields are the four numbers for each year under A*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) NW ($10,000) IF(1-T) $30,000 $30,000 $30,000 $30,000 A*T 4,800 8,640 6,912 5,530 Operating F $34,800 $38,640 $36,912 $35,530 Salvage value $60,000 NW recovery $10,000 Net F ($130,000) $34,800 $38,640 $36,912 $105,530 PV of F ($130,000) $31,636 $31,934 $27,733 $72,078 NPV $33,381 The formula Method Everything is the same as before, except that S = $60,000 and U 4 = $55,296, then you can doublecheck that you get NPV = $33,381.