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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

Capital Budeting with the Net Present Value Rule

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

Vietnam 2004

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

121

1

0

-100

Vietnam 2004

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

Net future value

+121

+100

0

1

-100

-110

Vietnam 2004

net present value
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

Vietnam 2004

internal rate of return
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

Vietnam 2004

irr versus npv
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

Vietnam 2004

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%

IRR: a general definition

Vietnam 2004

extension to several periods
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

Vietnam 2004

npv general formula
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

Vietnam 2004

npv calculation example
NPV calculation - example
  • Suppose r = 10%

Vietnam 2004

irr in multiperiod case
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

Vietnam 2004

constant perpetuity
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 =

Vietnam 2004

growing perpetuity
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 =

Vietnam 2004

constant annuity
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.

Vietnam 2004

growing annuity
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

Vietnam 2004

review general formula
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

Vietnam 2004

review shortcut formulas
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

Vietnam 2004

irr and npv example
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%

Vietnam 2004

npv profiles
NPV Profiles

Vietnam 2004

the payback period rule
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

Vietnam 2004

the payback period rule continued
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

Vietnam 2004

the profitability index pi rule
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

Vietnam 2004

incremental cash flows
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
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

Vietnam 2004

working capital requirement definition
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

Vietnam 2004

interest bearing debt definition
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
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

Vietnam 2004

slide28
(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)

Vietnam 2004

free cash flow
Free cash flow
  • FCF = (NI +DEP - WCR) - (INV)
  • = CFop + CFinv
  • From the statement of cash flows
  • FCF = - (SI + D - DIV) + CASH

Vietnam 2004

understanding fcf
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

Vietnam 2004

npv calculation example31
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%

Vietnam 2004

inflation
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

Vietnam 2004

scenario 2 inflation 100
Scenario 2 : Inflation = 100%

Nominal discount rate:

(1+10%) x (1+100%) = 2.20

Nominal rate = 120%

NPV now negative. Why?

Vietnam 2004

decomposition of npv
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