Any model we use has the potential to have error
How do we account for the uncertainty associated with our inputs into the model?

Stand Alone Risk
Views project in isolation
With-in firm (Corporate Risk)
Looks at the firms portfolio of projects and how they interact
Market Risk
Risk from the view of a well diversified investor.

Risk
Exposure to a chance of injury or loss
Probability
The likelihood an event occurs
Risk vs. Uncertainty
Risk – the probability of the outcome is known
Uncertainty – includes judgment concerning the probability

Objective Prob –can measure prob. precisely
Subjective Prob. – Includes judgment or opinion
Variation Risk – We want to look at a range of possible outcomes

1.
Stand Alone Risk is the easiest to measure
2.
Market Risk is the most important to the shareholder
3.
To evaluate risk you need three things i.
Standard deviation of the projects forecasted returns ii. Correlation of the projects forecasted returns with the firms other assets iii. Correlation of the projects forecasted returns with the market

4. Using the numbers in 3) you can find the corporate beta and market beta coefficient (equal to (( s/s)r)
5. M ost projects have a + correlation with other projects and a coefficient < 1
6. Most projects are positively correlated with the market with coefficient < 1
7. Corporate risk should also be examined
1. More important to small business
2. Investors may look at things other than market risk
3. Firm Stability is important to creditors, suppliers etc

The easiest approach to measuring stand alone risk is to use the standard deviation of the projects returns.
Just like security analysis you need to be careful looking at only standard deviation – don’t forget coefficient of variation

Sensitivity Analysis
Scenario Analysis
Monte Carlo Simulation

Looks at the change in your decision variable when one input changes.
Examples: what happens to the value of a project if sales are 10% higher than expected.
What happens to the cost of capital if the risk free rate increases.

Basic time value of money problem.
Assume you believe you need 2,000,000 when you retire and you are now 25 years old.
How much will you need to deposit each year at the end of the year if your account earns 8% each year?
$27,357.56

Example 1 Change ONLY
Expected return
What if your estimated rate of return is of by 10% of the base (What if your account earns 8.8% each year?
Or 7.2% each year)?
Rate
7.2%
8%
8.8%
Payment
$24,322.44
$27,357.56
$30,724.44

Example 2: change ONLY
Amount Needed for Retirement
Now assume that the amount you need for retiremetn may be off either way by 10%
Savings Needed
1,800,000
2,000,000
2,200,000
Payment
$24,621.80
$27,357.56
$30,093.31

Usually the results are represented in a table where the response of the decision variable to changes in more than one individual variable are reported.
Then you can compare across variables to see which one has the largest impact on your decision

Change in Payment needed for Retirement
Savings Needed Expected Return
+10% $24,621.80
$24322.44
Base
-10%
$27,357.56
$30,093.31
$27,357.56
$30,724.49

a.
b.
Benefits
Easy to Calculate and Understand
Measures risk associated with individual inputs
Weaknesses a.
b.
c.
Ignores probability of event
Ignores interaction among the variables
Ignores gains from diversification

Differences from Sensitivity Analysis
Allows you to change more than one variable at a time
Look at a group of scenarios (best case, base case, and worst case) for example worst case – what if all variables change against us by 20%….
Includes probability estimates of each scenario

Now let both the future cash flows and the cost of capital change.
Worst Case Scenario Best Case Scenario
(Savings h Return i ) (Savings i Return h )
Need $2,200,000
Return = 7.2%
PMT = 33,796.89
Need $1,800,000
Return = 8.8%
PMT = $21,890.20

Given the NPV and Probability you can find the expected NPV and standard deviation
Scenario NPV Prob. NPV(Prob)
Worst
Base
Best
$33,796.89
$27357.56
$21,890.20
Expected NPV
.33
.33
.33
$11,265.63
$ 9,119.17
$ 7,296.73
$27,681.55
Standard Deviation $ 5,959.95

The project has an expected return on
4204.94 with standard deviation of 741.38
This implies a 68 % confidence interval of
(3463.56 to 4946.32) a large range of possible outcomes
The coefficient of variation would be .1763
(you are accepting .1763 units of risk for each unit of return)

1.
2.
3.
Benefits
More than one variable changes at a time
Accounts for probability
Easy to perform
1.
2.
Weaknesses
Small number of scenarios is unrealistic
Probability distributions difficult to estimate

A more advanced form of scenario analysis
Utilizes the computer to make random choices for each variable input then calculate the expected return and standard deviation

4.
5.
1.
2.
3.
Construct a model of the firms cash flows and
NPV’s
Specify a probability distribution for each uncertain variable (characterized by mean and standard dev) and correlation among variables.
Allow computer to select a random draw form the distribution for each variable
Calculate NPV (this is one scenario).
Repeat 3) an 4) (10,000 or 100,000 times) equal chance of each scenario –Calculate expected NPV and standard deviation.

1.
2.
Benefits
More realistic selection of variables
Easy to understand results
1.
Weaknesses
Only as good as probability estimate and correlation of variables

Sensitivity Analysis Scenario Analysis and
Monte Carlo Simulation were all used to measure stand alone risk
Each is designed to provide more information about the uncertainty associated with the project – they do not provide a clear cut decision rule.