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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 = $0 Actual average gain from playing the game ___ times =
SkewnessSkewness
Skewness measures the symmetry of the distribution
– No skewness ==> symmetric
– Most loss distributions exhibit ________
Loss Forecasting: Component ApproachLoss Forecasting: Component Approach
Estimating the Annual Claim Distribution
Historical Claims Frequency Historical Claims Severity
Loss Development Adjustment Inflation Adjustment
Exposure Unit Adjustment
Frequency Probability Distribution Severity Probability Distribution
--------- Claim Distribution
Annual Claims are shared:
Firm Retains a Portion Transfers the Rest
Firm’s Loss Forecast Premium for Losses
Transferred
Loss Payment Pattern Premium Payment
Pattern
Mean and Variance impact on e.p.s.
Slip and Fall Claims at Well-Slip and Fall Claims at Well-Known Food ChainKnown Food Chain
YearRaw Claim Data by
Size ($) Number of ClaimsExposure Base:
$ or FootageAdjusted No. of
Claims
Claims Cost Price Index: Currrent Year =
100Adjusted Loss
Size ($)
1985 - 0 1,000,000 0 32.60 -1986 460.00 1 1,000,000 2 35.20 1,306.82 1987 590.00 2 1,000,000 4 37.90 1,556.73
520.00 1,372.03 1988 - 0 1,000,000 0 40.80 -1989 200.00 1 1,000,000 2 44.00 454.55 1990 - 0 1,000,000 0 47.40 -1991 - 0 2,000,000 0 51.10 -1992 775.00 1 2,000,000 1 55.00 1,409.09 1993 - 0 2,000,000 0 59.30 -1994 830.00 3 2,000,000 3 63.90 1,298.90
905.00 1,416.28 670.00 1,048.51
1995 - 0 2,000,000 0 68.90 -1996 1,080.00 1 2,000,000 1 74.20 1,455.53 1997 590.00 2 2,000,000 2 79.90 738.42
340.00 425.53 1998 - 0 2,000,000 0 86.10 -1999 - 0 2,000,000 0 100.00 -
Unadjusted Frequency Unadjusted Frequency DistributionDistribution
Number of Probability Cumulative
Claims of Claim Probability
0 .5333 .5333
1 _____ .8000
2 .1333 _____
3 .0667 1.0000
Unadjusted Frequency Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3
Number of Claims
Pro
babi
lity
Unadjusted Severity Unadjusted Severity DistributionDistribution
Interval Relative Cumulative
in Dollars Frequency Probability
200-375 .1818 .1818
___-___ .1818 .3636
551-725 .2727 .6363
726-900 _____ .9090
900-1100 .0910 1.0000
Severity Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
200-375 376-550 551-725 726-900 901-1100
Pro
ba
bil
ity
Annual Claim DistributionAnnual Claim Distribution
Combine the _______ and ______ distributions to obtain the annual claim distribution
Sometimes this can be done mathematicallyUsually it must be done using “brute force”
statistical procedures. An example of this follows.
Frequency DistributionFrequency Distribution
Number Probability
of Claims of Claim
0 .1
1 .6
2 .25
3 .05
Severity DistributionSeverity Distribution
Prob. Cum.
Amount of Loss Midpoint of Loss Prob.$0 to $2,000 $1,000 .2 .2
2,001 to 8,000 5,000 ___ ____
8,001 to 12,000 10,000 ___ ____
12,001 to 88,000 50,000 .06 .96
88,001 to 312,000 200,000 .03 .99
GT 312,000 500,000 .01 1.00
Annual Claim DistributionAnnual Claim Distribution
Cumulative
Claim Amount Probability $0 .1 .1
1 to 2,000 .13 .23
2,001 to 8,000 _____ _____
8,001 to 12,000 .2566 .7694
12,001 to 70,000 .17984 .94924
70,001 to 450,000 .038299 .987539
450,001 to 511,000 _______ .998759
GT 511,000 .001241 1.000000
________ ________ Loss when applied to:– severity distribution– annual claim distribution
Loss Forecasting Aggregate ApproachLoss Forecasting Aggregate Approach
Estimating the Annual Claim Distribution
Annual Claims: Raw Figures
Loss Development Adjustment
Inflation Adjustment
Exposure Unit Adjustment
Annual Claim Distribution
Loss Forecasting Aggregate ApproachLoss Forecasting Aggregate Approach
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.