Transcript Topic 4

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
occurring at the same time is _____ .

Collectively exhaustive (events)

Independent (events)
 At
events
least one of events must occur.
 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


Historical, empirical, posteriori probability


Number of times an event has occurred divided all possible times it
could have occurred.
Subjective probability



Number of possible equally likely occurrences divided by all
occurrences.
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

Variance

Standard deviation

Coefficient of variation
 cf.
Median, Mode
 Deviation from the mean
 Dispersion around the mean
 Square
root of the variance
 Standard deviation
 “Unitless” measure
divided by the mean
Probability of Loss

Chance of the loss or likelihood of the loss
Probability
of loss
Meaning


0
Impossible
1
Certain
A statement “Risk increases as probability of
a loss increases,” is ____________ .
Risk is not the same as probability of a loss.
Expected Value
Loss ($)
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
4,550
20,702,500
621,075
10,000
.02
200
9,550
91,202,500
1,824,050
1.00
450
Total
Standard Deviation = $1,627.11
Coefficient of Variation = 3.62
2,647,500
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
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
Mean
Probability Distribution
North
Co. A
Co. B
South
Mean
Mean A
Mean B
Application in Insurance

Loss Frequency


Probable number that may occur over a period of time
Loss Severity

Maximum possible loss


Maximum probable loss


Worst lost that is most likely to happen
Loss Frequency Distribution


Worst loss that could possible happen (worst scenario)
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

Loss amount
Probability
0
.85
1,000
.10
5,000
.03
10,000
.02
Maximum possible
loss
 10,000
 Independent
probability

of
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
# 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) =0.12
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.
$ 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
 What
is expected $ loss per loss? $1,125
 What is expected $ loss per auto? $135
Law of Large Numbers



The probability that an average outcome
differs from the expected value by more
than a small number approaches zero as the
number of exposures in the pool approaches
infinity.
The law of large numbers allows us to obtain
certainty from uncertainty and order from
chaos.
In short, the sample mean converges to the
distribution mean with probability 1.
Law of Large Numbers

Subject to
 Events
have to take place under same
conditions.
 Events can be expected to occur in the
future.
 The events are independent of one another
or uncorrelated.
Insurance Premium



Gross premium = premium charged by an insurer for a
particular loss exposure
Gross premium
= pure premium + risk charge + loading
Pure premium



A portion of the gross premium which is calculated as being
sufficient to pay for losses only.
Expected Loss (EL)
Pure premium must be estimated and the estimate may not
be sufficient to cover future losses.
Insurance Premium

Risk Charge
To reflect the estimation risk , insurers would add “risk charge” in
their premium calculation as a buffer.
 To deal with the fact that EL must be estimated, and the risk
charge covers the risk that actual outcome will be higher than
expected
 What determines the size/magnitude of the risk charge?



Amount of available past information to estimate EL
The level of confidence in the estimated EL.




Size/magnitude of the risk charge varies inversely with the level of
confidence in the estimated EL
Loss exposures with vast past information needs low risk charge and loss
exposures with little past information needs high risk charge.
Loss exposures with great deal of past information
Loss exposures with very little past information
Insurance Premium

Loading
 Expense

loading
Administrative expenses, including advertising,
underwriting, claim, general, agent’s commission,
etc …
 Profit
loading
Insurance Premium
Loss ($)
Prob.
Outcome
Weight
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
EL
450
Risk Charge = 495/450 = 10%
Risk Adjusted
Weight
Risk Adjusted
EL
495
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
Early
Detection
Sprinklers
Work?
No
(.01)
No
(.10)
Fire stop Probability
OK?
No
(.001)
10-6
$100 mil
Yes
(.999)
.000999
$10 mil
.099
$100 K
.90
0
Yes
(.99)
Fire
Loss
Yes
(.90)
What is the expected severity of a fire?
$19,990
Using Probabilistic Approach
What if there is no sprinkler system…
Early
Detection
Fire stop Probability
OK?
Loss
No
(.001)
10-4
$100 mil
Yes
(.999)
.0999
$10 mil
No
(.10)
Fire
Yes
(.90)
What is the expected severity of a fire?
.90
$1,009,000
0