So now we're going to take a closer look at
some book depreciation methods. These are going to be the straight line method, the declining
balance and double declining balance method, and then what we call a units of production methods.
So you can think of this as three different types of methods. Straight line, as you'll see, is
just what it sounds like. Declining balance, double declining balance are kind of percentage
based. And units of production is based on actual use. And although in the studies that we're going
to do as our engineering economic studies we're going to ultimately be looking at tax depreciation
methods 'cause we're thinking about cash flows and taxable income, how much are we paying in taxes
and what is our cash flow after taxes. The reason that we're going to look at book depreciation
methods, well there are a couple of them. First of all, tax depreciation methods are based on
book depreciation methods and so the principles that we're going to look at are the same. You
also need to be aware that in the reporting that companies do externally these are the methods
that they're using and you should understand what type of methods are used. Other countries and some
states, as well as equipment placed into service in the U.S. before 1981, the tax depreciation
for those is actually determined using some of these other methods and not makers, and so very
important, and we'll get to tax depreciation, but we're gonna look at these book depreciation
methods and kind of lay some foundation here. So lemme throw a few more terms at you. We are
going to talk about the depreciation rate and it's sometimes going to be designated as little
d, sometimes with the letter alpha and this is the percentage or the fraction of the first cost that
is going to be removed by depreciation each year. With straight line depreciation it's a constant
amount every year. With the other depreciation methods the rate changes. When we talk about the
depreciation charge, this is big letter D, so depreciation rate, little d or alpha depreciation
charge big D, this is the actual amount in dollars of the depreciation that we're going to subtract
from gross income. A couple more terms for you. We're going to talk about book value. So the
book value in any particular year is the cost basis less all of the depreciation charges taken
to date. These are usually updated at the end of the company's accounting year, which by the way,
may or may not be the same as a calendar year, and what this book value reflects at any point
in time is the undepreciated value of an asset. By the way, it may or may not be the same as the
market value because market value is the amount that we could actually get by selling this asset
to a willing buyer. I'm sure there's some other term there we could think of around that, but
oftentimes, book and market values can be very different. Think about computers, for example.
Companies invest in a lot of computers. You would invest in a computer too. Those things in
terms of the value on the market decrease pretty rapidly. However, we also saw that that kind of
equipment would be considered office equipment and the book value on those, the life on those
can go out for several years meaning the book value will stay higher. So computer equipment
is one where the market value may be much, much lower than the actual book value. Sometimes
though, if we look at real property, it's the other way around. Commercial buildings may tend to
increase in market value but their book value may actually be less than what their market value
is as we take depreciation. So, book value and market value are two different terms. I don't
think we're actually gonna come back to talking about market value. We're gonna talk about book
value, we've already talked about salvage value and those are the ones that we're gonna continue
to use here as we look at depreciation methods. So first depreciation method is straight line
depreciation and what we're assuming here is that the book value is going to decline in
a constant fashion, in a uniform manner, down to some salvage value over however long the
depreciable life is and so the depreciation rate is just going to be one divided by the number of
years. So, it's just going to be a constant rate going from our initial value down to our salvage
value and every year we are taking off the same amount in depreciation. So the depreciation rate
is just one over N and the depreciation amount is just going to be the difference between our
initial investment and our salvage value, either multiplied by the rate or divided by the number
of years. Our book value then is just going to be our initial investment, less our depreciation.
So the book value in year one would be minus the depreciation in year one. Book value in year
two would be less the depreciation for year two, et cetera, et cetera. And a handy computing
formula for book value is just the expression here on the end, the investment less the number
of years times the depreciation charge. So useful life, little N designates whatever year we're
talking about, the depreciation charge in a particular year is D sub N. The salvage value is
that number that we estimate at the beginning for what the value's gonna be at the end of the useful
life. And then the book value is just your initial investment less all of your depreciation charges
taken to date. So we have an argon gas processor that has a first cost of $20,000, a salvage
value after five years of $5,000. So I is 20,000. Salvage value is 5,000. N is equal to five
years. What is the book value at the end of year three? Let's also find the annual depreciation
charges and book values over the processor life and let's go ahead and create a plot here. And so
this is where we can either go to the calculating formulas that we have up here or we can start
thinking about how we're gonna work in these problems in tables. And so let's just take a quick
look at what this might be with these formulas here. So the depreciation rate, little d, is equal
to 1 over 5, or 0.20. The depreciation charge in year three is just going to be 0.20 X 20,000
5,000, or 0.2 times 15,000. And I get, I guess I'm gonna do that, 0.2 X 15,000. $3,000. Yes, I
just did do that. And so our book value in year three would just be our initial investment minus
n times our depreciation charge, or 20,000 - n, which is three years, x 3000 or let's see, 20,000
9,000, $11,000. So let's take a look at all of the depreciation charges though. So in year
zero we don't have a prior year book value. We're not applying any depreciation. But what we
have is an initial investment of $20,000. We've already figured out that our depreciation rate
is going to be 0.20 for each of these years and we have a constant amount depreciation charge
of 0.2 times the difference, or we said that was 3000. And so book value in n minus one in
year one's 20,000. 20,000 - 3000 gives us a book value of 17,000. 17,000 - 3000 gives us a
book value of 14,000. 14,000 - 3000 gives us a book value of 11,000. Yay, by the way, it matches
the number that I calculated up here using the formula. That's supposed to be a thousand. 11,000
3000 leaves us with 8,000. And then 8,000 minus the 3000 leaves us with 5,000. Another good sanity
check here, by the way, that's our salvage value. And so we have our book value for every year,
we have our depreciation charge for every year, and the next thing I'm asking you to do here is to
plot this over time. And the reason for this will be clear in a little bit because I'd like you to
try to compare it to some others. So versus time and then the book value. And we're gonna go
from, let's see, 20,000 10, 15, 5. So year one, well actually we'll even start at year zero, it's
20,000. Year one, my book value is about 17,000. Year two, it's about 14. 11. 8. And 5. And so can
you see why we call it straight line depreciation? Another type of depreciation is something we
call declining balance depreciation. This is a method that accelerates the depreciation that
we're taking. This allows us to take larger depreciation charges earlier in the life of an
asset and if you think about it we take larger depreciation charges that means we're lowering our
taxable income, we're minimizing our taxes and the result is that we would have an, considering the
time value of money over the life of the project, we would have an overall higher net present value.
And so when we look at the straight line method, it is just, if we're looking at it in terms
of a declining balance method, it's just alpha equal to one over N. We would refer to that as the
straight line rate. That's the lower boundary on a declining balance depreciation. The upper boundary
is two divided by N. That would be the double declining rate where our depreciation rate is two
divided by N, or considering our last example, that would be 2 / 5. Another common rate
that you will see used is something that is referred to as the 150% declining balance
rate that would be alpha equal to 1.5 over N, or the 150% declining balance. So straight
line rate. Another common one is the 150% declining balance rate. Another common one then
is also the double declining rate, 200%. And so these are methods that we can also apply for
book depreciation. And as I said, if you think about this in terms of your project and the cash
flows that are happening with it, with the greater depreciation amounts early on we are having
more worth in our projects, higher net present worth. So I drew a picture because I wanted you to
compare it to this picture here. If we're taking higher depreciation or larger depreciation
charges earlier in the life of our asset, then what's going to happen is that the book value
is going to decline more rapidly and that's what this DB line is here. And so the one that I've
drawn off here is just kind of an arbitrary one that is something faster than the straight line.
If we're talking about 150% declining balance we're removing 15% of the book value every year.
And remember we're doing that down to the salvage value. If it's double declining balance we're
removing 20% of the book value every year. So when we take a bigger depreciation charge earlier on
our book value decreases more rapidly and that's what we see going on here with the straight line
all the way to some salvage value at the end of its useful life. So the equations for declining
balance. The depreciation charge in year one is just simply the rate times the investment.
Depreciation charge in the second year is just, this is the book value times the depreciation
rate. And then we have a simplified computing formula here for any particular year to see
what the depreciation charge is. If you want to calculate the total depreciation without having to
go through it year by year, the total depreciation declining balance approach we use this calculating
formula. Similarly, when we're trying to determine the book value in a particular year, if we're
just doing successive year by year calculations, this formula on the left hand side is the one that
would be most advantageous to us. If we're trying to figure out the book value in any particular
year without doing year by year calculations, we will use the formula here on the right
hand side. All right, so let's take a look at an example. We have an engineer working with a
consulting firm in Dubai and her client is having some difficulty understanding the difference
between 150% declining balance depreciation and double declining balance depreciation and so
she's going to use an example to illustrate to him the difference. We have a piece of equipment,
the initial cost basis is $180,000. It can be sold for an estimated salvage value of 30,000.
A useful life of 12 years. What are the book values after five years for both methods? So we
know I, we know S, we know N, we know little n, yes, this is a little n and so what we wanna
determine is the book value for 150% method and for the double declining method. So if we're
gonna take a look at the 150% method first. So in this case alpha, our depreciation rate, is going
to be 1.5 divided by the useful life. 1.5 / 12, or 0.125. And the book value in, we're looking
at year five, so the book value in year five, if we just use the total formula over here would be
I times 1 minus alpha to the n, or what is this, 180,000? 1 - 0.125 to the 5, or we have a book
value here of 92,324. So if we elect to use a 150% declining balance approach, after five years, the
book value is going to be 92,000. If we're using the double declining balance, which is basically
the same as 200% declining balance. Alpha will be equal to 2 over N. Yes, that's an N. 2 / 12, or
0.167. And the book value in year five is going to be 180 times 1 - 0.167 to the 5, or 72,193.
So understanding the difference between the two, the net result is here after five years. If
I use the 150% approach I will have a value on my books of about 92,000. If I use the double
declining approach I will have a value on my books of about 72,000. And so we need to think about
which of these is realistic, perhaps for the life of the product that it's being used for, we
may need to take other things into consideration, but with the double declining balance we're
taking larger depreciation charges early on than we would be with the 150% declining balance
and so that gets us to a smaller book value since we've been taking the larger depreciation charges.
Let's take a look at another example here. We have a corn husking machine. Do we have any folks from
Nebraska out there? Any Nebraska Cornhusker fans? Anyway, we have a corn husking machine with
the first cost of 10,000, a salvage value of a nice round number of $778 after five years.
You'll see why we picked this one shortly. And for book depreciation calculations this particular
company uses a depreciation rate of 0.40. So let's find the annual depreciation charges and the book
values over the next five years of the useful life of the machine. So no book value in the minus one
year. We're not charging depreciation. We start off with an investment of $10,000. So 10,000.
Our depreciation rate is going to be the same every year. Every year it's going to be 40% of
whatever the book value is. So my depreciation charge here in the first year is going to be 4,000
leaving me a book value of 6,000. The next year I can take 40% of 6,000. 2,400. Which will give me
3,600 of book value. 3,600 in book value. I think I've got an extra zero there. Yeah, 3,600 in book
value, I can take 40% of that, or 1440, which will gimme a book value of 2160. 2160 and I take 40%
of that gives me 864 or a book value of 1296 at the end of year four. And then I can take another
40% which gives me a depreciation charge of 518 or the final book value of 778. So with our book
depreciation methods, this kind of reinforces one of the other points that I made, When we finish
our recovery life we've depreciated it all the way down to its salvage value. If you remember when
we start talking about tax depreciation, salvage value is gonna be zero. So that number's gonna
be zero. But anyway, this corn husking machine, we have depreciated over five years down to its
book value of $778. We've got the solution here, and as I mentioned, this is a very special
case where we have depreciated all the way down at the end of the life to the book
value being equal to the salvage value and this is case one of some situations that
we're going to look at later. Thank you.