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15. Risk and Information

15.1 Describing Risky Outcomes

15.2 Evaluating Risky Outcomes

15.3 Bearing and Eliminating Risk

15.4 Analyzing Risky Decisions

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15.1 Probability Terminology• When there are multiple outcomes,

probabilities can be assigned to the outcomes

Terminology:Sample Space – set of all possible outcomes

from a random experiment-ie S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}-ie E = {Pass exam, Fail exam, Fail horribly}

Event – a subset of the sample space-ie B = {3, 6, 9, 12} ε S-ie F = {Fail exam, Fail horribly} ε E

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15.1 Probability

Probability = the likelihood of an event occurring (between 0 and 1)

P(a) = Prob(a) = probability that event a will occur

P(Y=y) = probability that the random variable Y will take on value y

P(ylow < Y < yhigh) = probability that the rvariable Y takes on any value between ylow and yhigh

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15.1 Probability Extremes

If Prob(a) = 0, the event will never occur ie: Canada moves to Europeie: the price of cars drops below zeroie: your instructor turns into a giant llama

If Prob(b) = 1, the event will always occur ie: you will get a mark on your final examie: you will either marry your true love or

notie: the sun will rise tomorrow

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15.1 Probability Types

• There are two categories of probabilities:

Objective Probabilities:Probabilities that are (mathematically)

certainie: rolling a dice, drawing a card

Subjective Probabilities:Probabilities based on beliefs and

expectationsie: gambling, stocks, many investments

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15.1 Objective Probability –Card Example

Sample space = {A, 1, 2…J, Q, K} of each suit-or [Ax,Kx] where x ε {hearts, diamonds, spades, clubs}

Events:-drawing red card-drawing even card-drawing face card-drawing an ace-drawing a “one eyed jack”-drawing two cards of total value 15

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15.1 Objective Probability Examples

1) Probability of drawing a heart = ¼2) Probability of drawing less than 3 = 2/133) Probability of drawing a King or a heart = 13(hearts)+3(non-heart kings)/52 = 16/524) Probability of throwing a 13 = 05) Probability of tossing 6 heads in a row =

1/646) Probability of drawing a red or black card

=17) Probability of passing the course = ?

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15.1 Subjective Probability –Investment Example

You decide to invest in Risktek Inc.Sample space = {-$1000, -$500, +$3000}

Events:-losing $1000-losing $500-losing money-gaining $3000

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15.1 Subjective Probability Examples

Based on your subjective knowledge, probabilities are:

1) P {-$1000}=0.32) P {-$500}=0.53) P {$3000}=0.2

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15.1 Probability Density Functions

• The probability density function (pdf) summarizes probabilities associated with possible outcomes

f(y) = Prob (Y=y)0≤ f(y) ≤1Σf(y) = 1

-the sum of the probabilities of all possible outcomes is one

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15.1 Objective Dice Example

• The probabilities of rolling a number with the sum of two six-sided die

• Each number has different die combinations:

7={1+6, 2+5, 3+4, 4+3, 5+2, 6+1}

• Exercise: Construct a table with 1 4-sided and 1 8-sided die

y f(y) y f(y)

2 1/36 8 5/36

3 2/36 9 4/36

4 3/36 10 3/36

5 4/36 11 2/36

6 5/36 12 1/36

7 6/36

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15.1 Expected Values

Expected Value – measure of central tendency; center of the distribution; population mean- average outcome

)()( xxfxE

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15.1 Objective Example

What is the expected value from a dice roll?

E(W) = Σwf(w)=2(1/36)+3(2/36)+…+11(2/36)+12(1/36)

=7

Exercise: What is the expected value of rolling a 4-sided and an 8-sided die? A 6-sided and a 10-sided die?

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15.1 Subjective Example

What is the expected value from investing in Risktek?

Recall: P {-$1000}=0.3, P {-$500}=0.5P {$3000}=0.2

E($) = Σ$f($)= -$1000(0.3)-$500(0.5)+$3000(0.2)

= $50

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15.1 Properties of Expected Values

a) Constant PropertyE(a) = a if a is a constant or non-random

variableie: E($100)=$100

b) Constants and random variablesE(a+bW) = a+bE(W)If a and b are non-random and W is randomie: E[$100+2(investment)]

=$100+2E(investment)

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15.1 Variance

Consider the following 3 midterm exams:

1) Average = 70%; everyone gets 70%2) Average = 70%; the class is equally

distributed between 50% and 90%3) Average = 70%; most of the class

gets 70%, with a few 100%’s and a few 40%’s who became sociologists

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15.1 Variance

Variance – a measure of dispersion (how far a distribution is spread out)

Variance is a way of measuring risk

σY2= Var(Y)= Σ(y-E(Y))2f(y)

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15.1 VariancesExample 1:E(Y)=70Yi =70 for all i

Var(Y) = Σ(y-E(Y))2f(y)= Σ(70-70)2 (1)= Σ(0)(1)=0

If all outcomes are the same, there is no variance.

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15.1 VariancesExample 2:E(Y)=70Y= 50, 60, 70, 80 ,90

Var(Y) = Σ(y-E(Y))2f(y)= (50-70)2(1/5)+ (60-70)2(1/5)+

(70-70)2(1/5)+ (80-70)2(1/5)+ (90-70)2(1/5)+=400/5+100/5+0/5+100/5+400/5=1000/5=200

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15.1 VariancesExample 3:E(Y)=70Y= 40, 70, 70, 70 ,100

Var(Y) = Σ(y-E(Y))2f(y)= (40-70)2(1/5)+ (70-70)2(1/5)+

(70-70)2(1/5)+ (70-70)2(1/5)+ (100-70)2(1/5)+=900/5+0/5+0/5+0/5+900/5=1800/5=360

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15.1 Standard Deviation

Standard Deviation is more useful for a visual view of dispersion:

Standard Deviation = Variance1/2

sd(W)=[var(W)]1/2

σ= (σ2)1/2

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15.1 SD Examples

In our first example, σ =01/2=0No dispersion exists

In our second example, σ =2001/2≈14.1

In our third example, σ =3601/2=19.0

If you could choose an exam to take, the third exam would be the riskiest.

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15.1 Constant Property of Variance

Constant Property

Var(a) = 0 if a is a constantIe: Var($100)=0, the risk of having $100

(and not gambling) is zero.

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15.2 Risk and UtilityOption 1 – Government job. Wage = $50,000Option 2 – Start-Up Company. Wage = $10,000

Plus: $100,000 if successful (0.4)$0 otherwise (0.6)

E($) = Σ$f($)= $10,000(0.6)+$110,000(0.4)

= $50,000

Which should you choose?

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15.2 Expected Utility

Expected Utility – probability-weighted average of the utility from each outcome

E(U) = ΣUf(U)

If U=($)1/2,

Option 1:E(U) = (50,000)1/2 (1)E(U) = 224

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15.2 Expected Utility

If U=($)1/2, Option 2:

E(U) = ΣUf(U)E(U) = (10,000)1/2 (0.6)+($110,000)1/2(0.4)E(U) = 60 + 133E(U) = 193

Option 1 has a higher expected utility, (224>193) so you would choose option 1.

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15.2 Risk Characteristics

Different people would make different decisions given the above choices. Your choice depends on your RISK CHARACTERISTIC:

a)Risk Neutralb)Risk Aversec)Risk Loving

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15.2a Risk Neutral

Someone is RISK NEUTRAL if they will always choose the highest expected income.

A RISK NEUTRAL agent has CONSTANT MARGINAL UTILITY:

02

2

I

U

I

MU

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15.2a Risk Neutral Example

Ned’s Utility is U(I) = 5I. Ned could:

a) Work for Sony for $60,000 a yearb) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%)

000,60$($)

)1(000,60$($)

($)$($)

a

a

a

E

E

fE

000,46$($)

)9.0(000,40$)1.0(000,100$($)

($)$($)

b

b

b

E

E

fE

Ned would choose option a.

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15.2a Risk Neutral Example

Ned’s Utility is U(I) = 5I. Ned could:

a) Work for Sony for $60,000 a yearb) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%)

000,300)(

)1)(000,60($5)(

)()(

a

a

a

UE

UE

UUfUE

000,230)(

)9.0)(000,40(5)1.0)(000,100(5)(

)()(

b

b

b

UE

UE

UUfUE

Ned would choose option a.

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Income

U

Ned has a constant

marginal utility. Choosing the

highest expected value give him

the highest utility.

40K

U=5(I)

60K 100K

300K

230K

E(I)= 46K

0

I

MU

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15.2b Risk Averse

Someone is RISK AVERSE if they prefer a certain income to a risky income with the same expected value

A RISK AVERSE agent has DECREASING MARGINAL UTILITY:

02

2

I

U

I

MU

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15.2b Risk Averse Example

Averly’s Utility is U(I) = √I. She could:

a) Work for Sony for $46,000 a yearb) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%)

000,46$($)

)1(000,46$($)

($)$($)

a

a

a

E

E

fE

000,46$($)

)9.0(000,40$)1.0(000,100$($)

($)$($)

b

b

b

E

E

fE

Here both expected incomes are equal.

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15.2b Risk Averse Example

Averly’s Utility is U(I) = √I. She could:

a) Work for Sony for $46,000 a yearb) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%)

214)(

)1(000,46)(

)()(

a

a

a

UE

UE

UUfUE

212)(

)9.0(000,40)1.0(000,100)(

)()(

b

b

b

UE

UE

UUfUE

Averly would choose option a.

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Income

U

Averly has a decreasing marginal utility. She prefers the certain

income.

40K

U= √I

100K

214212

E(I)= 46K

II

MU

4

1

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15.2c Risk Loving

Someone is RISK LOVING if they prefer a risky income to a certain income with the same expected value

A RISK LOVING agent has INCREASING MARGINAL UTILITY:

02

2

I

U

I

MU

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15.2c Risk Loving Example

Lana’s Utility is U(I) = (I/1,000)2. She could:

a) Work for Sony for $46,000 a yearb) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%)

000,46$($)

)1(000,46$($)

($)$($)

a

a

a

E

E

fE

000,46$($)

)9.0(000,40$)1.0(000,100$($)

($)$($)

b

b

b

E

E

fE

Here both expected incomes are equal.

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15.2c Risk Loving Example

Lana’s Utility is U(I) = (I/1,000)2. She could:

a) Work for Sony for $46,000 a yearb) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%)

2116)(

)1(46)(

)()(2

a

a

a

UE

UE

UUfUE

2440)(

)9.0(40)1.0(100)(

)()(22

b

b

b

UE

UE

UUfUE

Lana would choose option b.

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Income

U

Lana has an increasing marginal utility. She

prefers the risky income.

40K

(U= I/1000)2

100K

2116

2440

E(I)= 46K

500

1

I

MU