Sensitivity and Scenario Analysis

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

Three Types of Risk 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.

Definitions Risk Probability Risk vs. Uncertainty 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

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

Issues in Risk Measurement Stand Alone Risk is the easiest to measure Market Risk is the most important to the shareholder To evaluate risk you need three things Standard deviation of the projects forecasted returns Correlation of the projects forecasted returns with the firms other assets Correlation of the projects forecasted returns with the market

Issues in Risk Management con’t Using the numbers in 3) you can find the corporate beta and market beta coefficient (equal to ((s/s)r) Most projects have a + correlation with other projects and a coefficient < 1 Most projects are positively correlated with the market with coefficient < 1 Corporate risk should also be examined More important to small business Investors may look at things other than market risk Firm Stability is important to creditors, suppliers etc

Stand Alone Risk (Review) 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

Measuring Stand Alone Risk Sensitivity Analysis Scenario Analysis Monte Carlo Simulation

Sensitivity Analysis 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.

Example 1 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 Payment 7.2% $24,322.44 8% $27,357.56 8.8% $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 Payment 1,800,000 $24,621.80 2,000,000 $27,357.56 2,200,000 $30,093.31

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

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

Scenario Analysis 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

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

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

Interpreting the Results 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)

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

Monte Carlo Simulation 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

Mont Carlo Simulation 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.

Monte Carlo Simulation Benefits More realistic selection of variables Easy to understand results Weaknesses Only as good as probability estimate and correlation of variables

Quick Review 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.