Decision-making under uncertainty

Introduction

Definition of risk

Attitudes toward risk

Avoiding risk:DiversificationInsurance

Uncertainty is everywhere

Car: how long will it last? House: will it be destroyed in an earthquake?

Company stocks: will they be profitable? And so on.

All these situations contain an element of risk which must be taken into account when making decisions.

Probabilities

Ex.: Tomorrow’s weather.

Two possible events:RainShine

Weather forecast: 30 % chance of rain.

We say that the probability that it will rain is 0.3 and the probability that it will not rain is 0.7 (= 1- 0.3 ).

Expected Value

Ex: Günther is a concert promoter

He is scheduling an outdoors concert (the rap band “le 83”). Two possibilities :

Shine πshine = 15,000 $Rain πrain = - 5,000 $

Forecast: prob. of rain is 0.5.

Expected value:

EV = Prob(shine) x πshine + Prob(rain) x πrain

Compute Günther’s EV is he books “le 83”.

EV=0.5*(-5,000$) + 0.5*(15,000$)=5,000$

Interpretation of EV :

The average value of the gains and losses expected from the uncertain event before the uncertainty is resolved, before the weather is known.

Expected Value (cont.)

Risk level (variability)Alternatively, Günther can promote a Justin Timberlake concert.Two possibilities :

Shine (50 %) πshine = 160,000 $ Rain (50 %) πrain = - 150,000 $

-Compute Günther’s EV if he books JT.EV=0.5*(-150,000$) + 0.5*(160,000$)=5,000$

- Compare with his EV of booking “le 83”.VE(83) = VE(JT)

-Comment on the spread of possible gains/losses.-The difference between potential gains and losses is much larger for the JT concert.

Standard deviation

A measure of risk is the standard deviation.

-Compute the standard deviation of booking JT.σ(JT) =[0.5*(-150K$-5K$)2 + 0.5*(160K$ - 5K$)2]1/2=155,000

-Compare with the standard deviation of booking le 83.σ(JT) =[0.5*(-150K$-5K$)2 + 0.5*(160K$ - 5K$)2]1/2=10,000

22 )(*)()(*)( EVrainPEVshineP rainshine

Decision under uncertainty

Will Günther choose to promote JT or “le 83”?

In other words, which bet is Günther willing to take?

It will depend on his attitude towards risk.

Fair bets

Ex.: Flip a coin: If Heads, win 100 $If Tails, lose 100 $

It is called a fair bet because EV = 0 $

Nevertheless, this gamble contains risk. Would you take such a bet?

Three attitudes towards risk

Risk aversion: always refuses a fair bet Risk preference: always accepts a fair bet

Risk neutrality: only cares about EV (indifferent between all fair bets)

*What are you?Choose a gamble (A, B, C or D)

Write down your choice and your name on a piece of paper, and hand it to your teacher.

Your teacher will flip a single coin, which will apply to everyone.

Warning !The payoffs/losses are percentage points of your participation grade

Tails Heads

A 0 0

B -10 +10

C -25 +20

D +11 -10

*What are you? (cont.)

What type of person would choose gamble A? B? C? D? Explain.

A and B are fair bets (see attitude definitions)

C is a risky wager with a negative expected value (EV), only a risk loving agent would choose to take part in this gamble.

D too is a risky wager but it’s expected value is positive. Risk seeker and risk neutral agents would be happy to take part in this gamble. It is unclear whether a risk averse agent would, it depends on his level of aversion.

Expected utility

Definition (in the rain/shine example):

EU = Prob(shine) x U(πshine)+ Prob(rain) x U(πrain)

Risk aversion (most people)You lose 100$. What is your disutility (loss of utility) if: You only owned 100$?You already owned 10,000$?

Conclusion: Your disutility depends on your current wealth.Draw the shape of your utility curve.

A: Utility loss associated with a 100$ loss for a revenue of 10,000$B:Utility loss associated with a 100$ loss for a revenue of 100$ 100

U

10k9900

A

B

An exampleJonathan just inherited a vase, of unknown value:

either a Ming vase (value: 700$), with a 50% chance,or a fake (value: 100$), with a 50% chance.

His utility is:

U(100$) = 60

U(260$) = 95

U(400$) = 110

U(700$) = 130 $

100

U

400260 700

60

130

11095

A decision to make

Someone wants to buy the vase from Jonathan for 400$. Will he accept? Why or why not?

EU(400$ no risk)= 1* U(400$)=110

EU(400$ with risk)=

0.5*U(100$) + 0.5*U(700$)=95

Risk premium

Notice that:

0,5 x U(700$) + 0,5 x U(100$) = U(260$)

What can you conclude?Jonathan is indifferent between an lower offer with no risk and a higher offer with some risk. He’s clearly risk averse because he’s willing to sacrifice expected gains to lower his risk.

We call risk premium the amount of money someone is willing to give up in order to get rid of risk.

What is Jonathan’s risk premium in this example?Risk Premium=400$-260$=140$, The difference between the sure offer and the risky offer with the same expected value.

Risk neutralityOnly EV matters : EU = U(EV)

Ex.:

U(100$) = 60

U(400$) = 95

U(700$) = 130 $

100

U

400 700

60

130

95

Risk preference

Would rather take the risk than receive 400 $:

EU > U(EV)

Ex.:

U(100$) = 60

U(400$) = 80

U(700$) = 130 $

100

U

400 700

60

130

80

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

Rain (50%) Shine (50%)

Raincoat sales $30,000 $12,000

Sunglasses sales $12,000 $30,000

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$What is your expected profit if you devote half your stock to sunglasses, and the other half to raincoats? VE(LI) = 0.5*(1/2*30K$ +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.

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$ VE=.25*80K$+.75*160K$=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.VE=0.25*(80K$+4*x$-x$)+0.75(160K$-x$)VE=0.25*(80K$)+0.75(160K$)+0.25*3x$+0.75*x$ - x$ Every dollar of insurance leaves the expected value of the house unaffected.Is this a fair bet? (YES) We say that the insurance policy is actuarially fair.A risk averse agent will always choose to fully insure when offered the possibility of buyning an actuarilly fair policy.

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).

k$

80

U

140 160

EUU(EV)

Conclusion

Attitudes towards risk (most of us are risk-averse)

Strategies for avoiding risk

Next: Asymmetric information