Net Present Value and Capital Budgeting
(Text reference: Chapter 7)
Topics
what to discount
the CCA system
total project cash flow vs. tax shield approach
detailed CCA calculations and examples
project interactions
AFM 271 - NPV and Capital Budgeting
Slide 1
What to Discount
some general principles:
1. Only cash flow is relevant
the NPV rule is stated in terms of cash flows
cash flow is a simple idea: dollars in - dollars out
don’t confuse cash flow with accounting income
(note that accounting income is needed in some
cases to calculate taxes)
2. Always estimate cash flows on an incremental basis
incremental cash flows are the additional cash
flows generated by the project. To identify them,
ask two questions:
What is the cash flow if the project is taken?
What is the cash flow if the project is not taken?
If the answers differ, then the cash flow is
incremental.
AFM 271 - NPV and Capital Budgeting
Slide 2
Cont’d
some things to watch for:
exclude sunk costs: it is incorrect to include costs
which have already been incurred and cannot be
recovered
include opportunity costs: e.g. a firm owns some land
worth $25 million which it is considering using as a
new factory site. If the firm builds the factory, it is
giving up the $25 million it could have received by
selling the land.
incorporate side effects: it is important to ensure that
all effects on the remainder of a firm’s operations are
taken into account (e.g. a new product line may reduce
sales of existing products)
AFM 271 - NPV and Capital Budgeting
Slide 3
Cont’d
include working capital requirements: most projects
will require an additional investment in working capital
(e.g. due to increased inventories, accounts
receivable, etc.); such investments are typically
recovered later on
allocated overhead costs: ensure that only those
charges which are actually due to a project are
allocated to it
interest expense: ignore this for now
AFM 271 - NPV and Capital Budgeting
Slide 4
Cont’d
3. Treat inflation consistently
nominal interest rate i
notation:
inflation rate
π
real interest rate
r
recall the “Fisher relation”
1 + i = (1 + r) × (1 + π ) ⇒ r =
1+i
−1
1+π
example: if i = 10% and π = 3.5%, what is the FV
after 2 years of $10,000 in real and nominal
terms? (assume the money is invested at i = 10%)
AFM 271 - NPV and Capital Budgeting
Slide 5
Cont’d
consistency requires that nominal cash flows be
discounted at a nominal discount rate and real
cash flows be discounted at a real discount rate.
For example, suppose that i = 8% and π = 3% and
that we have the following real cash flows:
C0
C1
C2
-1000 750 900
NPV in real terms:
NPV in nominal terms:
AFM 271 - NPV and Capital Budgeting
Slide 6
Cont’d
why does this work? Let C0 ,C1 ,C2 , . . . be real cash flows,
so:
PVnominal = C0 +
= C0 +
C1 (1 + π ) C2 (1 + π )2
+
+...
1+i
(1 + i)2
C1 (1 + π )
C2 (1 + π )2
+
+...
(1 + r)(1 + π ) (1 + r)2 (1 + π )2
it is important to understand how to use both real and
nominal discounting since you sometimes have to use
both approaches. This is because some cash flow
forecasts (e.g. sales revenue) are often made in real
terms, whereas others (e.g. depreciation tax shields) are
calculated in nominal terms.
AFM 271 - NPV and Capital Budgeting
Slide 7
Cont’d
for a single cash flow received at period n:
Cnominal = Creal × (1 + π )n
for a series of cash flows, the growth rate must also be
changed (e.g. a perpetuity which is constant in nominal
terms is actually decreasing in real terms due to inflation).
The relationship between real and nominal growth rates is:
(1 + gnominal ) = (1 + greal ) × (1 + π )
e.g. a perpetuity of $1,000 per year (nominal), with r = 10%
and π = 3%:
AFM 271 - NPV and Capital Budgeting
Slide 8
Cont’d
some examples (assume i = 10% and π = 5%):
1. perpetuity Cnominal = $100, 1st payment at t = 1, gnominal = 2%
2. perpetuity Cnominal = $100, 1st payment at t = 4, greal = 2%
3. 10 period annuity Creal = $500, 1st payment at t = 1, greal = 4%
AFM 271 - NPV and Capital Budgeting
Slide 9
The CCA System
depreciation or capital cost allowance (CCA) is not a cash flow,
but it has cash flow consequences because it is deductible
from taxable income
since CCA reduces taxable income, it increases cash flow
assets such as land or securities cannot be depreciated.
Others are assigned to various classes, with varying
depreciation rates. Table 7A.1 (text p. 219) provides some
common classes: e.g. manufacturing and processing
equipment is in Class 8 with a 20% rate, brick buildings are
in Class 1 with a 4% rate. The depreciation rate d reflects
the economic life of the asset.
note that all assets of a firm within a particular class are treated
as if they are a single asset (a “pool”)
AFM 271 - NPV and Capital Budgeting
Slide 10
Cont’d
how to calculate CCA:
CCA is calculated on a declining balance
allows for faster depreciation than straight line
receive cash flows from CCA tax shield faster
increases NPV of investments
the half year rule: in the first year of its life, an asset
is depreciated at half the normal rate, i.e. d/2
example: a firm purchases some Class 8 (d = 20%)
equipment for $150,000. What CCA can be claimed
for the first four years?
AFM 271 - NPV and Capital Budgeting
Slide 11
Cont’d
how to calculate PV of CCA tax shield:
assume for now that you keep the asset forever, and ignore
the half year rule. We have:
Year
Beginning UCC
CCA
Ending UCC
1
C
Cd
C(1 − d)
2
C(1 − d)
Cd(1 − d)
C(1 − d)2
3
C(1 − d)2
Cd(1 − d)2
C(1 − d)3
...
...
...
...
n
C(1 − d)n−1
Cd(1 − d)n−1
C(1 − d)n
...
...
...
...
the tax shields are just CCA multiplied by the corporate tax
rate Tc , so (using k as the discount rate):
PV =
AFM 271 - NPV and Capital Budgeting
CdTc
CdTc CdTc (1 − d) CdTc (1 − d)2
+
+
+
·
·
·
=
1+k
(1 + k)2
(1 + k)3
k+d
Slide 12
Cont’d
now incorporate the half year rule. We have two
perpetuities, decreasing at rate d, one with first
payment after one year, the other after two years:
1
2 CdTc
1
2 CdTc
1
+
×
k+d
k+d
1+k
"
#
1
CdTc 1
=
+ 2
k+d 2 1+k
"
#
1
1
CdTc 2 (1 + k) + 2
=
k+d
1+k
"
#
k
1
+
CdTc
2
=
k+d 1+k
PV =
AFM 271 - NPV and Capital Budgeting
Slide 13
Cont’d
now assume that the asset is sold for an amount S at
the end of year n. If the firm has other assets in this
CCA class, this might reduce the PV of the CCA tax
shield as follows:
"
#
k
CdTc 1 + 2
1
min(C, S)dTc
PV =
×
−
k+d 1+k
k+d
(1 + k)n
note that the above formula is only an example: it
and the similar equation (7.6) from p. 203 of the text
are not always applicable!
AFM 271 - NPV and Capital Budgeting
Slide 14
Cont’d
in general, the following steps apply when a firm sells a CCA
eligible asset:
1. The UCC in the asset class is reduced by the lesser of the sale price or the initial
cost.
2. If step 1 leaves a negative balance, this amount is added to taxable income
(recaptured depreciation), and the UCC of the asset class is reset to zero.
3. If step 1 leaves a positive balance and there are no other assets in the asset
class, this amount is deducted from taxable income (terminal loss), and the UCC
of the asset class is reset to zero.
4. If step 1 leaves a positive balance and there are assets left in the class, the
balance becomes the new UCC for the class.
5. If the asset is sold for more than its initial cost, the difference is a capital gain
(50% inclusion rate).
6. Suppose there is a new acquisition in the same year as an asset is sold. Define
net acquisitions as acquisitions less disposals. If net acquisitions are > 0, apply
the half year rule to net acquisitions; if net acquisitions are < 0, do not apply the
half year rule.
AFM 271 - NPV and Capital Budgeting
Slide 15
Cont’d
example: Atlantic Trucking Co. is starting up business and has
just purchased its first truck for $25,000. The truck is in Class
10 with a CCA rate of 30%. Calculate applicable CCA for years
1, 2, and 3:
Year
Beginning UCC
CCA
Ending UCC
1
2
3
now suppose that the firm buys a second truck for $35,000 in
year 2 and that it sells the first truck for $7,000 in year 3:
AFM 271 - NPV and Capital Budgeting
Slide 16
Cont’d
Year
Beginning UCC
CCA
Ending UCC
1
12,500
3,750
8,750
2
3
instead suppose that the firm buys a second truck for $35,000
in year 2 and that it sells the first truck for $7,000 in year 2:
Year
Beginning UCC
CCA
Ending UCC
1
12,500
3,750
8,750
2
3
AFM 271 - NPV and Capital Budgeting
Slide 17
Cont’d
to illustrate some asset disposition cases, consider the following four scenarios. Here
a firm purchased a number of assets in a single class some time ago for $117,000,
and at the beginning of the current year, this pool of assets had a UCC of $82,500.
Scenario
1
2
3
4
$82,500
$82,500
$82,500
$82,500
assets sold
$35,000
$117,000
$85,000
$117,000
Sale proceeds
$12,000
$100,000
$90,000
$50,000
Beginning UCC
Capital cost of
Capital gains
UCC after sale
Terminal loss
Recapture
Ending UCC
AFM 271 - NPV and Capital Budgeting
Slide 18
Total Project Cash Flow vs. Tax Shield Approach
to illustrate, we will consider in detail the MMCC
example from the text (pp. 188-193, 200-204). The
information provided includes:
Expected life of machine
Costs of test marketing
Current market value of factory site
Cost of machine
Salvage value after 8 years
Production in thousands of units (by year)
Unit price in first year
Growth rate in unit price after first year
Unit production costs in first year
Growth rate in production costs after first year
Corporate tax rate
Working capital: initial
Working capital: end
Working capital: during
Inflation:
Fixed costs:
8 years
$250,000
$0
$800,000
$150,000
6, 9, 12, 13, 12, 10, 8, 6
$100
2%
$64
5%
40%
$40,000
$0
15% of sales
5%
$50,000 per year
AFM 271 - NPV and Capital Budgeting
Slide 19
Cont’d
the total project cash flow approach:
the basic idea is to determine the project cash flows
year by year, add them up, and discount to obtain
NPV
some notes:
in general, cannot use convenient PV formulas
(annuities, perpetuities)
requires that all cash flows (for a given year) must
be either in real terms or in nominal terms
requires that all cash flows be discounted at the
same rate
must use this approach if we want to calculate
IRR, payback, or AAR
see spreadsheet handouts for MMCC example
AFM 271 - NPV and Capital Budgeting
Slide 20
Cont’d
the tax shield approach:
a “divide and conquer” method
recall that after tax operating cash flow = revenues expenses - taxes
since taxable income = revenues - expenses - CCA, we
have
taxes = Tc × ( revenues − expenses − CCA )
this implies
after tax operating cash flow =
revenues × (1 − Tc ) − expenses × (1 − Tc ) + Tc × CCA
see spreadsheet handout for MMCC example
AFM 271 - NPV and Capital Budgeting
Slide 21
Detailed CCA Calculations and Examples
assume that salvage takes place at the end of year n,
and consider the following formula for the PV of CCA
tax shields:
"
#
k
CdTc 1 + 2
∆UCCdTc
PV of CCA =
−
× (1 + k)−n
k+d 1+k
k+d
∆UCC depends on circumstances. Let the
undepreciated capital cost of the asset class just before
the asset is disposed of be UCCn .
if the firm only ever has one asset in the class, then
UCCn = C(1 − d/2)(1 − d)n−1
AFM 271 - NPV and Capital Budgeting
Slide 22
Cont’d
Calculate the value X = UCCn − min(C, S). Then:
1. If X < 0 ⇒ recaptured depreciation
add X to taxable income in year n:
PV of recapture =
−XTc
(1 + k)n
set UCC of asset class to zero (so
∆UCC = UCCn ):
"
#
k
CdTc 1 + 2
UCCn dTc
PV of CCA =
−
× (1 + k)−n
k+d 1+k
k+d
AFM 271 - NPV and Capital Budgeting
Slide 23
Cont’d
2. If X > 0 and there are no other assets in the class ⇒
terminal loss
subtract X from taxable income in year n:
PV of terminal loss =
XTc
(1 + k)n
set UCC of asset class to zero (so
∆UCC = UCCn ):
"
#
k
UCCn dTc
CdTc 1 + 2
−
× (1 + k)−n
PV of CCA =
k+d 1+k
k+d
AFM 271 - NPV and Capital Budgeting
Slide 24
Cont’d
3. If X > 0 and there are other assets in the class
X becomes the new UCC of the asset class (so
∆UCC = min(C, S)):
"
#
CdTc 1 + 2k
min(C, S)dTc
PV of CCA =
−
× (1 + k)−n
k+d 1+k
k+d
three further points:
1. If S > C, there is a capital gain:
PV of tax liability = −(S −C)Tc /2 × (1 + k)−n
2. Formulas for CCA tax shields are always given in
nominal terms.
AFM 271 - NPV and Capital Budgeting
Slide 25
Cont’d
3. The formulas given on the preceding slides have a half year rule adjustment
applied to the first term but not the second term. This may not always be the
case: the half year rule may instead be applied to both terms, neither term, or the
second term but not the first, depending on circumstances. In particular, the net
acquisitions rule works as follows. Recall that all assets of a given firm within a
single CCA class are treated as part of a common pool. In practice, firms often
buy and sell many assets in a single class within a year. Define net acquisitions
for an asset class as the total capital cost of all acquisitions (in a year) less the
total adjusted cost of all disposals within that class and in that year. If net
acquisitions is positive, apply the half year rule. If net acquisitions is negative,
there is no adjustment for the half year rule.
AFM 271 - NPV and Capital Budgeting
Slide 26
Cont’d
illustrative problem: you are considering whether to undertake a project that will
generate revenues of $50,000 per year for 8 years and expenses of $20,000 per year
for 8 years. The project requires an investment of $150,000 today in class 8 machinery
(d = 25%). Assume k = 12%, Tc = 40%, and all cash flows are nominal, and calculate
project NPV under the following scenarios:
1. You always have many other class 8 assets and a positive UCC in that class and
you can salvage the machinery at the end of the 9th year for (i) $10,000; and (ii)
$200,000. (Assume there are no other acquisitions or disposals of class 8 assets
in either the current year or in the 9th year.)
2. The machinery will always be in its own class and it can be salvaged in 9 years for
(i) $10,000; (ii) $20,000; and (iii) $200,000.
3. From this point on, the machinery will be in its own class and it can be salvaged in
9 years for $10,000. However, you purchased one other class 8 asset 5 years ago
for $200,000 which you have just sold for (i) $100,000; and (ii) $175,000.
AFM 271 - NPV and Capital Budgeting
Slide 27
Cont’d
1. (i)
Cost of machine:
-$150,000.00
PV after-tax operating revenues:
$149,029.19
PV after-tax operating expenses:
-$59,611.68
PV salvage:
PV perpetual tax shield on $150,000:
PV lost tax shield:
NPV:
AFM 271 - NPV and Capital Budgeting
$3,606.10
$38,368.73
-$974.62
-$19,582.28
Slide 28
Cont’d
1. (ii)
Cost of machine:
-$150,000.00
PV after-tax operating revenues:
$149,029.19
PV after-tax operating expenses:
-$59,611.68
PV salvage:
$72,122.00
PV capital gain tax:
-$3,606.10
PV perpetual tax shield on $150,000:
$38,368.73
PV lost tax shield:
-$14,619.33
NPV:
$31,682.81
AFM 271 - NPV and Capital Budgeting
Slide 29
Cont’d
2. (i)
Cost of machine:
-$150,000.00
PV after-tax operating revenues:
$149,029.19
PV after-tax operating expenses:
-$59,611.68
PV salvage:
$3,606.10
PV perpetual tax shield on $150,000:
$38,368.73
PV lost tax shield:
-$1,280.64
PV terminal loss:
NPV:
AFM 271 - NPV and Capital Budgeting
$452.90
-$19,435.40
Slide 30
Cont’d
2. (ii)
Cost of machine:
-$150,000.00
PV after-tax operating revenues:
$149,029.19
PV after-tax operating expenses:
-$59,611.68
PV salvage:
$7,212.20
PV perpetual tax shield on $150,000:
$38,368.73
PV lost tax shield:
-$1,280.64
PV recapture:
-$989.54
NPV:
-$17,271.74
AFM 271 - NPV and Capital Budgeting
Slide 31
Cont’d
2. (iii)
Cost of machine:
-$150,000.00
PV after-tax operating revenues:
$149,029.19
PV after-tax operating expenses:
-$59,611.68
PV salvage:
$72,122.10
PV capital gain tax:
-$3,606.10
PV perpetual tax shield on $150,000:
$38,368.73
PV lost tax shield:
-$1,280.64
PV recapture:
NPV:
AFM 271 - NPV and Capital Budgeting
-$19,741.26
$25,280.24
Slide 32
Cont’d
3. (i)
Cost of machine:
-$150,000.00
PV after-tax operating revenues:
$149,029.19
PV after-tax operating expenses:
-$59,611.68
PV salvage:
$3,606.10
PV perpetual tax shield on $50,000:
$12,789.58
PV continuing tax shield on $55,371.09
$14,965.16
PV lost tax shield:
-$832.08
PV recapture (year 9):
-$210.96
PV avoided recapture today:
NPV:
$17,851.56
-$12,413.13
AFM 271 - NPV and Capital Budgeting
Slide 33
Cont’d
3. (ii)
Cost of machine:
-$150,000.00
PV after-tax operating revenues:
$149,029.19
PV after-tax operating expenses:
-$59,611.68
PV salvage:
$3,606.10
PV perpetual tax shield on $30,371.09:
$8,208.40
PV lost tax shield:
-$222.25
PV recapture (year 9):
-$1,113.51
PV avoided recapture today:
$47,851.56
NPV:
-$2,252.19
AFM 271 - NPV and Capital Budgeting
Slide 34
Project Interactions
many projects have effects on others, e.g. it is important to
incorporate effects on other aspects of a firm’s business in
capital budgeting analysis (including opportunity costs), and
CCA pooling of assets can lead to other interactions between
projects
another aspect of this is choosing between investments of
unequal lives. Suppose that two machines produce the same
output but have the following after-tax operating costs per year:
Machine
C0
C1
C2
C3
A
-1,000
-500
-500
0
B
-2,000
-100
-100
-100
At a 10% discount rate, which is cheaper to operate?
AFM 271 - NPV and Capital Budgeting
Slide 35
Cont’d
the replacement chain:
why not just use NPV and choose the machine with
lower discounted costs?
assume that machines are needed forever
the matching cycle approach:
run the example for 6 years. A has 3 complete
cycles, B has 2:
PV of costs over 6 years:
AFM 271 - NPV and Capital Budgeting
Slide 36
Cont’d
the equivalent annual cost (EAC) approach:
idea is to convert PV of costs for machine into an
appropriate annuity:
firm is indifferent between PV of costs and EAC
since the project is assumed to continue forever,
the EAC lasts forever, so choose the machine with
lower EAC
AFM 271 - NPV and Capital Budgeting
Slide 37
Cont’d
when to replace an old machine?
1. Calculate EAC for new machine (EACnew )
2. Calculate cost of operating old machine for 1 more
year (Cold )
3. Replace just before Cold > EACnew
example: a firm has an existing machine, which could be salvaged for $2,800 today,
$2,100 after one year, $1,200 after two years, or zero after 3 years (at which point it
would have to be replaced). Maintenance costs on this old machine are $1,175 after
one year, $1,600 after two years, and $1,800 after three years. A new machine is
available which costs $5,000 and can be salvaged after four years for $1,800. Its
maintenance costs are $1,000 after one year, $1,250 after two years, $1,500 after
three years, and $2,000 after four years. The new machine produces the same output
as the old machine. When should the firm replace the old machine? Assume a
discount rate of 10%.
AFM 271 - NPV and Capital Budgeting
Slide 38
Cont’d
the EAC for the new machine is:
should the existing machine be replaced now?
AFM 271 - NPV and Capital Budgeting
Slide 39
Cont’d
what about replacing it after one year?
note that these types of decisions can easily get far
more complicated (e.g. different output levels for the
machines, different numbers of machines required,
anticipated new technology, etc.)
AFM 271 - NPV and Capital Budgeting
Slide 40