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Topic 4. Quantitative Methods BUS 200 Introduction to Risk Management and Insurance Jin Park

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Terminology Probability The likelihood of a particular event occurring The relative frequency of an event in the long run Non-negative Between 0 and 1 Probability distribution Representations of all possible events along with their associated probabilities

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Terminology Mutually exclusive (events) events that cannot happen together The probability of two mutually exclusive events occurring at the same time is _____. Collectively exhaustive (events) At least one of events must occur. Independent (events) the occurrence or non occurrence of one of the events does not affect the occurrence or non occurrence of the others

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Terminology Probability Theoretical, priori probability Number of possible equally likely occurrences divided by all occurrences. Historical, empirical, posteriori probability Number of times an event has occurred divided all possible times it could have occurred. Subjective probability Professional or trade skills and education Experience Random variable Number (or numeric outcome) whose value depends on some chance event or events The outcome of a coin toss (head or tail) Total number of points rolled with a pair of dice Total number of automobile accidents in a day in Illinois Total $ value of losses that do in fact occur

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Probability Distribution Random variable Total number of points rolled with a pair of dice Possible outcomes two to twelve OutcomeProbability 21/36 32/36 43/36 54/36 65/36 76/36 85/36 94/36 103/36 112/36 121/36

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Terminology Mean, average, expected value cf. Median, Mode Variance Deviation from the mean Dispersion around the mean Standard deviation Square root of the variance Coefficient of variation Standard deviation divided by the mean Unitless measure

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Probability of Loss Chance of the loss or likelihood of the loss A statement Risk increases as probability of a loss increases, is ____________. Risk is not the same as probability of a loss.

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Expected Value Loss ($)Prob.ELAL – EL(AL-EL) 2 (AL-EL) 2 ·Prob. 0.85 0-450202,500172,125 1,000.10100550302,50030,250 5,000.031504,55020,702,500621,075 10,000.022009,55091,202,5001,824,050 Total1.004502,647,500 Standard Deviation = $1,627.11 Coefficient of Variation = 3.62

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Variability Refer to your assigned reading South faces most risk because higher measure of dispersion as measured by the variance or the standard deviation. Another case Co. B faces most risk because higher measure of dispersion as measured by the variance or the standard deviation. According to the coefficient of variation, … NorthSouth Mean22 Variance0.81.3 Std. Dev.0.891.14 Coeff of Variation.445.57 Co. ACo. B Mean.501.00 Std. Dev..45.87 Coeff of Variation 0.90.87

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Probability Distribution

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Application in Insurance Loss Frequency Probable number that may occur over a period of time Loss Severity Maximum possible loss Worst loss that could possible happen (worst scenario) Maximum probable loss Worst lost that is most likely to happen Loss Frequency Distribution The distribution of the number of occurrences per a period of time Loss Severity Distribution The distribution of the dollar amount lost per occurrence per a period of time

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Application in Insurance Maximum possible loss 10,000 Independent of probability Maximum probable loss 98% chance that losses will be at most $5,000 95% chance that loss will be at most $1,000 Loss amountProbability 0.85 1,000.10 5,000.03 10,000.02

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Application in Insurance # of losses per auto # of autos with loss probabilityTotal # of loss 0900900/10000 18080/100080 22020/100040 Expected # of loss per auto (frequency) =0.12 Expected # of total loss = 120 1,000 rental cars

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Application in Insurance Case 1 If severity is not random. Let severity = $1,125 What is expected $ loss per auto? $1,125 x 0.12 = $135 What is expected $ loss for the rental company in a given time period? $135 x 1,000 cars = $135,000

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Application in Insurance Case 2 If severity is random with the following distribution. What is expected $ loss per loss? $1,125 What is expected $ loss per auto? $135 $ amount of loss # of lossesprobabilityTotal $ amount of losses 5003030/12015,000 1,0006060/12060,000 2,0003030/12060,000

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Law of Large Numbers The probability that an average outcome differs from the expected value by more than a small number approaches zero as the number of exposures in the pool approaches infinity. The law of large numbers allows us to obtain certainty from uncertainty and order from chaos. In short, the sample mean converges to the distribution mean with probability 1.

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Law of Large Numbers Subject to Events have to take place under same conditions. Events can be expected to occur in the future. The events are independent of one another or uncorrelated.

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Insurance Premium Gross premium = premium charged by an insurer for a particular loss exposure Gross premium = pure premium + risk charge + loading Pure premium A portion of the gross premium which is calculated as being sufficient to pay for losses only. Expected Loss (EL) Pure premium must be estimated and the estimate may not be sufficient to cover future losses.

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Insurance Premium Risk Charge To reflect the estimation risk, insurers would add risk charge in their premium calculation as a buffer. To deal with the fact that EL must be estimated, and the risk charge covers the risk that actual outcome will be higher than expected What determines the size/magnitude of the risk charge? Amount of available past information to estimate EL The level of confidence in the estimated EL. Size/magnitude of the risk charge varies inversely with the level of confidence in the estimated EL Loss exposures with vast past information needs low risk charge and loss exposures with little past information needs high risk charge. Loss exposures with great deal of past information Loss exposures with very little past information

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Insurance Premium Loading Expense loading Administrative expenses, including advertising, underwriting, claim, general, agents commission, etc … Profit loading

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Insurance Premium Loss ($)Prob. Outcome Weight EL Risk Adjusted Weight Risk Adjusted EL 0.851.0 00.0 0 1,000.101.01000.8 80 5,000.031.01501.1165 10,000.021.02001.25250 Total1.00450495 Risk Charge = 495/450 = 10%

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Insurance Premium Expected Loss (frequency) – 0.06 loss/exposure Expected $ Loss (severity) - $2,500 per loss Risk charge – 10% of pure premium All loadings - $100 Gross premium =

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Using Probabilistic Approach Simple example of event tree What is the expected severity of a fire? $19,990

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Using Probabilistic Approach What if there is no sprinkler system… What is the expected severity of a fire? $1,009,000

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