Topic 4. Quantitative Methods

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

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

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

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

Probability Distribution • Random variable • Total number of points rolled with a pair of dice • Possible outcomes • two to twelve

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

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.

Expected Value Standard Deviation = $1,627.11 Coefficient of Variation = 3.62

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 highermeasure of dispersion as measured by the variance or the standard deviation. • According to the coefficient of variation, …

Probability Distribution

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

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

Application in Insurance 1,000 rental cars Expected # of loss per auto (frequency) =0.12 Expected # of total loss = 120

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

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

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.

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.

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.

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

Insurance Premium • Loading • Expense loading • Administrative expenses, including advertising, underwriting, claim, general, agent’s commission, etc … • Profit loading

Insurance Premium Risk Charge = 495/450 = 10%

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 =

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

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

Topic 4. Quantitative Methods