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Statistical Analysis in Risk Management

- 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 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 Distribution

- Important property
- If Losses are normally distributed with
- Then
- Probability (Loss < ) = 0.95
- Probability (Loss < ) = 0.99

Using the Normal Distribution - An Example

- 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 Distribution

- Applies only to aggregate losses, not _______losses
- Thus, it cannot be used to analyze decisions about per occurrence deductibles and limits

Monte 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 Profile

1. E(L) forecast

a. single best estimate ……….

b. variations from this number will occur, therefore …

2. Example for a large company.(next slide)

mode, median

expected = $

Pr(L) > $11,500,000 =

Pr(L) > $14,000,000 =

B. Monte Carlo Steps

1. 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

IterationPercentileTotal Losses

1 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

Within LimitsAt Limits

,000 X BARSigmaX BARSigma

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 $

Simulation 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 - Assumptions

- Claim severity follows a
- expected value =
- standard deviation =
- note skewness

Simulation Example - Alternative Strategies

Policy 123

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 - Results

StatisticPolicy 1:Policy 2: Policy 3: No insurance

Mean 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

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