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