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7/30/2019 e2.Cost and Time Value Lecture2[1]
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Last time.
Basics of financial analysis
Estimating revenues and expenses iscrucial
Time value of money concept
The significance of present valuecomparisons
Conversion of cash flows to present values
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Profit Revisited
Profit = Revenues - Expenses
Expenses should include loss of value ofequipment with time due to
Wear and Tear
Obsolescence
Loss of value (expiration of assets) is thebasis of DEPRECIATION
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Depreciation and Taxes
Suppose a company has $10 million in profits onDecember 31, i.e.
Profits = Revenues - Expenses = $10,000,000 Corporate taxes are, in simplest terms, based on a
a percentage of profits
Suppose that as a way of beating taxes thecompany purchases $10 million worth of newequipment on December 31
Is the profit = 0?
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No! Profit is not zero
The company has merely converted oneasset (cash) to another (equipment). This iswhy Uncle Sam controls how equipment isexpensed-- i.e. you cannot declare items of
capital equipment as expenses whenpurchased. Instead, they are depreciated.
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Depreciation Calculations-
Information Needed
We need:
Price originally paid for the equipment or
asset Estimate of lifetime (IRS)
Salvage Value at the end of lifetime
Calculations to be shown neglect specialcircumstances, e.g. investment tax credits,additional first year allowances
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Depreciation
A new machine is not asgood as an old machine
Depreciation is a way toaccount for the expiration
of the machine, or any asset Many methods: straight lineversus accelerated
Has important tax
consequences,which need to be considered inpresent value calculations
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Mmmm. Math
Ci = Initial cost of an assetCs = Final salvage value of anasset
Cd =Depreciable cost =(Ci- Cs)m = lifetime for tax purposes
(often differs from actuallifetime)dk = fractional depreciation
in year kDk = Dollar amount of depreciation, year k
Dk = dk Cd , Book Value = Ci- CddkBook Value is often not true asset value.
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Depreciation Methods
Straight LineDk = Cd/m (same over lifetime)
Link to Summary of Depreciation Methods
Sum of the Years Digits
Dk = Cd (Useful years left = m-k +1)/
m + (m-1) + (m-2) + ... + 2 + 1k = current yearm = lifetime
Accelerated Depreciation
Double Declining BalanceD1=Ci(2/m) B1= Ci- D1D2= B1 (2/m) = [Ci(1-2/m)](2/m)],etc.
http://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_Depreciation_methods.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_Depreciation_methods.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_Depreciation_methods.doc7/30/2019 e2.Cost and Time Value Lecture2[1]
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After-Tax Interest Rate If we have an investment of $P yielding i interest per year,
at the end of one year we have:
P(1 + i) We have to pay taxes on earnings
Earnings = P(1 + i) - P = P i
Tax rate is T
Taxes = P i T Real Earnings = Pi - P i T
= Pi (1 - T)
Define after tax interest rate
iT = i(1 - T) So, real after tax earnings = PiT
We will use iT in after-tax comparisons
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Consider the Effect of Depreciation
and Taxes on Present Value (P)
If no depreciation & taxes, the decision toinvest $Ci in a piece of equipment at timezero is worth
P = -Ci Reflects that Ci of cash of unavailable for
other investments
Now, we need to consider the fact thatdepreciation gives us a tax savings each year
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Cash Flow Time Line for Investments
0 41 N2 3D1T DNTD2T D3T D4T
CS
Ci
Cash outflow is shown below the line
Savings and/or revenues above the line
Cs is salvage value
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Cash Flow Time Line for Investments
NN Nd dm T
m m
Tm 1 m 1T T
C CD T 1 1 (1 i )T T
N N i(1 i ) (1 i )
NT
s dNT
T
1 T 1 (1 i )P C 1 C 1
N i(1 i )
0 41 N2 3D1T DNTD2T D3T D4T
CS
Ci
Nsm
d s m Nm 1 T T
CD TP (C C )
(1 i ) (1 i )
Ci
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After-Tax Cost
Comparison Formulae
Link to After-Tax CostComparison Formulae
Eff f R i
http://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_after-tax_comparison_formulae.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_after-tax_comparison_formulae.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_after-tax_comparison_formulae.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_after-tax_comparison_formulae.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_after-tax_comparison_formulae.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_after-tax_comparison_formulae.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/EN90_Handout_after-tax_comparison_formulae.doc7/30/2019 e2.Cost and Time Value Lecture2[1]
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Effect of Revenues in
After Tax Comparisons
For every $R of revenue, a profit making firm pays $RTin tax where
T = fractional tax rate
Thus, the firm actually keeps($R - $RT) = $R(1 - T)
An after-tax cash flow time line would therefore haveamounts as shown
R(1 - T)
0 41 2 3
R(1 - T) R(1 - T) R(1 - T)R(1 - T) ...
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Expenses in After-Tax Comparisons
An expense of X in a particular tax year has two effectson cash flow
-the actual out-of-pocket payment of X-the reduction of taxes as a result of the expense
(XT) Profit Before Expense (p) - Expense (X)
= Profit After Expense (px)
Tax = pxT = pT - XT
Profit after Taxes = px - pxT= px(1 - T)
Therefore, Effect of Expense = -X(1 - T)
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After-Tax Cash Flow Time Line Showing
Revenues, Expenses and Depreciation
CS
R(1 - T) R(1 - T) R(1 - T)R(1 - T)
0 41 2 3
DT DT DT DT
Ci X(1 - T) X(1 - T) X(1 - T) X(1 - T)
Note! Depreciation is not a real cash flow intocompany. It has the effect of reducing taxes.
Note! No taxes associated with Ci or Cs terms.
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Profitability vs. Cash Flow Assume Companies A & B make the same product, in same
quantities and have the same revenues
R = $100,000/yr
Raw materials & labor $50,000/yr for both
A produces products on a machine worth $200,000 andconsumes 20% of its useful life/yr
Bs machine also costs $200,000, but they consume 15%/yrof its useful life
Assume actual maintenance costs are the same for A & B
Cash flow, before taxesFor A = $100,000 - $50,000 = $50,000/yrFor B = $100,000 - $50,000 = $50,000/yr
NO DIFFERENCE!Yet, we know that B is more profitable because it consumes
less of its capital assets.
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Profits (Including Depreciation) before Taxes
For A = $100,000 - $50,000 - (0.20)(200,000) = $10,000/yrFor B = $100,000 - $50,000 - (0.15)(200,000) = $20,000/yr
B shows itself to be better!
Taxes @ (50%) A = 0.50($10,000) = $5000B = 0.50($20,000) = $10,000
After-Tax Income(Before Tax Profit) - (Taxes)
A = 10,000 - 5000 = $5000B = 20,000 - 10,000 = $10,000
But after tax cash flow[R - X - Taxes]
A = $100,000 - $50,000 - $5000 = $45,000B = $100,000 - $50,000 - $10,000 = $40,000
Which company is better?
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Which company is better?
B is the better company!
A has turned more of its assets into cash,
but is using its assets less efficiently than B,as profit illustrates
Therefore, profitability = cash flow
D i i S f C h??
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Depreciation - a Source of Cash??
Sales
Uncle Sams perspective
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Profitability MeasuresPayout time / Payback period
- Many definitions of this- Generally
Payback Period (N, in years) =
Initial investment is Ci total investment for some people, onlyCd (depreciable investment) for others
Income/yr for some is average profit/yr, excluding depreciationand taxes, but some include depreciation and taxes
Basic question addressedHow soon do I recoup my original investment?
Initial Investment
Income/yr
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ROI (Return on Original Investment)
ROI =
Neither payback period nor ROI explicitlyconsiders the time value of money!
Income / yr
Initial investment
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Preferred Methods
Net Present Value (NPV)Also known as Venture Worth (VW)
Discounted Cash Flow Rate of Return(DCFRR)
Same as NPV = 0, solve for iT
Iw = working capital (similar to initial investment
in treatment)
tN N
sk k wi wk k N Nk 1 k 1T T T T
CD T (R X) (1 T) IP C I(1 i ) (1 i ) (1 i ) (1 i )
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Which Method is Better? Net Present Value
Requires setting a value of iT before you start
Any NPV > 0 means a worthwhile project
In choosing between alternatives with unequallifetimes, need to choose on an annualized
income basis (i.e. convert P X at end)
DCFRR
No need to have same time basis or to choose iT apriori
Go down list from highest iT to lowest (down to aminimum acceptable iT)
E l T C i
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Example - Two Competing
Investment Opportunities
Opportunity 1 Opportunity 2
Revenues ($/yr)
Costs ($/yr)
Salvage Value at End ($)
Project Life (yrs)
60,000 75,000
10,000 15,000
130,000 150,00010,000 30,000
Required Investment ($)
Depreciation Lifetime (yrs)
4
33
5
After tax interest rate = 0.10/yr = iTCombined Fed/State tax rate = 0.48 = T
Depreciation method = Straight line
C h Fl Ti Li
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Cash Flow Time Lines
(Amounts in 1000s)
Opportunity 1
60(1-T)
0 41 2 3
130 10(1- T)
-T)
0 51 2 3
40T 40T40T
60(1-T)-T) 60(1-T)-T) 60(1-T)-T) 60(1-T)-T)
10(1- T) 10(1- T)10(1- T)10(1- T)
10
Note: d i s
D D
C C C 130 10D 40
N N 3
ND = depreciation lifetime = N = Project Lifetime
C h Fl Ti Li
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Cash Flow Time Lines
(Amounts in 1000s)
Opportunity 2
75(1-T)
0 41 2 3
150 15(1- T)
-T)
0 1 2 3
40T 40T40T
75(1-T)-T) 75(1-T)-T) 75(1-T)-T)
15(1- T) 15(1- T)15(1- T)
30
Note: D =150 30
403
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Present Value Calculations
5 3T T
1 5T TT
5 3
5
Cs 1 (1 i ) 1 (1 i )
P Ci (R X)(1 T) DTi i(1 i )
10 1 (1 0.1) 1 (1 0.1)130 (60 10)(1 0.48) 40(0.48)
0.1 0.1(1 0.1)
22.52 (thousands of dollars)
4 3
2 4
30 1 (1 0.1) 1 (1 0.1)P 150 (75 15)(1 0.48) 40(0.48)
0.1 0.1(1 0.1)
17.14 (thousands of dollars)
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Present Value Calculations cont
Since P1 > 0 and P2 > 0, do both projects, if possible
If can only choose one or the other
Choose Opportunity 1 over Opportunity 2 (X1 > X2)
Note, if P1 had been just a bit less, could have hadP1 > P2 but X1 < X2 . In this case, would choose
Opportunity 2 instead.
31 5
32 4
0.1 22.52X 22.52 $5.94x10 / yr
3.791 (1 0.1)
0.1 17.14X 17.14 $5.42x10 / yr
3.161 (1 0.1)
DCFRR
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DCFRR Let P1 = 0 and solve for iT
Need a root finding technique
Know iT > 0.1 / yr In this case
(iT)1 from
(iT)2 from
Choose projects based on iT, highest to lowest until yourun out of money to invest (Here, choose 1 over 2)
Use a graphical or numerical approach to solve for iT
5 3T T
5T T
T
T
10 1 (1 i ) 1 (1 i )0 130 (50)(.52) 40(0.48)
i i(1 i )
I 17%
4 3T T
4T T
T
T
30 1 (1 i ) 1 (1 i )0 150 (60)(.52) 40(0.48)
i i(1 i )
I 15%
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Continuous Interest and
Discounting
Treats compounding in a continuous manner,as if in every infinitesimal time period, interest
accrues (instead of only at year end):1+ iannual = (1 + icont/k)
k
where there are k compounding periods peryear.
Now let k, (1 + icont/k)k eicont
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Continuous Discounting
Thus
iannual = eicont -1
and
S = P (1 + iannual )n = P (1 + e icont -1)n
= P e i n
where it is now understood that in these typesof calculations, i = icont
Link to Continuous Interest Formulae
http://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/Continuous_interest_formulae.dochttp://c/WINDOWS/Temporary%20Internet%20Files/Content.IE5/Word_Excel_Files/Continuous_interest_formulae.doc