EMGT 269 - Elements of Problem Solving and Decision Making
13. RISK ATTITUDES
Solving a Decision Problem using EMV is convenient but
may lead to counter intuitive decisions.
Example:
Max Profit
$50
Game 1
0.5
-$1
0.5
0.5
Game 2
$2000
-$1900
0.5
WHICH GAME WOULD YOU PREFER?
Most people prefer Game 1 over Game 2.
But:
GAME 2
GAME 1
Prob
0.5
0.5
Payoff
Prob*Payoff
$30.00
$15.00
-$1.00
-$0.50
EMV=
$14.50
Prob
Payoff
Prob*Payoff
0.5 $2,000.00
$1,000.00
0.5 -$1,900.00
-$950.00
EMV=
$50.00
Why?
 Expected Monetary Value =(EMV) is a long term average,
whereas the games are only played once.
 EMV does not take into account the risk (=fear) involved
with loosing larger amounts of money.
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 160
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Solution?
Capture RISK ATTITUDES when
solving a Decision Problem
A. RISK
A
$1000
10-2
0
0.99
B
$100000
10-4
0
0.9999
 Somebody who is Risk Neutral is indifferent between A
and B
 Somebody who is Risk Averse prefers A
 Somebody who is Risk Seeking prefers B
Capture behavior above through use of Utility Functions
 A method of modeling risk attitude by transforming $ into
Utility Units (=Utils)
UTILITY FUNCTION: U(X) (in Utils)
in Dollars
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 161
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
B. RISK ATTITUDES
Risk Averse U2
Utility
1
Risk Neutral U1
Risk Seeking U3
0
0 103
Min
105 Monetary Value
Max
Note:
 Utility of Best Case = 1
 Utility of Worst Case = 0.
Risk Neutral:
5

5
2
3
 2 U 1 (10 )
U
(
10
)

EU ( A)  10 U1 (10 )  10
U1 (103 )  1 2   1
10 2 A ~ B
10

EU1 ( B)  10 4 U1 (105 )

Risk Averse:
5

5
2
3
 2 U 2 (10 )
U
(
10
)

EU
(
A
)

10

U
(
10
)

10
2
U 2 (103 )  2 2   2
10 2 A > B
10

4
5

EU 2 ( B)  10  U 2 (10 )

Risk Seeking:
5

2
3
 2 U 3 (10 )
U (10 )
 EU ( A)  10  U 3 (10 )  10
U 3 (10 3 )  3 2   3
10 2 A < B
10

EU 3 ( B)  10 4  U 3 (105 )

5
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 162
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Utility
Concave  Risk Averse
1
0
Min
Risk Attitude 1:
Utility
Max
X
Linear Line  Risk Neutral
1
0
Min
Risk Attitude 2:
Utility
Max
X
Convex  Risk Seeking
1
0
Risk Attitude 3:
Min
X
Max
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 163
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Note:
 All Utility Functions are upward sloping indicating more
wealth is better
 Curvature (=Concave, Linear, Convex) indicates Risk
Attitude.
Utility Function may be specified in different formats:
1. Graphical Format:
Utility
U(X)
0
Min
X
2. Tabular Format:
Wealth
0
400
600
1000
1500
2500
Utility
0.15
0.47
0.65
0.93
1.24
1.50
Use straight-line approximation to determine risk attitude.
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 164
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
3. Functional Format:
x
U ( x )  Log ( x ), U ( x )  1  exp(  ), U ( x )  x
R
Plot Utility Function to determine risk attitude.
EXAMPLE:
You own the following bet or game:
$500
0.5
-$500
0.5
You friend approaches you and asks whether you would like
to trade the game. You are faced with the following decision
problem.
Max Profit
$500
0.5
-$500
0.5
$X ?
HOW MUCH SHOULD YOU CHARGE or SHOULD YOU
GIVE IT AWAY FOR FREE or WOULD YOU PAY YOUR
FRIEND TO ACCEPT THE GAME?
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 165
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Answer depends on your Risk Attitude:
The EMV of your game is $0
You are Risk Neutral: You give the bet away for free
You are Risk Seeking: You charge your friend an amount.
You are Risk Averse: You are willing to give the bet away
and pay your friend to accept the bet.
CAN WE THINK OF SUCH AN EXAMPLE IN REAL LIFE?
Max Profit
No
Insurance
$0
1-p
LARGE LOSS
p
INSURANCE PREMIUM
Take Insurance
Insurance charges more than the EMV. If you pay for
insurance you are risk averse.
USING UTILITY FUNCTIONS IN DECISION TREES
1. Convert Dollar Amounts to Utils using Utility Function
2. Make Decision based on maximizing expected utility (EU)
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 166
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
STOCK MARKET EXAMPLE:
Max Profit
Up (0.5)
EMV =
580
1500K
Prob
Same (0.3)
High Risk Stock
100K
Down (0.2)
EMV =
540
Low Risk Stock
Up (0.5)
1000K
200K
Saving Account
Payoff
Prob*Payoff
1,500.00
750.00
100.00
30.00
-1,000.00
-200.00
EMV=
$580.00
0.5
0.3
0.2
Payoff
Prob*Payoff
1,000.00
500.00
200.00
60.00
-100.00
-20.00
EMV=
$540.00
-1000K
Same (0.3)
High (0.2)
0.5
0.3
0.2
Prob
-100K
500K
You lean towards investing in High Risk Stock, but you are
concerned about possibly loosing $1000. You decide to
model your risk attitude resulting in the following utility table.
Payoff
-1000
-100
100
200
500
1000
1500
Utility
0.00
0.33
0.46
0.52
0.65
0.86
1.00
You replace the dollars amounts by their Utilities and solve
the problem again based on Expected Utility.
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 167
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Max Utility
EU =
0.638
Up (0.5)
1.00
Prob
High Risk Stock
0.46
Down (0.2)
EU =
0.652
Low Risk Stock
Up (0.5)
0
0.86
Same (0.3)
0.52
High (0.2)
Saving Account
Utility
0.5
0.3
0.2
Same (0.3)
Prob
1.00
0.46
0.00
EU=
Prob*Utility
0.500
0.138
0.000
0.638
0.86
0.52
0.33
EU=
Prob*Utility
0.430
0.156
0.066
0.652
Utility
0.5
0.3
0.2
0.33
0.65
Conclusion:
Based on your risk attitude you decide to put your money in
the low risk stock.
C. EXPECTED UTILITY, CERTAINTY EQUIVALENTS AND
RISK PREMIUMS
Definition:
CERTAINTY EQUIVALENT =
Amount of money for which you
are willing to trade a bet.
STOCK MARKET EXAMPLE CONTINUED:
Low Risk Stick Bet:
EU(Low Risk Stock) > EU(Savings Account) 
CE(Low Risk Stock) > CE(Savings Account) = $500
High Risk Stock Bet:
U($200) < EU(High Risk Stock) < EU(Savings Account) 
$200< CE(High Risk Stock) < $500
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 168
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Can we determine CE(LRS) and CE(HRS) exactly?
CE(LOW RISK STOCK):
Utility
EU =
0.652
-1000
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-500
CE =
520
0
500
1000
1500
Payoff
CE(HIGH RISK STOCK):
Utility
EU =
0.638
-1000
-500
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
CE =
450
0
500
1000
1500
Payoff
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 169
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Summarizing:
LOW RISK STOCK
SAVING ACCOUNT
HIGH RISK STOCK
UTILITY
0.652
0.650
0.638
CE
$520.00
$500.00
$450.00
Conclusion:
 You get the same ranking of alternatives when you rank
according to Certainty Equivalents in stead off Utilities.
 CE are giving in $, Utilities are given in Utils
 CE is easier to interpret than Utility.
CE(Bet 1) = CE(Bet 2) 
U(Bet 1) = U(Bet 2)
Definition:
RISK PREMIUM = EMV – CE.
Interpretation:
The Risk Premium is the amount of money your are willing to
give up to avoid risk (= uncertainty).
Steps in determining Risk Premium of a Bet:
1. Calculate EMV of Bet
2. Calculate EU of Bet
3. Determine CE of BET
4. RP = EMV-CE
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 170
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Risk Averse U2
Utility
1
Risk Neutral U1
EU
Risk Seeking U3
0
CE2
RP2>0




EMV = CE1
RP1=0
CE3
RP3<0
Risk Averse Risk Premium is Positive
Risk SeekingRisk Premium is Negative
Risk NeutralRisk Premium is 0.
CE, EU, RP depend on Utility Function U(X) and
Probability distribution of the payoffs.
STOCK MARKET EXAMPLE CONTINUED:
High Risk Stock:
EMV = 580, CE = 450, RP = 580 – 450 = 130
Low Risk Stock:
EMV = 540, CE = 520, RP = 540 – 520 = 20
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 171
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
D. UTILITY FUNCTION ASSESSMENT
 Risk Attitude is personal Use of Subjective Judgment
1. Assessment using CE
STEP 1: Set U(Min) = 0, U(Max) = 1
Example: Suppose you are encountering an investment
decision which a payoff that ranges from $10 to $100. Then:
U($10)=0, U($100)=1
STEP 2: Asses utility for several intermediate values using
reference lotteries that ask for CE.
Example:
Max Profit
$100
0.5
$ 10
0.5
$X?
For how much money are you willing to trade?
Answer: $30 U($30)
= 0.5U($10) + 0.5U($100)
= 0.50 + 0.51 = 0.5
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 172
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
In general: Suppose you know U(Y), U(Z). Ask for the CE
using the following reference lottery.
Max Profit
$Y
0.5
$Z
0.5
$X?
Then:
U(X) = 0.5U(Y) + 0.5U(Z)
Example: Suppose you want to know the utility of an
amount between $30 and $100.
Note: U($100) = 1, U($30)=0.5
Max Profit
$100
0.5
$ 30
0.5
$X?
For how much money are you willing to trade?
Answer: $50 U($50)
= 0.5U($30) + 0.5U($100)
= 0.50.5 + 0.51 = 0.75
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 173
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Suppose you want to know the utility of an amount between
$10 and $30.
Note: U($10) = 0, U($30)=0.5
Max Profit
$30
0.5
$ 10
0.5
$X?
For how much money are you willing to trade?
Answer: $18 U($18)
= 0.5U($10) + 0.5U($30)
= 0.50 + 0.50.5 = 0.25
STEP 3: Approximate Utility Function using Straight Line
Approximation.
Example:
X
$10.00
$18.00
$30.00
$50.00
$100.00
U(X)
0.00
0.25
0.50
0.75
1.00
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 174
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
1.00
0.75
0.50
0.25
$100.00
$90.00
$80.00
$70.00
$60.00
$50.00
$40.00
$30.00
$20.00
$10.00
0.00
$0.00
U(X)
EMGT 269 - Elements of Problem Solving and Decision Making
X
2. Assessment using Probabilities
STEP 1: Set U(Min) = 0, U(Max) = 1
Example: Suppose you are encountering an investment
decision which a payoff that ranges from $10 to $100. Then:
U($10)=0, U($100)=1
STEP 2: Asses utility for several intermediate values using
reference lotteries that ask for indifference using probability
mechanism.
Example: Suppose you want to assess the utility of $65.
You offer a reference lottery and ask for the probability for
which you are indifferent using a probability mechanism.
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 175
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Max Profit
$100
p
$ 0
1-p
$65
Suppose you answer p, then:
U(65)=(1-p) U($10) +p U($100) =(1-p)0 + p1 = p
Repeat the above procedure for other values between $10
and $100.
STEP 3: Approximate Utility function using the straight line
approximation.
3. Assessment using Theoretical Utility Models
STEP 1: Asses a Theoretical Utility Model e.g.
x
U ( x )  Log ( x ), U ( x )  1  exp(  ), U ( x )  x
R
Theoretical Utility Models have underlying properties, e.g.
constant risk aversion, increasing risk aversion etc. The
choice of such a theoretical utility model should be based on
these properties. An in-depth discussion of these properties
is beyond the scope of this course.
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 176
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Example: Exponential Utility Function
U ( x )  1  exp( 
X
)
R
1.000
0.800
0.600
0.400
U(X)
0.200
0.000
-5
0
5
10
15
20
-0.200
-0.400
-0.600
-0.800
X
R=10
R=20
R=30
 R is referred to as the Risk Tolerance Parameter
 As R increases the more tolerant you are with respect to
Risk, the less Risk averse you are.
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 177
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
STEP 2: Asses the parameters of the Utility Model.
Example: Exponential Utility Function
U ( X )  1  exp( 
X
)
R
To estimate R use the following reference lottery.
Max Profit
$X
0.5
-$X/2
0.5
$0
For what value of $X are you indifferent?
Answer: $60 R  60, Why?
60
60
U
(
60
)

1

exp(

)

1

exp(

)  0.6321

R
60
30
30
U
(

30
)

1

exp(
)

1

exp(
)  0.6487

R
60
0
U
(
0
)

1

exp(

)  11  0

R
 0.5 * U (60)  0.5 * U ( 30)  0.5 * 0.6321  0.5 * 0.6487
 0.008  0
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 178
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Risk Premium and CE using Exponential Utility Function
U ( X )  1  exp( 
X
), R  $900
R
Consider the following bet and assessment of its CE
Max Profit
$2000
0.4
0.4
0.2
$1000
$ 500
$X?
Steps in calculating RP of a bet:
1.
2.
3.
4.
Calculate EMV of the Bet
Calculate EU of the Bet
Determine CE of bet
RP = EMV-CE
STEP 1: Calculate EMV
Prob
0.4
0.4
0.2
Pay-Off
$2,000.00
$1,000.00
$500.00
EMV=
Prob*Pay-Off
$800.00
$400.00
$100.00
$1,300.00
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 179
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
STEP 2: Calculate EU
2000
U
(
2000
)

1

exp(

)  0.892

900
1000
U
(
1000
)

1

exp(

)  0.671

900
500
U
(
500
)

1

exp(

)  0.426

900
Prob
0.4
0.4
0.2
Utility
0.892
0.671
0.426
EU=
Prob*Utility
0.357
0.268
0.085
0.710
STEP 3: Calculate CE
Calculate $X, such that U(X) = 0.710
U ( X )  1  exp( 

X
X
)  0.710  exp( 
)  0.290 
900
900
X
 Ln(0.290)  X  900  Ln(0.290)  $1114.785
900
Thus, CE=$1114.79
STEP 5: Calculate RP
RP = EMV-CE = $1300 - $1114.79 = $185.21
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 180
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
DECREASING RISK AVERSION
Suppose you have $10 in your wallet. You have a lottery
ticket in your hand that pays $30 if you win. Your objective is
to maximize your total wealth and your utility function for
total wealth X is given by:
U(X) = Ln(X)
What is the CE equivalent in terms of total wealth for the
lottery ticket and how much are you willing to give up in
terms of total wealth, i.e. what is the Risk Premium of the
ticket.
Max Total Wealth
+ $30
Current Wealth =
$10
$40
0.50
+ $0
$10
0.50
$CE
STEP 1: EMV = 0.50$40 + 0.50$10 = $25
STEP 2: U($10) = Ln(10)=2.303, U($40)=Ln(40)=3.689
Prob
0.5
0.5
Utility
2.303
3.689
EU=
Prob*Utility
1.151
1.844
2.996
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 181
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
STEP 3: Calculate CE
Solve for X such that: U(X) = 2.996 LN (X) = 2.996
X = EXP(2.996) = $20, Interpretation?
STEP 4: Calculate RP
RP = EMV - CE = $25 - $20 = $5, Interpretation?
 Suppose I would have done the above analysis with $20,
$30, $40, $500 in my wallet.
Current Wealth
10
20
30
40
500
Expected Value
$25.00
$35.00
$45.00
$55.00
$515.00
CE
$20.00
$31.62
$42.43
$52.92
$514.78
RP
$5.00
$3.38
$2.57
$2.08
$0.22
Conclusion:
Risk Premium decreases as my current wealth increases. As
my current wealth approaches infinity, I approach a Risk
Neutral behavior.
WHAT IF WE WOULD HAVE DONE THE
ABOVE ANALYSIS WITH AN EXPONENTIAL UTILITY
FUNCTION WITH RISK TOLERANCE R=$20.80?
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 182
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
CONSTANT RISK AVERSION
Suppose you have $10 in your wallet. You have a lottery
ticket in your hand that pays $30 if you win. Your objective is
to maximize your total wealth and your utility function for
total wealth X is given by:
U ( X )  1  exp( 
X
)
20.8
What is the CE equivalent in terms of total wealth for the
lottery ticket and how much are you willing to give up in
terms of total wealth, i.e. what is the Risk Premium of the
ticket.
Max Total Wealth
+ $30
Current Wealth =
$10
$40
0.50
+ $0
$10
0.50
$CE
STEP 1: EMV = 0.50$40 + 0.50$10 = $25
STEP 2:
10
U
(
10
)

1

exp(

)  0.382

20.8
40
U
(
40
)

1

exp(

)  0.854

20.8
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 183
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Prob
0.5
0.5
Utility
0.382
0.854
EU=
Prob*Utility
0.191
0.427
0.618
STEP 3: Calculate CE
Solve for X such that: U(X) = 0.618 

U ( X )  1  exp( 

X
X
)  0.618  exp( 
)  0.382 
20.80
20.80
X
 Ln(0.382)  X  20.80  Ln(0.382)  $20
20.80
STEP 4: Calculate RP
RP = EMV - CE = $25 - $20 = $5
 Suppose I would have done the above analysis with $20,
$30, $40, $500 in my wallet.
Current Wealth
10
20
30
40
500
Expected Value
$25.00
$35.00
$45.00
$55.00
$515.00
CE
$20.00
$30.00
$40.00
$50.00
$510.00
RP
$5.00
$5.00
$5.00
$5.00
$5.00
Conclusion:
Risk Premium stays constant. As my current wealth
increases, my Risk Behavior remains the same.
Lecture notes by: Dr. J. Rene van Dorp
Session 11 - Page 184
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen