Chapter Outline Chapter Outline 3.1 THE PERVASIVENESS OF RISK Risks Faced by an Automobile Manufacturer Risks Faced by Students 3.2 BASIC CONCEPTS FROM PROBABILITY AND STATISTICS Random Variables and Probability Distributions Characteristics of Probability Distributions Expected Value Variance and Standard Deviation Sample Mean and Sample Standard Deviation Skewness Correlation 3.3 RISK REDUCTION THROUGH POOLING INDEPENDENT LOSSES 3.4 POOLING ARRANGEMENTS WITH CORRELATED LOSSES Other Examples of Diversification 3.5 SUMMARY
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Chapter Outline 3.1THE PERVASIVENESS OF RISK Risks Faced by an Automobile Manufacturer Risks Faced by Students 3.2BASIC CONCEPTS FROM PROBABILITY AND STATISTICS.
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Chapter OutlineChapter Outline3.1 THE PERVASIVENESS OF RISK
Risks Faced by an Automobile ManufacturerRisks Faced by Students
3.2 BASIC CONCEPTS FROM PROBABILITY AND STATISTICSRandom Variables and Probability DistributionsCharacteristics of Probability DistributionsExpected ValueVariance and Standard DeviationSample Mean and Sample Standard DeviationSkewnessCorrelation
3.3RISK REDUCTION THROUGH POOLING INDEPENDENT LOSSES
3.4POOLING ARRANGEMENTS WITH CORRELATED LOSSESOther Examples of Diversification
3.5SUMMARY
Appendix OutlineAppendix Outline
APPENDIX: MORE ON RISK MEASUREMENT AND RISK REDUCTION
The Concept of Covariance and More about Correlation
Expected Value and Standard Deviation of Combinations of Random Variables
Expected Value of a Constant times a Random VariableStandard Deviation and Variance of a Constant times a
Random
Variable
Expected Value of a Sum of Random Variables
Variance and Standard Deviation of the Average of Homogeneous
Random Variables
Probability DistributionsProbability Distributions
Probability distributions
– Listing of all possible outcomes and their associated probabilities
– Sum of the probabilities must ________
– Two types of distributions:
discrete
continuous
Presenting Probability DistributionsPresenting 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
Example of a Discrete Example of a Discrete Probability DistributionProbability Distribution
– Random variable = damage from auto accidents
Possible Outcomes for Damages Probability
$0 0.50
$200 ____
$_____ 0.10
$5,000 ____
$10,000 0.04
Example of a Discrete Example of a Discrete Probability DistributionProbability Distribution
0
0.2
0.4
0.6
0.8
1
0 200 1000 5000 10000
Damages
Pro
bab
ilit
y
Example of a Continuous Example of a Continuous Probability DistributionProbability Distribution
Probability Distribution for Auto Maker's Profits
-20,000 0 20,000 40,000Profits
Pro
bab
ility
Continuous DistributionsContinuous Distributions
Important characteristic
– Area under the entire curve equals ____
– Area under the curve between ___ points gives the probability of outcomes falling within that given range
Probabilities with Continuous Probabilities with Continuous DistributionsDistributions
Find the probability that the loss > $______ Find the probability that the loss < $______ Find the probability that $2,000 < loss < $5,000
Possible Losses
Probability
$5,000$2,000
Expected ValueExpected Value– Formula for a discrete distribution:
Expected Value = x1 p1 + x2 p2 + … + xM pM .
– Example:
Possible Outcomes for Damages Probability Product$0 0.50 0$200 0.30 60$1,000 0.10 100$5,000 0.06 300$10,000 0.04 400
$860Expected Value =
Expected ValueExpected Value
Comparing the Expected Values of Two Distributions Visually
0 3000 6000 9000 12000 15000 18000 21000
Outcomes
Pro
ba
bil
ity
B A
Standard Deviation and Standard Deviation and VarianceVariance
– 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 ______deviation from the expected value
probabilities of the ______ outcomes increase
Standard Deviation and Standard Deviation and VarianceVariance
– 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
_____ ____ _____ ____ _____ ___
$750 0.33 $1000 0.33 $1000 0.4
Standard Deviation and Standard Deviation and VarianceVarianceComparing the Standard Deviations of two
Distributions
0 500 1000 1500 2000 2500Outcomes
Pro
ba
bil
ity
A
B
Sample Mean and Standard DeviationSample 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 headsX = -$1 if tails
Expected average gain from game = $0 Actual average gain from playing the game ___ times =
SkewnessSkewness
Skewness measures the symmetry of the distribution
– No skewness ==> symmetric
– Most loss distributions exhibit ________
Loss Forecasting: Component ApproachLoss Forecasting: Component Approach
Estimating the Annual Claim Distribution
Historical Claims Frequency Historical Claims Severity
Loss Development Adjustment Inflation Adjustment
Exposure Unit Adjustment
Frequency Probability Distribution Severity Probability Distribution
--------- Claim Distribution
Annual Claims are shared:
Firm Retains a Portion Transfers the Rest
Firm’s Loss Forecast Premium for Losses
Transferred
Loss Payment Pattern Premium Payment
Pattern
Mean and Variance impact on e.p.s.
Slip and Fall Claims at Well-Slip and Fall Claims at Well-Known Food ChainKnown Food Chain