Chapter Outline

1. 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

2. Appendix 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 Variable Standard 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

3. Probability Distributions • Probability distributions • Listing of all possible outcomes and their associated probabilities • Sum of the probabilities must ________ • Two types of distributions: • discrete • continuous

4. Presenting 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

5. Example of a Discrete Probability Distribution • Random variable = damage from auto accidents Possible Outcomes for Damages Probability $0 0.50 $200 ____ $_____ 0.10 $5,000 ____ $10,000 0.04

6. Example of a Discrete Probability Distribution

7. Example of a Continuous Probability Distribution

8. Continuous 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

9. Probabilities with Continuous Distributions • Find the probability that the loss > $______ • Find the probability that the loss < $______ • Find the probability that $2,000 < loss < $5,000 Probability Possible Losses $5,000 $2,000

10. Expected Value • Formula for a discrete distribution: • Expected Value = x1 p1 + x2 p2 + … + xM pM . • Example: Possible Outcomes for DamagesProbabilityProduct $0 0.50 0 $200 0.30 60 $1,000 0.10 100 $5,000 0.06 300 $10,000 0.04 400 $860 Expected Value =

11. Expected Value

12. Standard Deviation and Variance • 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

13. Standard Deviation and Variance • 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

14. Standard Deviation and Variance

15. 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 ___ times =

16. Skewness • Skewness measures the symmetry of the distribution • No skewness ==> symmetric • Most loss distributions exhibit ________

17. Loss 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

18. 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.

19. Slip and Fall Claims at Well-Known Food Chain

20. Unadjusted Frequency Distribution Number of Probability Cumulative Claimsof ClaimProbability 0 .5333 .5333 1 _____ .8000 2 .1333 _____ 3 .0667 1.0000

22. Unadjusted Severity Distribution Interval Relative Cumulative in DollarsFrequency Probability 200-375 .1818 .1818 ___-___ .1818 .3636 551-725 .2727 .6363 726-900 _____ .9090 900-1100 .0910 1.0000

24. Annual Claim Distribution • Combine the _______ and ______ distributions to obtain the annual claim distribution • Sometimes this can be done mathematically • Usually it must be done using “brute force” statistical procedures. An example of this follows.

25. Frequency Distribution Number Probability of Claimsof Claim 0 .1 1 .6 2 .25 3 .05

26. Severity Distribution Prob. Cum. Amount of LossMidpointof LossProb. $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

27. Annual Claim Distribution Cumulative Claim AmountProbability $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

28. ________ ________ Loss when applied to: • severity distribution • annual claim distribution

29. Loss Forecasting Aggregate Approach • Estimating the Annual Claim Distribution Annual Claims: Raw Figures  Loss Development Adjustment  Inflation Adjustment  Exposure Unit Adjustment  Annual Claim Distribution

30. Loss 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.