**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 | D_{new} |
D_{old} |
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 |

YEAR | DCF |

1 | 199,000 |

2 | 255,400 |

3 | 194,300 |

4 | 161,400 |

5 | 156,700 |

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:

YEAR | DCF |

0 | -776,000 |

1 | 199,000 |

2 | 255,400 |

3 | 194,300 |

4 | 161,400 |

5 | 271,900 |

NPV = - outlay + DCF_{1}/(1 + k)^{1}
+ DCF_{2}/(1 + k)^{2} + ...
+ DCF_{n}/(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 + DCF_{1}/(1 + r)^{1}
+ DCF_{2}/(1 + r)^{2} + ...
+ DCF_{n}/(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.