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Borrowing, Depreciation, Taxes in Cash Flow Problems
H. Scott Matthews12-706 / 19-702 /73-359Lecture 4
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Textbook Spreadsheets
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http://www.uq.edu.au/economics/bca/
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You can download spreadsheet files referenced in Campbell book chapters
Theme: Cash Flows
Streams of benefits (revenues) and costs over time => “cash flows”
This is our focus for next few classesWe need to know what to do with them in terms
of finding NPV of projectsDifferent 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 publicCash flows come from: operation, financing,
taxes
Notes on Tax deductibilityReason we care about financing and depreciation:
they affect taxes owedFor 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 IncomeTaxable Income = Gross Income - Operating
Expenses - Depreciation - Loan Interest - Bond Dividends
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
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 end up selling asset for salvage value, no further tax implications If we end up selling asset for higher than salvage value, we may pay additional taxes since we received depreciation expense benefits (but we will generally ignore this)
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
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**
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
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
Results - Straight Line Dep.
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”
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
Uniform Values - Theory
Find present value of n payment of UAssume end of period valuesP = U/(1+i) +U/(1+i)2 + ..+ U/(1+i)n
(P|U,i,n) = $1[(1+i)-1+(1+i)-2 + ..+ (1+i)-n][(1+i)-1+(1+i)-2 + ..+ (1+i)-n] = “A”[1+(1+i)-1+(1+i)-2 + ..+ (1+i)1-n] = A(1+i)A(1+i) - A = [1 - (1+i)-n]A = [1 - (1+i)-n] / i, ….soP = U*(P|U,i,n) = U*[1 - (1+i)-n] / i
Uniform Values - Application
Recall $1000 / year for 5 years example
P = U*(P|U,i,n) = U*[1 - (1+i)-n] / i(P|U,5%,5) = 4.329P = 1000*4.329 = $4,329
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 loanSource 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.
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
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)
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
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
Repayment Options
Single Loan, Single payment at end of loan
Single Loan, Yearly PaymentsMultiple Loans, One repayment
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
Notes
Mixed funds problem - buy computer Below: Operating cash flows At Four 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)
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
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
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
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
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 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)
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
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%
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
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
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