Borrowing, Depreciation, Taxes in
Cash Flow Problems
H. Scott Matthews
12-706 / 19-702
Theme: Cash Flows
Streams of benefits (revenues) and costs over
time => “cash flows”
We need to know what to do with them in terms
of finding NPV of projects
Different perspectives: private and public
We will start with private since its easier
Why “private..both because they are usually of
companies, and they tend not to make studies public
Cash flows come from: operation, financing,
taxes
Without taxes, cash flows simple
A = B - C
Cash flow = benefits - costs
Or.. Revenues - expenses
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: Cash Flows after taxes
(CFAT)
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
Most scenarios (and all problems we will look at) only deal with
one or two of these issues at a time
Investment types
Debt financing: using a bank or investor’s
money (loan or bond)
DFD:disbursement (payments)
DFR:receipts (income)
DFI: portion tax deductible (only non-principal)
Equity financing: using own money (no
borrowing)
Why Finance?
Time shift revenues and expenses construction expenses paid up front,
nuclear power plant decommissioning at
end.
“Finance” is also used to refer to plans to
obtain sufficient revenue for a project.
Borrowing
Numerous arrangements possible:
bonds and notes (pay dividends)
bank loans and line of credit (pay interest)
municipal bonds (with tax exempt interest)
Lenders require a real return - borrowing
interest rate exceeds inflation rate.
Issues
Security of loan - piece of equipment,
construction, company, government. More
security implies lower interest rate.
Project, program or organization funding
possible. (Note: role of “junk bonds” and rating
agencies.
Variable versus fixed interest rates: uncertainty
in inflation rates encourages variable rates.
Issues (cont.)
Flexibility of loan - can loan be repaid
early (makes re-finance attractive when
interest rates drop). Issue of
contingencies.
Up-front expenses: lawyer fees, taxes,
marketing bonds, etc.- 10% common
Term of loan
Source of funds
Sinking Funds
Act as reverse borrowing - save revenues
to cover end-of-life costs to restore mined
lands or decommission nuclear plants.
Low risk investments are used, so return
rate is lower.
Recall: Annuities (a.k.a uniform
values)
 Consider the PV of getting the same amount ($1) for many years
 Lottery pays $A / yr for n yrs at i=5%
A
A
A
A
PV  1i
 (1i)


..
2
(1i)3
(1i)n
A
A
A
PV * (1 i)  A  (1i)
 (1i)

..
2
(1i)n1
 ----- Subtract above 2 equations.. ------A
PV * (1 i)  PV  A  (1i)
n
n
1
PV *(i)  A(1 (1i)
)

A(1
(1
i)
)
n
PV 
A (1(1i) n )
i
 When A=1 the right hand side is called the “annuity factor”
Uniform Values - Application
Note Annual (A) values also sometimes
referred to as Uniform (U) ..
$1000 / year for 5 years example
P = U*(P|U,i,n) = (P|U,5%,5) = 4.329
P = 1000*4.329 = $4,329
Relevance for loans?
Borrowing
Sometimes we don’t have the money to
undertake - need to get loan
i=specified interest rate
At=cash flow at end of period t (+ for loan
receipt, - for payments)
Rt=loan balance at end of period t
It=interest accrued during t for Rt-1
Qt=amount added to unpaid balance
At t=n, loan balance must be zero
Equations
i=specified interest rate
At=cash flow at end of period t (+ for
loan receipt, - for payments)
It=i * Rt-1
Qt= At + It
Rt= Rt-1 + Qt <=> Rt= Rt-1 + At + It
 Rt= Rt-1 + At + (i * Rt-1)
Annual, or Uniform, payments
Assume a payment of U each year for n
years on a principal of P
Rn=-U[1+(1+i)+…+(1+i)n-1]+P(1+i)n
Rn=-U[((1+i)n-1)/i] + P(1+i)n
Uniform payment functions in Excel
Same basic idea as earlier slide
Example
Borrow $200 at 10%, pay $115.24 at end
of each of first 2 years
R0=A0=$200
A1= -$115.24, I1=R0*i = (200)*(.10)=20
Q1=A1 + I1 = -95.24
R1=R0+Qt = 104.76
I2=10.48; Q2=-104.76; R2=0
Various Repayment Options
Single Loan, Single payment at end of
loan
Single Loan, Yearly Payments
Multiple Loans, One repayment
Notes
 Mixed funds problem - buy computer
 Below: Operating cash flows At
 Four financing options (at 8%) in At section below
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’
Why? These funds could earn 10% !
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
Bonds
Done similar to loans, but mechanically
different
Usually pay annual interest only, then
repay interest and entire principal in yr. n
Similar to financing option #3 in previous
slides
There are other, less common bond methods
Tax Effects of Financing
 Companies deduct interest expense
 Bt=total pre-tax operating benefits
Excluding loan receipts
 Ct=total operating pre-tax expenses
Excluding loan payments
 At= Bt- Ct = net pre-tax operating cash flow
 A,B,C: financing cash flows
 A*,B*,C*: pre-tax totals / all sources
Depreciation
Decline in value of assets over time
Buildings, equipment, etc.
Accounting entry - no actual cash flow
Systematic cost allocation over time
Main emphasis is to reduce our tax burden
Government sets dep. Allowance
P=asset cost, S=salvage,N=est. life
Dt= Depreciation amount in year t
Tt= accumulated (sum of) dep. up to t
Bt= Book Value = Undep. amount = P - Tt
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
Depreciation Example
 Simple/straight line dep: Dt= (P-S)/N
Equal expense for every year
$16k compressor, $2k salvage at 7 yrs.
Dt= (P-S)/N = $14k/7 = $2k
Bt= 16,000-2t, e.g. B1=$14k, B7=$2k
 Salvage Value is an investing activity that is considered
outside the context of our income tax calculation
If we sell asset for salvage value, no further tax implications (IN
THIS COURSE WE ASSUME THIS TO SIMPLIFY)
If we sell asset for higher than salvage value, we pay taxes
since we received depreciation expense benefits (but we will
generally ignore this since its not the focus of the course)
We show salvage value on separate lines to emphasize this.
Accelerated Dep’n Methods
Depreciation greater in early years
Sum of Years Digits (SOYD)
Let Z=1+2+…+N = N(N+1)/2
Dt= (P-S)*[N-(t-1)]/Z, e.g. D1=(N/Z)*(P-S)
D1=(7/28)*$14k=$3,500, D7=(1/28)*$14k
Declining balance: Dt= Bt-1*r (where r is rate)
Bt=P*(1-r)t, Dt= P*r*(1-r)t-1
Requires us to keep an eye on B
Typically r=2/N - aka double dec. balance
Ex: Double Declining Balance
Could solve P(1-r)N = S (find nth root)
t
0
1
2
3
4
5
6
7
Dt
(2/7)*$16k=$4,571.43
(2/7)*$11,428=$3265.31
$2332.36
$1,665.97
$1,189.98
$849.99
$607.13**
Bt
$16,000
$11,428.57
$8,163.26
$5,830.90
$4,164.93
$2,974.95
$2,124.96
$1,517.83**
Notes on Example
Last year would need to be adjusted to
consider salvage, D7=$124.96
We get high allowable depreciation
‘expenses’ early - tax benefit
We will assume taxes are simple and
based on cash flows (profits)
Realistically, they are more complex
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
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