Chapter 11 – Introduction to Risk Analysis u Why do individuals, companies, and stockholders take risks?

Terminology u Risk - possibility of an undesired outcome u Probability - expected relative frequency of an event

Terminology u Risk and uncertainty u Risk - probabilities of outcomes is known -- casino u Uncertainty - outcomes not known with certainty – reality u Probability distributions - u discrete - number of probability occurrences is finite u continuous - infinite number of occurrence - range of outcomes

Terminology u Subjective versus objective probability u Subjective - someone’s opinion u Objective - can be measured

Terminology u Variation versus event risk u Event risk - probability of a certain, such as bankruptcy u Variation risk- probability of a range of outcomes around an event u typically measured by standard deviation

Terminology u Diversifiable versus nondiversifiable risk u Diversifiable risk -- risk that can be reduced or eliminated by combining one investment with another u Must have a correlation less than +1 u Nondiversifiable risk -- risk that remains after combining large numbers of projects

Probability Rules u Mutually exclusive events - u add the probabilities of the events u Independent events u build a table of possible combinations of events u multiply the probabilities to get table values u Dependent events u build a table where the probabilities of outcomes for the second event are dependent on the first event

Stages of Risk Measurement u Stage 1 -- Descriptive and Subjective u listing of things that might go wrong u good for identifying the important variables u Stage 2 -- Sensitivity analysis u look at possible outcomes over a range of values for a critical variable (such as sales) u do not attempt to assign probabilities u example -- breakeven analysis

Stages of Risk Measurement u Stage 3 -- Event probability u assign probabilities to the various outcomes u one in ten chance of bankruptcy u Stage 4 -- Summary measures of probability distributions u Measures of central tendency u Measures of dispersion

Summary Measures: Central Tendency u Expected value: possibilities time probabilities u Median: Center outcome; probability of outcome above median equals probability of outcome below median. u Mode: Most common outcome u Geometric mean (Pi = probability of outcome i): [(1+ return 1 )^P 1 ][(1+ return 2 )^P 2 ]....

Summary Measures: Dispersion u Variance u Multiply squared distances from the expected value by the probability, then sum u Standard deviation u Square root of the variance u Same unit of measure as the original problem u Coefficient of variation u standard deviation/ expected value u adjust for the scale of the project

Summary Measures: Dispersion u Semivariance u computed like variance, but considers only outcomes below the expected value u used when the distribution is not normal (skewed) u Quartile range u There is a 25% probability of a value greater than X and a 25% probability of a value less than Y

Summary Measures u Normal distributions and standard deviations u using a z-table you can find the area under the normal curve (probability of a range of outcomes)

Utility Theory u Assumptions u Completeness -- you can judge your preference in all situations u Rational -- consistent in judgements order of presentation does not matter u Transitivity -- if A is preferred over B and B is preferred over C then A is preferred over C

Utility Theory u Types of utility functions u Increasing -- risk seeker or lover -- will pay to take the riskier project -- casinos and lottery tickets u Constant -- risk neutral -- is indifferent to risk -- will accept the same expected return for risky as well as safe projects u Decreasing -- risk averse -- prefer safety to risk and must be compensated for accepting additional risk

Utility Theory u Problems with utility functions in reality u Hard to measure u Whose utility should we measure? u Once measured then the decision can be made by the analyst u Utility theory is important to arbitrage pricing theory u equal expected utilities should have equal prices

Risk Perspectives u Single investment perspective u Proposing manager -- Chapter 12 u Company perspective u Senior management and board -- Chapter 13 u Shareholder perspective u Shareholder -- Chapter 14 u Contingent claims u Option writer, debt-holder -- Chapter 15 u Overall economy u Everybody