C-2: Loss SimulationC-2: Loss Simulation

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”

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:

Using the Normal DistributionUsing the Normal Distribution

Important property

– If Losses are normally distributed with

– Then

Probability (Loss < ) = 0.95

Probability (Loss < ) = 0.99

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

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

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

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 =

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

3. Uses of Total Loss Profile

a. Evaluate and loss limits

b.

c.

d. MPL (MPY)

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

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 $ $ $

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

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

D. Interpretation of ResultsD. Interpretation of Results

1. Look at summary statistics: mean, sigma, percentiles

2.

3.

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 $

E. Aggregates – Recap using text E. Aggregates – Recap using text

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.

Simulation Example - AssumptionsSimulation Example - Assumptions

– Claim severity follows a

expected value = standard deviation = note skewness

Simulation Example - Simulation Example - AssumptionsAssumptions

Frequency Distribution with Expected Value Equal to 30

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 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

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

Simulation Example - ResultsSimulation Example - Results No Insurance

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

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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

$500,000 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

$6 Million Aggregate Annual 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

$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

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