Economics of Insurance 2.

1
Economics of Insurance 2.
Lecture 8
The Critique of Insurance Demand Theory, its
Rebuttal and State Contingent Income Transfer
Theory
2
The critique of risk theory and insurance
Outline
We will
Look again at the Kahneman and Tversky critique
of conventional risk theory
Consider an implication of the critique for
insurance theory
Consider Nymans reconciliation of K/T and
conventional theory when applied to insurance
Demonstrate Nymans argument
Consider the interpretation of the demand for
insurance used by Nyman
3
The critique of risk theory and insurance
Kahneman and Tversky
Kahneman and Tversky
want to replace utility maximising hypothesis
with
BEHAVIOURAL theories of choice in the face of risk
Such theories are based directly on psychological
experiments that appear to contradict the
expected utility maximising hypothesis.
In particular, K and Ts experiments appear to
show that peoples behaviour towards risk is
different according to whether -
they perceive themselves as GAINING from the
risks, in which case conventional EUH theories
are reasonable
or whether people perceive of the risk in terms
of LOSSES
in which case, in many circumstances they act as
if they PREFER risk
K Ts theroies are often (confusingly) called
PROSPECT THEORIES
4
The critique of risk theory and insurance
Kahneman and Tversky
K T think that people
Make decisions in CONTEXT they have a frame or
norm from which they make judgements
Tend to favour the STATUS QUO
Are LOSS averters, not simple risk averters
Have a LOSS FUNCTION
Loss function is CONCAVE from below.
Consciously or mistakenly misestimate
probabilities
Have a probability weighting function
5
The critique of risk theory and insurance
A Kahnemann/Tversky type critque of conventional
insurance theory
KT are often illustrated using a modified
utility function, sometimes called a value
diagram or a loss function
A FULL FAIR premium is paid for a loss due to a
bad event
The small subjective loss of utility due to
paying the premium is GREATER THAN the EXPECTED
SUBJECTIVE LOSS due to the bad event.
Subjective value of assets
So the client will not insure
Standard utility function for GAINS in utility
LOSS in bad event
premium
Money value of assets
THIS MUCH
This is because, in the KT theory, people PREFER
small RISK OF LOSS in bad event to the CERTAINTY
of making a small loss (premium payment) in
either event
Subjective loss due to premium payment
BUT THE EXPECTED SUBJECTIVE LOSS MUST BE
Subjective value of loss in bad event
KT utility LOSS function
Because people hate making losses more than they
like making gains (they are LOSS AVERTERS) the
loss function has the shape illustrated above.
6
The critique of risk theory and insurance
Nymans Income Transfer Theory
David Nyman believes that although we should
accept K/T evidence about risk taking,
K/T theory does not invalidate conventional
insurance theory as much as has been suggested in
the previous slides
But the two theories are compatible only if we -
RE-INTERPRET the conventional expected utility
hypothesis in a new and radical way
Why people buy insurance
In Nymans view, people buy insurance not so much
to ensure (a greater amount of) financial
CERTAINTY (conventional view)
BUT MORE TO ENSURE THAT WHEN A BAD EVENT OCCURS,
SUCH AS ILLNESS OR A CAR CRASH, THEY HAVE ACCESS
TO A HELPFUL FLOW OF CASH
Insurance, from a demand side perspective is thus
about transferring income from the good state of
the world (when we need it less has lower
utility) to the bad state of the world (when we
need it more has higher utility).
K/T and conventional theories
If Nymans interpretation of demand for insurance
is correct, we can interpret the demand for
insurance in TERMS OF GAINS, not losses
So we may be able to use the conventional part of
the K/T diagram, in the positive quadrant, where
people are conventional risk averters and
conventional theory is applicable
We show this to be so on the following slides
7
The critique of risk theory and insurance
Applying Nymans theory
Take the example of full insurance
Lets look at the gains to an individual insurance
client
BAD event
GOOD event
in the
when they are
Not insured
Insured
Insured
Not insured
e.g. crashing their car
e.g. driving safely
8
The critique of risk theory and insurance
Applying Nymans theory
GOOD event
BAD event
Not insured
Not insured
Insured
Insured
Value of assets
X1
G

X0
L
R
L
X0
- (X0 X1)
So
expected value of assets
qX1
premium has been paid, thus losing income
compensation is received from the insurance
company
BUT G L (full insurance)
income lost repairing car
and expected UTILITY of assets
qU(X1) or
qU1
X0 value of asset undamaged
R premium
X1 value of assets if fully insured
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
L money value of damage to asset in bad event
9
The critique of risk theory and insurance
Applying Nymans theory
BAD event
GOOD event
Insured
Not insured
Not insured
Insured
Value of assets
X1
Value of assets
X0 - L
X2
G

X0
L
R
L
X0
- (X0 X1)
So
So
expected value of assets
X0 (X0 X2)
Value of asset undamaged - cost of damage
expected value of assets
qX1
qX2
and expected UTILITY of assets
and expected UTILITY of assets
qU(X1) or
qU1
qU(X2) or
qU2
X0 value of asset undamaged
X1 value of assets if fully insured
R premium
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
X2 value of assets in bad event if UNinsured
L money value of damage to asset in bad event
10
The critique of risk theory and insurance
Applying Nymans theory
BAD event
GOOD event
Insured
Not insured
Not insured
Insured
Value of assets
X1
Value of assets
X0 - L
X2
G

X0
L
R
L
X0
- (X0 X1)
So
So
expected value of assets
expected value of assets
qX1
qX2
and expected UTILITY of assets
and expected UTILITY of assets
qU(X1) or
qU1
qU(X2) or
qU2
So the GAIN from insuring if the bad event occurs
is
qX1
- qX2
as expected value
or
qU1
- qU2
as expected utility
X0 value of asset undamaged
X1 value of assets if fully insured
R premium
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
X2 value of assets in bad event if UNinsured
L money value of damage to asset in bad event
11
The critique of risk theory and insurance
Applying Nymans theory
BAD event
GOOD event
Insured
Not insured
Not insured
Insured
Value of assets
X1
Value of assets
X0 - L
X2
Value of assets
- R
G

X0
X1
X0
L
R
L
X0
- (X0 X1)
So
So
expected value of assets
expected value of assets
qX1
expected value of assets
Value of asset undamaged
- premium payment to insurance company
qX2
X0 (X0 X1)
pX1
and expected UTILITY of assets
and expected UTILITY of assets
and expected UTILITY of assets pU1
qU(X1) or
qU1
qU(X2) or
qU2
So the GAIN from insuring if the bad event occurs
is
qX1
- qX2
as expected value
or
qU1
- qU2
as expected utility
X0 value of asset undamaged
X1 value of assets if fully insured
R premium
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
X2 value of assets in bad event if UNinsured
L money value of damage to asset in bad event
p probability of good event
12
The critique of risk theory and insurance
Applying Nymans theory
BAD event
GOOD event
Insured
Not insured
Not insured
Insured
Value of assets
X1
Value of assets
X0 - L
X2
Value of assets
X0
Value of assets
G

X1
X0
L
R
L
X0
- (X0 X1)
So
So
expected value of assets
expected value of assets
qX1
expected value of assets
expected value of assets
Value of asset undamaged
qX2
pX1
pX0
and expected UTILITY of assets
and expected UTILITY of assets
and expected UTILITY of assets pU1
and expected UTILITY of assets pU0
qU(X1) or
qU1
qU(X2) or
qU2
So the GAIN from insuring if the bad event occurs
is
qX1
- qX2
as expected value
or
qU1
- qU2
as expected utility
X0 value of asset undamaged
X1 value of assets if fully insured
R premium
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
X2 value of assets in bad event if UNinsured
L money value of damage to asset in bad event
p probability of good event
13
The critique of risk theory and insurance
Applying Nymans theory
BAD event
GOOD event
Insured
Not insured
Not insured
Insured
Value of assets
X1
Value of assets
X0 - L
X2
Value of assets
X0
Value of assets
G

X1
X0
L
R
L
X0
- (X0 X1)
So
So
expected value of assets
expected value of assets
qX1
So expected value of assets
expected value of assets
qX2
pX1
pX0
and expected UTILITY of assets
and expected UTILITY of assets
and expected UTILITY of assets pU1
and expected UTILITY of assets pU0
qU(X1) or
qU1
qU(X2) or
qU2
So the GAIN from NOT insuring if the good event
occurs is
So the GAIN from insuring if the bad event occurs
is
qX1
- qX2
as expected value
or
pX1
- pX0
as expected value
or
qU1
- qU2
as expected utility
pU1
- pU0
as expected utility
X0 value of asset undamaged
X1 value of assets if fully insured
R premium
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
X2 value of assets in bad event if UNinsured
L money value of damage to asset in bad event
p probability of good event
14
The critique of risk theory and insurance
Applying Nymans theory
BAD event
GOOD event
Insured
Not insured
Not insured
Insured
Value of assets
X1
Value of assets
X0 - L
X2
Value of assets
X0
Value of assets
- R
G

X0
X1
X0
L
R
L
X0
- (X0 X1)
So
So
expected value of assets
expected value of assets
qX1
So expected value of assets
expected value of assets
qX2
pX1
pX0
and expected UTILITY of assets
and expected UTILITY of assets
and expected UTILITY of assets pU1
and expected UTILITY of assets pU0
qU(X1) or
qU1
qU(X2) or
qU2
So the GAIN from NOT insuring if the good event
occurs is
So the GAIN from insuring if the bad event occurs
is
qX1
- qX2
as expected value
- pX1
pX0
or
as expected value
or
- qU2
qU1
- qU2
qU1
as expected utility
- pU1
pU0
- pU1
pU0
as expected utility
The potential insurance client will insure
gt
expected utility gains from not insuring in good
event
if expected utility GAINS from insurance in the
bad event
gt
X0 value of asset undamaged
X1 value of assets if fully insured
R premium
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
X2 value of assets in bad event if UNinsured
L money value of damage to asset in bad event
p probability of good event
15
The critique of risk theory and insurance
Applying Nymans theory
- qU2
pU0
qU1
- pU1
qU2
pU1
The potential insurance client will insure
gt
expected utility gains from not insuring in good
event
if expected utility GAINS from insurance in the
bad event
gt
qU1
- qU2
- pU1
pU0
X0 value of asset undamaged
X1 value of assets if fully insured
R premium
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
X2 value of assets in bad event if UNinsured
L money value of damage to asset in bad event
p probability of good event
16
The critique of risk theory and insurance
Applying Nymans theory
The potential insurance client will insure
gt
expected utility gains from not insuring in good
event
if expected utility GAINS from insurance in the
bad event
gt
pU1
qU1
pU0
qU2
Bu this is U1 as p q 1
This is the expected utility of the initial
prospect i.e. uninsured prospect
qU2
pU0
U1
gt
which is identical with the criterion for
insurance from the expected utility hypothesis
(and which will always hold true for fair
insurance offered to a risk averter)
In other words are we can interpret the expected
utility hypothesis when applied to insurance in a
way that is not incompatible with the
Kahnemann/Tversky results
provided that people are interpreted as gaining
utility from insurance BECAUSE OF THE TRANSFER OF
INCOME TO THEM IF THEY ENCOUNTER THE BAD STATE OF
THE WORLD
rather than simply seeking to avoid risky
financial situations.
This interpretation has important policy
implications as mentioned in the lecture
X0 value of asset undamaged
X1 value of assets if fully insured
R premium
G gross compensation
U(Xi) or Ui is the utility asset of money value X
in state i
q probability of bad event
X2 value of assets in bad event if UNinsured
L money value of damage to asset in bad event
p probability of good event