e2.Cost and Time Value Lecture2[1]

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.doc


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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.doc


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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

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e2.Cost and Time Value Lecture2[1]