Last time….

• Basics of financial analysis• Estimating revenues and expenses is

crucial• Time value of money concept• The significance of present value

comparisons• Conversion of cash flows to present values

Profit Revisited

• Profit = Revenues - Expenses

• Expenses should include loss of value of equipment with time due to

• Wear and Tear

• Obsolescence

• Loss of value (“expiration of assets”) is the basis of DEPRECIATION

Depreciation and Taxes

• Suppose a company has $10 million in profits on December 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” the company purchases $10 million worth of new equipment on December 31

• Is the profit = 0?

No! Profit is not zero• The company has merely converted one

asset (cash) to another (equipment). This is why Uncle Sam controls how equipment is “expensed”-- i.e. you cannot declare items of capital equipment as expenses when purchased. Instead, they are depreciated.

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 special

circumstances, e.g. investment tax credits, additional first year allowances

Depreciation

• A new machine is not as good as an old machine

• Depreciation is a way to account for the expiration of the machine, or any asset

• Many methods: straight line versus accelerated

• Has important tax consequences, which need to be considered in present value calculations

Mmmm…. MathCi = Initial cost of an assetCs = Final salvage value of an

assetCd =Depreciable cost =(Ci- Cs)m = lifetime for tax purposes

(often differs from actual lifetime)

dk = fractional depreciation in year k

Dk = Dollar amount of depreciation, year k

Dk = dk Cd , Book Value = Ci- Cddk Book Value is often not true asset value.

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 Balance

D1 = Ci (2/m) B1 = Ci - D1

D2 = B1 (2/m) = [Ci (1-2/m)](2/m)],etc.

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 earningsEarnings = 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

Consider the Effect of Depreciation and Taxes on Present Value (P)

• If no depreciation & taxes, the decision to invest $C i in a piece of equipment at time zero is worth

P = -Ci

• Reflects that Ci of cash of unavailable for other investments

• Now, we need to consider the fact that depreciation gives us a tax savings each year

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

Cash Flow Time Line for Investments

NN Nd dm T

m mTm 1 m 1T T

C CD T 1 1 (1 i ) T T

N N i(1 i ) (1 i )

NT

s dNTT

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

After-Tax Cost Comparison Formulae

Link to After-Tax Cost Comparison Formulae

Effect of Revenues in After Tax Comparisons

• For every $R of revenue, a profit making firm pays $RT in 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 have

amounts as shown

R(1 - T)

0 41 2 3

R(1 - T) R(1 - T) R(1 - T)R(1 - T) ...

Expenses in After-Tax Comparisons

• An expense of X in a particular tax year has two effects on cash flow

-the actual out-of-pocket payment of X-the reduction of taxes as a result of the expense

(XT)• Profit Before Expense () - Expense (X)

= Profit After Expense (x)

• Tax = xT = T - XT

• Profit after Taxes = x - xT= x(1 - T)

• Therefore, Effect of Expense = -X(1 - T)

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 into company. It has the effect of reducing taxes.

Note! No taxes associated with Ci or Cs terms.

Profitability vs. Cash Flow• Assume Companies A & B make the same product, in same

quantities and have the same revenuesR = $100,000/yr

• Raw materials & labor $50,000/yr for both• A produces products on a machine worth $200,000 and

“consumes” 20% of its useful life/yr• B’s machine also costs $200,000, but they consume 15%/yr

of 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.

Profits (Including Depreciation) before Taxes• For A = $100,000 - $50,000 - (0.20)(200,000) = $10,000/yr

For B = $100,000 - $50,000 - (0.15)(200,000) = $20,000/yrB 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?

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

Depreciation - a “Source” of Cash??

Sales

Other

inco

me

Variable Costs (ra

w materia

ls, labor, e

tc.)

Fixed Costs (excluding depreciation)

Depreciation (paid to yourself)Profit

TaxesBuildup of cash

Real cash buildup

Buy new equipment

Uncle Sam’s perspective

Profitability MeasuresPayout time / Payback period

- Many definitions of this- Generally

Payback Period (N, in years) =

• Initial investment is Ci total investment for some people, only Cd (depreciable investment) for others

• Income/yr for some is average profit/yr, excluding depreciation and taxes, but some include depreciation and taxes

• Basic question addressedHow soon do I recoup my original investment?

Initial Investment

Income/yr

ROI (Return on Original Investment)

ROI =

• Neither payback period nor ROI explicitly considers the time value of money!

Income / yr

Initial investment

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 Nsk k w

i 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 )

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 unequal

lifetimes, 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 a

priori

– Go down list from highest iT to lowest (down to a minimum acceptable iT)

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

130,000 150,000

10,000 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

Cash Flow Time Lines(Amounts in 1000’s)

• 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

Cash Flow Time Lines(Amounts in 1000’s)

• 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

Present Value Calculations5 3

T T1 5

T TT

5 3

5

Cs 1 (1 i ) 1 (1 i )P Ci (R X)(1 T) DT

i 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)

Present Value Calculations con’t• 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 had P1 > 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• 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 you run out of money to invest (Here, choose 1 over 2)

• Use a graphical or numerical approach to solve for iT

5 3T T

5T TT

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 TT

T

30 1 (1 i ) 1 (1 i )0 150 (60)(.52) 40(0.48)

i i(1 i )

I 15%

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 per year.

Now let k, (1 + icont/k)k e icont

Continuous Discounting

Thus

iannual = e icont -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 types of calculations, i = icont

Link to Continuous Interest Formulae