Transcript Review of Prob & Stat

Risk Identification and Measurement
 Risk identification
 Probability distribution
 Expected Value and Standard Deviation
 Other Loss Measures
 Correlations
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Identifying Business Risk
Exposures
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Property
Business income
Liability
Human resource
External economic forces
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Identifying Business Exposures
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Property
Business income
Liability
Human resource
External economic forces
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Identifying Individual Exposures
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Earnings
Physical assets
Financial assets
Medical expenses
Longevity
Liability
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Probability Distribution
 A probability distribution identifies all the possible
outcomes for the random variable and the
probability of the outcomes
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Probability Distribution: Example1
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Random variable = damage from auto accidents
Possible Outcomes for Damages
$0
$200
$1,000
$5,000
$10,000
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Probability
0.50
0.30
0.10
0.06
0.04
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Graph of Example1
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Probability Distribution: Example2
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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
$5,000
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Possible
Losses
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Continuous Distribution
 Important characteristic of density functions
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Area under the entire curve equals one
Area under the curve between two points gives the
probability of outcomes falling within that given
range
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Risk Management & Probability
Distributions
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Ideally, a risk manager would know the probability
distribution of losses
Then assess how different risk management
approaches would change the probability
distribution
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Which distribution would you rather have?
Prob
Cost
Cost
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Frequency of Loss and Severity of
Loss
 Frequency of loss measures the number of losses
in a given period of time
 Severity of loss measures the magnitude of loss
per occurrence
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Expected Loss
 When the frequency and severity of losses
is uncorrelated with each other, then:
Expected Loss = Frequency * Severity
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Example
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50,000 employees in each of the past five years
1,500 injuries over the five-year period
$3 million in total injury costs
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Frequency of injury per year = 1500 / 50000 = 0.03
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Average severity of injury = $3 m/ 1500 = $2,000
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Annual expected loss per employee = 0.03 x $2,000 = $60
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Expected Value
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Formula for a discrete distribution:
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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 =
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Variance and Standard Deviation
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Variance measures the probable variation in
outcomes around the expected value
N
Variance   pi ( xi   ) 2
i 1
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Standard deviation is the square root of the
variance
Standard deviation (variance) is higher when
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when the outcomes have a greater deviation from the
expected value
probabilities of the extreme outcomes increase
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Variance and Standard Deviation
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Comparing standard deviation for three discrete
distributions
Distribution 1
Distribution 2
Distribution 3
Outcome
$250
$500
$750
Outcome
$0
$500
$1000
Outcome
$0
$500
$1000
Prob
0.33
0.34
0.33
Prob
0.33
0.34
0.33
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Prob
0.4
0.2
0.4
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Standard Deviation and Variance
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Sample Mean and Standard
Deviation
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Sample mean and standard deviation can and usually
will differ from population expected value and
standard deviation
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Coin flipping example
$1 if heads
X=
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-$1 if tails
Expected average gain from game = $0
Actual average gain from playing the game 5 times =
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Skewness
 Skewness measures the symmetry of the
distribution
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No skewness ==> symmetric
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Most loss distributions exhibit skewness
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Maximum Probable Loss
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Maximum Probable Loss at the 95% level is the number,
MPL, that satisfies the equation:
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Probability (Loss < MPL) < 0.95
Losses will be less than MPL 95 percent of the time
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Value at Risk (VAR)
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VAR is essentially the same concept as maximum
probable loss, except it is usually applied to the
value of a portfolio
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If the Value at Risk at the 5% level for the next
week equals $20 million, then
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Prob(change in portfolio value < -$20 million) = 0.05
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In words, there is 5% chance that the portfolio will
lose more $20 million over the next week
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Value at Risk
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Example:
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Assume VAR at the 5% level =$5 million
And VAR at the 1% level = $7 million
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Important Properties of the Normal
Distribution
 Often analysts use the following properties of the
normal distribution to calculate VAR:
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Assume X is normally distributed with mean  and
standard deviation . Then
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Prob (X >  +2.33) = 0.01
Prob (X <  -2.33) = 0.01
Prob (X >  +1.645) = 0.05
Prob (X <  -1.645) = 0.05
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Example
 Company A estimates the expected value and
standard deviation of its total property loss as $20
million and $5 million. Assume the total property
loss is normally distributed, what is the predicted
maximum probable loss at the 95 percent level? at
the 99 percent level?
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Correlation
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Correlation identifies the relationship between two
probability distributions
 Uncorrelated (Independent)
 Positively Correlated
 Negatively Correlated
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Exercise: Expected value and standard
deviation
Outcome
250
300
400
500
probability
0.05
0.25
0.55
0.15
Calculate the mean and standard deviation of the loss
distribution.
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