Borrowing, Depreciation, Taxes in Cash Flow Problems Scott Matthews 12-706 / 19-702.
Project Finance (part 2) H. Scott Matthews 12-706/73-359 Lecture 12 - Oct. 8, 2003.
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Transcript of Project Finance (part 2) H. Scott Matthews 12-706/73-359 Lecture 12 - Oct. 8, 2003.
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Project Finance (part 2)
H. Scott Matthews12-706/73-359Lecture 12 - Oct. 8, 2003
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Admin Issues
Pipeline case out - read for Monday Brief discussion on preparing cases
HW 2 back todayProject groups/ideas due today
Short (1/2 page) description of project, client
Midterm
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Notes on Notation
PV = $FV / (1+i)n = $FV * [1 / (1+i)n ] But [1 / (1+i)n ] is only function of i,n $1, i=5%, n=5, [1/(1.05)5 ]= 0.784 = (P|F,i,n) Would see tables like this in ‘old’ textbooks
As shorthand: Future value of Present: (P|F,i,n)
So PV of $500, 5%,5 yrs = $500*0.784 = $392
Present value of Future: (F|P,i,n) We’ll see similar notations for other types
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Timing of Future Values
Noted last time that we assume ‘end of period’ values
What is relative difference?Consider comparative case:
$1000/yr (uniform) Benefit for 5 years @ 5%
Assume case 1: received beginning Assume case 2: received end
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Timing of BenefitsDraw 2 cash flow diagramsNPV1 = 1000 + 1000/1.05 + 1000/1.052 +
1000/1.053 + 1000/1.054
NPV1 =1000 + 952 + 907 + 864 + 823 = $4,545
NPV2 = 1000/1.05 + 1000/1.052 + 1000/1.053 + 1000/1.054 + 1000/1.055
NPV2 = 952 + 907 + 864 + 823 + 784 = $4,329
NPV1 - NPV2 ~ $216Notation: (P|U,i,n)
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Relative NPV AnalysisIf comparing, can just find ‘relative’ NPV
compared to a single option E.g. beginning/end timing problem Net difference was $216
Alternatively consider ‘net amounts’ NPV1 =1000 + 952 + 907 + 864 + 823 = $4,545 NPV2 = 952 + 907 + 864 + 823 + 784 = $4,329 ‘Cancel out’ intermediates, just find ends NPV1 is $216 greater than NPV2
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Uniform Values - Theory
Assume ‘end of period’ valuesA = U/(1+i) +U/(1+i)2 + ..+ U/(1+i)n
A = U*[(1+i)-1+(1+i)-2 + ..+ (1+i)-n]A(1+i)=U*[1+(1+i)-1+(1+i)-2 + ..+
(1+i)1-n]A(1+i) - A = U*[1 - (1+i)-n]A = U*[1 - (1+i)-n] / i = U*(P|U,i,n)
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Uniform Values - Application
Recall $1000 / year for 5 years example
Stream = U*(P|U,i,n) = U*[1 - (1+i)-n] / i(P|U,5%,5) = 4.329Stream = 1000*4.329 = $4,329 =
NPV2
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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.
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Borrowing
Numerous arrangements possible: bonds and notes bank loans and line of credit municipal bonds (with tax exempt
interest)Lenders require a real return -
borrowing interest rate exceeds inflation rate.
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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.
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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 loanSource of funds
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Borrowing
Sometimes we don’t have the money to undertake - need to get loan
i=specified interest rateAt=cash flow at end of period t (+ for loan
receipt, - for payments)Rt=loan balance at end of period tIt=interest accrued during t for Rt-1
Qt=amount added to unpaid balanceAt t=n, loan balance must be zero
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Equations
i=specified interest rateAt=cash flow at end of period t (+ for
loan receipt, - for payments)It=i * Rt-1
Qt= At + ItRt= Rt-1 + Qt <=> Rt= Rt-1 + At + It Rt= Rt-1 + At + (i * Rt-1)
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Option: 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 ExcelSame basic idea as earlier slide
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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
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Repayment Options
Single Loan, Single payment at end of loan
Single Loan, Yearly PaymentsMultiple Loans, One repayment
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Note on Taxes
Companies pay tax on net incomeIncome = Revenues - ExpensesThere are several types of expenses
that we care about Interest expense of borrowing Depreciation (can only do if you own
asset) These are also called ‘tax shields’
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Depreciation
Decline in value of assets over time Buildings, equipment, etc. Accounting entry - no actual cash flow Systematic cost allocation over time
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
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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
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Accelerated Dep’n Methods
Depreciation greater in early yearsSum 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-1r (r is rate) Bt=P(1-r)t, Dt= Pr(1-r)t-1
Requires us to keep an eye on B Typically r=2/N - aka double dec. balance
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Ex: Double Declining Balance
Could solve P(1-r)N = S (find nth root)
t Dt Bt0 - $16,0001 (2/7)*$16k=$4,571.43 $11,428.572 (2/7)*$11,428=$3265.31 $8,163.263 $2332.36 $5,830.904 $1,665.97 $4,164.935 $1,189.98 $2,974.956 $849.99 $2,124.967 $607.13** $1,517.83**
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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
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Tax Effects of Financing
Companies deduct interest expenseBt=total pre-tax operating benefits
Excluding loan receipts
Ct=total operating pre-tax expenses Excluding loan payments
At=net pre-tax operating cash flow A,B,C: financing cash flowsA*,B*,C*: pre-tax totals / all sources
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Notes
Mixed funds problem - buy computerBelow: Operating cash flows AtFour financing options in At
t At(Operation)
0 -22,000 10,000 10,000 10,000 10,0001 6,000 -2,505 -800 -2,8002 6,000 -2,505 -800 -2,6403 6,000 -2,505 -800 -2,4804 6,000 -2,505 -800 -2,3205 6,000 -14,693 -2,505 -10,800 -2,160
2,000
At(Financing)
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Further Analysis (still no tax)t At
8% (Operation)0 -22,000 10,000 10,000 10,000 10,000 -12,000 -12,000 -12,000 -12,0001 6,000 -2,505 -800 -2,800 6,000 3,495 5,200 3,2002 6,000 -2,505 -800 -2,640 6,000 3,495 5,200 3,3603 6,000 -2,505 -800 -2,480 6,000 3,495 5,200 3,5204 6,000 -2,505 -800 -2,320 6,000 3,495 5,200 3,6805 6,000 -14,693 -2,505 -10,800 -2,160 -8,693 3,495 -4,800 3,840
2,000 2,000 2,000 2,000 2,000NPV 3317.427 0.1911 -1.7386 0 1E-12 3317.62 3315.69 3317.4 3317.43
At(Financing at 8%)
A*(Total pre-tax)
MARR (disc rate) equals borrowing rate, so financing plans equivalent.
When wholly funded by borrowing, can set MARR to interest rate
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Effect of other MARRs (e.g. 10%)t At
10% (Operation)0 -22,000 10,000 10,000 10,000 10,000 -12,000 -12,000 -12,000 -12,0001 6,000 -2,505 -800 -2,800 6,000 3,495 5,200 3,2002 6,000 -2,505 -800 -2,640 6,000 3,495 5,200 3,3603 6,000 -2,505 -800 -2,480 6,000 3,495 5,200 3,5204 6,000 -2,505 -800 -2,320 6,000 3,495 5,200 3,6805 6,000 -14,693 -2,505 -10,800 -2,160 -8,693 3,495 -4,800 3,840
2,000 2,000 2,000 2,000 2,000NPV 1986.563 876.8 504.08 758.16 483.69 2863.37 2490.64 2744.7 2470.25
At A*(Financing at 8%) (Total pre-tax)
‘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
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Effect of other MARRs (e.g. 6%)t At
6% (Operation)0 -22,000 10,000 10,000 10,000 10,000 -12,000 -12,000 -12,000 -12,0001 6,000 -2,505 -800 -2,800 6,000 3,495 5,200 3,2002 6,000 -2,505 -800 -2,640 6,000 3,495 5,200 3,3603 6,000 -2,505 -800 -2,480 6,000 3,495 5,200 3,5204 6,000 -2,505 -800 -2,320 6,000 3,495 5,200 3,6805 6,000 -14,693 -2,505 -10,800 -2,160 -8,693 3,495 -4,800 3,840
2,000 2,000 2,000 2,000 2,000NPV 4768.699 -979.46 -551.97 -842.5 -525.1 3789.23 4216.73 3926.2 4243.61
At A*(Financing at 8%) (Total pre-tax)
Now reverse is true Why? Internal funds only earn 6% ! First option now worst
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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
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Equations
Dt= Depreciation allowance in tIt= 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) ORYt= At + At - Xt (At-Dt -It)
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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
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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
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Results - SOYD
D1=(6/21)*$42k = $12,000SOYD really reduces taxable income!
t At SOYD Tax Income Inc Tax Aft-Tax6% (Pre-tax) Dt Wt Tt Yt
0 -46,000 -46,0001 15,000 12,000 3,000 1,500 13,5002 13,000 10,000 3,000 1,500 11,5003 11,000 8,000 3,000 1,500 9,5004 9,000 6,000 3,000 1,500 7,5005 7,000 4,000 3,000 1,500 5,5006 5,000 2,000 3,000 1,500 3,500
4,000 4,000NPV 7661.004 285.02
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Results - Straight Line Dep.
Now NPV is negative - shows effect of depreciation method on decision Negative tax? Typically a credit not cash back
t At SL Tax Income Inc Tax Aft-Tax6% (Pre-tax) Dt Wt Tt Yt
0 -46,000 -46,0001 15,000 7,000 8,000 4,000 11,0002 13,000 7,000 6,000 3,000 10,0003 11,000 7,000 4,000 2,000 9,0004 9,000 7,000 2,000 1,000 8,0005 7,000 7,000 0 0 7,0006 5,000 7,000 -2,000 -1,000 6,000
4,000 4,000NPV 7661.004 -548.9
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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%
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t At At Bt Dt Rt It Wt Tt Yt8% (Operation) (Loan 8%)
0 -22,000 10,000 22,000 10000 -12,0001 6,000 13,200 8,800 10800 800 -3,600 -1800 7,8002 6,000 7,920 5,280 11664 864 -144 -72 6,0723 6,000 4,752 3,168 12597 933 1,899 949.44 5,0514 6,000 2,851 1,901 13605 1,008 3,091 1545.7 4,4545 6,000 -14,693 2,000 851 14693 1,088 4,061 2030.3 -10,723
2,000 2,000NPV 3317.427 0.19109 1774.38
Single Repayment
Had to ‘manually adjust’ Dt in yr. 5Note loan balance keeps increasing
Only additional interest noted in It as interest expense
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Uniform paymentst At At Bt Dt Rt It Wt Tt Yt
8% (Operation) (Loan 8%)0 -22,000 10,000 22,000 10000 -12,0001 6,000 -2,505 13,200 8,800 8295 800 -3,600 -1800 5,2952 6,000 -2,505 7,920 5,280 6453.6 664 56 28.2 3,4673 6,000 -2,505 4,752 3,168 4464.9 516 2,316 1157.9 2,3374 6,000 -2,505 2,851 1,901 2317.1 357 3,742 1871 1,6245 6,000 -2,505 2,000 851 -2.555 185 4,964 2481.8 1,013
2,000 2,000NPV 3317.427 -1.7386 974.707
Note loan balance keeps decreasingNPV of this option is lower - should
choose previous (single repayment at end).. not a general result