1. 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:
Actuarially Fair Insurance
Risk Premium

2. 15.3 Actuarially Fair Insurance
-insurance where the premium is equal to the expected value of the payout

3. 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

4. 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

5. 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

6. Actuarially Fair Insurance
Utility
AFI gives you the expected income of a risky situation U
Uinsure •
Uno insure D
25K
$40K=E(I)
100K
Income
Chapter Fifteen

7. 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?

8. 15.3 Risk Premium Risk Premium
-Maximum amount of money that a risk- averse 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.

9. Risk Premium • • E(U) Risk premium = horizontal distance ED E D
Utility
Risk premium = horizontal distance ED U E • •
E(U) D Is
E(I)
Income
Chapter Fifteen

10. Calculating Risk Premium
Calculate E(I) of risky choice.
Calculate E(U) of risky choice
Calculate sure income Is of E(U)
Risk Premium = E(I)- Is
Conclude

11. Risk Premium Example
U=√I Risky Income: p($100,000 )=0.2, p($25,000)=0.8
1) Calculate E(I) of risky choice
2) Calculate E(U) of risky choice

12. 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)
4) Calculate Risk Premium

13. 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

14. Risk Premium • • E(U) Risk premium = horizontal distance $4,014 E D
Utility
Risk premium = horizontal distance $4,014 U E • •
E(U)
4,014 D
25K IS
E(I)
100K
Income
Chapter Fifteen

15. 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.

16. 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

17. 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:
Hidden actions – Moral Hazard
Hidden information – Adverse Selection

18. 15.3 Moral Hazard
If people have insurance, their actions may change in two ways:
They are riskier (take laptop to beach, eat unhealthy – health insurance)
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:

19. This causes repair expenditures of area A.
Moral Hazard P
Without insurance, repairs cost P0 and Q0 repairs are made (where S=D).
This causes repair expenditures of area A.
S=MC (constant) P0 A B P1 Q Q0 Q1
D=MB
With insurance, repairs cost P1 and Q1 are made (where new S=D). This causes repair expenditures of Area A +B (expenditures increase).

20. This overcomsumption causes deadweight loss where MC>MB:
Moral Hazard P
This overcomsumption causes deadweight loss where MC>MB:
S=MC (constant) P0
DWL P1 Q Q0 Q1
D=MB
The insurers are forced to cover waste, therefore insurance premiums increase.

21. 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:”
The problem is caused by improper, rough or careless treatment;
The problem is caused by a fire or other natural calamity;

22. 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!”

23. 15.3 Fighting Moral Hazard
B) Introducing a cost to claim the warranty/insurance.
ie:
Deductible
Shipping Costs
Cost of time
Long repair time
Hard-to-get-to repair location

24. 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)

25. 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

26. 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

27. 15.3 Adverse Selection
5 Issues can keep Adverse Selection from killing a private insurance market:
Risk Aversion
Group Insurance
Insurance Choice
Risk Categories/Risk Profiling
Mandatory Insurance

28. i) Risk Aversion ii) Group Insurance
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 self-employed

29. 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)

30. 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

31. 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)
If there is statistical evidence?

32. 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)

33. 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:
Stock Market – buying a variety of stocks
Sales – selling a variety of products
Insurance – buying a variety of good without the warranty.

34. 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

35. 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)

36. 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:

37. 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).

38. 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

39. 15.4 Risk and Game Trees Example 1
Now new best responses lead to an overall Equilibrium

40. 15.4 Risk and Game Trees Example 2
Sometimes the process takes multiple steps

41. 15.4 Risk and Game Trees Example 21)
Backward induction of E and F
2) Expected return of B and C

42. 15.4 Risk and Game Trees Example 2
3) Expected Return of D

43. 15.4 Risk and Game Trees Example 2
4) Final Backward Induction

44. 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

45. 15.4 Value of Information Examples
People pay money for information in a variety of ways:
New house inspections
Car inspections
Consumer Report subscriptions
Online dating sites
Etc.

46. Chapter 15 Conclusions
P(a) = Prob(a) = probability that event a will occur
E($) = Σ$f($)
E(U) = ΣUf(U)
People can be risk averse, risk neutral, or risk loving depending upon their preferences between certain and uncertain incomes.
Actuarially Fair Insurance=E(loss)
Most people are willing to pay a RISK PREMIUM above Actuarially Fair Insurance

47. 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.