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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

Net future value

- NFV 121 - 100 ? 1.10 11
- C1 - I (1r)
- Decision rule invest if NFVgt0
- Justification takes into cost of capital
- cost of financing
- opportunity cost

121

100

0

1

-100

-110

Net Present Value

- NPV - 100 121/1.10
- 10
- - I C1/(1r)
- - I C1 ? DF1
- DF1 1-year discount factor
- a market price
- C1 ? DF1 PV(C1)
- Decision rule invest if NPVgt0
- NPVgt0 ? NFVgt0

121

110

-100

-121

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 gt r

IRR versus NPV

- In this simple setting, the NPV rule and the Rate

of Return Rule lead to the same decision - NPV -IC1/(1r) gt0
- ? C1gtI(1r)
- ? (C1-I)/Igtr
- ? IRRgtr

IRR a general definition

- The Internal Rate of Return is the discount rate

such that the NPV is equal to zero. - -I C1/(1IRR) ? 0
- In our example
- -100 121/(1IRR)0
- ? IRR21

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

lt0 - 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/(1r)5 - a market price

NPV general formula

- Cash flows C0 C1 C2 Ct CT
- t-year discount factor DFt 1/(1r)t
- NPV C0 C1 DF1 Ct DFt CT DFT

NPV calculation - example

- Suppose r 10

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

Constant perpetuity

Proof PV C d C d² C d3 PV(1r) C C

d C d² PV(1r) 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

Growing perpetuity

- Ct C1 (1g)t-1 for t1, 2, 3, .....

rgtg - 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

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.

Growing annuity

- Ct C1 (1g)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 T1 with first cash flow C1

(1g)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

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

Review Shortcut formulas

- Constant perpetuity Ct C for all t
- Growing perpetuity Ct Ct-1(1g)
- rgtg t 1 to 8
- Constant annuity CtC t1 to T
- Growing annuity Ct Ct-1(1g)
- t 1 to T

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

NPV Profiles

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

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

The Profitability Index (PI) Rule

- PI Total Present Value of future CFs / Initial

Investment - Minimum Acceptance Criteria Accept if PI gt 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

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

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

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

Interest-bearing debt definition

- Long-term debt
- Current maturities of long term debt
- Notes payable to banks

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

- (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)

Free cash flow

- FCF (NI DEP - ?WCR) - (INV)
- CFop CFinv
- From the statement of cash flows
- FCF - (SI ?D - DIV) ?CASH

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

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

Scenario 1 no 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

Scenario 2 Inflation 100

Nominal discount rate (110) x (1100)

2.20 Nominal rate 120

NPV now negative. Why?

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