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

2. Overview Terminology Application in Risk Management & Insurance Insurance Premium Using Probabilistic Approach

3. Terminology Probability  The likelihood of an event  The relative frequency of an event in the long run  Range 0 to 1, inclusive  Non-negative

4. 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. Not a true probability  Subjective probability Professional or trade skills and education Experience Random variable (or r.v.)  A number (or numeric outcome) whose value depends on some chance event or events

5. Terminology Mutually exclusive (events)  The probability of two mutually exclusive events occurring at the same time is ____. Collectively exhaustive (events) Independent (events)

6. Terminology Probability Distribution  Representations of all possible events along with their associated probabilities  Example; Total number of points rolled with a pair of dice. OutcomeProbability 21/36 32/36 43/36 54/36 65/36 76/36 85/36 94/36 103/36 112/36 121/36

7. Terminology Measure of central tendency  Mean, Median, Mode Measure of variability (risk)  Difference (Min, Max)  Variance  Standard deviation  Coefficient of variation “Unitless” measure

8. Examples LossProb.Loss x ProbLoss – EL(Loss-EL) 2 (Loss-EL) 2 ·Prob. 0.85 0-.45 0.2025 0.172125 1.10.55 0.3025 0.03025 5.03.154.55 20.7025 0.621075 10.02.209.55 91.2025 1.82405 Total1.00.452.6475 Standard Deviation = 1.6271 Variance Coefficient of Variation = 3.62 Loss Distribution Expected Loss, Mean

9. Which one faces more risk? Probability Distribution for the # of robbery per month for Store A and B # of Robbery Store A Probability Store B Probability 0.05.10 1.20.25 2.50.30 3.20.25 4.05.10

10. Decision Store B faces more risk because the higher measure of variance or the standard deviation. Another case Store AStore B Mean22 Variance0.81.3 Std. Dev.0.891.14 Coeff of Variation.445.57 Co. XCo. Y Mean.501.00 Std. Dev..45.87 Coeff of Variation 0.90.87

11. Probability Distribution

12. Application in RMI Loss Frequency Loss Severity  Maximum possible loss  Maximum probable loss Loss Frequency Distribution Loss Severity Distribution Total Loss Distribution

13. Application in RMI 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

14. Application in RMI - Frequency # of accidents per auto# of autosprobabilityTotal # of loss 0900900/10000 18080/100080 22020/100040 Expected # of accidents per auto (frequency) = Expected total # of losses = 120 A rental company with 1,000 rental cars

15. Application in RMI – Severity Case 1 - Severity per accident is not random.  Let severity = $1,125 1. What is expected $ loss per auto?  $1,125 x 0.12 = $135 2. What is expected $ loss for the rental company in a given time period?  $135 x 1,000 cars = $135,000

16. Application in RMI Case 2 - Severity is random with the following distribution.  What is expected $ loss per accident? $1,125  What is expected $ loss per auto? $135 Loss ($)# of accidentsProbabilityTotal losses ($) 5003030/120 =.2515,000 1,0006060/120 =.5060,000 2,0003030/120 =.2560,000

17. Insurance Premium Gross premium  premium charged by an insurer for a particular loss exposure = pure premium + risk charge + other loadings  Pure premium = Expected Loss (EL) A portion of the gross premium which is calculated as being sufficient to pay for losses only. Pure premium must be estimated.

18. Insurance Premium Risk Charge (Risk Loading)  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.  The higher the level of confidence in the estimated EL, the _____ the risk charge. The number of loss exposures insured by the insurer The size of loss exposures Example:  Risk charge for terrorism coverage would be _______.  Risk charge for personal automobile insurance would be _______.

19. Insurance Premium Other Loadings  Expense loading Administrative expenses, including advertising, underwriting, claims, general expenses, agent’s commission, etc …  Profit loading

20. Insurance Premium Expected Loss (frequency)  0.06 loss/exposure Expected $ Loss (severity)  $2,500 per loss Risk charge - 10% of pure premium Profit loading – 5% of pure premium Expense loading - $60 Gross premium =

21. 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%

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

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