Economics of Insurance 1

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Economics of Insurance 1
Lecture 2
Why people want to insure a brief introduction
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Why people want to insure a brief introduction
We will cover
States of the world, contingency and events
Risk Pooling
Expected value
Risk aversion
Types of premium
Types of compensation
Premium rates as prices
Probabilities and Odds
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Why people want to insure a brief introduction
Why People Insure
Faced with a risky world,
many people wish to reduce the effect of that
risk by insuring.
But what is risk?
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Why people want to insure a brief introduction
Risk
Risk means multiplicity of possibilities, each
possibility having a known or plausible
PROBABILTY.
Driving a car
Some examples of risky activities
A lottery or other gamble
buying a house
Negative equity
Buying a share
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Why people want to insure a brief introduction
States and events
Notice that risk involves our assets or
activities having more than one possible value or
outcome
To analyse this we distinguish between different
states of the world or event
My house burning down is an EVENT, .
the fact that my house is burned down is a STATE
of the WORLD
Each event or state of the world has its OPPOSITE

another event or state in which the bad thing
DID NOT HAPPEN
ie my house not burning down, or not being burned
down
We often use numbers to show which event or state
we mean
i.e. 2 means house burned down,
1 means house not burned down
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Why people want to insure a brief introduction
Multiple possible outcomes
So in a risky world
Decisions or processes always have
MORE THAN ONE POTENTIAL OUTCOME
When we talk about outcomes
or their potential money value
We must distinguish between
and
ANALYSIS
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Assets
Why people want to insure a brief introduction
Financial consequences are considered through the
concept of ASSETS that have monetary values
ie my house is my ASSET, worth 120000
But my house is worth 120000 only if not burned
down, that is in STATE of the World 1 It is worth
just 20000 if burned down, that is in STATE of
the world 2
So in a risky world, assets have more than a
single value, each individual value being
attached to a possible event
My house is worth 120,000 in state 1,
OR
20,000 in state 2
Monetary asset values will be shown in these
lectures as X with events or states as subscripts.
So the value of my house could be described as
(X1, X2)
where X1 is
120,000 X2 is 20,000
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Probabilities and Odds
Why people want to insure a brief introduction
In our example we know the two possible values of
the house, each corresponding to a state of the
world
But riskiness involves chance - probability
which we can measure as a decimal or
So suppose there was a 0.01, or 1, probability
of the house burning down in any year
This implies a 0.99 or 99 chance of the house
not burning down
Note a probability is always between 0 and 1
The house is expected to burn down once in a
hundred years
What does a probability of 1 mean
What does a probability of 0 mean?
What is the TOTAL of all probabilities describing
a situation?
Odds
It is sometimes useful to measure probabilities
in terms of ODDS rather than ABSOLUTE
PROBABILITIES.
Odds can be defined as P/(1-P) where P is the
absolute probability of the event insured against
NOT happening. In our example the probability of
the house NOT burning down is 0.99, whilst the
probability of it burning down is 0.01. The odds
are thus 0.99/0.01 99. This is usually
expressed as 99 to 1 against burning down.
Odds are RELATIVE probabilities and can take any
positive value
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Prospects, contingency and loss
Why people want to insure a brief introduction
If we knew or could guess the probabilities
involved we could write all we knew about a risky
asset as a PROSPECT
this is a list of possible asset values and their
probabilities
In the previous example
20000
(120000,
0.99,
0.01)
the houses value can be written as
This PROSPECT is the most accurate way of
expressing the houses value.
If the house doesnt burn down
--------- If the house burns down ---------
It takes into account the fact that our house has
two values
Each of the values DEPENDS ON is CONTINGENT ON
- the state of the world
If there were 3 possibilities or states (house
undamaged, house burned but land value remains,
house burned and landed poisoned and so
unsellable) there would be 3 events and the
prospect would look like this (X1, X2, X3
P1, P2, P3)
In the event of fire the LOSS the house owner
makes is 120000 - 20000 100000
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Summarising the value of risky assets
Why people want to insure a brief introduction
Having suggested that assets have a multiplicity
of values, it has to be noted that, if used with
care, a single value can demonstrate many of the
characteristics of a risky asset.
Such a value is
EXPECTED VALUE
(weighted average, arithmetic mean, often written
as E(V) )

This is calculated as S
Pi Xi where P and X are probabilities and asset
values respectively and i is the STATE of the
WORLD
i
In the case of our house, its average value is (
P1 x X1 ) (P2 x X2)

( 0.99 x 120000) (0.01
x 20000)
118800
200

( 120000, 20000 0.99, 0.01)
So houses
E(V)
119000
i.e. you would expect the house to be burned down
once every 100 years, so on average it is worth
99 x 120000 (99 years when not burned AND SO
worth 120000)1 x 20000 (1 year when burned so
worth 20000) 119000
100 ( the
number of years)
or you would need a fund of 100,000 once in
every 100 year period to replace your burned
house. 1000 saved each year would produce that
fund. Thus the NET value of your assets each year
is 120,000 - 1000 119000
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Why people want to insure a brief introduction
What happens if a person doesnt insure
Now for the key question.
Why would an ordinary person insure a house
against fire ?
First consider how well-off a houseowner would be
if they DIDNT insure
The risk of a house being burned down is 0.01.
The house is worth 120,000 or 20000 if burned
down
i.e. the house is an asset whose prospect
(120000,200000.99,0.01)
The expected value of the house is thus-
(0.01 x 20000) (0.99 x 120000) 119000
This is a MORE ACCURATE valuation of the house
than 120000
(though less accurate than the prospect!)
Because the house WILL probably burn down once in
100 years and the owner WILL therefore make a
loss of 100000
So on average the homeowner DOES only have an
asset worth 119000
This DOES average out to 1000 per year
But this manifests itself as 99 years asset worth
120000
1 Year asset worth 20000
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Why people want to insure a brief introduction
What happens if a person doesnt insure
In this theory
The person will take account of
BOTH
value (120000) in the good years
and
when his asset is worth only 20000 in the
disastrous year
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Mutual Insurance and Risk Pooling
Why people want to insure a brief introduction
Now our individual joins a MUTUAL INSURANCE CLUB
Suppose 100 people including our home owner each
contributes 1000 each year to a communal
insurance-against-fire fund. That fund
called a pool -
would have an income of 100000 each year.
Each year the fund would expect to pay out
compensation for houses destroyed by fire. If the
conditions of the fund are that each contributor
receives full compensation for damage,
the fund would expect to pay-out 0.01 x 100 x
100000 100000 each year.
one person in a hundred
In an average year
claims
100000
Thus on average the fund is just big enough to
cover claims for compensation.
SO OUR HOMEOWNER WILL ALWAYS HAVE ASSETS WORTH
120000 - 1000 119000
(mustnt forget the houseowners contribution to
the fund)
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Why people want to insure a brief introduction
Insurance and Risk
Notice that
With insurance our house owner actually has an
asset worth 119000
ASSET VALUE WILL NOT CHANGE
THERE IS NO FINANCIAL RISK
Insurance has removed the houseowners risk
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Why people want to insure a brief introduction
Will the insurance be bought
Perhaps not!
119000
No Insurance
Expected value
Apparently no difference
119000
Insurance
Actual value
EXCEPT THAT
THERE IS A RISK THAT VALUE MIGHT BE 20000
WITH NO INSURANCE
WITH INSURANCE
VALUE OF ASSET IS A SURE 119000
i.e. LOWER VALUE than uninsured in good state,
BUT with NO RISK
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Why people want to insure a brief introduction
Comparing being insured with being uninsured
So to insure on this deal, the house owner will
compare
99 chance of 120000
but 1 chance being only 20000
(not insured)
with
i.e.119000 with NO RISK
CERTAINTY OF 119000
(insured)
TO CHOOSE TO INSURE the home owner must DISLIKE
THE RISK
more than
LOSING 1000 of value in the good state
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Why people want to insure a brief introduction
Risk aversion
SUCH A PERSON IS
RISK AVERSE
Risk AVERTERS dislike risk AS SUCH
Consequently they will pay to avoid risk
(In our example they would pay 1000 per year to
avoid risk)
THAT IS WHY PEOPLE CHOOSE TO INSURE
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Fair Insurance
Why people want to insure a brief introduction
An insurance deal is FAIR if
in our example
e.g.
1000 0.01 x 100000
In what sense fair?
Because over 100 years the client would
make premium payments to the insurance company
of 100 x 1000
100000
100000
receive compensation payments from the company
of 1 x 100000
i.e. the client receives back from the insurance
company exactly what they paid in
If insurance is fair then-
1) Premium/gross compensation q
2) A RISK AVERTER WILL ALWAYS BUY FAIR INSURANCE
3) Will an insurance company sell it?
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Types of Compensation
Why people want to insure a brief introduction
Our house owner, if their house is burned-down,
receives 100000. This is called the Gross
Compensation. Insurance companies use this
measure of compensation in their contractual
terms. However, in some ways it is a misleading
measure. This is because it fails to take into
account the fact that the client has already had
to pay a premium. Net Compensation is a measure
that takes this into account.
 

In this example
99000                           
100000         -    1000
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Premium Rates and the Price of Insurance
Why people want to insure a brief introduction
Insurance companies often quote a price for
insurance in terms of premium per 1 of
compensation.
This is known as the premium rate and is, of
course, the ratio Premium/Compensation payable
But since there are two type of compensation, we
must distinguish
from
 
 
In this example
GPR 1000/100000 1/100 i.e. 1/100 per 1
gross compensation 1 penny per 1 gross
compensation
NPR 1000/99000 1/99 i.e. 1/99 per 1 net
compensation 1.0101 pence per 1 net
compensation
Note that the premium rate is a common sense
definition of price. It is the RATE at which the
client is being charged for the service
contingent payment of compensation offered by
the insurance company.
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Types of Insurance Contract
Why people want to insure a brief introduction
Full v Partial
Full insurance covers the whole of any loss
within the insurance scheme
gross compensation the whole loss
More precisely
This is the case of our example
Loss 100000
so gross compensation 100000
Partial insurance covers only part of any loss
within the scheme. The client is thus covering
part of the loss himself or herself.
More precisely
gross compensation a part of the loss
If the contract in our example had have been for
50 compensation
Loss 100000
so gross compensation 0.5 x 100000
50000
Fixed v Flexible
A fixed contract is one in which the insurance
company determines the amount of cover as well as
the premium rate. The only choice the client has
is whether or not to buy the insurance from that
company at all.
A flexible contract is one in which the client
determines the amount of cover (for example full
or partial cover) whilst the insurance company
determines the premium rate.