The Problem
Step 1:  Computing the Incremental Cash Flows
Step 2:  Computing the Initial Outlay
Step 3:  Computing the NPV and IRR

The Problem
Old equipment: book value = $600,000 with a remaining life of 5 years
Expected salvage value in 5 years = 0
Market value today = $265,000
Straight-line depreciation

New equipment: purchase price = $1,175,000 and a MACRS life of 5 years
Expected salvage value in 5 years = $145,000
Annual savings = $255,000

Other information: 40% marginal tax rate and a 12% hurdle rate

Step 1:  Incremental Cash Flows
DCF = (DS - DC - DD)(1 - t) + DD
Let's first find DD. The depreciation on the old equipment is easy since it's straight-line. D = 600000/5 = 120000 per year. The new machine is being depreciated using MACRS and it's in the 5 year class. Look in the text for a table of the percentages to use each year. For a 5 year asset, the percentages are 20%, 32%, 19%, 12% and 11%. That's a total of 94%. That means the book value at the end of the 5th year is 6%. Remember, with MACRS a 5 year asset really means 5.5 years and the remaining 6% is a half year's depreciation. Applying these rates to the price of 1,175,000 gives the following annual depreciation for the new equipment: .20*1,175,000 = 235,000; .32*1,175,000 = 376,000; .19*1,175,000 = 223,250; .12*1,175,000 = 141,000; .11*1,175,000 = 129,250 with a remaining book value of .06*1,175,000 = 70,500. Merely subtract the depreciation on the old equipment of $120,000 from the depreciation on the new equipment to get each year's DD.

YEAR Dnew Dold DD
1 235,000 120,000 115,000
2 376,000 120,000 256,000
3 223,250 120,000 103,250
4 141,000 120,000 21,000
5 129,250 120,000 9,250
We now can compute the incremental cash flows for each year by using the equation DCF = (DS - DC - DD)(1 - t)+ DD where DS = 0, DC = -255,000 and DD is obtained from the above table. For year 1 DCF = (0 -(-255,000) - 115,000)*(1 - .40) + 115,000 = 199,000. Repeat the process for the other 4 years.
1 199,000
2 255,400
3 194,300
4 161,400
5 156,700
In the terminal fifth year, lots of special things often happen. First is the return of any incremental net-working-capital. In this problem, there isn't any, so forget it. Usually you need to include the return of any net-working-capital as a cash inflow.

What can the old machine be sold for? Zero. So forget that, too. Usually you need to include any proceeds from selling the old machine as a cash inflow. Since the old machine is being depreciated using straight-line, it is being depreciated to its predicted salvage value. Thus, for nearly all problems, when using straight-line the machine's predicted market value and its book value will both equal the expected salvage value and, hence, there won't be any tax effect. But in this problem, the expected salvage value and the book value are both 0.

The new machine can be sold for $145,000 but its book value at the end of the 5th year is 70,500. Whenever any machine is sold for a price other than its book value, there is a tax effect that you must consider. This could pertain to the old machine or the new machine. The net proceeds from the sale is always MV - t(MV - BV). Be careful of the signs--but this always works. In this example, 145,000 - .4*(145,000 - 70,500) = 115,200. In the fifth year, there is this cash inflow of 115,200 from selling the new machine. This will make the total cash flows in the firth year 115,200 (nonoperating) + 156,700 (operating) = 271,900. Now let's solve for the initial outlay.

Step 2:  Initial Outlay
The initial outlay usually has three parts: cost of the new machine, the proceeds from selling the old machine (if there is one) and any additional net-working-capital that must be raised for the project. In this problem, the new machine has a purchase price of $1,175,000. There is no additional net-working-capital to be raised. If there were any, you would add it to the outlay. The proceeds from selling the old machine is usually the most complicated. In the example, it can be sold for $265,000 which is a lot less than its current book value of $600,000. Let's use the formula we just talked about: net proceeds = MV - t(MV - BV). Here it's 265,000 - .4*(265,500 - 600,000) = 399,000. There is a big capital loss on the sale (suggesting that it hasn't been depreciated enough to this point and the IRS is giving them some of the loss back). The firm will come away from the sale with $399,000. This will lower the outlay for the new machine to 1,175,000 - 399,000 = 776,000. Same formula as we used in the section above for the terminal cash flow, only this time we apply it to the sale of the old machine. Above we applied it to the sale of the proposed new machine 5 years from now when it is scrapped.

It's the same formula and it's applied for the exact same reason: you're selling a machine at a market value different from its book value. You gotta pay additional taxes if it's a gain and you get a tax refund if the sale is at a loss. There's another way of looking at the tax effect on the proposed new machine. Let's pretend that the firm buys the proposed new machine today and the forecasts all hold. 5 years from now they are considering buying another proposed new machine and you need to consider the initial outlay of that purchase. The "old" machine (the one you bought today) has a book value of 70,500. It can be sold for 145,000. The net proceeds is equal to 145,000 - .4*(145,000 - 70,500) = 115,200. Exactly what you did in the cash flow section above but then it was the "new" machine being sold 5 years from now. In this section it has now become the "old" machine and it is being sold 5 years from today because the firm is considering buying a third machine.

Now let's combine the answers from Steps 1 and 2 to see if the replacement is worthwhile.

Step 3:  NPV and IRR
Here's a table of our cash flows:

0 -776,000
1 199,000
2 255,400
3 194,300
4 161,400
5 271,900
The formula for NPV is:

NPV = - outlay + DCF1/(1 + k)1 + DCF2/(1 + k)2 + ... + DCFn/(1 + k)n

NPV = - 776,000 + 199,000/(1.12)1 + 255,400/(1.12)2 + 194,300/(1.12)3 + 161,400/(1.12)4 + 271,900/(1.12)5

NPV = 436.77 acceptable

Use the cash flow keys of your calculator or click on the button below for the Fin 125 Cash Flow Calculator.

The formula for IRR is:

0 = - outlay + DCF1/(1 + r)1 + DCF2/(1 + r)2 + ... + DCFn/(1 + r)n

where r is the IRR of the project.

0 = - 776,000 + 199,000/(1 + r)1 + 255,400/(1 + r)2 + 194,300/(1 + r)3 + 161,400/(1 + r)4 + 271,900/(1 + r)5

r = 12.02% acceptable

We knew it was going to be slightly larger than 12% because when we calculated the NPV using 12%, it was a very small $436.77.

Verify the answer using either your calculator or click the above button.