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 Europe
ie: the price of cars drops below zero
ie: 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 exam
ie: you will either marry your true love or not
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ie: the sun will rise tomorrow
15.1 Probability Types
• There are two categories of probabilities:
Objective Probabilities:
Probabilities that are (mathematically) certain
ie: rolling a dice, drawing a card
Subjective Probabilities:
Probabilities based on beliefs and expectations
ie: 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”
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-drawing two cards of total value 15
15.1 Objective Probability Examples
1) Probability of drawing a heart = ¼
2) Probability of drawing less than 3 = 2/13
3) Probability of drawing a King or a heart
= 13(hearts)+3(non-heart kings)/52 = 16/52
4) Probability of throwing a 13 = 0
5) Probability of tossing 6 heads in a row = 1/64
6) Probability of drawing a red or black card =1
7) 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}=0.3
2) P {-$500}=0.5
3) 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) ≤1
Σf(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 sixsided 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
E ( x)   xf ( x)
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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 10sided 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}=0.5
P {$3000}=0.2
E($) = Σ$f($)
= -$1000(0.3)-$500(0.5)+$3000(0.2)
= $50
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15.1 Properties of Expected Values
a) Constant Property
E(a) = a if a is a constant or non-random variable
ie: E($100)=$100
b) Constants and random variables
E(a+bW) = a+bE(W)
If a and b are non-random and W is random
ie: 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 Variances
Example 1:
E(Y)=70
Yi =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 Variances
Example 2:
E(Y)=70
Y= 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 Variances
Example 3:
E(Y)=70
Y= 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=0
No 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 constant
Ie: Var($100)=0, the risk of having $100 (and
not gambling) is zero.
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15.2 Risk and Utility
Option 1 – Government job. Wage = $50,000
Option 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 + 133
E(U) = 193
Option 1 has a higher expected utility, (224>193)
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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 Neutral
b)Risk Averse
c)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:
MU  U
 2 0
I
I
2
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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 year
b) Work for Risky for $100,000 a year (10%) or
$40,000 a year (90%)
E ($) a   $ f ($)
E ($) b   $ f ($)
E ($) a  $60,000(1)
E ($) b  $100,000(0.1)  $40,000(0.9)
E ($) a  $60,000
E ($) b  $46,000
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 year
b) Work for Risky for $100,000 a year (10%) or
$40,000 a year (90%)
E (U ) a  Uf (U )
E (U )b  Uf (U )
E (U ) a  5($60,000)(1) E (U )b  5(100,000)(0.1)  5(40,000)(0.9)
E (U ) a  300,000
E (U )b  230,000
Ned would choose option a.
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U
U=5(I)
MU
0
I
Ned has a
constant
marginal utility.
Choosing the
highest expected
value give him
the highest
utility.
300K
230K
Income
40K 60K 100K
E(I)= 46K
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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:
MU  U
 2 0
I
I
2
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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 year
b) Work for Risky for $100,000 a year (10%) or
$40,000 a year (90%)
E ($) a   $ f ($)
E ($) b   $ f ($)
E ($) a  $46,000(1)
E ($) b  $100,000(0.1)  $40,000(0.9)
E ($) a  $46,000
E ($) b  $46,000
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 year
b) Work for Risky for $100,000 a year (10%) or
$40,000 a year (90%)
E (U ) a  Uf (U )
E (U )b  Uf (U )
E (U ) a  46,000 (1)
E (U )b  100,000 (0.1)  40,000 (0.9)
E (U ) a  214
E (U )b  212
Averly would choose option a.
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U
MU
1

I
4 I
Averly has a decreasing
marginal utility. She
prefers the certain
income.
U= √I
214
212
Income
40K
100K
E(I)= 46K
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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:
MU  U
 2 0
I
I
2
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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 year
b) Work for Risky for $100,000 a year (10%) or
$40,000 a year (90%)
E ($) a   $ f ($)
E ($) b   $ f ($)
E ($) a  $46,000(1)
E ($) b  $100,000(0.1)  $40,000(0.9)
E ($) a  $46,000
E ($) b  $46,000
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 year
b) Work for Risky for $100,000 a year (10%) or
$40,000 a year (90%)
E (U ) a   Uf (U )
E (U ) b   Uf (U )
E (U ) a  46 2 (1)
E (U ) b  100 2 (0.1)  40 2 (0.9)
E (U ) a  2116
E (U ) b  2440
Lana would choose option b.
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U
MU
1

I
500
Lana has an increasing
marginal utility. She
prefers the risky
income.
(U= I/1000)2
2440
2116
Income
40K
100K
E(I)= 46K
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