Risk Identification and Measurement

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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

• Property
• Business income
• Liability
• Human resource
• External economic forces

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Identifying Business Exposures

• Property
• Business income
• Liability
• Human resource
• External economic forces

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Identifying Individual Exposures

• 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

• Random variable damage from auto accidents
• Possible Outcomes for Damages Probability
• 0 0.50
• 200 0.30
• 1,000 0.10
• 5,000 0.06
• 10,000 0.04

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Graph of Example1
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Probability Distribution Example2

• Find the probability that the loss gt 5,000
• Find the probability that the loss lt 2,000
• Find the probability that 2,000 lt loss lt 5,000

Probability
Possible Losses
5,000
2,000
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Continuous Distribution

• 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

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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

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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

• 50,000 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 1500 / 50000
  0.03
• Average severity of injury 3 m/ 1500 2,000
• Annual expected loss per employee 0.03 x 2,000
  60

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Expected Value

• Formula for a discrete distribution
• Expected Value x1 p1 x2 p2 xM pM .
• Example
• Possible Outcomes for Damages Probability
• 0 0.50
• 200 0.30
• 1,000 0.10
• 5,000 0.06
• 10,000 0.04
• Expected Value

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Variance and Standard Deviation

• Variance measures the probable variation in
  outcomes around the expected value
• Standard deviation is the square root of the
  variance
• Standard deviation (variance) is higher when
• 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

• Comparing standard deviation for three discrete
  distributions
• Distribution 1 Distribution 2 Distribution 3
• Outcome Prob Outcome Prob Outcome Prob
• 250 0.33 0 0.33 0 0.4
• 500 0.34 500 0.34 500 0.2
• 750 0.33 1000 0.33 1000 0.4

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Standard Deviation and Variance
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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
  

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Skewness

• Skewness measures the symmetry of the
  distribution
• No skewness gt symmetric
• Most loss distributions exhibit skewness

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Maximum Probable Loss

• Maximum Probable Loss at the 95 level is the
  number, MPL, that satisfies the equation
• Probability (Loss lt MPL) lt 0.95
• Losses will be less than MPL 95 percent of the
  time

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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 lt -20 million)
  0.05
• 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

• Example
• 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
• Assume X is normally distributed with mean ? and
  standard deviation ?. Then
• Prob (X gt ? 2.33?) 0.01
• Prob (X lt ? -2.33?) 0.01
• Prob (X gt ? 1.645?) 0.05
• Prob (X lt ? -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

• 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 probability
• 250 0.05
• 300 0.25
• 400 0.55
• 500 0.15
• Calculate the mean and standard deviation of the
  loss distribution.