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