Capital Budeting with the Net Present Value Rule

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Capital Budeting with the Net Present Value Rule. Professor André Farber Solvay Business School Université Libre de Bruxelles. Time value of money: introduction. Consider simple investment project: Interest rate r = 10%. 121. 1. 0. -100. NFV = +121 - 100  1.10 = 11

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### Capital Budeting with the Net Present Value Rule

Professor André Farber

Université Libre de Bruxelles

Vietnam 2004

Time value of money: introduction
• Consider simple investment project:
• Interest rate r = 10%

121

1

0

-100

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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

Net future value

+121

+100

0

1

-100

-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
• NPV>0  NFV>0

+121

+110

-100

-121

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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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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%

IRR: a general definition

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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 C1C2 … 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%

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IRR in multiperiod case
• Reinvestment assumption: the IRR calculation assumes that all future cash flows are reinvested at the IRR
• Does not distinguish between investing and financing
• IRR may not exist or there may be multiple IRR
• Problems with mutually exclusive investments
• Easy to understand and communicate

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Constant perpetuity

Proof:

PV = C d + C d² + C d3 + …

PV(1+r) = C + C d + C d² + …

PV(1+r)– PV = C

PV = C/r

• 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 =

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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 =

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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

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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

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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=Ct=1 to T
• Growing annuity: Ct = Ct-1(1+g)

t = 1 to T

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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

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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)
• 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
• 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
• Problems with mutually exclusive investments
• 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:

DSE = STOCK ISSUE + RETAINED EARNINGS

= SI + NET INCOME - DIVIDENDS

DFA = 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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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?

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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

Vietnam 2004