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