Borrowing, Depreciation, Taxes in
Cash Flow Problems
H. Scott Matthews
12-706 / 19-702 /73-359
Lecture 5
Admin Issues
Who bought Boardman book? Can return.
Campbell Textbook due today. First 20
names will be sent
TA Office Hours in CEE Lounge (PH 118
hallway near my office)
Joe and Paulina will swap coverage next
week (i.e, go to office hours in lounge, but
look for Joe instead)
Notes on Tax deductibility
Reason we care about financing and
depreciation: they affect taxes owed
For personal income taxes, we deduct items like
IRA contributions, mortgage interest, etc.
Private entities (eg businesses) have similar
rules: pay tax on net income
Income = Revenues - Expenses
There are several types of expenses that we
care about
Interest expense of borrowing
Depreciation (can only do if own the asset)
These are also called ‘tax shields’
Goal: Find Cash Flows after taxes
 Master equation conceptually:
 CFAT = -equity financed investment + gross income operating expenses + salvage value - taxes + (debt
financing receipts - disbursements) + equity financing
receipts
 Where “taxes” = Tax Rate * Taxable Income
 Taxable Income = Gross Income - Operating Expenses Depreciation - Loan Interest - Bond Dividends
After-tax cash flows
Dt= Depreciation allowance in t
It= Interest accrued in t
+ on unpaid balance, - overpayment
Qt= available for reducing balance in t
Wt= taxable income in t; Xt= tax rate
Tt= income tax in t
Yt= net after-tax cash flow
Equations
Dt= Depreciation allowance in t
It= Interest accrued in t
Qt= available for reducing balance in t
So At = Qt - It
Wt= At-Dt -It (Operating - expenses)
Tt= Xt Wt
Yt= A*t - Xt Wt (pre tax flow - tax) OR
Yt= At + At - Xt (At-Dt -It)
Simple example
Firm: $500k revenues, $300k expense
Depreciation on equipment $20k
No financing, and tax rate = 50%
Yt= At + At - Xt (At-Dt -It)
Yt=($500k-$300k)+0-0.5 ($200k-$20k)
Yt= $110k
Notes
 Mixed funds problem - buy computer
 Below: Operating cash flows At
 Four financing options in At
t
0
1
2
3
4
5
At
(Operation)
-22,000
6,0 00
6,0 00
6,0 00
6,0 00
6,0 00
2,0 00
10,000
-14,693
At
(Fin ancing)
10,000
10,000
-2,5 05
-800
-2,5 05
-800
-2,5 05
-800
-2,5 05
-800
-2,5 05
-10,800
10,000
-2,8 00
-2,6 40
-2,4 80
-2,3 20
-2,1 60
Further Analysis (still no tax)
t
At
8% (Opera tion )
0
-22 ,000
1
6,000
2
6,000
3
6,000
4
6,000
5
6,000
2,000
NPV 33 17.4 27
10 ,000
-14 ,693
0.1911
At
(Fi nancing at 8%)
10 ,000
10 ,000
-2,505
-80 0
-2,505
-80 0
-2,505
-80 0
-2,505
-80 0
-2,505 -10 ,800
-1.7386
0
10 ,000
-2,800
-2,640
-2,480
-2,320
-2,160
1E-1 2
-12 ,000
6,000
6,000
6,000
6,000
-8,693
2,000
33 17.6 2
A*
(To tal pre-ta x)
-12 ,000
-12 ,000
3,495
5,200
3,495
5,200
3,495
5,200
3,495
5,200
3,495
-4,800
2,000
2,000
33 15.6 9
33 17.4
 MARR (disc rate) equals borrowing rate, so financing
plans equivalent.
 When wholly funded by borrowing, can set MARR to
interest rate
-12 ,000
3,200
3,360
3,520
3,680
3,840
2,000
33 17.4 3
Effect of other MARRs (e.g. 10%)
t
At
10 % (Opera tion )
0
-22 ,000
1
6,000
2
6,000
3
6,000
4
6,000
5
6,000
2,000
NPV 19 86.5 63
10 ,000
-14 ,693
87 6.8
At
(Fi nancing at 8%)
10 ,000
10 ,000
-2,505
-80 0
-2,505
-80 0
-2,505
-80 0
-2,505
-80 0
-2,505 -10 ,800
50 4.08
75 8.16
10 ,000
-2,800
-2,640
-2,480
-2,320
-2,160
48 3.69
-12 ,000
6,000
6,000
6,000
6,000
-8,693
2,000
28 63.3 7
A*
(To tal pre-ta x)
-12 ,000
-12 ,000
3,495
5,200
3,495
5,200
3,495
5,200
3,495
5,200
3,495
-4,800
2,000
2,000
24 90.6 4
27 44.7
 ‘Total’ NPV higher than operation alone for all options
All preferable to ‘internal funding’ (equity financing)
Why? Internal funds could earn 10%, we’re only paying 8%
First option ‘gets most of loan’, is best
-12 ,000
3,200
3,360
3,520
3,680
3,840
2,000
24 70.2 5
Effect of other MARRs (e.g. 6%)
t
At
6% (Opera tion )
0
-22 ,000
1
6,000
2
6,000
3
6,000
4
6,000
5
6,000
2,000
NPV 47 68.6 99
-14 ,693
At
(Fi nancing at 8%)
10 ,000
10 ,000
-2,505
-80 0
-2,505
-80 0
-2,505
-80 0
-2,505
-80 0
-2,505 -10 ,800
10 ,000
-2,800
-2,640
-2,480
-2,320
-2,160
-97 9.46
-55 1.97
-52 5.1
10 ,000
-84 2.5
-12 ,000
6,000
6,000
6,000
6,000
-8,693
2,000
37 89.2 3
A*
(To tal pre-ta x)
-12 ,000
-12 ,000
3,495
5,200
3,495
5,200
3,495
5,200
3,495
5,200
3,495
-4,800
2,000
2,000
42 16.7 3
39 26.2
Now reverse is true
Why? Internal funds only earn 6% !
First option now worst
-12 ,000
3,200
3,360
3,520
3,680
3,840
2,000
42 43.6 1
First Complex Example
Firm will buy $46k equipment
Yr 1: Expects pre-tax benefit of $15k
Yrs 2-6: $2k less per year ($13k..$5k)
Salvage value $4k at end of 6 years
No borrowing, tax=50%, MARR=6%
Use SOYD and SL depreciation
Results - SOYD
t
At
6% (Pre-ta x)
0
-46,000
1
15,000
2
13,000
3
11,000
4
9,0 00
5
7,0 00
6
5,0 00
4,0 00
NPV 766 1.00 4
SOYD
Dt
12,000
10,000
8,0 00
6,0 00
4,0 00
2,0 00
Tax Inco me
Wt
3,0 00
3,0 00
3,0 00
3,0 00
3,0 00
3,0 00
Inc Ta x
Tt
1,5 00
1,5 00
1,5 00
1,5 00
1,5 00
1,5 00
Aft-Tax
Yt
-46,000
13,500
11,500
9,5 00
7,5 00
5,5 00
3,5 00
4,0 00
285 .02
D1=(6/21)*$42k = $12,000
SOYD really reduces taxable income!
Results - Straight Line Dep.
t
At
6% (Pre-ta x)
0
-46,000
1
15,000
2
13,000
3
11,000
4
9,0 00
5
7,0 00
6
5,0 00
4,0 00
NPV 766 1.00 4
SL
Dt
7,0 00
7,0 00
7,0 00
7,0 00
7,0 00
7,0 00
Tax Inco me
Wt
8,0 00
6,0 00
4,0 00
2,0 00
0
-2,0 00
Inc Ta x
Tt
Aft-Tax
Yt
-46,000
4,0 00
11,000
3,0 00
10,000
2,0 00
9,0 00
1,0 00
8,0 00
0
7,0 00
-1,0 00
6,0 00
4,0 00
-548 .9
 NPV negative - shows effect of depreciation (why lower?)
Negative tax? Typically treat as credit not cash back
Projects are usually small compared to overall size of company this project would “create tax benefits”
Let’s Add in Interest - Computer
Again
Price $22k, $6k/yr benefits for 5 yrs, $2k
salvage after year 5
Borrow $10k of the $22k price
Consider single payment at end and uniform
yearly repayments
Depreciation: Double-declining balance
Income tax rate=50%
MARR 8%
Single Repayment
t
At
At
8% (Operation) (Loa n 8% )
0
-22,000
10,000
1
6,0 00
2
6,0 00
3
6,0 00
4
6,0 00
5
6,0 00
-14,693
2,0 00
NPV 331 7.42 7
0.1 9109
Bt
22,000
13,200
7,9 20
4,7 52
2,8 51
2,0 00
Dt
8,8 00
5,2 80
3,1 68
1,9 01
851
Rt
100 00
108 00
116 64
125 97
136 05
146 93
It
800
864
933
1,0 08
1,0 88
Wt
-3,6 00
-144
1,8 99
3,0 91
4,0 61
Tt
-180 0
-72
949 .44
154 5.7
203 0.3
Yt
-12,000
7,8 00
6,0 72
5,0 51
4,4 54
-10,723
2,0 00
177 4.38
Had to ‘manually adjust’ Dt in yr. 5
Note loan balance keeps increasing
Only additional interest noted in It as interest
expense
Uniform payments
t
At
At
8% (Operation) (Loa n 8% )
0
-22,000
10,000
1
6,0 00
-2,5 05
2
6,0 00
-2,5 05
3
6,0 00
-2,5 05
4
6,0 00
-2,5 05
5
6,0 00
-2,5 05
2,0 00
NPV 331 7.42 7
-1.7 386
Bt
22,000
13,200
7,9 20
4,7 52
2,8 51
2,0 00
Dt
8,8 00
5,2 80
3,1 68
1,9 01
851
Rt
100 00
829 5
645 3.6
446 4.9
231 7.1
-2.5 55
It
Wt
800
664
516
357
185
-3,6 00
56
2,3 16
3,7 42
4,9 64
Tt
-180 0
28.2
115 7.9
187 1
248 1.8
Note loan balance keeps decreasing
NPV of this option is lower - should
choose previous (single repayment at
end).. not a general result
Yt
-12,000
5,2 95
3,4 67
2,3 37
1,6 24
1,0 13
2,0 00
974 .707
Leasing
‘Make payments to owner’ instead of
actually purchasing the asset
Since you do not own it, you can not take
depreciation expense
Lease payments are just a standard expense
(i.e., part of the Ct stream)
At= Bt - Ct ; Yt= At - At Xt
Tradeoff is lower expenses vs. loss of
depreciation/interest tax benefits
Social Discount Rate
Rate used to make investment decisions for
society
Discounting rooted in consumer preference
We tend to prefer current, rather than future,
consumption
Marginal rate of time preference (MRTP)
Face opportunity cost (of foregone interest)
when we spend not save
Marginal rate of investment return
Intergenerational effects
We have tended to discuss only short
term investment analyses (e.g. 5 yrs)
What about effects in distant future?
Called intergenerational effects
Economists agree that discounting should
be done for public projects
Do not agree on positive discount rate
Discounting handout
How much do/should we care about
people born after we die?
Higher the discount rate, the less future
values will count compared to today
Ethically, no one’s interests should count
more than another’s
Implies there is no justification for
discounting across long time periods
Called ‘equal standing’
Climate Change
Discussions ongoing about how best to
manage global CO2 emissions to limit
effects of global change
Should we sacrifice short-run economic
growth to do something to improve
environment and leave resources for the
future?
Really asking 2 separate questions!
Two Questions
What duty do we have to make sacrifices
for future generations?
If we sacrifice, what is the optimal policy
to maximize benefit?
So we should compare global change
proposals with alternatives
Perhaps higher R&D spending on science or
medicing would have higher benefits!
Hume’s Law
Thus discounting issues are normative vs.
positive battles
Hume noted that facts alone cannot tell us what
we should do
Any recommendation embodies ethics and judgment
E.g. focusing on ‘highest NPV’ implies net benefits is
only goal for society
Some evidence
Cropper et al surveyed 3000 homes
Asked about saving lives in the future
Found a 4% discount rate for lives 100 years
per now
Equal standing does not imply different
generations have equal claims to present
resources!
Harsanyi says only do so if their marginal
gain is higher than our loss
More evidence
If future generations will be better off than us
anyway
Then we might have no reason to make additional
sacrifices
There might be ‘special standing’ in addition to
‘equal standing’
Immediate relatives vs. distant relatives
Different discount rates over time
Why do we care so much about future and ignore some
present needs (poverty)
Based on these arguments, what discount rates
should we use for policy problems (eg climate)
Government Discount Rates
 US Government Office of Management and Budget (OMB)
Circular A-94
 http://www.whitehouse.gov/omb/circulars/a094/a094.html
Discusses how to do BCA and related performance studies
Match real values with real discount rates, etc
How to do sensitivity analysis / which inputs to vary
What discount, inflation, etc. rates to use
Basically says “use this rate, but do sensitivity analysis with nearby
rates”
OMB Circular A-94, Appendix C
 Provides the current suggested values to use for federal
government analyses
 http://www.whitehouse.gov/omb/circulars/a094/a94_appx-c.html
Revised yearly, usually “good until January of the next year”
How would the government decide its discount rates?
What is the government’s MARR?
Historic Nominal Interest Rates
(from OMB A-94)
Real Discount Rates (from A-94)
Effect of these Discount Rates
These are ‘effectively zero’
What does this mean for projects and
project selection decisions?
What does it say about intergenerational
effects?
What are implications of zero or negative
discount rates?
Next Up: Sensitivity Analysis
Skim Clemen Chapter 5
Refers to decision/trees, etc that we
have not done yet.