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Risk

- Review
- Risk aversion
- Expected Value
- Standard error
- Expected utility

- Diversification (« hedging »)
- Insurrance

Avoiding risk : Diversification

Ex : You have the option of selling sunglasses and/or raincoats. Below are the corresponding profits:

Diversification (cont.)

If you choose to sell only sunglasses or only raincoats, what is your expected profit ?

What is your expected profit if you devote half your stock to sunglasses, and the other half to raincoats?

Compare the risk levels of the two scenarios above. Conclude on the ability to reduce risk via diversification.

Diversification (cont.)

If you choose to sell only sunglasses or only raincoats, what is your expected profit ?

VE(L) = 0.5*30K$ + 0.5*12K$ = 21K$

VE(I) = 0.5*12K$ + 0.5*30K$ = 21K$

Diversification (cont.)

What is your expected profit if you devote half your stock to sunglasses, and the other half to raincoats?

VE(L + I) =

0.5*(1/2*30K$ +1/2 *12K$) + 0.5*(1/2*12K$ +1/2 *30K$)

= 21K$

The expected profits are the same whether it rains or shines.

Compare the risk levels of the two scenarios above. Conclude on the ability to reduce risk via diversification.

By diversifying the investment, the exposure to uncertain outcomes is diminished. In this example, diversification completly eradicates the risk because rain and sun are two perfectly negatively correlated events.

Diversification (Example)

- You currently own stocks in Campbell Inc., the company that produces Campbell soup cans. Your broker advizes you to diversify your holdings in order to protect yourself against the risk of a downtrun in the World’s economy.
- You want to add one stock to your portfolio and you are contemplating dividing your holdings between Campbell and one of two companies.
- Experts are predicting a financial crisis but they do not agree on its magnitude; half of them predict a recession, the other half, a depression.
- The following table indicates the expected return of the tree assests under each scenario.

Diversification (Example)

- Which stock should you add to your portfolio to lower its volatility (risk)? Justify.

Avoiding risk: Insurance

You buy a house in the woods:

- 25% chance of a forest fire value = 80,000$
- 75 % chance of no fire value = 160,000$
- EV = 0.25*80K$+0.75*160K$ = 140$
An insurance company offers you the following contract: For each dollar paid to the company, it will reimburse you 4$ in the event of a fire.

Is this a fair bet? We say that the insurance policy is actuarially fair.

Avoiding risk: Insurance

For each dollar paid to the company, it will reimburse you 4$ in the event of a fire. Is this a fair bet? We say that the insurance policy is actuarially fair.

- EV= 0.25*(80K$+4*x$-x$)+0.75(160K$-x$)
- EV= 0.25*(80K$) + 0.75(160K$) + 0.25*3x$ - 0.75*x$
- EV= 0.25*(80K$)+0.75(160K$)
Money invested in the insurance policy leaves the expected value of the house unaffected. Why?

Because the policy has an EV of 0, it is a fair bet.

Insurance (cont.)

If you pay a $20,000 insurance premium, are you sufficiently covered?

Consider both cases:

- If fire: - value of house = +80,000
- insurance premium = -20,000

- compensation = +(4*20,000)

- total = 140,000

- If no fire: - value of house = 160,000
- insurance premium = 20,000

- compensation = 0

- total = 140,000

*Insurance (end)

You receive 140,000$ in each case, which is better for you than an expected value of 140,000$ (because you are risk-averse).

U

U(EV)

EU

k$

80

140

160

Insurance (Example)

The chances that a security guard from Blackwater Corp. gets injured while working in Irak are estimated at 20%. BlueCross, an insurance company, offers the guards a policy that grants 15$ of compensation for every 3$ of policy purchased.

- Is this policy actuarially fair?
- Which of the following guards would purchase this policy?
- Adam: U(I) = 1.5 I 2/3
- Brenda: U(I) = 9 I
- Claude: U(I) = 0.75 I 3/2

Introduction

- Markets with asymmetric information
- Adverse selection
- Possible solutions:
- Signalling
- Screening

Asymmetric information

Asymmetric information: When a person has access to economically relevant information which is not known by all.

Examples of markets with AI :

- _______________

- _______________

- …

Asymmetric information

Asymmetric information: When a person has access to economically relevant information which is not known by all.

Examples of markets with AI :

- Used cars

- Restaurants

- Students

The market for used cars

lemon: A cursed car which looks good but breaks down all the time.

Ex: 1,000 resellers of good cars (GC) and 1,000 resellers of lemons (L). Many potential buyers, each willing to pay:

- $1,000 for an L

- $2,000 for a GC

The resellers’ reserve price is:

- $750 for an L

- $1,750 for a GC

Perfect information

If everyone (buyers and sellers) knows the quality of each car:

- How many Ls will be sold? Why?

- How many GCs will be sold? Why?

- Is the resulting allocation efficient? Why or why not?

Perfect information

If everyone (buyers and sellers) knows the quality of each car:

- How many Ls will be sold? 1000
- Why? Price between 750 and 1000 creates gains from trade.
- How many GCs will be sold? 1000
- Why? Price between 1750 and 2000 creates gains from trade.
- Is the resulting allocation efficient? Yes, gains from trade are maximized.

Total ignorance

If no one knows the quality of the cars. What is the EV of a car for a buyer?

What is the EV of a car for a seller?

How many cars will be sold if everyone is risk-neutral?

Total ignorance

If no one knows the quality of the cars. What is the EV of a car for a buyer?

EVA = 0.5 * 1000$ + 0.5 * 2000$ = 1500$

What is the EV of a car for a seller?

EVV = 0.5 * 750$ + 0.5 * 1750$ = 1250$

How many cars will be sold if everyone is risk-neutral?

2000 cars will be sold at a price between 1250$ and 1500$

Asymmetric information

If only resellers know the quality of the cars, will the resellers of GCs agree to sell at a price equal to a buyer’s EV?

Consequence: adverse selection

All the sellers of GCs will exit the market. Buyers will realize that, and will know that the only cars on the market are Ls.

This phenomenon is called adverse selection.

Therefore, in equilibrium, all Ls are sold (at a price between $750 and $1,000) and not a single GC is sold. Is the market efficient?

Consequence: adverse selection

All the sellers of GCs will exit the market. Buyers will realize that, and will know that the only cars on the market are Ls.

This phenomenon is called adverse selection.

Therefore, in equilibrium, all Ls are sold (at a price between $750 and $1,000) and not a single GC is sold. Is the market efficient?

No because there are unrealized gains from trade.

A solution: Signalling

For GCs to sell, sellers must be able to credibly signal the car’s quality.

E.g.: Take and pass an inspection

Credible because more costly for a L (e.g. $1,200) than for a GC (e.g. $100) to pass the inspection.

Signalling (cont.)

A GC reseller, if he passes the inspection, can sell his car for up to $2,000. His net profit is:

2000 – 100 – 1750 = 150 $ > 0 $ (if he doesn’t sell)

An L reseller, if he passes the inspection, can also sell his car for up to $2,000. But his net profit will be:

2000 – 1200 – 750 = 50 $

Whereas, if he sold it for $1,000 (no inspection), his net profit would be 1000 – 750 = 250 $

In equilibrium

Resellers of Ls choose to not get their car inspected, and resellers of GC take and pass the inspection.

Because different types of sellers behave differently, we call this a separating equilibrium.

In equilibrium, all cars are sold, and efficiency is recovered thanks to signaling.

Signalling v. Screening

Essentially the same thing, but:

- Signalling comes from the intiative of the informed party (e.g.: a reseller)
- Warranty, inspection,...

- Screening comes from the initiative of the uninformed party (e.g.: an insurer)
- Providing reference letters, diplomas,...

Conclusions

- Asymmetric information adverse selection
- Markets with AI:
- used cars
- insurance
- …

- Solutions: signalling, screening
- Next: moral hazard and contracts

Example (Q.8, Ch 17)

- Two used car dealerships compete side by side on a main road. The first, Harry’s Cars, always sells high-quality cars that it carefully inspects and, if necessary, services. On average, it costs Harry $8,000 to buy and service each car that it sells. The second dealership, Lew’s Motors, always sells lower-quality cars. On average, it costs Lew only $5,000 for each car that it sells. If consumers knew the quality of the used cars they were buying, they would gladly pay $10,000 on average for Harry’s cars, but only $7,000 on average for Lew’s cars.

Example (Q.8, Ch 17)

- Without more information, consumers do not know the quality of each dealership’s cars and estimate that they have a 50-50 chance of ending up with a high-quality car, and are thus willing to pay $8,500 for a car, the EV of the car.
- Harry has an idea: He will offer a bumper-to-bumper warranty for every cars he sells. He knows that a warranty lasting Y years will cost $500Y on average, and he also knows that if Lew tries to offer the same warranty, it will cost him $1000Y on average.
- (Note: If Harry’s and Lew’s offer the same level of warranty, consumers will be unable to differentiate the quality of their cars and will pay $8,500 for a car, regardless of the seller.)

Example (Q.8, Ch 17)

- Suppose Harry offers a one-year warranty on all cars it sells.
- What is Lew’s profit if it does not offer a one-year warranty? If it does offer a one-year warranty?
- What is Harry’s profit if Lew’s does not offer a one-year warranty? If it does offer a one-year warranty?
- Will Lew’s match Harry’s one-year warranty?
- Is it a good idea for Harry’s to offer a one-year warranty?

Example (Q.8, Ch 17)

One year of warranty by Harry

πL(No warranty)=7000-5000=2000$

πL(Warranty)=8500-5000-1000=2500$

πH(No warranty by Lew)=10000-8000-500=1500$

πH(Warranty by Lew)=8500-8000-500=0$

Example (Q.8, Ch 17)

No warranty by Harry

πL(No warranty)=8500-5000=3500$

πL(Warranty)=10000-5000-1000=4000$

πH(No warranty by Lew)=8500-8000=500$

πH(Warranty by Lew)=7000-8000=-1000$

Example (Q.8, Ch 17)

- Will Lew’s match Harry’s one-year warranty?
- Is it a good idea for Harry’s to offer a one-year warranty?

Example (Q.8, Ch 17)

Will Lew’s match Harry’s one-year warranty?

YES! (it is his best-response)

Is it a good idea for Harry’s to offer a one-year warranty?

NO!

Example (Q.8, Ch 17)

- What if Harry offers a two-year warranty? Will this generate a credible signal of quality?

Example (Q.8, Ch 17)

2 year waranty by Harry

πL(No Warranty)=7000-5000=2000$

πL(Warranty)=8500-5000-2000=1500$

πH(No Warranty from Lew)=10000-8000-1000=1000$

πH(Warranty from Lew)=8500-8000-1000=-500$

The offer generates a credible signal about the quality of Harry’s cars because it is too costly for Lew to immitate. The signal generates a separating equilibrium.

Example (Q.8, Ch 17)

- If you were advising Harry, how long a warranty would you urge him to offer if he could offer fractions of years? Explain why.

Example (Q.8, Ch 17)

- If you were advising Harry, how long a warranty would you urge him to offer if he could offer fractions of years? Explain why.
The objective is to identify the shortest warranty period that makes it too costly for Lew to immitate, that makes the signal credible and generates a seperating equilibrium

Example (Q.8, Ch 17)

We want Lew’s profits to be higher without the warranty than with so he doesn’t immitate Harry.

πL(No warranty) < πL(warranty)

7000-5000 < 8500 – 5000 – 1000t

2000 < 3500 -1000t

1500 < 1000t

1.5 < t

A warranty of a year and a half is sufficient to generate a seperating equilibrium.

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