Chap 3
Net Present Value
Net Present Value
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Net present value is the single most widely used tool for large
investments made by corporations.
Klammer reported a survey of over 100 large companies
indicating that in 1959 only 19 percent used NPV techniques, but
by 1970, 57 percent used them.
Roughly a decade later, Schall, Sundem, and Geijsbeek sampled
424 large firms and found that 86 percent of those responding
used NPV.
It took over two decades for NPV to be widely accepted.
Undoubtedly, this rate of adoption was affected by the
introduction of pocket calculators and desktop personal
computers.
Net Present Value
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We want to review NPV carefully because it is the foundation for
real options analysis.
” The separation principle ” : it shows that the shareholders of a
firm will, regardless of their individual rates of time preference,
unanimously agree that the managers of the firm should
maximize shareholders’ wealth by taking investments that earn at
least the market-determined opportunity cost of capital.
Free cash flows of the project, the weighted average cost of
capital,
The equivalence of the risk-adjusted and the certainty-equivalent
methods of estimating the NPV of a project.
The separation principle
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Simple stated, the separation principle is the useful result that
shareholders of a firm will agree about the decision rule they
want managers to execute on their behalf – namely to undertake
investments until the marginal return on the last dollar invested is
greater than or equal to the market-determined opportunity cost
of capital .
Shareholders do not have to take a vote – they will unanimously
agree.
This is a critical keystone in the theory of decision making
because we do not have to construct a complicated rule for the
managers that requires that they acquire and use individual owner
( shareholder ) preferences.
Shows the rate of exchange between consumption today
and consumption at the end of the year that will leave
Mr. Crusoe with the same total utility.
 In effect, his marginal rate of substitution is his
subjective price of units of consumption tomorrow for
units of consumption today.
 He always requires extra units of consumption
tomorrow in return for giving up a unit of consumption
today.

 The
slope of a line drawn tangent to it is called
the marginal rate of transformation.
 Marginal
rate of substitution equals his marginal
rate of transformation.
 He will decide to remain at point B, where he
consumes
C0* today and C1* at the end of the period, and
invests (C0*  C1* ) today.
 Won’t they all choose different optimal
consumption/production points because they all
have different preferences for consumption over
time, and therefore difference indifference
curves? The answer is no.
C1
W0 
 C0
1 r
 Note that the production output at point B, which is
where the market line is just tangent to the production
opportunity set, provides him with the greatest feasible
*
W
wealth, 0 .
 It also provides him with the highest possible total
utility, because he will produce the output at point B,
then borrow against it at rate r to reach point C, a
*
W
bundle of consumption with wealth 0 .
 At
point C, his marginal rate of substitution is
the slope of the market line [ i.e., - (1 + r)].
 Also, the same market line is tangent to the
production opportunity set at point B.
 Therefore, if he maximizes his wealth, and his
total utility, he will choose to produce the
combination at point B, then borrow to move to
point C.
 At point C, his marginal rate of substitution
equals the slope of the market line, which in turn
equals his marginal rate of transformation.
 Therefore,
all individuals, regardless of their
time preferences for consumption today versus
consumption tomorrow, will choose to invest
until the marginal rate of return on the last unit
of investment is just equal to the market rate ( at
point B ).
 This separation principle means that the wealthmaximizing rule for investment is separate from
any information about individual utility
functions.
Estimating free cash flows
 The
first step will always be to estimate the
present value of the project without flexibility.
 The free cash flows that are payable to both
sources of capital – debt and equity.
$225 $225
S

 $750
ks
.30
$50
B
 $500
.10
 Adding
these values together gives the value of
the firm, $1,250.
 The shareholders receive $1,250 in year 5 but
must pay $500 to bondholders.
The weighted average cost of capital
( WACC )
The weighted average cost of capital is the weighted
average of the after-tax marginal costs of capital.
 It is appropriate for discounting entity or project cash
flows because these cash flows are available for
payment to both sources of capital – debt and equity.
 Market value weights are used because the market value
of capital committed, not the book value, determines the
cash flow required on investment.

B
S
WACC  kb (1  T )
 ks
BS
BS
500
750
 .10(1  0.5)
 .30
500  750
500  750
 .02  .18  20%
 We
find other debt with the same risk and
assume that its yield to maturity is the same as
ours.
 Comparable securities in the capital markets to
estimate the capital for our project.
 One
of the advantages of discounting the firm’s
free cash flows at the after-tax weight average
cost of capital is that this technique separates the
investment decisions of the firm from its
financing decisions.
 The definition of free cash flows shows what the
firm will earn after taxes assuming that it has no
debt capital.
 Thus,
changes in the assumed debt-to-equity
ratio have no effect on the definition of cash
flows for capital budgeting purposes.
 The effect of financial decisions is reflected in
the cost of capital.
Certainty-equivalent approach to net
present value
It is possible to estimate the value of a project either by
taking its expected future free cash flows and
discounting them at a risk-adjusted weighted average
cost of capital, or to risk-adjust the cash flows and
discount them at the risk-free rate.
 The answer should be the same either way.
 The certainty-equivalent approach is a common method
for valuing options in a lattice.
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 Consider
a simple one-period example.
 A project’s expected cash flows are $1,000, the
risk-free rate is10 percent, the expected rate of
return on the market is 17 percent, and the
project’s beta is 1.5.
 If it is an all-equity firm, then its present value is
E ( FCF )
PV 
1  R f  [ E ( Rm )  R f ] j
$1, 000
$1, 000


 $829.88
1  .10  (.17  .10)1.5 1, 205
 If
the investment outlay is $800, then its net
present value is
NPV = PV – I = $829.88 - $800 = $29.88
j 
COV ( R j , Rm )
VAR( Rm )
FCF  PV FCF
Rj 

1
PV
PV
 FCF

COV 
 1, Rm 
1  COV ( FCF , Rm ) 
PV


j 



VAR( Rm )
PV  VAR( Rm )

  E ( Rm  R f )  VAR ( Rm ) is
the market price of risk of
risk in the capital asset pricing model.
 This approach adjusts for risk by subtracting a
penalty from expected cash flows to first obtain
certainty-equivalent cash flows, then it discounts
them at the risk-free rate.
 $1,000 - $87.13 = $912.87
$1, 000
E ( FCF )
PV 

 $829.88
1.205 1  risk  adjusted rate
$912.87 E ( FCF )  risk premium
PV 

 $829.88
1.10
1  riskfree rate
 That
we can obtain the same answer using either
a risk-adjusted or a risk-neutral approach.
Differences between the net present value
and the real options approaches
N
E ( FCFt )
NPV   I  
t
t 1 (1  WACC )
 Note
that the uncertainty of cash flows is not
explicitly modeled in the NPV approach.
 One merely discounts expected cash flows.
NPV rule : MAX (at t  0)[0, E0 (VT  X )]
 The
problem solution is to compare all possible
mutually exclusive routes to determine their
value, E0 (VT  X ) , then to choose the best
among them.
 Mathematically,
a call option is an expectation
of maximums ( not a maximum of
expectations ) :
ROA rule : E0 MAX (at t  T )[0,VT  X ]