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.