Vietnam 2004 1
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
-100
0 1
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Net future value
• NFV = +121 - 100 1.10 = 11
• = + C1 - I (1+r)
• Decision rule: invest if NFV>0
• Justification: takes into cost of capital
– cost of financing
– opportunity cost
-100
+100+121
-110
0 1
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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
• NPV>0 NFV>0
-100
+121
-121
+110
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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% -25.0-20.0
-15.0-10.0
-5.00.05.0
10.015.0
20.025.0
0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%
Discount rateNe
t Pre
sent
Val
ue
IRR
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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 3Cash flow -100 30 60 40Discount Factor 1 0.9091 0.8264 0.7513PresentValue -100.0 27.3 49.6 30.1NPV 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, .....
• Examples: Preferred stock (Stock paying a fixed dividend)
• Suppose r =10% Yearly dividend = 50
• Market value P0?
• Note: expected price next year =
• Expected return =
50010.
501 P
r
CPV
Proof:PV = C d + C d² + C d3 + …PV(1+r) = C + C d + C d² + …PV(1+r)– PV = CPV = C/r
50010.
500 P
%10500
)500500(50)(
0
011
P
PPdiv
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Growing perpetuity
• Ct =C1 (1+g)t-1 for t=1, 2, 3, ..... r>g
• Example: Stock valuation based on: Next dividend div1, long term growth of dividend g
• If r = 10%, div1 = 50, g = 5%
• Note: expected price next year =
• Expected return =
gr
CPV
1
000,105.10.
500
P
050,105.10.
5.521
P
%10000,1
)000,1050,1(50)(
0
011
P
PPdiv
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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
• 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
T
r
g
gr
CPV
1
111
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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
TT
Tt
t
t
r
C
r
C
r
C
r
CPV
)1(...
)1(...
)1()1( 22
2
1
1
TT
tt
r
C
r
C
r
C
r
CPV
)1(...
)1(...
)1()1( 221
If r1 = r2 = ...=r
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Review: Shortcut formulas
• Constant perpetuity: Ct = C for all t
• Growing perpetuity: Ct = Ct-1(1+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
r
CPV
gr
CPV
1
))1(
11(
Trr
CPV
))1(
)1(1(1
T
T
r
g
gr
CPV
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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 Project A Project B
0 -$200 -$150
1 $200 $50
2 $800 $100
3 -$800 $150
NPV 42 91
IRR 0%, 100% 36%
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NPV Profiles
-150.0
-100.0-50.0
0.0
50.0
100.0150.0
200.0
Project A 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 = CASHCF from operation + CF from investment + CF from financing = CASH
Cash flow from operation
Cash flow from investment
Cash flow from financing
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 0 1 2 3Sales 100 100Cost of sales 50 50EBITD 50 50Depreciation 30 30EBIT 20 20Taxes 8 8 8Net Income 12 12 -8
Net Income 12 12 -8+ Depreciation 30 30-DWCR 25 0 -25Investment -60 20Free cash flow -60 17 42 37
NPV 17.96 IRR 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%
Year 0 1 2 3Sales 200 400Cost of sales 100 200EBITD 100 200Depreciation 30 30EBIT 70 170Taxes 28 68 64Net Income 42 102 -64
Net Income 42 102 -8+ Depreciation 30 30-DWCR 50 50 -100Investment -60 160Free cash flow -60 22 82 196
NPV -14.65 IRR 94%
Nominal discount rate:
(1+10%) x (1+100%) = 2.20
Nominal rate = 120%
NPV now negative. Why?
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Decomposition of NPV
– EBITDA after taxes 52.07 52.07
– Depreciation tax shield 20.83 7.93 WCR -3.94 -23.67
– Investment -60 -60
– Resale value after taxes 9.02 9.02
– NPV 17.96 14.65