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Chapter 11 –
Introduction to Risk Analysis
 Why
do individuals, companies, and
stockholders take risks?
 Risk
- possibility of an undesired outcome
 Probability - expected relative frequency of an
 Risk
and uncertainty
 Risk
- probabilities of outcomes is known -- casino
 Uncertainty - outcomes not known with certainty –
 Probability
 discrete
distributions -
- number of probability occurrences is finite
 continuous - infinite number of occurrence
- range of outcomes
 Subjective
versus objective probability
- someone’s opinion
 Objective - can be measured
 Subjective
 Variation
versus event risk
 Event
risk - probability of a certain , such as
 Variation risk- probability of a range of outcomes
around an event
typically measured by standard deviation
Diversifiable versus nondiversifiable risk
 Diversifiable risk
-- risk that can be reduced or
eliminated by combining one investment with
Must have a correlation less than +1
 Nondiversifiable
risk -- risk that remains after
combining large numbers of projects
Probability Rules
 Mutually
 add
exclusive events -
the probabilities of the events
 Independent
 build
a table of possible combinations of events
 multiply the probabilities to get table values
 Dependent
 build
a table where the probabilities of outcomes for
the second event are dependent on the first event
Stages 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 Measurement
 Stage
3 -- Event probability
 assign
probabilities to the various outcomes
 one in ten chance of bankruptcy
 Stage
4 -- Summary measures of probability
 Measures
of central tendency
 Measures of dispersion
Summary Measures: Central
 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: Dispersion
 Variance
 Multiply
squared distances from the expected value
by the probability, then sum
 Standard
 Square
root of the variance
 Same unit of measure as the original problem
 Coefficient
 standard
of variation
deviation/ expected value
 adjust for the scale of the project
Summary Measures: Dispersion
 Semivariance
 computed
like variance, but considers only outcomes
below the expected value
 used when the distribution is not normal (skewed)
 Quartile
 There
is a 25% probability of a value greater than X
and a 25% probability of a value less than Y
Summary Measures
 Normal
 using
distributions and standard deviations
a z-table you can find the area under the normal
curve (probability of a range of outcomes)
Utility Theory
 Assumptions
 Completeness
-- you can judge your preference in all
 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 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 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
 Utility
theory is important to arbitrage pricing
 equal
expected utilities should have equal prices
Risk Perspectives
Single investment perspective
Proposing manager -- Chapter 12
Senior management and board -- Chapter 13
Shareholder -- Chapter 14
Option writer, debt-holder -- Chapter 15