Chapter 11 – Chapter 11 – Introduction to Risk Analysis Introduction to Risk Analysis

Why do individuals, companies, and stockholders take risks?

TerminologyTerminology

Risk - possibility of an undesired outcome Probability - expected relative frequency of an

event

TerminologyTerminology

Risk and uncertainty Risk - probabilities of outcomes is known -- casino Uncertainty - outcomes not known with certainty –

reality Probability distributions -

discrete - number of probability occurrences is finite continuous - infinite number of occurrence

- range of outcomes

TerminologyTerminology

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

TerminologyTerminology

Variation versus event risk Event risk - probability of a certain , such as

bankruptcy Variation risk- probability of a range of outcomes

around an event typically measured by standard deviation

TerminologyTerminology

Diversifiable versus nondiversifiable risk Diversifiable risk -- risk that can be reduced or

eliminated by combining one investment with another

Must have a correlation less than +1 Nondiversifiable risk -- risk that remains after

combining large numbers of projects

Probability RulesProbability Rules

Mutually exclusive events - add the probabilities of the events

Independent events build a table of possible combinations of events multiply the probabilities to get table values

Dependent events build a table where the probabilities of outcomes for

the second event are dependent on the first event

Stages of Risk MeasurementStages of Risk Measurement

Stage 1 -- Descriptive and Subjective listing of things that might go wrong good for identifying the important variables

Stage 2 -- Sensitivity analysis look at possible outcomes over a range of values for

a critical variable (such as sales) do not attempt to assign probabilities example -- breakeven analysis

Stages of Risk MeasurementStages of Risk Measurement

Stage 3 -- Event probability assign probabilities to the various outcomes one in ten chance of bankruptcy

Stage 4 -- Summary measures of probability distributions Measures of central tendency Measures of dispersion

Summary Measures: Central Summary Measures: Central TendencyTendency

Expected value: possibilities time probabilities Median: Center outcome; probability of

outcome above median equals probability of outcome below median.

Mode: Most common outcome Geometric mean (Pi = probability of outcome i):

[(1+ return1)^P1][(1+ return2)^P2] . . . .

Summary Measures: DispersionSummary Measures: Dispersion

Variance Multiply squared distances from the expected value

by the probability, then sum Standard deviation

Square root of the variance Same unit of measure as the original problem

Coefficient of variation standard deviation/ expected value adjust for the scale of the project

Summary Measures: DispersionSummary Measures: Dispersion

Semivariance computed like variance, but considers only outcomes

below the expected value used when the distribution is not normal (skewed)

Quartile range There is a 25% probability of a value greater than X

and a 25% probability of a value less than Y

Summary MeasuresSummary Measures

Normal distributions and standard deviations using a z-table you can find the area under the

normal curve (probability of a range of outcomes)

Utility TheoryUtility Theory

Assumptions Completeness -- you can judge your preference in all

situations Rational -- consistent in judgements order of

presentation does not matter Transitivity -- if A is preferred over B and B is

preferred over C then A is preferred over C

Utility TheoryUtility Theory

Types of utility functions Increasing -- risk seeker or lover -- will pay to take

the riskier project -- casinos and lottery tickets Constant -- risk neutral -- is indifferent to risk -- will

accept the same expected return for risky as well as safe projects

Decreasing -- risk averse -- prefer safety to risk and must be compensated for accepting additional risk

Utility TheoryUtility Theory

Problems with utility functions in reality Hard to measure Whose utility should we measure? Once measured then the decision can be made by the

analyst Utility theory is important to arbitrage pricing

theory equal expected utilities should have equal prices

Risk PerspectivesRisk Perspectives Single investment perspective

Proposing manager -- Chapter 12

Company perspective Senior management and board -- Chapter 13

Shareholder perspective Shareholder -- Chapter 14

Contingent claims Option writer, debt-holder -- Chapter 15

Overall economy Everybody