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Chapter 3
Risk Identification
and Measurement
McGraw-Hill/Irwin
Copyright © 2004 by the McGraw-Hill Companies, Inc. All rights reserved.
Identifying Business Risk Exposures
• Property
• Business income
• Liability
• Human resource
• External economic forces
H&N, Ch. 3
T3.2
Identifying Individual Exposures
• Earnings
• Physical assets
• Financial assets
• Medical expenses
• Longevity
• Liability
H&N, Ch. 3
T3.3
Probability Distributions
• Probability distributions
• Listing of all possible outcomes and their
associated probabilities
• Sum of the probabilities must equal 1
• Two types of distributions:
• discrete
• continuous
H&N, Ch. 3
T3.4
Presenting Probability Distributions
• Two ways of presenting discrete distributions:
• Numerical listing of outcomes and probabilities
• Graphically
• Two ways of presenting continuous distributions:
• Density function (not used in this course)
• Graphically
H&N, Ch. 3
T3.5
Example of a Discrete Probability Distribution
• Random variable = damage from auto accidents
Possible Outcomes for Damages
$0
$200
$1,000
$5,000
$10,000
H&N, Ch. 3
Probability
0.50
0.30
0.10
0.06
0.04
T3.6
Example of a Discrete Probability Distribution
H&N, Ch. 3
T3.7
Example of a Continuous Distribution
H&N, Ch. 3
T3.8
Continuous Distributions
• Important characteristic of density functions
• Area under the entire curve equals one
• Area under the curve between two points gives
the probability of outcomes falling within that given
range
H&N, Ch. 3
T3.9
Probabilities with Continuous Distributions
• Find the probability that the loss > $5,000
• Find the probability that the loss < $2,000
• Find the probability that $2,000 < loss < $5,000
Probability
$2,000
H&N, Ch. 3
$5,000
Possible
Losses
T3.10
Risk Management & Probability Distributions
• Ideally, a risk manager would know the probability
distribution of losses
• Then assess how different risk management
approaches would change the probability distribution
• Example: Which distribution would you rather have?
Insurance
No RM
Prob
Accident
Cost
H&N, Ch. 3
Accident Cost +
Insurance Cost
T3.11
Summary Measures of Loss Distributions
• Instead comparing entire distributions, managers often
work with summary measures of distributions:
• Frequency
• Severity
• Expected loss
• Standard deviation or variance
• Maximum probable loss (Value at Risk)
• Then ask: How does RM affect each of these measures?
H&N, Ch. 3
T3.12
Expected Value
• Formula for a discrete distribution:
• Expected Value = x1 p1 + x2 p2 + … + xM pM .
• Example:
Possible Outcomes for Damages
$0
$200
$1,000
$5,000
$10,000
Probability
0.50
0.30
0.10
0.06
0.04
Expected Value =
H&N, Ch. 3
T3.13
Expected Value
H&N, Ch. 3
T3.14
Standard Deviation and Variance
• Standard deviation indicates the expected
magnitude of the error from using the expected
value as a predictor of the outcome
• Variance = (standard deviation) 2
• Standard deviation (variance) is higher when
• when the outcomes have a greater deviation from
the expected value
• probabilities of the extreme outcomes increase
H&N, Ch. 3
T3.15
Standard Deviation and Variance
• Comparing standard deviation for three discrete
distributions
H&N, Ch. 3
Distribution 1
Distribution 2
Distribution 3
Outcome Prob
$250
0.33
$500
0.34
$750
0.33
Outcome Prob
$0
0.33
$500
0.34
$1000 0.33
Outcome Prob
$0
0.4
$500
0.2
$1000 0.4
T3.16
Standard Deviation and Variance
H&N, Ch. 3
T3.17
Sample Mean and Standard Deviation
• Sample mean and standard deviation can and
usually will differ from population expected value and
standard deviation
• Coin flipping example
$1 if heads
X=
-$1 if tails
• Expected average gain from game = $0
• Actual average gain from playing the game 5 times =
H&N, Ch. 3
T3.18
Skewness
• Skewness measures the symmetry of the
distribution
• No skewness ==> symmetric
• Most loss distributions exhibit skewness
H&N, Ch. 3
T3.19
Maximum Probable Loss
• Maximum Probable Loss at the 95% level is the number,
MPL, that satisfies the equation:
• Probability (Loss < MPL) < 0.95
• Losses will be less than MPL 95 percent of the time
H&N, Ch. 3
T3.20
Value at Risk (VAR)
• VAR is essentially the same concept as maximum
probable loss, except it is usually applied to the
value of a portfolio
• If the Value at Risk at the 5% level for the next
week equals $20 million, then
• Prob(change in portfolio value < -$20 million) = 0.05
• In words, there is 5% chance that the portfolio will
lose more $20 million over the next week
H&N, Ch. 3
T3.21
Value at Risk
• Example:
• Assume VAR at the 5% level =$5 million
• And VAR at the 1% level = $7 million
H&N, Ch. 3
T3.22
Important Properties of the Normal Distribution
• Often analysts use the following properties of
the normal distribution to calculate VAR:
• Assume X is normally distributed with mean  and
standard deviation . Then
• Prob (X >  -2.33) = 0.01
• Prob (X >  -1.645) = 0.05
H&N, Ch. 3
T3.23
Correlation
• Correlation identifies the relationship between two
probability distributions
• Uncorrelated (Independent)
• Positively Correlated
• Negatively Correlated
H&N, Ch. 3
T3.24
Calculating the Frequency and Severity of Loss
• Example:
• 10,000 employees in each of the past five years
• 1,500 injuries over the five-year period
• $3 million in total injury costs
• Frequency of injury per year = 1.500 / 50,000 = 0.03
• Average severity of injury = $3 m/ 1,500 = $2,000
• Annual expected loss per employee = 0.03 x $2,000 = $60
H&N, Ch. 3
T3.25