Chapter 3
Risk Assessment and Pooling
Insurable Loss Exposures
• Estimation of financial impact of each risk identified
previously.
• Two key statistical measures:
• Frequency with which losses occur.
• Their severity.
Basic Statistical Concepts ‐ 1
• Random Variable:
Future value is not
known with
certainty.
• Probability
Distribution: Shows
all possible
outcomes for a
Random Variable.
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Basic Statistical Concepts ‐ 2
• Expected Value: Sum of the multiplication of each possible
outcome of the variable with its probability.
E[R] = Σ Ri * Pi
• Variance and Standard Deviation:
σ=
The Expected Value
• Can be calculated by multiplying the expected losses with their
probability and calculating the sum of all outcomes
• Is a starting point for calculating an insurance premium or how
much a firm should set aside each year to cover the losses
Average Loss
• Loss Frequency = Total Amount of Losses divided by Total Number
of Accidents
• Loss Severity = Total Number of Accidents divided by Total Units
Analyzed.
• Average Loss = Average Loss Frequency X Average Loss Severity.
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Measures of Risk
• Value at Risk (VaR) and Maximum Probable Annual Loss
(MPAL)
• VaR: The worst‐case scenario dollar value loss (up to a specified
probability level) that could occur for a company exposed to a
specific set of risks.
• It can measure risks in any single security or an entire portfolio as
long sufficient historical data is available.
• In the context of pure risk exposures, MPAL looks at a probability
distribution and then picks the selected lower percentile value as
the MPAL.
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
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
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Value at Risk
 Example:
Assume VAR at the 5% level =$5 million
And VAR at the 1% level = $7.5 million
Risk Pooling
• Risk can be reduced through diversification.
• The creation of a pool of many (exposure) units helps the insurer
to better predict any individual unit’s risk of loss.
• The Probability Distribution matters!
Pooling Arrangements
•Basic Idea:
• Replace your loss with the average loss of a group
•Issues:
• What happens to each person’s
• Expected loss
• Standard deviation of loss
• Maximum probable loss
• How do these results change with
• More participants
• Correlation of losses
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Risk Pooling with 2 People
•Two people with same distribution
Outcome
$10,000
Probability
0.05
Loss =
$0
0.95
• Assume losses are uncorrelated
• Expected value = $500
• Standard deviation = $2179.44
Risk Pooling with 2 People
•Pooling changes distribution of costs
Loss to:
Outcome
Neither $0
Ann
5,000
Bob
5,000
Both 10,000
Probability
(.95)(.95)
(.05)(.95)
(.95)(.05)
(.05)(.05)
Risk Pooling with 2 People
•Effect on E(Loss):
• w/o pooling, E(Loss) = $500
• w/ pooling, E(Loss) = $500
•Effect on Std. Dev.
• w/o pooling, std. dev. = $2,179
• w/ pooling, std. dev. = $1,541
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Risk Pooling with 4 People
•Pooling Arrangement between 4 people:
Outcome
$10,000
$7,500
Loss = $5,000
$2,500
$0
Probability
0.000006
0.000475
0.014
0.171
0.815
• E(Loss) = $500
• Std. dev. = $1,089
Risk Pooling: Main Points
•Pooling arrangements
•
•
•
•
•
•
do not change expected loss
reduce uncertainty
variance decreases
losses become more predictable
maximum probable loss declines
distribution of costs becomes more symmetric
• less skewness
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Normal Probability Distribution
• “Bell Curve”:
Confidence Intervals 1
• Assuming Normal Distribution:
Estimated Mean Loss ± (k) * Estimated σ
• Where:
• (k) = Specified number of standard deviations which reflect the
uncertainty.
• σ = Standard Deviation calculated using loss data from past.
• This is the Confidence Interval.
Confidence Intervals 2
• (k) * Estimated σ is also called the Risk Charge.
• It represents the margin of error.
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Practical Considerations
• Insurers sort consumers into homogeneous categories:
• Age.
• Gender.
• Etc.
• Yet still independent of each other.
• Insurers will not insure when these assumptions are violated.
Effect of Correlated Losses
•Now allow correlation in losses
• Result: uncertainty is not reduced as much
•Intuition:
• What happens to one person happens to others
• One person’s large loss does not tend to be offset by others’ small
losses
• Therefore pooling does not reduce risk as much
Correlation and Risk Reduction
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Risk Pooling: Main Points
•Main Points:
• Pooling reduces each participant’s risk
• i.e., costs from loss exposure become more predictable
• Predictability increases with the number of participants
• Predictability decreases with correlation in losses
Diversification
•Pooling is diversification
• Own “portfolio” of pure risks
•Other applications
• stock market diversification
• diversification across lines of business within a firm
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Convolution
• Calculates all possible combinations of losses indicated by the
frequency and severity loss distributions, as well as their
corresponding probabilities of occurring.
• Often done by computer simulation due to complexity of
calculations.
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Chapter 2
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