Risk and Insurance
Vani Borooah
University of Ulster
Gambles
An action with more than one possible
outcome, such that with each outcome
there is an associated probability of that
outcome occurring. If the outcomes are
good (G) and bad (B), denote the
associated probabilities by pG and pB
Payoffs and Utilities
With each outcome is associated a “payoff”
which can be expressed in terms of money: $cG
and $cB
With each payoff is associated a “utility”, u(c):
u(cG) is the utility in the good situation u(cB) is
the utility in the bad situation. We assume that
utility increases with payoff
Note: a payoff is different from the utility from the
payoff
Expected Return and Utility
• Expected Return: The expected return
from the gamble is: ER=pGcG+pBcB
• Expected Utility: The expected utility
from the gamble is: EU=pGu(cG)+pBu(cB)
Note: The return expected from a gamble
is different from the utility expected from
the gamble
Facing a Gamble
You are faced with a gamble:
If you accept the gamble you will, in
exchange for $W (the amount “staked”),
receive CG with probability pG and CB with
probability pB
If you reject the gamble you will keep your
$W
You have to decide whether or not to
accept the gamble?
Expected Utility Rule
• If you accept the gamble, your expected
utility is EU=pGu(cG)+pBu(cB)
• If you reject the gamble, your (certain)
utility is u(W)
• The expected utility rule requires you to
compare EU and u(W) and:
 accept if EU>u(W)
reject if EU<u(W)
indifferent if EU=u(W)
Certainty Equivalent
• How much should the stake be to make
you indifferent between accepting and
rejecting the gamble?
• Or what value of W will equate:
U(W) = EU=pGu(cG)+pBu(cB)
• Suppose W* solves the above equation
• Then W* is known as the certainty
equivalent of the gamble
• it expresses the worth of the gamble: $W*
Choice Using Certainty Equivalent
If the certainty equivalent is W* and W is
the stake, you will:
1. Accept the gamble if W < W*
2. Reject the gamble if W > W*
3. Indifferent to the gamble if W = W*
Risk Premium
• The risk premium associated with a
gamble is the maximum amount a person
is prepared to pay to avoid the gamble
RP = ER - CE
An Example
Suppose you have to pay $2 to enter a
competition. The prize is $19 and the probability
of winning is 1/3. You have a utility function
u(x)=log x and your current wealth is $10.
What is the certainty equivalent of this
competition?
What is the risk premium?
Should you enter the competition?
Answer:I
1
2
EU  log(10  2  19)  log(10  2)
3
3
1
2
 log(27)  log(8)
3
3
1/ 3
2/3
 log(27 )  log(8 )
 log(3)  log(4)
 log(12)  CE  12
The gamble is worth $12 to him. But, if he rejects the
gamble, he has only $10. So, he will accept the
gamble.
Answer:II
The expected wealth from the lottery is:
1
2
ER   (10  2  19)   (10  2)
3
3
1
2
1
  27   8  14
3
3
3
1
1
So, RP  14  12  2
3
3
Attitudes to Risk
• Intuitively, whether someone accepts a
gamble or not depends on his attitude to
risk
• Again intuitively, we would accept
“adventurous” persons to accept gambles
that more “cautious” persons would reject
• To make these concepts more precise we
define three broad attitudes to risk
Three Attitudes to Risk
•
•
•
•
The Risk Averse Person
The Risk Neutral Person
The Risk Loving Person
To define these attitudes, we use the
concept of a fair gamble
• In essence, a fair gamble allows you
receive the same amount of money
through two distinct ways:
• Gambling or not gambling
A Fair Gamble
• A fair gamble is one in which the sum that
is bet (W) is equal to the expected return:
W = ER = pGcG+pBcB
• You are offered a gamble in which you bet
W=$500 and receive:
• $250 with pB = 0.5 or $750 with pG= 0.5
• ER=$500=W: fair gamble
An Unfair Gamble
• An unfair gamble is one in which the sum
that is bet (W) is different (usually less)
from the expected return: W < ER =
pGcG+pBcB
• You are offered a gamble in which you bet
W=$500 and receive:
• $250 with pB = 0.6 or $750 with pG= 0.4
• ER=$450<W: unfair gamble
Attitudes to Risk and Fair Gambles
• A risk averse person will never accept a
fair gamble
• A risk loving person will always accept a
fair gamble
• A risk neutral person will be indifferent
towards a fair gamble
What Does This Mean?
• Given the choice between earning the
same amount of money through a gamble
or through certainty
The risk averse person will opt for
certainty
The risk loving person will opt for the
gamble
The risk neutral person will be indifferent
Diminishing Marginal Utility
• Why does the risk averse person reject the
fair gamble?
• Answer: because her marginal utility of
money diminishes
Example
• Your wealth is $10. I toss a coin and offer you $1
if it is heads and take $1 from you if it is tails
• This is a fair gamble: 0.511+0.59=10, but you
reject it
• Because, your gain in utility from another $1 is
less than your loss in utility from losing $1
• Your MU diminishes, you are risk averse
• Conversely, if you are risk averse, your MU
diminishes
A risk averse person / with diminishing MU / with a concave utility
function will reject a fair gamble
u(c)
u(750)
u(500)
EU
u(250)
250
400
500
750
c
The certainty equivalent of the gamble is $400; the risk premium is $100
A risk neutral person / with constant MU / with a linear utility function
will be indifferent between accepting/rejecting a fair gamble
u(c)
u(750)
u(500)
=EU
u(250)
250
400
500
750
c
The certainty equivalent of the gamble is $500; the risk premium is $0
A risk loving person / with increasing MU / with a convex utility function
will accept a fair gamble
u(c)
u(250)
u(750)
EU
u(500)
250
500 600
750
c
The certainty equivalent of the gamble is $600; the risk premium is -$100
Contingent Commodities
• With contingent commodities, the nature of
the commodity depends upon the
contingency:
A house before a storm is a different good
after a storm
A car before an accident is a different
good after an accident
 A holiday in sunshine is a different good
from a holiday during which it rains
Trade in Contingent Markets
• The risk of a gamble is the difference between the payoff in
the good state (CG) and that in the bad state (CB): Risk = CGCB
• When we buy insurance we try to reduce risk by trading
between two contingent states: “good” and “bad”
• We do this by buying wealth in the bad state and paying for it
from wealth in the good state
• The rate at which we can make this exchange depends on the
premium $ (per $ of insurance bought) charged by the
insurance company
• $(1-) of additional CB can be bought by giving up $ of CG
• So $1 of additional CB can be bought by giving up (/1-) of
CG
The Insurance Budget Line
Z is amount of insurance
CG = CG- Z < CG
CB =CB- Z + Z =CB + (1-)Z > CB
The slope of the budget
line is -/(1-): as
insurance gets cheaper,
the BL becomes flatter
CG
No Insurance point: Z=0
CG
450 line: CG = CB or full insurance
CB
CB
The Contingent Consumption
Indifference Curves
On each curve, different combinations of CG and CB give
the same level of Expected Utility: pGU(CG) + pBU(CB)
CG
Higher EU on black
curve than on red
CB
Equilibrium in the Insurance
Market
Given the terms offered by the
insurance company, consumer
maximises EU at point A
CG
X: no insurance point
A: equilibrium point
CG
CG*

Z = CB*-CB is amount of
insurance bought
CB CB
*
CB
Different Types of Equilibrium in
the Insurance Market
Given, the terms offered by the insurance company, consumer maximises
EU at point X or at Y or at some point in between X and Y
CG
X: no insurance equilibrium, Z=0,
insurance “too expensive”
CG
Y: full insurance equilibrium,
insurance “cheap”
CG*
CB CG
*
CB
Condition for Equilibrium
• Indifference Curve should be tangential to
budget line
• This means that the slope of indifference curve
equals slope of budget line
• Slope of indifference curve is marginal rate of
substitution:
 how much of wealth in the good state you are
prepared to give up to get another $ of wealth
in the bad state and still be on the same IC
• Slope of budget line is rate of exchange:
 how much of wealth in the good state you have
to give up to get another $ of wealth in the bad
state
Interpreting Equilibrium
• MRSBG = /(1-)
pB
u(CB )



1  pB u(CG ) 1  
Note: pG = 1 - pB
An Actuarially Fair Premium
• An actuarially fair premium is one which
is equal to the probability of the adverse
contingency:  = pB
• When the premium is actuarially fair:
u(CB)=u(CG)
• So, under diminishing marginal utility:
CB= C G
• Implying full insurance
Under what market conditions will an
actuarially fair premium be charged?
• The expected profit of an insurance
company is:
Z - pBZ  0
• When the insurance industry is
competitive, free entry of new firms will
compete away excess profits:
Z - pBZ = 0
• Which implies:  = pB