1 15. risk and information 15.1 describing risky outcomes 15.2 evaluating risky outcomes 15.3...
TRANSCRIPT
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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