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McGraw-Hill/Irwin Copyright © 2004 by the McGraw-Hill Companies, Inc. All rights reserved. Chapter 3 Risk Identification and Measurement

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T3.2 H&N, Ch. 3 Identifying Business Risk Exposures Property Business income Liability Human resource External economic forces

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T3.3 H&N, Ch. 3 Identifying Individual Exposures Earnings Physical assets Financial assets Medical expenses Longevity Liability

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T3.4 H&N, Ch. 3 Probability Distributions Probability distributions Listing of all possible outcomes and their associated probabilities Sum of the probabilities must equal 1 Two types of distributions: discrete continuous

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T3.5 H&N, Ch. 3 Presenting Probability Distributions Two ways of presenting discrete distributions: Numerical listing of outcomes and probabilities Graphically Two ways of presenting continuous distributions: Density function (not used in this course) Graphically

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T3.6 H&N, Ch. 3 Example of a Discrete Probability Distribution Random variable = damage from auto accidents Possible Outcomes for DamagesProbability $00.50 $2000.30 $1,0000.10 $5,0000.06 $10,0000.04

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T3.7 H&N, Ch. 3 Example of a Discrete Probability Distribution

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T3.8 H&N, Ch. 3 Example of a Continuous Distribution

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T3.9 H&N, Ch. 3 Continuous Distributions Important characteristic of density functions Area under the entire curve equals one Area under the curve between two points gives the probability of outcomes falling within that given range

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T3.10 H&N, Ch. 3 Probabilities with Continuous Distributions Find the probability that the loss > $5,000 Find the probability that the loss < $2,000 Find the probability that $2,000 < loss < $5,000 Possible Losses Probability $5,000 $2,000

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T3.11 H&N, Ch. 3 Risk Management & Probability Distributions Ideally, a risk manager would know the probability distribution of losses Then assess how different risk management approaches would change the probability distribution Example: Which distribution would you rather have? Accident Cost No RM Prob Insurance Accident Cost + Insurance Cost

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T3.12 H&N, Ch. 3 Summary Measures of Loss Distributions Instead comparing entire distributions, managers often work with summary measures of distributions: Frequency Severity Expected loss Standard deviation or variance Maximum probable loss (Value at Risk) Then ask: How does RM affect each of these measures?

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T3.13 H&N, Ch. 3 Expected Value Formula for a discrete distribution: Expected Value = x 1 p 1 + x 2 p 2 + … + x M p M. Example: Possible Outcomes for DamagesProbability $00.50 $2000.30 $1,0000.10 $5,0000.06 $10,0000.04 Expected Value =

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T3.14 H&N, Ch. 3 Expected Value

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T3.15 H&N, Ch. 3 Standard Deviation and Variance Standard deviation indicates the expected magnitude of the error from using the expected value as a predictor of the outcome Variance = (standard deviation) 2 Standard deviation (variance) is higher when when the outcomes have a greater deviation from the expected value probabilities of the extreme outcomes increase

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T3.16 H&N, Ch. 3 Standard Deviation and Variance Comparing standard deviation for three discrete distributions Distribution 1Distribution 2Distribution 3 Outcome ProbOutcome ProbOutcome Prob $2500.33$00.33$00.4 $5000.34$5000.34$5000.2 $7500.33$10000.33$10000.4

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T3.17 H&N, Ch. 3 Standard Deviation and Variance

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T3.18 H&N, Ch. 3 Sample Mean and Standard Deviation Sample mean and standard deviation can and usually will differ from population expected value and standard deviation Coin flipping example $1 if heads X = -$1 if tails Expected average gain from game = $0 Actual average gain from playing the game 5 times =

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T3.19 H&N, Ch. 3 Skewness Skewness measures the symmetry of the distribution No skewness ==> symmetric Most loss distributions exhibit skewness

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T3.20 H&N, Ch. 3 Maximum Probable Loss Maximum Probable Loss at the 95% level is the number, MPL, that satisfies the equation: Probability (Loss < MPL) < 0.95 Losses will be less than MPL 95 percent of the time

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T3.21 H&N, Ch. 3 Value at Risk (VAR) VAR is essentially the same concept as maximum probable loss, except it is usually applied to the value of a portfolio If the Value at Risk at the 5% level for the next week equals $20 million, then Prob(change in portfolio value < -$20 million) = 0.05 In words, there is 5% chance that the portfolio will lose more $20 million over the next week

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T3.22 H&N, Ch. 3 Value at Risk Example: Assume VAR at the 5% level =$5 million And VAR at the 1% level = $7 million

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T3.23 H&N, Ch. 3 Important Properties of the Normal Distribution Often analysts use the following properties of the normal distribution to calculate VAR: Assume X is normally distributed with mean and standard deviation . Then Prob (X > -2.33 ) = 0.01 Prob (X > -1.645 ) = 0.05

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T3.24 H&N, Ch. 3 Correlation Correlation identifies the relationship between two probability distributions Uncorrelated (Independent) Positively Correlated Negatively Correlated

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T3.25 H&N, Ch. 3 Calculating the Frequency and Severity of Loss Example: 10,000 employees in each of the past five years 1,500 injuries over the five-year period $3 million in total injury costs Frequency of injury per year = 1.500 / 50,000 = 0.03 Average severity of injury = $3 m/ 1,500 = $2,000 Annual expected loss per employee = 0.03 x $2,000 = $60

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