Chapter 14
Risk and Managerial (Real) Options in Capital Budgeting
Learning Objectives
After studying Chapter 14, you should be able to:• Define the "riskiness" of a capital investment project. • Understand how cash-flow riskiness for a particular period
is measured, including the concepts of expected value, standard deviation, and coefficient of variation.
• Describe methods for assessing total project risk, including a probability approach and a simulation approach.
• Judge projects with respect to their contribution to total firm risk (a firm-portfolio approach).
• Understand how the presence of managerial (real) options enhances the worth of an investment project.
• List, discuss, and value different types of managerial (real) options.
• The Problem of Project Risk
• Total Project Risk
• Contribution to Total Firm Risk: Firm-Portfolio Approach
• Managerial (Real) Options
• The Problem of Project Risk
• Total Project Risk
• Contribution to Total Firm Risk: Firm-Portfolio Approach
• Managerial (Real) Options
Topics
ANNUAL CASH FLOWS: YEAR 1PROPOSAL APROPOSAL A
State ProbabilityProbability Cash FlowCash Flow
Deep Recession .10 $ 3,000Mild Recession .20 3,500Normal .40 4,000Minor Boom .20 4,500Major Boom .10 5,000
ANNUAL CASH FLOWS: YEAR 1PROPOSAL APROPOSAL A
State ProbabilityProbability Cash FlowCash Flow
Deep Recession .10 $ 3,000Mild Recession .20 3,500Normal .40 4,000Minor Boom .20 4,500Major Boom .10 5,000
An Illustration of Total Risk (Discrete Distribution)
.40
.10
.20
Pro
babi
lity
3,000 4,000 5,000
Cash Flow ($)
Probability Distribution of Year 1 Cash Flows (Proposal A)
CFCF11 PP11 (CFCF11)()(PP11))
$ -3,000 .10 $ 300 1,000 .20 700 5,000 .40 1,600 9,000 .20 900 13,000 .10 500
=1.001.00 CFCF11=$4,000$4,000
CFCF11 PP11 (CFCF11)()(PP11))
$ -3,000 .10 $ 300 1,000 .20 700 5,000 .40 1,600 9,000 .20 900 13,000 .10 500
=1.001.00 CFCF11=$4,000$4,000
Expected Value of Year 1 Cash Flows (Proposal A)
(CFCF11)()(PP11)) ( (CFCF11 - - CFCF11))22((PP11) )
$ 300 ( 3,000 - 4,000)22 ( (.10.10)= 100,000)= 100,000 700 ( 3,500 - 4,000)22 ( (.20.20)= 50,000)= 50,000
1,600 ( 4,000 - 4,000)22 ( (.40.40)= 0)= 0 900 ( 4,500 - 4,000)22 ( (.20.20)= 50,000)= 50,000
500 ( 5,000 - 4,000)22 ( (.10.10)= 100,000)= 100,000 $4,000$4,000 300,000300,000
(CFCF11)()(PP11)) ( (CFCF11 - - CFCF11))22((PP11) )
$ 300 ( 3,000 - 4,000)22 ( (.10.10)= 100,000)= 100,000 700 ( 3,500 - 4,000)22 ( (.20.20)= 50,000)= 50,000
1,600 ( 4,000 - 4,000)22 ( (.40.40)= 0)= 0 900 ( 4,500 - 4,000)22 ( (.20.20)= 50,000)= 50,000
500 ( 5,000 - 4,000)22 ( (.10.10)= 100,000)= 100,000 $4,000$4,000 300,000300,000
Variance of Year 1 Cash Flows (Proposal A)
Summary of Proposal A
Standard deviation Standard deviation = SQRT (300,000)= $548$548
Expected cash flow Expected cash flow = $4,000$4,000
Coefficient of Variation (CV)Coefficient of Variation (CV) = $548 / $4,000 == $548 / $4,000 = 0.140.14
CV is a measure of CV is a measure of relativerelative risk and is the ratio of standard risk and is the ratio of standard deviation to the mean of the distribution.deviation to the mean of the distribution.
ANNUAL CASH FLOWS: YEAR 1PROPOSAL BPROPOSAL B
State ProbabilityProbability Cash FlowCash Flow
Deep Recession .10 $ 2,000Mild Recession .20 3,000Normal .40 4,000Minor Boom .20 5,000Major Boom .10 6,000
ANNUAL CASH FLOWS: YEAR 1PROPOSAL BPROPOSAL B
State ProbabilityProbability Cash FlowCash Flow
Deep Recession .10 $ 2,000Mild Recession .20 3,000Normal .40 4,000Minor Boom .20 5,000Major Boom .10 6,000
An Illustration of Total Risk (Discrete Distribution)
.40
.10
.20
Pro
babi
lity
2,000 3,000 4,000 5,000 6,000
Cash Flow ($)
Probability Distribution of Year 1 Cash Flows (Proposal B)
CFCF11 PP11 (CFCF11)()(PP11))
$ 2,000 .10 $ 200 3,000 .20 600 4,000 .40 1,600 5,000 .20 1,000 6,000 .10 600
=1.001.00 CFCF11=$4,000$4,000
CFCF11 PP11 (CFCF11)()(PP11))
$ 2,000 .10 $ 200 3,000 .20 600 4,000 .40 1,600 5,000 .20 1,000 6,000 .10 600
=1.001.00 CFCF11=$4,000$4,000
Expected Value of Year 1 Cash Flows (Proposal B)
(CFCF11)()(PP11)) ((CFCF11 - - CFCF11))22((PP11))
$ 200 ( 2,000 - 4,000)22 ( (.10.10) = 400,000 ) = 400,000 600 ( 3,000 - 4,000)22 ( (.20.20) = 200,000) = 200,000 1,600 ( 4,000 - 4,000)22 ( (.40.40) = 0) = 0 1,000 ( 5,000 - 4,000)22 ( (.20.20) = 200,000) = 200,000 600 ( 6,000 - 4,000)22 ( (.10.10) = 400,000) = 400,000 $4,000$4,000 1,200,000 1,200,000
(CFCF11)()(PP11)) ((CFCF11 - - CFCF11))22((PP11))
$ 200 ( 2,000 - 4,000)22 ( (.10.10) = 400,000 ) = 400,000 600 ( 3,000 - 4,000)22 ( (.20.20) = 200,000) = 200,000 1,600 ( 4,000 - 4,000)22 ( (.40.40) = 0) = 0 1,000 ( 5,000 - 4,000)22 ( (.20.20) = 200,000) = 200,000 600 ( 6,000 - 4,000)22 ( (.10.10) = 400,000) = 400,000 $4,000$4,000 1,200,000 1,200,000
Variance of Year 1 Cash Flows (Proposal B)
Summary of Proposal B
Standard deviation Standard deviation = SQRT (1,200,000) = $1,095$1,095
Expected cash flow Expected cash flow = $4,000$4,000
Coefficient of Variation (CV)Coefficient of Variation (CV) = $1,095 / $4,000 = = $1,095 / $4,000 = 0.270.27
Comparison of Proposal A & B
The standard deviation of B B > > AA ( ($1,095 $1,095 > > $548$548), so “), so “BB” is ” is moremore risky than “A”. risky than “A”.
The coefficient of variation of B > A (The coefficient of variation of B > A (0.27 0.27 < < 0.140.14), so “), so “BB” has ” has higherhigher relative risk than “ relative risk than “AA”.”.
Proposal A Proposal B
Standard deviation Standard deviation $548$548 $1,095$1,095
Expected cash flow Expected cash flow $4,000$4,000 $4,000$4,000
Coefficient of Variation (CV) Coefficient of Variation (CV) 0.140.14 0.270.27
Total Project Risk
Projects have risk that may change from period to period.
Projects are more likely to have
continuous, rather than discrete distributions.
Cas
h F
low
($)
11 22 33 Year
Probability Tree Approach
A graphic or tabular approach for organizing the possible cash-flow streams generated by an investment. The presentation resembles the branches of a tree. Each complete branch represents one possible cash-flow sequence.
Probability Tree Approach
Basket Wonders is examining a project that will have an initial cost initial cost today of $240$240. Uncertainty surrounding the first year cash flows creates three possible cash-flow scenarios in Year 1Year 1.
-$240-$240
Probability Tree Approach
Node 1: 25% chance of a $500$500 cash-flow.
Node 2: 50% chance of a $200$200 cash-flow.
Node 3: 25% chance of a -$100$100 cash-flow.
-$240-$240
(.25) $500$500
(.25) -$100100
(.50) $200$200
Year 1Year 1
11
22
33
Probability Tree Approach
Each node in Year 2 Year 2 represents a branchbranch of our probability tree.
The probabilities are said to be conditional conditional probabilitiesprobabilities.
-$240-$240
(.25.25) $500$500
(.25.25) -$100-$100
(.5050) $200$200
Year 1Year 1
11
22
33
(.40) $500$500
(.20) $200$200
(.40) $800$800
(.20) $ 500$ 500
(.60) $ 200$ 200
(.20) -$ 100 -$ 100
(.20) $ 200$ 200
(.40) -$ 100-$ 100
(.40) -$ 400-$ 400
Year 2Year 2
Joint Probabilities [P(1,2)]
.10 Branch 1
.10 Branch 2
.05 Branch 3
.10 Branch 4
.30 Branch 5
.10 Branch 6
.05 Branch 7
.10 Branch 8
.10 Branch 9
-$240-$240
(.25.25) $500$500
(.25.25) -$100-$100
(.5050) $200$200
Year 1Year 1
11
22
33
(.40) $500$500
(.20) $200$200
(.40) $800$800
(.20) $500$500
(.60) $400$400
(.20) - -$100$100
(.20) $200$200
(.40) -$100-$100
(.40) -$400-$400
Year 2Year 2
Project NPV Based on Probability Tree
The probability tree accounts for the distribution of
cash flows. Therefore,
discount all cash flows at only the risk-freerisk-free rate of
return.
The NPV for branch i NPV for branch i of the probability tree for two years
of cash flows is
NPV = (NPVNPVii)(PPii)
NPVNPVii = CFCF11
(1 + RRff )11 (1 + RRff )22
CFCF22
- ICOICO
+
i = 1
z
NPV for Each Cash-Flow Stream at 8% Risk-Free Rate
$ 909
$ 652
$ 394
$ 374
$ 117
-$ 141
-$ 161
-$ 418
-$ 676
-$240-$240
(.25.25) $500$500
(.25.25) -$100-$100
(.5050) $200$200
Year 1Year 1
11
22
33
(.40) $ 500$ 500
(.20) $ 200$ 200
(.40) $ 800$ 800
(.20) $ 500$ 500
(.60) $ 200$ 200
(.20) -$ 100 -$ 100
(.20) $ 200$ 200
(.40) -$ 100-$ 100
(.40) -$ 400-$ 400
Year 2Year 2
Calculating the Expected Net Present Value (NPV)
Branch NPV NPVii
Branch 1 $ 909Branch 2 $ 652Branch 3 $ 394Branch 4 $ 374Branch 5 $ 117Branch 6 -$ 141Branch 7 -$ 161Branch 8 -$ 418Branch 9 -$ 676
P(1,2) P(1,2) NPVNPVii * P(1,2) P(1,2) .10 $ 91 .10 $ 65 .05 $ 20 .10 $ 37 .30 $ 35 .10 -$ 14 .05 -$ 8 .10 -$ 42 .10 -$ 68
Expected Net Present Value Expected Net Present Value = $116$116
Calculating the Variance of the Net Present Value
NPVNPVii
$ 909 $ 652 $ 394 $ 374 $ 117-$ 141-$ 161-$ 418-$ 676
P(1,2) P(1,2) ((NPVNPVii - NPVNPV )2[P(1,2)P(1,2)] .10 $ 62,884.90 .10 $ 28,729.60 .05 $ 3,864.20 .10 $ 6,656.40 .30 $ 0.30 .10 $ 6,604.90 .05 $ 3,836.45 .10 $ 28,515.60 .10 $ 62,726.40
Variance Variance = $203,818.75$203,818.75
Calculating the Variance of the Net Present Value
Prob(CF1) CF1
Prob(CF2) CF2
Joint Prob. NPV EV(NPV) Var(NPV)
0.25 500 0.4 800 0.1 $908.83 $90.88 62755.08
0.25 500 0.4 500 0.1 $651.63 $65.16 28620.30
0.25 500 0.2 200 0.05 $394.43 $19.72 3858.02
0.5 200 0.2 500 0.1 $373.85 $37.39 6615.27
0.5 200 0.6 200 0.3 $116.65 $35.00 0.00
0.5 200 0.2 -100 0.1 ($140.55) ($14.05) 6615.27
0.25 -100 0.2 200 0.05 ($161.12) ($8.06) 3858.02
0.25 -100 0.4 -100 0.1 ($418.33) ($41.83) 28620.30
0.25 -100 0.4 -400 0.1 ($675.53) ($67.55) 62755.08
$116.65 $203,697.35
Summary of the Decision Tree Analysis
Standard deviation Standard deviation = SQRT ($203,697) = $451.33$451.33
Expected NPV Expected NPV = $116.65$116.65
Simulation Approach
An approach that allows us to test the possible results of an investment proposal before it is accepted. Testing is based on a model coupled with probabilistic information.
Simulation Approach
– Market analysisMarket analysis• Market size, selling price, market
growth rate, and market share
– Investment cost analysisInvestment cost analysis• Investment required, useful life of facilities,
and residual value– Operating and fixed costsOperating and fixed costs
• Operating costs and fixed costs
Factors we might consider in a model:
Simulation Approach
Each variable is assigned an appropriate probability distribution. The distribution for the selling price of baskets created by Basket Wonders might look like:
$20 $25 $30 $35 $40 $45 $50.02 .08 .22 .36 .22 .08 .02
The resulting proposal value is dependent on the distribution and interaction of EVERY variable.
Simulation Approach
Each proposal will generate an internal rate of internal rate of returnreturn. The process of generating many, many simulations results in a large set of internal rates of return. The distributiondistribution might look like the following:
INTERNAL RATE OF RETURN (%)
PR
OB
AB
ILIT
YO
F O
CC
UR
RE
NC
E
Combining projects in this manner reduces the firm risk due to diversificationdiversification.
Combining projects in this manner reduces the firm risk due to diversificationdiversification.
CA
SH
FLO
W
TIME TIMETIME
Proposal AProposal A Proposal BProposal BCombination of Combination of
Proposals Proposals AA andand BB
Contribution to Total Firm Risk: Firm-Portfolio Approach
NPVP is the expected portfolio NPV,
NPVj is the expected NPV of the jth NPV that the firm undertakes,
m is the total number of projects in the firm portfolio.
NPVP is the expected portfolio NPV,
NPVj is the expected NPV of the jth NPV that the firm undertakes,
m is the total number of projects in the firm portfolio.
Determining the Expected NPV for a Portfolio of Projects
m
jjP NPVNPV
1
jk is the covariance between possible NPVs for projects
j and k
jk = j k rrjk .
j is the standard deviation of project j,
k is the standard deviation of project k,
rjk is the correlation coefficient between projects j & k.
jk is the covariance between possible NPVs for projects
j and k
jk = j k rrjk .
j is the standard deviation of project j,
k is the standard deviation of project k,
rjk is the correlation coefficient between projects j & k.
Determining Portfolio Standard Deviation
m
j
m
kjkp
1 1
E: Existing ProjectsE: Existing Projects
8 Combinations
EE EE + 1 EE + 1 + 2 EE + 2 EE + 1 + 3EE + 3 EE + 2 + 3
EE + 1 + 2 + 3
AA, BB, and CC are dominatingdominating combinations from the eight
possible.
Combinations of Risky Investments
A
B
C
E
Standard Deviation
Exp
ecte
d V
alue
of
NP
V
Managerial (Real) Options
Management flexibility to make future decisions that affect a project’s expected cash flows, life, or future acceptance.
Project Worth = NPV + Option(s) Value
Managerial (Real) Options
Expand (or contract)Expand (or contract)
– Allows the firm to expand (contract) production if conditions become favorable (unfavorable).
AbandonAbandon
– Allows the project to be terminated early.
PostponePostpone
– Allows the firm to delay undertaking a project (reduces uncertainty via new information).
Probability Tree Approach
Node 1: 25% chance of a $1M$1M cash-flow.
Node 2: 50% chance of a $2M$2M cash-flow.
Node 3: 25% chance of a $3M$3M cash-flow.
-$3M-$3M
(.25) $1M$1M
(.25) $3M3M
(.50) $2M$2M
Year 1Year 1
11
22
33
Example without Project Abandonment
Assume that this project can be abandoned at the end of the first year for $1.5M$1.5M.
What is the project worthproject worth?
-$900-$900
(.25.25) $1M$1M
(.25.25) $3M$3M
(.5050) $2M$2M
Year 1Year 1
11
22
33
(.50) $ 1M$ 1M
(.25) $ 2M$ 2M
(.25) $ 0$ 0
(.25) $ 1M$ 1M
(.50) $ 2M$ 2M
(.25) $ 3M$ 3M
(.25) $ 2M$ 2M
(.50) $ 3M$ 3M
(.25) $ 3.5M$ 3.5M
Year 2Year 2
Example without Project Abandonment
Prob(CF1) CF1
Prob(CF2) CF2
Joint Prob. NPV EV(NPV) Var(NPV)
0.25 1000000 0.25 0 0.0625 ($2,090,909.09) ($130,681.82) 402005778400.550.25 1000000 0.5 1000000 0.125 ($1,264,462.81) ($158,057.85) 365388853433.850.25 1000000 0.25 2000000 0.0625 ($438,016.53) ($27,376.03) 48759756953.93
0.5 2000000 0.25 1000000 0.125 ($355,371.90) ($44,421.49) 80124014966.530.5 2000000 0.5 2000000 0.25 $471,074.38 $117,768.60 166751331.880.5 2000000 0.25 3000000 0.125 $1,297,520.66 $162,190.08 90796100206.61
0.25 3000000 0.25 2000000 0.0625 $1,380,165.29 $86,260.33 54629403835.970.25 3000000 0.5 3000000 0.125 $2,206,611.57 $275,826.45 387800232438.020.25 3000000 0.25 3500000 0.0625 $2,619,834.71 $163,739.67 295551728130.76
$445,247.93 1725222619698.11
$1,313,477.30
Decision Trees
• Decision tree graphically displays all decisions in a complex project and all the possible outcomes with their probabilities.
Decision Node
D1
D2
DX
Chance Node
C1
C2
CY
p1
p2
py
Outcome Node
Pruned Branch
Decision Tree (14.3)(New Product Development – with Abandonment)
1. Build New Product
2. Volume forNew Product
3. $0
No
YesFirst cost=$3M
7. CF1=$1.5M
Low Volume P=0.25
Med. Vol.
P=0.5
High Volume P=0.25
Terminate
Continue
t=0 t=1 t=2, …,
6. CF1=$3M
8.CF2=$0 (.25)
9.CF2=$1M (.5)
10.CF2=$2M (.25)
11.CF2=$1 (.25)
12.CF2=$2M (.5)
13.CF2=$3M (.25)Continue
14.CF2=$2 (.25)
15.CF2=$3M (.5)
16.CF2=$3.5M (.25)
5. CF1=$2M
4. CF1=$1M
Expand
Continue
Decision Tree (14.4)(New Product Development – with Abandonment)
1. Build New Product
2. Volume forNew Product
3. $0
No
YesFirst cost=$3M
7. CF1=$1.5M
Low Volume P=0.25
Med. Vol.
P=0.5
High Volume P=0.25
Terminate
Continue
t=0 t=1 t=2, …,
6. CF1=$3M
8.CF2=$0 (.25)
9.CF2=$1M (.5)
10.CF2=$2M (.25)
11.CF2=$1 (.25)
12.CF2=$2M (.5)
13.CF2=$3M (.25)Continue
14.CF2=$2 (.25)
15.CF2=$3M (.5)
16.CF2=$3.5M (.25)
5. CF1=$2M
4. CF1=$1M
Expand
Continue
EV(CF2)=1MEV(CF1)=.909M
Example with Project Abandonment
Prob(CF1) CF1
Prob(CF2) CF2
Joint Prob. NPV EV(NPV) Var(NPV)
0.25 2500000 1 0 0.25 ($727,272.73) ($181,818.18) 426943440082.65
0.5 2000000 0.25 1000000 0.125 ($355,371.90) ($44,421.49) 109258807671.95
0.5 2000000 0.5 2000000 0.25 $471,074.38 $117,768.60 2941493494.30
0.5 2000000 0.25 3000000 0.125 $1,297,520.66 $162,190.08 64436049663.62
0.25 3000000 0.25 2000000 0.0625 $1,380,165.29 $86,260.33 40062007483.27
0.25 3000000 0.5 3000000 0.125 $2,206,611.57 $275,826.45 330918018108.39
0.25 3000000 0.25 3500000 0.0625 $2,619,834.71 $163,739.67 260173765559.90
$579,545.45 1234733582064.07
$ 1,111,185.66