Topic 4. Quantitative MethodsBUS 200Introduction to Risk Management and Insurance

Jin Park

Overview

Terminology Application in Risk Management &

Insurance Insurance Premium Using Probabilistic Approach

Terminology

Probability The likelihood of an

event The relative

frequency of an event in the long run

Range 0 to 1, inclusive

Non-negative

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

Terminology

Mutually exclusive (events)The probability of two mutually

exclusive events occurring at the same time is ____ .

Collectively exhaustive (events) Independent (events)

Terminology

Probability Distribution Representations of all

possible events along with their associated probabilities

Example;Total number of points rolled with a pair of dice.

Outcome Probability

2 1/36

3 2/36

4 3/36

5 4/36

6 5/36

7 6/36

8 5/36

9 4/36

10 3/36

11 2/36

12 1/36

Terminology

Measure of central tendencyMean, Median, Mode

Measure of variability (risk) Difference (Min, Max)VarianceStandard deviationCoefficient of variation

“Unitless” measure

Examples

Loss Prob.Loss x Prob

Loss – EL

(Loss-EL)2 (Loss-EL)2·Prob.

0 .85 0 -.45 0.2025 0.172125

1 .10 .10 .55 0.3025 0.03025

5 .03 .15 4.55 20.7025 0.621075

10 .02 .20 9.55 91.2025 1.82405

Total 1.00 .45 2.6475

Standard Deviation = 1.6271 VarianceCoefficient of Variation = 3.62

Loss Distribution

ExpectedLoss, Mean

Which one faces more risk?

• Probability Distribution for the # of robbery per month for Store A and B

# of RobberyStore A

ProbabilityStore B

Probability

0 .05 .10

1 .20 .25

2 .50 .30

3 .20 .25

4 .05 .10

Decision

Store B faces more risk because the higher measure of variance or the standard deviation.

Another case

Store A Store B

Mean 2 2

Variance 0.8 1.3

Std. Dev.

0.89 1.14

Coeff of Variation

.445 .57

Co. X Co. Y

Mean .50 1.00

Std. Dev. .45 .87

Coeff of Variation

0.9 0.87

Probability Distribution

Mean

North

South

Co. A Co. B

Mean A Mean B

Application in RMI

Loss Frequency Loss Severity

Maximum possible lossMaximum probable loss

Loss Frequency Distribution Loss Severity Distribution Total Loss Distribution

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 amount

Probability

0 .85

1,000 .10

5,000 .03

10,000 .02

Application in RMI - Frequency

# of accidents per auto

# of autos probability Total # of loss

0 900 900/1000 0

1 80 80/1000 80

2 20 20/1000 40

Expected # of accidents per auto (frequency) =Expected total # of losses = 120

A rental company with 1,000 rental cars

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

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 accidents

Probability Total losses ($)

500 30 30/120 = .25

15,000

1,000 60 60/120 = .50

60,000

2,000 30 30/120 = .25

60,000

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.

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

_______.

Insurance Premium

Other LoadingsExpense loading

Administrative expenses, including advertising, underwriting, claims, general expenses, agent’s commission, etc …

Profit loading

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 =

Insurance Premium

Loss ($) Prob.Outcom

e Weight

ELRisk Adjusted

WeightRisk Adjusted

EL

0 .85 1.0 0 0.0 0

1,000 .10 1.0 100 0.8 80

5,000 .03 1.0 150 1.1 165

10,000 .02 1.0 200 1.25 250

Total 1.00 450 495

Risk Charge = 495/450 = 10%

Using Probabilistic Approach

N o(.1 0 )

Y e s(.9 0 )

F ire

N o(.0 1 )

Y e s(.9 9 )

N o(.0 0 1 )

Y e s(.9 9 9 )

E a rlyD e te c tion

S prink le rs W ork ?

F ire stop O K ?

P roba bili ty

1 0 -6 $ 1 0 0 m il

.0 0 0 9 9 9 $ 1 0 m il

.0 9 9 $ 1 0 0 K

.9 0 0

L oss

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…

N o(.1 0 )

Y e s(.9 0 )

F ire

N o(.0 0 1 )

Y e s(.9 9 9 )

E a rlyD e te c tion

F ire stop O K ?

P roba bili ty

1 0 -4 $ 1 0 0 m il

.0 9 9 9 $ 1 0 m il

.9 0 0

L oss

What is the expected severity of a fire? $1,009,000