15.3 Bearing and Eliminating Risk
•Why do people buy insurance?
•Why do people buy extended warranties?
•Why are extended warranties so expensive?
•What is a reasonable extended warranty?
These questions are answered by:
1) Actuarially Fair Insurance
2) Risk Premium
1
15.3 Actuarially Fair Insurance
Actuarially Fair Insurance
-insurance where the premium is equal to
the expected value of the payout
AFI  E ( payout )
AFI  ( payout ) f ( payout )
2
Actuarially Fair Insurance Example
Assume that you could buy fire insurance. You
have a $100,000 job, and an 80% chance to lose
$75,000 (house fire). Your utility is U=√I.
Risky Income: p($100,000 )=0.2, p($25,000)=0.8
1) Calculate Actuarially Fair Insurance Premium
AFI  E ( payout )
AFI  ($75,000)(0.8)
AFI  $60,000
3
Actuarially Fair Insurance Example
If you didn’t get insurance, your utility would be:
U=√I
Risky Income: p($100,000 )=0.2, p($25,000)=0.8
2) Utility without Insurance
E (U )   Uf (U )
E (U )  100,0001/ 2 (0.2)  25,0001/ 2 (0.8)
E (U )  189.7
4
Actuarially Fair Insurance Example
With fair insurance, your utility would be:
U=√I
Risky Income: p($100,000 )=0.2, p($25,000)=0.8
Insurance: $60,000
2) Utility with Insurance
E (U )   Uf (U )
E (U )  (100,000  60,000)1/ 2 (0.2)
 (25,000  60,000  75,000)1/ 2 (0.8)
E (U )  (40,000)1/ 2 (1)
E (U )  200
5
Actuarially Fair Insurance
Utility
AFI gives you the expected income of a risky situation
U
Uinsure
•D
Uno insure
0
25K
$40K=E(I)
100K
Income
6
Chapter Fifteen
15.3 Is Insurance ever Fair?
Actual insurance premiums are rarely actuarially
fair, partially due to a firm making profit, but also
due to other factors:
•administration
•moral hazard
•adverse selection
(which will be covered later)
What is the maximum amount someone will pay
above actuarially fair premiums?
7
15.3 Risk Premium
Risk Premium
-Maximum amount of money that a riskaverse person will pay to avoid taking a risk
-Maximum amount a person will pay in
premiums above actuarially fair premiums
Note: Even risk loving people consider
themselves risk averse for large purchases.
8
Risk Premium
Utility
Risk premium = horizontal distance ED
U
E(U)
0
E
•
Is
•D
E(I)
Income
9
Chapter Fifteen
Calculating Risk Premium
1) Calculate E(I) of risky choice.
2) Calculate E(U) of risky choice
3) Calculate sure income Is of E(U)
4) Risk Premium = E(I)- Is
5) Conclude
10
Risk Premium Example
U=√I
Risky Income: p($100,000 )=0.2, p($25,000)=0.8
1) Calculate E(I) of risky choice
E ( I )   If ( I )
E ( I )  $100,000(0.2)  $25,000(0.8)
E ( I )  $40,000
2) Calculate E(U) of risky choice
E (U )   Uf (U )
E (U )  100,0001/ 2 (0.2)  25,0001/ 2 (0.8)
E (U )  189.7
11
Risk Premium Example
U=√I
Risky Income: p($100,000 )=0.2, p($25,000)=0.8
E(I) = $40,000
E(U) = 189.7
3) Calculate Is of E(U)
E (U )  I s
189.7  I s
$35,986  I s
4) Calculate Risk Premium
RP  E ( I )  I S
RP  $40,000  $35,986
RP  $4,014
12
Risk Premium Example
This person would spend a maximum of $4,014 above
actuarially fair insurance premiums to avoid the risk in
his job.
This person would accept a job paying at least $35,986
instead of taking the risky job.
•This person is willing to buy additional insurance
against his risky job
13
Risk Premium
Utility
Risk premium = horizontal distance $4,014
U
E
•
E(U)
0
25K
IS
4,014
•D
E(I)
100K
Income
14
Chapter Fifteen
15.3 Administration and Profit
Providing insurance isn’t free, there are
administration costs:
•Paying employees
•Overhead
•Legal Costs
•Etc
Insurance firms also desire profits. Many
extended warranties carry 40%-80% profit
margins.
15
15.3 Loading Fees
Loading Fee = Actual Premium – Actuarially Fair
Premium
-Average loading ratio (actual premium/fair
premium) for private US insurance companies
is 1.2 (Phelps 2003)
-(typical laptop service plan is $200 for 3 years,
working out to a Loading Ratio of 4.0 – 10%
failure rate in year 2 and 3 for $500 laptop)
-keep in mind administration costs
15.3 Asymmetric Information
Part of the additional costs of insurance, as well
as items such as deductibles and mandatory
insurance, arises from:
ASYMMETRIC INFORMATION – when one
party has information not available to another
party
•Typically, the person being insured has
information the insurance company doesn’t:
1) Hidden actions – Moral Hazard
2) Hidden information – Adverse Selection 17
15.3 Moral Hazard
If people have insurance, their actions may
change in two ways:
1) They are riskier (take laptop to beach, eat
unhealthy – health insurance)
2) They over consume insurance since it’s free
(Send laptop to be fixed, ask for unneeded
tests based on “House” – health insurance)
This second effect can be shown through supply
and demand:
P
Without
insurance, repairs
cost P0 and Q0
repairs are made
S=MC (constant)
(where S=D).
This causes repair
expenditures of area
A.
P0
P1
AB
Q0
Q1 D=MB
Q
With insurance, repairs cost P1 and Q1 are made
(where new S=D). This causes repair expenditures
19
of Area A +B (expenditures increase).
P
This
overcomsumption
causes
deadweight loss
S=MC (constant)
where MC>MB:
DWL
P0
P1
Q0
Q1 D=MB
Q
The insurers are forced to cover waste, therefore
insurance premiums increase.
20
15.3 Fighting Moral Hazard
Moral Hazard can be decreased by:
A) Including “reckless” situations that invalidate
warranty ie: Casio Calculator Warranty:
“The customer shall NOT have any claim under
this warranty for repair or adjustment
expenses if:”
1) The problem is caused by improper, rough or
careless treatment;
2) The problem is caused by a fire or other
natural calamity;
15.3 Fighting Moral Hazard
3) The problem is caused by improper repair or
adjustment made by anyone other than a
CASIO Service Center;
4) The problem is caused by battery leakage,
bending of the unit, broken display or key;
5) The battery is damaged or worn…
7) The proof of purchase is not presented when
requesting service
-although it can be hard to prove that a customer
has been “reckless”: “Of course I didn’t drop
my ipad!”
15.3 Fighting Moral Hazard
B) Introducing a cost to claim the
warranty/insurance.
ie:
1) Deductible
2) Shipping Costs
3) Cost of time
 Long repair time
 Hard-to-get-to repair location
15.3 Adverse Selection
Insurance can break down due to Adverse
Selection – an increase in insurance premium
increases the average risk of the insured
Assume there are 3 laptop purchasers:
Bill has a laptop failure rate of 10% (he’s a
computer technician)
Charles has a laptop failure rate of 20% (he’s
average)
Denis has a laptop failure rate of 30% (he
clicks on all the “you won” pop-ups)
15.3 Adverse Selection
Recall that actuarially fair insurance just
charges enough to over expected repairs
AFI = ($500xP(failure)):
Bill: $50
Charles $100
Denis $150
If you charge:
$50 – Charles and Denis cause a loss
$100 – Bill doesn’t want insurance and Denis
causes a loss
$150 – Charles and Bill don’t want insurance
15.3 Adverse Selection
If insurance is optional, those with higher risk
would buy
This leads to more expensive claims
This leads to higher premiums
This leads to more people not buying
insurance
The end result would be UNDERPROVISION of
insurance
15.3 Adverse Selection
5 Issues can keep Adverse Selection from
killing a private insurance market:
i) Risk Aversion
ii) Group Insurance
iii) Insurance Choice
iv) Risk Categories/Risk Profiling
v) Mandatory Insurance
i) Risk Aversion
Because people are risk averse, they are willing
to pay a RISK PREMIUM above the actuarially
fair premium.
This may keep more people in the market
ii) Group Insurance
Larger companies can offer group
insurance plans that automatically cover
everyone (high and low risk)
 This doesn’t help small firms or the selfemployed
iii) Insurance Choice
If different levels of insurance at different costs
are offered, people will self-sort themselves
into different categories:
Denis will pay $150+ for the full insurance (ie:
Product Replacement Plan)
Charles will pay $100+ for partial insurance (ie:
Product Service Plan)
Bill will pay $50+ for limited insurance (ie:
manufacturer warranty included in price)
iv) Risk Categories
Adverse selection occurs due to asymmetric
info – inability to know a person’s risk
HOWEVER, a company can charge premiums
based on OBSERVABLE characteristics
statistically linked to UNOBSERVABLE risk
ie: Male 20-year olds pay more for auto
insurance because they are STATISTICALLY
more likely to have an accident than Female
20-year Olds
iv) Risk Profiling?
The Supreme Court of Canada ruled this does
not violate the Canadian Charter of Rights and
Freedoms because there is statistical evidence
that 20-year-old males do have higher loss
probabilities
Some ask how long until we are charged based
on:
Ethnicity
Religion
Sexual Orientation (marital status already
applies)
v) Manditory Insurance
Public Health Insurance, Car insurance, etc is
MANDATORY, and therefore Adverse selection
is avoided since the low risk individuals can’t
drop out
PRO’s:
Mid and High-risk individuals are covered at a
reasonable rate ($100 in our example)
Con’s:
Low risk individuals would rather not be
covered at a high rate (for them)
15.3 Diversification – Insurance Alternative
Risk can also be managed through:
Diversification – Reducing risk by allocating
resources to a variety of activities whose
outcomes are not closely related
ie:
a) Stock Market – buying a variety of stocks
b) Sales – selling a variety of products
c) Insurance – buying a variety of good without
the warranty.
15.3 Law of Large Numbers
Diversification works because of:
Law of Large Numbers – as the number of
samples increases, the average of these
samples is likely to reach the mean of the
whole population (investopedia)
ie: Stock has 50% fail rate
Full fail chance of 1 stock = 50%
Full fail chance of 2 stocks* = 25%
Full fail chance of 8 stocks* = 0.39%
*stocks must be unrelated
-extreme outcomes reduce, expected outcome
increases
15.3 Diversification – Extended Warranties
Assume: You spend $5000 on electronics over
10 years, with an average FULL failure rate of
10%
No Extended Warranty: You spend $500 on
repairs and replacement
E(repair)=cost * f(cost)
E(repair)=$5000 * 0.10 = $500
Extended Warranty: You spend $1000 on
extended warranties (assume 50% profit
margin)
15.4 Risk and Game Trees
Probabilities can be combined with Game trees
from chapter 14
A player who MAKES decision is replaced by
an outcome that is chosen by chance
These game trees or decision trees can be
FOLDED BACK in a method similar to
backward induction to reduce the tree to the
simple trees seen in chapter 14:
15.4 Risk and Game Trees Example 1
Circles represent CHANCE NODES (choices
made by chance), while squares represent
DECISION NODES (choices made by players).
15.4 Risk and Game Trees Example 1
Chance Nodes are FOLDED BACK by replacing
them with the expected payoff:
E(B)= Σ$f($)=0.5($50)+0.5($10)=$30
15.4 Risk and Game Trees Example 1
Now new best responses lead to an overall
Equilibrium
15.4 Risk and Game Trees Example 2
Sometimes the process takes multiple steps
15.4 Risk and Game Trees Example 2
E ($)   $ f ($)
E ( I )  $50(0.5)  $10(0.5)
E ( I )  $30
E ($)   $ f ($)
E ( I )  $30(0.5)  $20(0.5)
E ( I )  $25
1)
Backward
induction of
E and F
2) Expected
return of B
and C
15.4 Risk and Game Trees Example 2
E ($)   $ f ($)
E ( I )  $50(0.5)  $10(0.5)
E ( I )  $30
E ($)   $ f ($)
E ( I )  $30(0.5)  $20(0.5)
E ( I )  $25
E ($)   $ f ($)
E ( I )  $50(0.5)  $20(0.5)
E ( I )  $35
3) Expected Return of D
15.4 Risk and Game Trees Example 2
4) Final Backward Induction
15.4 Value of Information
This previous example highlights the VALUE OF
INFORMATION
The firm expected return increases by $5
(million) if it is able to do a free test
The firm will pay up to $5 million for the test
Value of Perfect Information – increase in a
decision maker’s expected payoff when they
can conduct a costless test to determine the
outcome of a risky event
VPI = E($)with test- E($)without test
15.4 Value of Information Examples
People pay money for information in a variety
of ways:
1) New house inspections
2) Car inspections
3) Consumer Report subscriptions
4) Online dating sites
5) Etc.
Chapter 15 Conclusions
1) P(a) = Prob(a) = probability that event a will
occur
2) E($) = Σ$f($)
3) E(U) = ΣUf(U)
4) People can be risk averse, risk neutral, or risk
loving depending upon their preferences
between certain and uncertain incomes.
5) Actuarially Fair Insurance=E(loss)
6) Most people are willing to pay a RISK
PREMIUM above Actuarially Fair Insurance
Chapter 15 Conclusions
7) Insurance Premiums are increased by
Asymmetric Information (Moral Hazard and
Adverse Selection), which can be reduced but
never eliminated.
8) Diversification is an alternative to insurance
9) Game trees including risky outcomes can be
“Folded Back” using expected values and
analyzed normally
10) Information is valuable
11) Unless you can’t sleep at night without one,
say “no” to the extended warranty.