Numerical Simulation of Biodiversity Loss: Comparison of ...
C-2: Loss Simulation
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Transcript of C-2: Loss Simulation
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C-2: Loss SimulationC-2: Loss Simulation
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Statistical Analysis in Risk Statistical Analysis in Risk ManagementManagement
– Two main approaches:
– Maximum probable loss (or MPY)
if $5 million is the maximum probable loss at the _______percent level, then the firm’s losses will be less than $_____million with probability 0.95.
Same concept as “Value at risk”
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When to Use the Normal DistributionWhen to Use the Normal Distribution– Most loss distributions are not normal
– From the __________ theorem, using the normal distribution will nevertheless be appropriate when
– Example where it might be appropriate:
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Using the Normal DistributionUsing the Normal Distribution
Important property
– If Losses are normally distributed with
– Then
Probability (Loss < ) = 0.95
Probability (Loss < ) = 0.99
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Using the Normal Distribution - An Using the Normal Distribution - An ExampleExample
– Worker compensation losses for Stallone Steel
sample mean loss per worker = $_____ sample standard deviation per worker = $20,000 number of workers = ________
– Assume total losses are normally distributed with mean = $3 million standard deviation =
– Then maximum probable loss at the 95 percent level is
$3 million + = $6.3 million
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A Limitation of the Normal DistributionA Limitation of the Normal Distribution
Applies only to aggregate losses, not _______losses
Thus, it cannot be used to analyze decisions about per occurrence deductibles and limits
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Monte Carlo SimulationMonte Carlo Simulation– Overcomes some of the shortcomings of the normal
distribution approach
– Overview:
Make assumptions about distributions for ________ and _______ of individual losses
Randomly draw from each distribution and calculate the firm’s total losses under alternative risk management strategies
Redo step two many times to obtain a distribution for total losses
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A. Total Loss ProfileA. Total Loss Profile1. E(L) forecast
a. single best estimate ……….b. variations from this number will occur, therefore …
2. Example for a large company.(next slide)mode, medianexpected = $ Pr(L) > $11,500,000 = Pr(L) > $14,000,000 =
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Unlimited Loss Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
9 10 11 12 13 14 15 16 17 18 19
Total Losses (Millions)
Pro
bab
ilit
y
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3. Uses of Total Loss Profile
a. Evaluate and loss limits
b.
c.
d. MPL (MPY)
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B. Monte Carlo StepsB. Monte Carlo Steps1. Select frequency distribution
2. Select severity distribution
3. Draw from ________ distribution => N1 losses
4. Draw N1 severity values from severity distribution
5. Repeat steps____and ____ for 1000 or more iterations
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Iteration Number 1 2 1,000
N i 70 23 … 43
S1 $ 600 $ 94,000 $ _____
S2 $ 18,400 $ 150 $ 970 …
S10 $ _____ $ 2,600 $ 500 …
S23 $ 19,500 $ 1,350 $ 32,150 …
S43 $ 3,750 NA $182,000 …
S70 $ 54,000 NA NA
Total $ $ $
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Rank Order the Total Losses
Iteration Percentile Total Losses1 0.1 $ 143,000.100 10 1,790,000.500 50 2,280,000.700 70 ________.900 90 3,130,000.950 95 ________.1,000 100 3,970,000
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Draw LT 1,0001,000-4,999
5,000-9,999
10000-49,999
50,000-99,999
GE 100,000
Total
1 625 625 …98 ________ 2,050 _________…
251 999 4,000 _________..
730 789 789 …
980 999 4,000 5,000 40,000 50,000 10,001 110,000 Total 920,000 450,000 414,000 180,000 119,000 47,000 2,130,000
Horizontal Layering: From One Iteration
Layers for the 438th Iteration that produced 980 Severity Values
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D. Interpretation of ResultsD. Interpretation of Results
1. Look at summary statistics: mean, sigma, percentiles
2.
3.
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Within Limits At Limits
,000 X BAR Sigma X BAR Sigma
1 - 10 $ $ $ $
10 25 $ 612 $ 88 $ 2,655 $ 176
25 - 50 $ 326 $ 92 $ 2,981 $ 239
50 - 75 $ 128 $ 55 $ 3,109 $ 275
75 - 100 $ 65 $ 41 $ 3,174 $ 298
100 - 150 $ 60 $ 53 $ 3,234 $ 325
150 - 200 $ 26 $ 32 $ 3,260 $ 340
200 - 250 $ 15 $ 23 $ 3,275 $ 350
250 - 500 $ 23 $ 60 $ 3,298 $ 370
500 - 1,000 $ 9 $ 62 $ 3,307 $ 400
> 1,000 $ 1 $ 8 $ 3,307 $ 404 $
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E. Aggregates – Recap using text E. Aggregates – Recap using text
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Simulation Example - AssumptionsSimulation Example - Assumptions
– Claim frequency follows a Poisson distribution
Important property: Poisson distribution gives the probability of 0 claims, 1 claim, 2 claims, etc.
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Simulation Example - AssumptionsSimulation Example - Assumptions
– Claim severity follows a
expected value = standard deviation = note skewness
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Simulation Example - Simulation Example - AssumptionsAssumptions
Frequency Distribution with Expected Value Equal to 30
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0.05
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0.15
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0.25
0 6 12
18
24
30
36
42
48
54
Number of Claims
PR
OB
AB
ILIT
Y Sample Frequency Distribution with Uncertain
Expected Value (1000 trials)
0
0.05
0.1
0.15
0.2
0.25
0 6 12
18
24
30
36
42
48
54
Number of Claims
PR
OB
AB
ILIT
Y
Sample Loss Severity Distribution(1000 trials)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.0075 0.6 1.2 1.8 2.4 3
Loss in Millions
PR
OB
AB
ILIT
Y
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Simulation Example - Alternative Simulation Example - Alternative StrategiesStrategies
Policy 1 2 3
Per Occurrence Deductible $500,000 $1,000,000 none
Per Occurrence Policy Limit $5,000,000 $5,000,000 none
Aggregate Deductible none none $6,000,000
Aggregate Policy Limit none none $10,000,000
Premium $780,000 $415,000 $165,000
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Simulation Example - ResultsSimulation Example - Results No Insurance
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0.02
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0.06
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0.1
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0.18
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0 1. 5 3 4. 5 6 7. 5 9 10 .5 12
13 .5
Values in Millions
PR
OB
AB
ILIT
Y
$500,000 per Occurrence Retention
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0.02
0.04
0.06
0.08
0.1
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0 1. 5 3 4. 5 6 7. 5 9 10 .5 12
13 .5
Values in Millions
PR
OB
AB
ILIT
Y
$6 Million Aggregate Annual Retention
0
0.02
0.04
0.06
0.08
0.1
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0 1. 5 3 4. 5 6 7. 5 9 10 .5 12
13 .5
Values in Millions
PR
OB
AB
ILIT
Y
$1 Million per Occurrence Retention
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 1. 5 3 4. 5 6 7. 5 9 10 .5 12
13 .5
Values in Millions
PR
OB
AB
ILIT
Y
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Simulation Example - ResultsSimulation Example - ResultsStatistic Policy 1: Policy 2: Policy 3: No
insuranceMean value of retained losses $______ $2,716 $2,925 $3,042
Standard deviation of retained losses 1,065 1,293 1,494 1,839
Maximum probable retained loss at 95% level 4,254 5,003 ______ 6,462
Maximum value of retained losses 11,325 12,125 7,899 18,898
Probability that losses exceed policy limits 1.1% 0.7% 0.1% n.a.
Probability that retained losses $6 million 99.7% ____% 99.9% 92.7%
Premium $780 $415 $165 $0
Mean total cost 3,194 3,131 3,090 3,042
Maximum probable total cost at 95% level 5,034 5,418 6,165 6,462