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Introduction to Risk Management and Insurance: Quantitative Methods and Probabilistic Approaches

This overview provides an introduction to quantitative methods in risk management and insurance, focusing on key terminology and applications. It discusses probability concepts, such as theoretical and historical probability, random variables, and measures of central tendency and variability. The text elaborates on how these concepts apply to insurance pricing, including expected loss, risk charge, and premium calculations. Through practical examples, we explore loss distributions, frequency, severity, and the use of probabilistic approaches to assess risks in insurance contexts.

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Introduction to Risk Management and Insurance: Quantitative Methods and Probabilistic Approaches

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  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. Probability Distribution Representations of all possible events along with their associated probabilities Example; Total number of points rolled with a pair of dice. Terminology

  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 Loss Distribution Expected Loss, Mean Standard Deviation = 1.6271 Variance Coefficient of Variation = 3.62

  9. Which one faces more risk? • Probability Distribution for the # of robbery per month for Store A and B

  10. Decision • Store B faces more risk because the higher measure of variance or the standard deviation. • Another case

  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

  14. Application in RMI - Frequency A rental company with 1,000 rental cars Expected # of accidents per auto (frequency) = Expected total # of losses = 120

  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

  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 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

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