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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