Topic 4. Quantitative MethodsBUS 200Introduction 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

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

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.

P ro b a b ility 0 1o f lo s s

M e a ning Im p o s s ib le C e rta in

Expected ValueLoss ($) Prob. EL AL – EL (AL-EL)2 (AL-EL)2·Prob.

0 .85 0 -450 202,500 172,125

1,000 .10 100 550 302,500 30,250

5,000 .03 150 4,55020,702,50

0621,075

10,000 .02 200 9,55091,202,50

01,824,050

Total 1.00 450 2,647,500

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

higher measure of dispersion as measured by the variance or the standard deviation.

According to the coefficient of variation, …

North South

Mean 2 2

Variance 0.8 1.3

Std. Dev.

0.89 1.14

Coeff of Variation

.445 .57

Co. A Co. B

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 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

Loss amount Probability

0 .85

1,000 .10

5,000 .03

10,000 .02

Application in Insurance

# of losses per auto

# of autos with loss

probability Total # of loss

0 900 900/1000 0

1 80 80/1000 80

2 20 20/1000 40

Expected # of loss per auto (frequency) =0.12Expected # of total loss = 120

1,000 rental cars

Application in Insurance

Case 1 If severity is not random. Let severity =

$1,125What is expected $ loss per auto?

$1,125 x 0.12 = $135What 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

$ amount of loss

# of losses

probability Total $ amount of losses

500 30 30/120 15,000

1,000 60 60/120 60,000

2,000 30 30/120 60,000

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 toEvents 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

LoadingExpense loading

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

Profit loading

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%

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

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