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 CalculationsInformation 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…. Math
Ci = Initial cost of an asset
Cs = Final salvage value of an
asset
Cd =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- Cddk
Book Value is often not true asset value.
Depreciation Methods
• Straight Line
Dk = Cd/m (same over lifetime)
• Accelerated Depreciation
– Double Declining Balance
D1 = Ci (2/m)
B1 = Ci - D1
D2 = B1 (2/m) = [Ci (1-2/m)](2/m)],etc.
– Sum of the Years Digits
Dk = Cd (Useful years left = m-k +1)/
m + (m-1) + (m-2) + ... + 2 + 1
k = current year
m = lifetime
Link to Summary of Depreciation Methods
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
Consider the Effect of Depreciation
and Taxes on Present Value (P)
• If no depreciation & taxes, the decision to
invest $Ci 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
D 1T
0
1
D 2T
D 3T
D 4T
2
3
4
•••
CS
DNT
N
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
D 1T
0
1
D 2T
D 3T
D 4T
2
3
4
•••
CS
DNT
N
Ci
P  (Cd  Cs ) 
N
DmT
 (1 i
m 1
m
)
T

Cs
(1  iT )N
Ci
Cd N
Cd 1  (1  iT )N 
1

 (1 i )m  N T  (1 i )m  N T  i


T
m 1
m 1
T
T
N
DmT
 T 1  (1  i )N  

1 
T
P  Cs 1 

C
1





d
N
iT
 (1  iT ) 
 
 N 
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
R(1 - T)
1
R(1 - T)
2
R(1 - T)
3
R(1 - T)
4
...
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 (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)
After-Tax Cash Flow Time Line Showing
Revenues, Expenses and Depreciation
DT
R(1 - T)
0
Ci
1
X(1 - T)
DT
DT
CS
DT
R(1 - T)
R(1 - T)
R(1 - T)
2
3
X(1 - T)
X(1 - T)
4
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 revenues
R = $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 taxes
For A = $100,000 - $50,000 = $50,000/yr
For 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/yr
B shows itself to be better!
• Taxes @ (50%)
A = 0.50($10,000) = $5000
B = 0.50($20,000) = $10,000
• After-Tax Income
(Before Tax Profit) - (Taxes)
A = 10,000 - 5000 = $5000
B = 20,000 - 10,000 = $10,000
• But after tax cash flow
[R - X - Taxes]
A = $100,000 - $50,000 - $5000 = $45,000
B = $100,000 - $50,000 - $10,000 = $40,000
Which company is better?
Which company is better?
• B is the better company!
• A has “turned” more of its assets into cash,
but is using its assets less efficiently than B,
as profit illustrates
• Therefore, profitability = cash flow
Depreciation - a “Source” of Cash??
Sales
Uncle Sam’s perspective
Profitability Measures
Payout time / Payback period
- Many definitions of this
- Generally
Payback Period (N, in years) =
Initial Investment
Income/yr
• Initial investment is Ci total investment for some people, only
Cd (depreciable investment) for others
• Income/yr for some is average profit/yr, excluding depreciation
and taxes, but some include depreciation and taxes
• Basic question addressed
How soon do I recoup my original investment?
ROI (Return on Original Investment)
ROI =
Income / yr
Initial investment
• Neither payback period nor ROI explicitly
considers the time value of money!
Preferred Methods
• Net Present Value (NPV)
Also known as Venture Worth (VW)
Nt
P  Ci  
Dk T
k
(1

i
)
k 1
T
N
(R  X)k (1  T)
k 1
(1  iT )k


Cs
Iw

I

w
(1  iT )N
(1  iT )N
• Discounted Cash Flow Rate of Return
(DCFRR)
Same as NPV = 0, solve for iT
• Iw = working capital (similar to initial investment
in treatment)
Which Method is “Better”?
• Net Present Value
– Requires setting a value of iT before you start
– Any NPV > 0 means a worthwhile project
– In choosing between alternatives with unequal
lifetimes, need to choose on an annualized
income basis (i.e. convert P
X at end)
• DCFRR
– No need to have same time basis or to choose iT a
priori
– Go down list from highest iT to lowest (down to a
minimum acceptable iT)
Example - Two Competing
Investment Opportunities
Revenues ($/yr)
Costs ($/yr)
Required Investment ($)
Salvage Value at End ($)
Project Life (yrs)
Depreciation Lifetime (yrs)
Opportunity 1 Opportunity 2
75,000
60,000
10,000
15,000
130,000
150,000
10,000
30,000
5
4
3
3
After tax interest rate = 0.10/yr = iT
Combined Fed/State tax rate = 0.48 = T
Depreciation method = Straight line
Cash Flow Time Lines
(Amounts in 1000’s)
• Opportunity 1
40T
40T
40T
10
60(1- T)
0
130
60(1- T) 60(1- T) 60(1- T)
60(1- T)
1
2
3
4
5
10(1- T) 10(1- T) 10(1- T) 10(1- T) 10(1- T)
Cd Ci  Cs 130  10
Note: D 


 40
ND
ND
3
ND = depreciation lifetime = N = Project Lifetime
Cash Flow Time Lines
(Amounts in 1000’s)
• Opportunity 2
40T
40T
40T
75(1- T)
0
150
30
75(1- T) 75(1- T) 75(1- T)
1
2
3
4
15(1- T) 15(1- T) 15(1- T) 15(1- T)
150  30
 40
Note: D =
3
Present Value Calculations
1  (1  iT )5 
1  (1  iT )3 
P1  Ci 
 (R  X)(1  T) 
  DT 

5
iT
iT
(1  iT )




Cs
1  (1  0.1)5 
1  (1  0.1)3 
 130 
 (60  10)(1  0.48) 
  40(0.48) 

5
0.1
0.1
(1  0.1)




10
 22.52 (thousands of dollars)
1  (1  0.1)4 
1  (1  0.1)3 
P2  150 
 (75  15)(1  0.48) 
  40(0.48) 

4
0.1
0.1
(1  0.1)




30
 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

 22.52
0.1
3
X1  22.52 


$5.94x10
/ yr
5 
1  (1  0.1)  3.79

 17.14
0.1
3
X2  17.14 


$5.42x10
/ yr

4
1  (1  0.1)  3.16
• 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.
DCFRR
• Let P1 = 0 and solve for iT
• Need a root finding technique
Know iT > 0.1 / yr
• In this case
1  (1  iT )5 
1  (1  iT )3 
0  130 
 (50)(.52) 
  40(0.48) 

5
(iT)1 from
i
i
(1  iT )




T
T
IT  17%
10
1  (1  iT )4 
1  (1  iT )3 
0  150 
 (60)(.52) 
  40(0.48) 

4
iT
iT
(iT)2 from
(1  iT )




IT  15%
30
• 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
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 ein
where it is now understood that in these types
of calculations, i = icont
Link to Continuous Interest Formulae