Download Review of Prob & Stat

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
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
Probability 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 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
Probability Distribution
Random variable = damage from auto accidents
Possible Outcomes for Damages
$0
$200
$1,000
$5,000
$10,000
Probability
Example of a Discrete
Probability Distribution
1
Probability
0.8
0.6
0.4
0.2
0
0
200
1000
Dam ages
5000
10000
Example of a Continuous
Probability Distribution
Probability
Probability Distribution for Auto Maker's
Profits
-20,000
0
20,000
Profits
40,000
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
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
$2,000
$5,000
Possible
Losses
Expected Value
– Formula for a discrete distribution:

Expected Value = x1 p1 + x2 p2 + … + xM pM .
– Example:
Possible Outcomes for Damages
$0
$200
$1,000
$5,000
$10,000
Expected Value =
Probability
Product
Expected Value
Comparing the Expected Values of Two
Distributions Visually
A
Probability
B
0
3000
6000
9000
12000
Outcomes
15000
18000
21000
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


Standard Deviation and
Variance
– Comparing standard deviation for three discrete
distributions
Distribution 1
Distribution 2
Distribution 3
Outcome Prob
$250
0.33
_____ ____
$750
0.33
Outcome Prob
$0
0.33
_____ ____
$1000 0.33
Outcome Prob
$0
0.4
_____ ___
$1000 0.4
Standard Deviation and
Variance
Comparing the Standard Deviations of two
Distributions
Probability
A
B
0
500
1000
1500
Outcomes
2000
2500
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 =
Skewness

Skewness measures the symmetry of the
distribution
– No skewness ==> symmetric
– Most loss distributions exhibit ________
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

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-Known Food Chain
Exposure
Base: $ or
Footage
Claims Cost
Price Index:
Adjusted No.
Currrent
of Claims
Year = 100
Year
Raw Claim
Data by Size
($)
Number of
Claims
1988
-
0
1,000,000
0
32.60
1989
460.00
1
1,000,000
2
35.20
1990
590.00
2
1,000,000
4
37.90
0
1,000,000
0
40.80
1
1,000,000
2
44.00
520.00
1991
1992
200.00
1993
-
0
1,000,000
0
47.40
1994
-
0
2,000,000
0
51.10
1
2,000,000
1
55.00
0
2,000,000
0
59.30
3
2,000,000
3
63.90
0
2,000,000
0
68.90
1995
1996
1997
775.00
830.00
905.00
670.00
1998
-
1999
1,080.00
1
2,000,000
1
74.20
2000
590.00
2
2,000,000
2
79.90
340.00
2001
-
0
2,000,000
0
86.10
2002
-
0
2,000,000
0
100.00
Unadjusted Frequency
Distribution
Number of
Claims
0
1
2
3
Probability
of Claim
.5333
_____
.1333
.0667
Cumulative
Probability
1.0000
Unadjusted Frequency Distribution
0.6
Probability
0.5
0.4
0.3
0.2
0.1
0
0
1
2
Number of Claims
3
Unadjusted Severity
Distribution
Interval
in Dollars
200-375
___-___
551-725
726-900
900-1100
Relative
Frequency
.1818
.1818
.2727
_____
.0910
Cumulative
Probability
1.0000
Severity Distribution
0.3
Probability
0.25
0.2
0.15
0.1
0.05
0
200-375
376-550
551-725
726-900
901-1100
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.
Frequency Distribution
Number
of Claims
0
1
2
3
Probability
of Claim
.1
.6
.25
.05
Severity Distribution
Prob.
Amount of Loss Midpoint of Loss
$0
to
2,001 to
8,001 to
12,001 to
88,001 to
GT 312,000
$2,000
$1,000
8,000
5,000
12,000
10,000
88,000
50,000
312,000 200,000
500,000
.2
___
___
.06
.03
.01
Cum.
Prob.
.2
____
____
.96
.99
1.00
Annual Claim Distribution
Claim Amount
1
2,001
8,001
12,001
70,001
450,001
GT
$0
to 2,000
to 8,000
to 12,000
to 70,000
to 450,000
to 511,000
511,000
Cumulative
Probability
.1
.13
_____
.2566
.17984
.038299
_______
.001241
.1
.23
_____
.7694
.94924
.987539
.998759
1.000000

________ ________ Loss when applied to:
– severity distribution
– annual claim distribution
Loss 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 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.