1 15. Risk and Information 15.1 Describing Risky Outcomes 15.2 Evaluating Risky Outcomes 15.3...

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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

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

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

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

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

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

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 = ?

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

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

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

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

15.1 Expected Values

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

)()( xxfxE

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)

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

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)

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)

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

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)

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.

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

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

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

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.

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.

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?

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

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.

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

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:

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$($)

($)$($)

000,46$($)

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

($)$($)

Ned would choose option a.

15.2a Risk Neutral Example

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

000,300)(

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

000,230)(

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

Ned would choose option a.

Income

Ned has a constant

marginal utility. Choosing the

highest expected value give him

the highest utility.

U=5(I)

60K 100K

E(I)= 46K

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:

15.2b Risk Averse Example

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

000,46$($)

)1(000,46$($)

($)$($)

000,46$($)

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

($)$($)

Here both expected incomes are equal.

15.2b Risk Averse Example

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

)1(000,46)(

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

Averly would choose option a.

Income

Averly has a decreasing marginal utility. She prefers the certain

income.

U= √I

214212

E(I)= 46K

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:

15.2c Risk Loving Example

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

000,46$($)

)1(000,46$($)

($)$($)

000,46$($)

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

($)$($)

Here both expected incomes are equal.

15.2c Risk Loving Example

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

2116)(

)1(46)(

2440)(

)9.0(40)1.0(100)(

)()(22

Lana would choose option b.

Income

Lana has an increasing marginal utility. She

prefers the risky income.

(U= I/1000)2

E(I)= 46K

1 15. Risk and Information 15.1 Describing Risky Outcomes 15.2 Evaluating Risky Outcomes 15.3...

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