Transcript Luxembourg Scool of Finance - M.S. in Banking & Finance

Capital Budeting with the Net Present Value Rule
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
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Time value of money: introduction
• Consider simple investment project:
• Interest rate r = 10%
121
0
1
-100
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Net future value
•
•
NFV = +121 - 100  1.10 = 11
= + C1 - I (1+r)
•
Decision rule: invest if NFV>0
+121
+100
•
Justification: takes into cost of capital
– cost of financing
– opportunity cost
0
-100
1
-110
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Net Present Value
•
•
•
•
NPV = - 100 + 121/1.10
= + 10
= - I + C1/(1+r)
= - I + C1  DF1
•
•
•
DF1 = 1-year discount factor
a market price
C1  DF1 =PV(C1)
•
Decision rule: invest if NPV>0
+110
+121
-100
-121
•
NPV>0  NFV>0
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Internal Rate of Return
• Alternative rule: compare the internal rate of return for the project to the
opportunity cost of capital
• Definition of the Internal Rate of Return IRR : (1-period)
IRR = (C1 - I)/I
• In our example:
IRR = (121 - 100)/100 = 21%
• The Rate of Return Rule: Invest if IRR > r
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IRR versus NPV
• In this simple setting, the NPV rule and the Rate of Return Rule lead to the
same decision:
• NPV = -I+C1/(1+r) >0
•  C1>I(1+r)
•  (C1-I)/I>r
•  IRR>r
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IRR: a general definition
The Internal Rate of Return is the
discount rate such that the NPV is
equal to zero.
•
-I + C1/(1+IRR)  0
•
•
•
In our example:
-100 + 121/(1+IRR)=0
 IRR=21%
Net Present Value
•
25.0
20.0
15.0
IRR
10.0
5.0
0.0
-5.0 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%
-10.0
-15.0
-20.0
-25.0
Discount rate
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Extension to several periods
• Investment project: -100 in year 0, + 150 in year 5.
• Net future value calculation:
NFV5 = +150 - 100  (1.10)5 = +150 - 161 = -11 <0
Compound interest
• Net present value calculation:
NPV = - 100 + 150/(1.10)5
= - 100 + 150  0.621 = - 6.86
0.621 is the 5-year discount factor DF5 = 1/(1+r)5
a market price
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NPV: general formula
• Cash flows:
C0 C1 C2 … Ct … CT
• t-year discount factor: DFt = 1/(1+r)t
• NPV = C0 + C1 DF1 + … + Ct DFt + … + CT DFT
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NPV calculation - example
• Suppose r = 10%
t
0
1
2
3
Cash flow
-100
30
60
40
Discount Factor
1 0.9091 0.8264 0.7513
PresentValue
-100.0 27.3 49.6 30.1
NPV
6.9
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IRR in multiperiod case
• Reinvestment assumption: the IRR calculation assumes that all future cash
flows are reinvested at the IRR
• Disadvantages:
– Does not distinguish between investing and financing
– IRR may not exist or there may be multiple IRR
– Problems with mutually exclusive investments
• Advantages:
– Easy to understand and communicate
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Constant perpetuity
• Ct =C for t =1, 2, 3, .....
Proof:
PV = C d + C d² + C d3 + …
PV(1+r) = C + C d + C d² + …
PV(1+r)– PV = C
PV = C/r
C
PV 
r
• Examples: Preferred stock (Stock paying a fixed dividend)
• Suppose r =10%
Yearly dividend =50
50
P0 
 500
• Market value P0?
.10
• Note: expected price next year =
• Expected return =
P1 
50
 500
.10
div1  ( P1  P0 )
50  (500 500)

 10%
P0
500
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Growing perpetuity
• Ct =C1 (1+g)t-1 for t=1, 2, 3, .....
r>g
C1
PV 
rg
• Example: Stock valuation based on:
 Next dividend div1, long term growth of dividend g
• If r = 10%, div1 = 50, g = 5%
P0 
• Note: expected price next year =
• Expected return =
50
 1,000
.10  .05
P1 
52 .5
 1,050
.10  .05
div1  ( P1  P0 ) 50  (1,050 1,000)

 10%
P0
1,000
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Constant annuity
• A level stream of cash flows for a fixed numbers of periods
• C1 = C2 = … = CT = C
• Examples:
 Equal-payment house mortgage
 Installment credit agreements
• PV = C * DF1 + C * DF2 + … + C * DFT +
•
= C * [DF1 + DF2 + … + DFT]
•
= C * Annuity Factor
• Annuity Factor = present value of €1 paid at the end of each T periods.
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Growing annuity
• Ct = C1 (1+g)t-1 for t = 1, 2, …, T
r≠g
T

C1
1 g  
PV 
 
1  
r  g   1  r  
• This is again the difference between two growing annuities:
– Starting at t = 1, first cash flow = C1
– Starting at t = T+1 with first cash flow = C1 (1+g)T
• Example: What is the NPV of the following project if r = 10%?
Initial investment = 100, C1 = 20, g = 8%, T = 10
NPV= – 100 + [20/(10% - 8%)]*[1 – (1.08/1.10)10]
= – 100 + 167.64
= + 67.64
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Review: general formula
• Cash flows:
C1, C2, C3, … ,Ct, … CT
• Discount factors:
DF1, DF2, … ,DFt, … , DFT
• Present value:
PV = C1 × DF1 + C2 × DF2 + … + CT × DFT
If r1 = r2 = ...=r
PV 
Ct
C1
C2
CT


...


...

(1  r1 ) (1  r2 ) 2
(1  rt ) t
(1  rT )T
PV 
Ct
C1
C2
CT


...


...

(1  r ) (1  r ) 2
(1  r ) t
(1  r )T
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Review: Shortcut formulas
•
Constant perpetuity: Ct = C for all t
PV 
C
r
•
Growing perpetuity: Ct = Ct-1(1+g)
PV 
C1
rg
r>g
•
•
t = 1 to ∞
Constant annuity: Ct=C
t=1 to T
Growing annuity: Ct = Ct-1(1+g)
t = 1 to T
PV 
C
1
(1 
)
T
r
(1  r )
C1
(1  g )T
PV 
(1 
)
T
rg
(1  r )
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IRR and NPV - Example
Compute the IRR and NPV for the following two projects. Assume the required
return is 10%.
Year
0
1
2
3
NPV
IRR
Project A
-$200
$200
$800
-$800
42
0%, 100%
Project B
-$150
$50
$100
$150
91
36%
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Project A
1.
4
1.
2
1
0.
8
0.
6
0.
4
0.
2
200.0
150.0
100.0
50.0
0.0
-50.0
-100.0
-150.0
0
NPV Profiles
Project B
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The Payback Period Rule
• How long does it take the project to “pay back” its initial investment?
• Payback Period = # of years to recover initial costs
• Minimum Acceptance Criteria: set by management
• Ranking Criteria: set by management
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The Payback Period Rule (continued)
• Disadvantages:
–
–
–
–
–
–
Ignores the time value of money
Ignores CF after payback period
Biased against long-term projects
Payback period may not exist or multiple payback periods
Requires an arbitrary acceptance criteria
A project accepted based on the payback criteria may not have a positive NPV
• Advantages:
– Easy to understand
– Biased toward liquidity
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The Profitability Index (PI) Rule
•
•
•
•
PI = Total Present Value of future CF’s / Initial Investment
Minimum Acceptance Criteria: Accept if PI > 1
Ranking Criteria: Select alternative with highest PI
Disadvantages:
– Problems with mutually exclusive investments
• Advantages:
– May be useful when available investment funds are limited
– Easy to understand and communicate
– Correct decision when evaluating independent projects
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Incremental Cash Flows
• Cash, Cash, Cash, CASH
• Incremental
– Sunk Costs
– Opportunity Costs
– Side Effects
• Tax and Inflation
• Estimating Cash Flows
– Cash flows from operation
– Net capital spending
– Changes in net working capital
• Interest Expense
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Summarized balance sheet
• Assets
 Fixed assets (FA)
 Working capital requirement (WCR)
 Cash (Cash)
• Liabilities
 Stockholders' equity (SE)
 Interest-bearing debt (D)
• FA + WCR + Cash = SE + D
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Working capital requirement : definition
•
•
•
•
•
•
•
+
+
+
Accounts receivable
Inventories
Prepaid expenses
Account payable
Accrued payroll and other expenses
(WCR sometimes named "operating working capital")
– Copeland, Koller and Murrin Valuation: Measuring and Managing the
Value of Companies, 2d ed. John Wiley 1994
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Interest-bearing debt: definition
• +
• +
• +
Long-term debt
Current maturities of long term debt
Notes payable to banks
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The Cash Flow Statement
• Let us start from the balance sheet identity:
• FA + WCR + CASH = SE + D
• Over a period:
• FA + WCR + CASH = SE + D
• But:
SE =
STOCK ISSUE + RETAINED EARNINGS
= SI + NET INCOME - DIVIDENDS
FA =
INVESTMENT - DEPRECIATION
• (INV - DEP) + WCR + CASH = (SI + NI - DIV) + D
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•
•
•
•
•
•
•
•
•
•
(NI +DEP - WCR) - (INV) + (SI + D - DIV) = CASH

Net cash flows from
operating activities (CFop)

Cash flow from
investing activities (CFinv)

Cash flow from
financing activities (CFfin)
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Free cash flow
• FCF = (NI +DEP - WCR) - (INV)
•
= CFop + CFinv
• From the statement of cash flows
• FCF = - (SI + D - DIV) + CASH
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Understanding FCF
CF from operation + CF from investment + CF from financing = CASH
Cash flow from
operation
Cash flow
from financing
Cash flow from
investment
Cash
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NPV calculation: example
•
•
•
•
•
•
•
•
•
Length of investment : 2 years
Investment
: 60 (t = 0)
Resale value
: 20 (t = 3, constant price)
Depreciation
: linear over 2 years
Revenue
: 100/year (constant price)
Cost of sales
: 50/year (constant price)
WCR/Sales
: 25%
Real discount rate
: 10%
Corporate tax rate
: 40%
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Scenario 1: no inflation
Year
Sales
Cost of sales
EBITD
Depreciation
EBIT
Taxes
Net Income
0
1
100
50
50
30
20
8
12
2
100
50
50
30
20
8
12
12
30
25
12
30
0
17
42
Net Income
+ Depreciation
-DWCR
Investment
-60
Free cash flow -60
NPV
17.96 IRR
3
8
-8
-8
-25
20
37
24%
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Inflation
• Use nominal cash flow
• Use nominal discount rate
• Nominal versus Real Rate (The Fisher Relation)
(1 + Nominal Rate) = (1 + Real Rate) x (1 + Inflation Rate)
•
•
•
•
•
•
•
Example:
Real cash flow year 1 = 110
Real discount rate = 10%
Inflation = 20%
Nominal cash flow = 110 x 1.20
Nominal discount rate = 1.10 x 1.20 - 1
NPV = (110 x 1.20)/(1.10 x 1.20) = 110/1.10 = 100
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Scenario 2 : Inflation = 100%
Nominal discount rate:
(1+10%) x (1+100%) = 2.20
Nominal rate = 120%
NPV now negative. Why?
Year
Sales
Cost of sales
EBITD
Depreciation
EBIT
Taxes
Net Income
0
1
200
100
100
30
70
28
42
2
400
200
200
30
170
68
102
42
30
50
102
30
50
22
82
Net Income
+ Depreciation
-DWCR
Investment
-60
Free cash flow -60
NPV
-14.65 IRR
3
64
-64
-8
-100
160
196
94%
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Decomposition of NPV
–
–
–
–
–
–
EBITDA after taxes
Depreciation tax shield
WCR
Investment
Resale value after taxes
NPV
52.07
20.83
-3.94
-60
9.02
17.96
52.07
7.93
-23.67
-60
9.02
14.65
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