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C-2: Loss Simulation

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# C-2: Loss Simulation - PowerPoint PPT Presentation

C-2: Loss Simulation. 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.

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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 =

3. Uses of Total Loss Profile

a. Evaluate and loss limits

b.

c.

d. MPL (MPY)

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

D. Interpretation of Results

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

2.

3.

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