uum-bwrr3033-risk management--chapter 04 risk measurement
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Chapter 4: Risk Measurement
Frequency and severity of losses
Probability
Probability distribution Fault tree
Pooling arrangement and diversification of
risk
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Learning Objective
Review the concepts of probability and statistics
Apply mathematical concepts to understand the
frequency and severity of losses
Understand the concepts of expected value andvariance of random variables
Distinguish between binomial distribution and
poisson distribution, and which is more appropriatefor different situations
Show how pooling of independent loss exposures
reduces risk
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RISK MEASUREMENT
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Random Variables & Probability Distribution
BINOMIAL
DISTRIBUTION
POISSONDISTRIBUTION
PROBABILITY TREE
DIAGRAM
RV is a variable which outcome is uncertain. Example: Flipping a coin.
Each flips result is uncertain.
Probability Distribution identifies all of the
possible outcomes, associates a probability with
each outcome.
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0
0.1
0.2
0.3
0.4
0.5
0.6
RM0 RM500 RM1,000 RM5,000 RM10,000
Probability
Probability
BINOMIAL
DISTRIBUTION
POISSONDISTRIBUTION
Possible Outcomes for Damages()
Probability
1 RM0 0.50
2 RM500 0.30
3
RM1,000 0.10
4 RM5,000 0.06
5 RM10,000 0.04
The question now
How we make decision based on these data?
Use
Expected Value
Variance or Standard Deviation
Skewness
Correlation
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Expected Value
it is where the outcomes tend to occur, on
average.
,
=
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Expected Value
Probability
Outcome
B A
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Variance & Standard Deviation
=
Variance
Standard Deviation
=
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Expected Value
Probability
Outcome
B
A
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Expected Value & Skew
Probability
Outcome
D
C
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MPL : Worst loss in worst scenario
PML: Worst loss in normal scenario
Maximum Probable Loss
Vs
Probable Maximum Loss
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Maximum Probable Loss
(Value At Risk)
Probability
Annual Liability Loss RM20m
area= 0.05
Area = 0.01
RM30m
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Area: 0.05
Value at Risk
Probability
-RM7.5m -RM5m
Area: 0.01
0
Monthly change in value of portfolio
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When discussing about many types of risk, it is
important to study the relationship among
random variables.
Correlation = 0the random variables are
not related.
:: outcome of one random variables will not
give info about the other random variables.
:: random variables are said to be
independent or uncorrelated.
Correlation
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REVISION
INDEPENDENT
EVENTS
MUTUALLY
EXCLUSIVE
EVENTS
COLLECTIVE
EXHAUSTED
EVENTS
ALTERNATIVE
EVENTSJOINT EVENTS
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Pooling Arrangements and
Diversification of Risk Basic Idea:
Replace your loss with the average loss of a group
Issues:
What happens to each persons Expected loss
Standard deviation of loss
Maximum probable loss
How do these results change with More participants
Correlation in losses among the participants increases
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Uncorrelated Risk Pooling Example
with 2 People Two people with same distribution (w/o pooling)
Outcome Probability
$2,500 0.20
Loss =$0 0.80
Assume losses are uncorrelatedExpected value =
Standard deviation =
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Uncorrelated Risk Pooling Example
with 2 People Two people with same distribution (w/o pooling)
Outcome Probability
$2,500 0.20
Loss =$0 0.80
Assume losses are uncorrelatedExpected value =
2500 0.2 + 00.8
$500Standard deviation =
2500500 0.2 + 0 500 0.8 $1000
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Uncorrelated Risk Pooling Example
with 2 People
Loss
No
Loss
Loss
Loss
No
Loss
No
Loss
0.2
0.8
0.2
0.2
0.8
0.8
Tree Diagram Loss Amount to be shared
2500 x 2/2 = 2500
2500 / 2 = 1250
2500 / 2 = 1250
0
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Uncorrelated Risk Pooling Example
with 2 People Pooling Arrangement changes distribution of
accident costs for each individualOutcome Probability
$0
Cost = 1,250
2,500
Expected Cost =
SD =
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Uncorrelated Risk Pooling Example
with 2 People Pooling Arrangement changes distribution of
accident costs for each individualOutcome Probability
$0 0.64
Cost = 1,250 0.32
2,500 0.04
Expected Cost = $500
SD = $707
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Effect on Expected Loss
w/o pooling, expected loss = $500
with pooling, expected loss = $500
Effect on Standard Deviation
w/o pooling, standard. deviation = $1,000
with pooling, standard. deviation = $707
Risk Pooling Example with 2 People
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Risk Pooling Example with
More People
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The Effect of Risk Pooling Arrangements on Probability
Distributions for a large number of small business
Without Pooling
With Pooling
P
robabilityDistributions
Cost paid by each business20000 40000 60000BWRR3033_Rodziah(c) 24
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Risk Pooling of Uncorrelated Losses
Main Points: Pooling arrangements
do not change expected loss
reduce uncertainty (variance decreases,losses become more predictable, maximum
probable loss declines)
distribution of costs becomes moresymmetric (less skewness)
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Effect of positive correlation on Risk Reduction
Pro
babilityDistrib
utions
Loss
Positive Correlated
Uncorrelated
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Effect of positive correlation on Risk Reduction
Standard
Deviation
ofAverageLoss
Number of participants in risk pooling
Uncorrelated
Less than perfect correlation
Perfect Correlation
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RISK MEASUREMENT (Analyze)DEFINITION
(What?)
ESTIMATION
(How)
LOSS
FREQUENCY
Number of times loss occurs
during a specific time on record.
nil/ slight/ moderate/ ultimate
Very likely/ likely/ unlikely
LOSS
SEVERITY
Size of loss per occurrences. Critical/ important/
unimportant fatal,/ major/ minor/
negligible
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RISK MEASUREMENT (Evaluate)
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