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
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)

5. Example of a Discrete Probability Distribution
Random variable = damage from auto accidents
Possible Outcomes for Damages Probability $0
$200
$1,000
$5, $10,000

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 Damages Probability Product $0
$200
$1,000
$5,000
$10,000
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 (variance) is higher when

13. Standard Deviation and Variance
Comparing standard deviation for three discrete distributions
Distribution 1 Distribution 2 Distribution 3
Outcome Prob Outcome Prob Outcome Prob
$ $ $0 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
Claims of Claim Probability
1 _____

22. Unadjusted Severity Distribution
Interval Relative Cumulative
in Dollars Frequency Probability
___-___
_____

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 Claims of Claim

26. Severity Distribution
Prob. Cum.
Amount of Loss Midpoint of Loss Prob.
$0 to $2, $1,
2,001 to 8, , ___ ____
8,001 to 12, , ___ ____
12,001 to 88, ,
88,001 to 312, ,
GT 312, ,

27. Annual Claim Distribution
Cumulative
Claim Amount Probability $
1 to 2,
2, to 8, _____ _____
8, to 12,
12, to 70,
70, to 450,
450,001 to 511, _______
GT 511,

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.