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R I S K A N A LY S I S , R E A L O P T I O N S , A N D C A P I TA L B U D G E T I N G

Chapter 9

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KEY CONCEPTS AND SKILLS

• Grasp and execute decision trees• Apply scenario and sensitivity analysis• Comprehend and utilize the various forms of

break-even analysis• Conceptualize Monte Carlo simulation• Practically apply real options in capital budgeting

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CHAPTER OUTLINE

9.1 Decision Trees9.2 Sensitivity Analysis, Scenario

Analysis, and Break-Even Analysis

9.3 Monte Carlo Simulation9.4 Real Options

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9.1 DECISION TREES

•Graphical representation of the alternatives available in each period and the likely consequences of our choices

•This graphical representation helps identify the best course of action.

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EXAMPLE OF A DECISION TREE

Do not study

Study finance

Squares represent decisions to be made. Circles represent receipt of information, e.g., a test score.

The lines leading away from the squares represent the alternatives.

“C”

“A”

“B”

“F”

“D”

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STEWART PHARMACEUTICALS

• Stewart Pharmaceuticals Corporation is considering investing in the development of a drug that cures the common cold.

• A corporate planning group, including representatives from production, marketing, and engineering, has recommended that the firm go ahead with the test and development phase.

• This preliminary phase will last two years and cost $1.2 billion. Furthermore, the group believes that there is a 60% chance that tests will prove successful.

• If the initial tests are successful, Stewart Pharmaceu-ticals can go ahead with full-scale production, starting at t = 2. This investment phase will cost $1.6 billion. Production will occur over the following 4 years.

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NPV FOLLOWING SUCCESSFUL TEST

Note that the NPV is calculated as of t = 2

years, the date at which the investment of $1,600 million is

made. Later we bring this number back to

time 0. Assume a cost of capital (or required

return) of 10%.

Investment Year 2 Yrs. 3-6Revenues $7,000(Variable Costs) (3,000)(Fixed Costs) (1,800)(Deprec’n Exp.) (400)Taxable Inc. $1,800(Tax Exp. (34%)) (612)Net Income $1,188

Cash Flow −$1,600 $1,588

NPV2 = −1600 +1588/0.10∙(1−1/1.104)

= 3433.75

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NPV AFTER UNSUCCESSFUL TEST

Investment Year 2 Yrs. 3-6Revenues $4,050(Variable Costs) (1,735)(Fixed Costs) (1,800)(Deprec’n Exp.) (400)Taxable Inc. $115(Tax Exp. (34%)) (39.10)Net Income $75.90

Cash Flow −$1,600 $475.90

NPV2 = −1600 +475.90/0.10∙(1−1/1.104)

= −91.46

Note that the NPV is calculated as of t = 2

years, the date at which the investment of $1,600 million is

made. Later we bring this number back to

time 0. Assume a cost of capital (or required

return) of 10%.

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DECISION TREE FOR STEWART

Do not test

Test

Failure

Success

Do not invest

Invest

Invest

The firm has two decisions to make:

To test or not to test.

To invest or not to invest.

NPV =$3433.75 M

NPV = $0

NPV = –$91.46 M

t=0

t=2

t=2

t=2

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DECISION WHETHER TO TEST

• Let’s move back to the first stage, where the decision boils down to the simple question: should we invest?

• The expected payoff evaluated at t = 2 is:

= 0.60x3433.75 + 0.40x0 = 2060.25,where, more precisely, the 3433.75 represents the MAX( 3433.75 , 0 ) and the 0 represents the MAX ( 0 , −91.46 ).The NPV evaluated at date 0 is:

So, weshould test.

NPV = −1200 + 2060.25/1.102 = 702.69.

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9.2 SENSITIVITY, SCENARIO, AND BREAK-EVEN

• Sensitivity Analysis:• Also called “What if” analysis

• Allows the calculation of a range of NPVs based on the probability of changes in NPV variables

• TIP: When working with spreadsheets:• build the model so variables can be adjusted in

a single cell;• And the NPV calculations update automatically.

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SENSITIVITY ANALYSIS: STEWART• We might want to know how sensitive NPV is to %Δs

in revenues. In the Stewart Pharmaceuticals example, a 5% drop in revenue leads to a 51.67% drop in NPV.

For every 1% drop in revenue*, we can expect roughly a 10% drop in NPV:

*Note: the drop in revenue is presumed due to a drop in drug’s price, not in its quantity.

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SCENARIO ANALYSIS: STEWART• A variation on sensitivity analysis is scenario

analysis.• For example, the following three scenarios

could apply to Stewart Pharmaceuticals:1. The next years each have heavy cold seasons,

and revenues exceed expectations, but labor costs (part of variable costs) skyrocket.

2. The next years are normal, and sales meet expectations.

3. The next years each have lighter than normal cold seasons, so sales fail to meet expectations.

• Other scenarios could apply to FDA approval.• For each scenario, calculate the NPV.

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BREAK-EVEN ANALYSIS

• Common tool for analyzing the relationship between sales volume and profitability

• There are three common break-even measures• Accounting break-even: sales volume at which net

income = 0• Cash break-even: sales volume at which operating cash

flow = 0• Financial break-even: sales volume at which net present

value = 0

• This discussion is most concerned with financial break-even

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BREAK-EVEN ANALYSIS: STEWART

• Stewart Pharmaceuticals might want to know the financial break-even revenue, financial break-even sales volume, or financial break-even price assuming a successful test outcome.

• To find any of these, we first determine the break-even operating cash flow. That is, we solve for C that sets NPV = 0:NPV = 0 = −1600 + C/0.10∙(1−1/1.104)

… thus, break-even C = 504.75

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BREAK-EVEN ANALYSIS: STEWART

• The project requires an investment of $1,600.

• In order to cover our cost of capital (financial break even), the project needs to generate a cash flow of $504.75 each year for four years.

• This is the project’s break-even operating cash flow, OCFBE.

PMT

I/Y

FV

N

− 504.75

10

0

1,600

4

PV

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BREAK-EVEN REVENUE: STEWART

Work backwards from OCFBE to Break-Even Revenue

Revenue $5,358.71

Variable cost $3,000

Fixed cost $1,800

Depreciation $400

EBT and EBIT $158.71

Net Income $104.75

$104.75

1−0.34

+D+FC

+ VC

Financial Break-even Net Income= Financial Break-even OCF − Deprec’n Exp. = 504.75 − 400.00 = 104.75

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BREAK-EVEN ANALYSIS: PBE

• Now that we have break-even revenue of $5,358.71 million, we can calculate break-even price.

• The original plan was to generate revenues of $7 billion by selling the cold cure at $10 per dose and selling 700 million doses per year,

• We can reach break-even revenue with a price of only:

$5,358.71 million = 700 million × PBE

PBE = = $7.66 / dose700

$5,358.71

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9.4 REAL OPTIONS

• One of the fundamental insights of modern finance theory is that options have value.

• The phrase “We are out of options” is surely a sign of trouble.

• Corporations make decisions in a dynamic environment • Choice of options should be considered in project

valuation.

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REAL OPTIONS

• The Option to Expand• Has value if demand turns out to be higher than expected

• The Option to Abandon• Has value if demand turns out to be lower than expected

• The Option to Delay• Has value if the underlying variables are changing with a

favorable trend

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DISCOUNTED CF AND OPTIONS

• We can calculate the market value of a project as the sum of the NPV of the project without options and the value of the managerial options implicit in the project.

M = NPV + Opt• We can also infer an option’s value if we know

the project’s NPV with the option (i.e., if we know M) and if we know what NPV is in the absence of an option.

A good example would be comparing the desirability of a specialized machine versus a more versatile machine. If they both cost about the same and last the same amount of time, the more versatile machine is more valuable because it comes with options.

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THE OPTION TO ABANDON: EXAMPLE

• Suppose we are drilling an oil well. The drilling rig costs $300 today, and in one year the well is either a success or a failure.

• A successful drill is 45% likely; a failure is 55% likely. The discount rate is 10%.

• The PV of the successful payoff at time one is $575.

• The PV of the unsuccessful payoff at time one is $0.

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THE OPTION TO ABANDON: EXAMPLE

Ignoring the time-1 abandonment option would lead to rejection of the project.

NPV = = –$64.771.10

$258.75–$300 +

Expected Payoff*

= (0.45×$575) + (0.55×$0) = $258.75

=Expected Payoff*

Prob. Success

× Successful Payoff

+ Prob. Failure

×Failure Payoff

* We could also call this variable “expected NPV at t = 1”

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THE OPTION TO ABANDON: EXAMPLE

The firm has two decisions to make: drill or not, abandon or stay.

Do not drill

Drill

Failure

Sell the rig; salvage value

= $250

Sit on rig; stare at empty hole:

PV = $0.

Traditional NPV analysis overlooks the option to abandon.

t=0

t=1

t=1

Success

t=1

Continue;PV = $575

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THE OPTION TO ABANDON: EXAMPLE

• When we include the value of the option to abandon, the drilling project should proceed:

NPV = = $60.231.10

$396.25–$300 +

Expected Payoff

= (0.45×$575) + (0.55×$250) = $396.25

=Expected Payoff

Prob. Success

×Successful Payoff

+ Prob. Failure

×Failure Payoff

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VALUING THE OPTION TO ABANDON

• Recall that we can calculate the market value of a project as the sum of the NPV of the project without the abandonment option and the value of the abandonment option implicit in the project.

M = NPV + Opt$60.23 = –$64.77 + Opt$60.23 + $64.77 = Opt

We can this infer the Option’s Value: $125.00

Intution: The option has a 55% chance of being exercised at t=1. If exercised, it adds $250.00 value ($250.00 minus $0). So, the option’s expected time-1 value is $137.50. Discount-ing this time-1 value back to time 0 at 10% gives the same

answer.

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THE OPTION TO DELAY: EXAMPLE

• Consider the above project, which can be undertaken in any of the next 4 years. The discount rate is 10 percent. The present value of the benefits at the time the project is launched remains constant at $25,000, but since costs are declining, the NPV at the time of launch steadily rises.

• The best time to launch the project is in year 2—this schedule yields the highest NPV when judged today.

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QUICK QUIZ

• What information does a decision tree provide?• What are sensitivity analysis, scenario analysis,

break-even analysis, and simulation?• Why are these analyses important, and how

should they be used?• How do real options affect the value of capital

projects?