1. Risk Attitude Dr. Yan Liu
Department of Biomedical, Industrial & Human Factors Engineering
Wright State University

2. Which game would you choose, game 1 or game 2?
Introduction
Game 1
(0.5)
Payoff
$30
-$1
$2,000
-$1,900
Game 2
EMV=$14.5
Which game would you choose, game 1 or game 2?
EMV=$50
If EMV is the basis for the decision, you should choose Game 2. Most of us, however, may consider Game 2 to be too risky and thus choose Game 1.
This example illustrates that EMV analysis does not capture risk attitudes of decision makers. Individuals who are afraid of risk or are sensitive to risk are called risk-averse.

3. Utility Function
Utility functions are models of an individual’s attitude toward risk
Utility functions translate dollars into utility units
Graph
Table
Mathematical expression
Utility
Dollars x
U(x)
A utility function that displays risk-aversion
(upward sloping and concave)

4. Risk Attitude Risk-Averse: Afraid or Sensitive to Risk
Would trade a gamble for a sure amount that is less than the expected value of the gamble
U(x) is a concave curve
(continuous)
(discrete)
Risk-Seeking: Willing to Accept More Risk
Would play a state lottery
U(x) is a convex curve
(continuous)
(discrete)
Risk-Neutral: An EMV Decision Maker
Maximizing utility is the same as maximizing EMV
U(x) is a straight line
is constant (continuous)
(discrete)

5. Risk Attitude (Cont.)
Dollars
Utility
Risk-Averse
Risk-Neutral
Risk-Seeking
Shapes of Utility Functions of Three Different Risk Attitudes

6. Some Terminologies Expected Utility (EU) Certainty equivalent (CE)
Weighted average of utilities of all possible states
Certainty equivalent (CE)
Amount of money equivalent to the situation that involves uncertainty
Risk Premium
Difference between the EMV and the CE, i.e., the amount you would pay to avoid the risk
You have a lottery which has 0.3 probability of winning $200 and 0.7 probability of losing $10, and you are willing to sell it for $30.
Your certainty equivalent for this lottery is $30
The risk premium of the lottery is:
(0.3* *(-10))-30 =53-30=$23

7. Certainty Equivalent (CE)
EMV
Risk Premium
Utility Curve
Expected Utility (EU)
Utility
Dollar
U(CE) = EU
Graphical Representation of Expected Utility, Certainty Equivalent, and Risk Premium
For a risk-seeking person, CE would be on the right side of EMV on the horizontal axis

8. Utility Assessment
Assessing a Utility Function is a Subjective Judgment
Different people have different risk attitudes toward risk and are willing to accept different levels of risk
Two Methods
Assessment using certainty equivalent
Assessment using probabilities

9. Utility Function Assessment Via Certainty Equivalence
Assess several certainty equivalents from which the utility function is derived
Step 1: set the utility of the best payoff to 1 and the utility of the worst payoff to 0
Step 2: Construct a situation that involves uncertainty and find its CE using reference lottery.
Step 3: Calculate the expected utility of the lottery, EU. Because EU is equal to U(CE), we get another point (CE, EU) on the utility curve
Step 3: Repeat Steps 2 and 3 until getting enough points to plot the utility curve

10. Suppose your CE in this lottery is $30
You face an uncertain situation in which you may earn $10 in the worst case, $100 in the best case, or some amount in between. You have a variety of options, each of which leads to some uncertain payoff between $10 and $100. To evaluate the alternatives, you must assess your utility for payoffs from $10 to $100.
Step 1: let U(10)=0 and U(100)=1
Step 2: imagine you have the opportunity to play the following reference lottery A
(0.5)
$100
$10 B CE
Suppose your CE in this lottery is $30
Step 3: Calculate EU of lottery A, which is 0.5∙U(100)+0.5∙U(10)=0.5. Therefore, U(30)=0.5 and you have found a third point on your utility curve.
Step 4: To find another point, you can take a different reference lottery, say using $100 and $30 as two equally likely outcomes in lottery A, and then follow steps 2 and 3. Continue with the same procedures until you have enough points to plot the utility function.

11. Suppose you now have five points on your utility curves: U(10)=0, U(18)=0.25, U(30)=0.5, U(50)=0.75, and U(100)=1, you can plot the utility function
x (Dollars)
U(x)

12. Utility Function Assessment Via Probability-Equivalent
Assess the utility of a selected dollar amount directly
Adjust the probability in the reference lottery C
(p)
(1-p)
$100
$10 D
$65
U(65) = p∙U(100) + (1-p)∙U(10) = p∙1+(1-p)∙0 = p

13. Gambles, Lotteries, and Investments
Framing utility assessment in terms of gambles or lotteries may evoke images of carnival games or gambling which seem irrelevant to decision making or even distasteful
An alternative is to think in terms of risky investment, particularly for investment decisions
Whether you should make a particular investment

14. Risk Tolerance and Exponential Utility Function
R is risk tolerance, showing how risk-adverse the function is. Larger R means less risk-adversion and makes the utility function flatter
Exponential Utility Functions with Three Different Risk Tolerances
x ↑ => U(x) →1
x =0 => U(x) = 0

15. Risk Tolerance and Exponential Utility Function (Cont.)
Assess Risk Tolerance R E
(0.5) $Y
– $Y/2 F $0
The largest Y for which you prefer to take gamble E is approximately equal to your risk tolerance
Suppose you decide Y is $900,
then R=900, and your utility function is

16. Risk Tolerance and Exponential Utility Function (Cont.)
Find CE of Given Uncertain Event
First calculate the expected utility (EU) of the uncertain event
Since U(CE)=EU, you can solve the equation to get CE
If you estimate the expected value, ,and variance, , of the payoffs, then CE can be approximately calculated as

17. Suppose you face the following gamble: 1) win $2000 with probability 0
Suppose you face the following gamble: 1) win $2000 with probability 0.4; 2) win $1000 with probability 0.4, or 3) win $500 with probability 0.2, and your utility can be modeled as an exponential function with R=900. What is your CE of this gamble?
The expected utility of the gamble is:
EU = 0.4∙U($2000)+0.4 ∙U($1000)+ 0.2∙U($500)
= 0.4∙(1-e-2000/900) +0.4 ∙(1-e-1000/900)+ 0.2∙(1-e1-500/900) = 0.710
Solve 0.710=1-e-CE/900 for CE, you can get CE=$

18. Constant and Decreasing Risk-Aversion
Constant Risk-Aversion
The risk premium for a gamble does not depend on the initial wealth held,
Can be represented using an exponential utility function
You have $x in your pocket, and you are facing a bet: 1) win $15 with probability 0.5, or 2) lose $15 with probability 0.5. Suppose your utility function can be modeled as an exponential function with risk tolerance R=35.
(0.5)
$x+15
$x –15
$ in Pocket
Probability Tree of the Bet

19. When x=$25
EU=0.5∙U($10)+0.5∙U($40)=0.5∙(1-e-10/35)+0.5∙(1-e-40/35)=0.4648
EU=U(CE) 0.4648=1-e-CE/35CE=$21.88
EMV= 0.5∙10+0.5∙40=$25
Risk Premium=EMV-CE= =$3.12 $x
EV ($)
CE ($)
Risk Premium ($) 25
21.88
3.12 35
31.88 45
41.88 55
51.88

20. Constant and Decreasing Risk-Aversion (Cont.)
Typically, people’s attitude towards risk changes with their initial wealth
The risk premium for a gamble decreases along with the increase of wealth held
Can be represented with the logarithmic utility function

21. In the previous example of betting, suppose your utility function can be modeled as a logarithmic function
When x=$25
EU=0.5∙U($10)+0.5∙U($40)=0.5∙ln(10)+0.5∙ln(40)=2.9957
EU=U(CE) 2.9957=ln(CE)CE=$20
EMV= 0.5∙10+0.5∙40=$25
Risk Premium=EMV-CE=25-20=$5 $x
EV ($)
CE ($)
Risk Premium ($) 25 20
5.00 35
31.62
3.38 45
42.43
2.57 55
52.92
2.08

22. Some Caveats Utilities DO NOT Add Up
U(A+B)≠U(A)+U(B) (why?)
Utility Difference Does Not Express Strength of Preferences
U(A1)-U(A2) > U(B1)-U(B2) does not mean we would rather go from A1 to A2 instead of from B1 to B2
Utility only provides a numerical scale for ordering preferences, not a measure of their strengths
Utilities are Not Comparable from Person to Person
A utility function is a subjective personal statement of an individual’s preference

23. Exercise
An investor with assets of $10,000 has an opportunity to invest $5,000 in a venture that is equally likely to pay either $15,000 or nothing. The investor’s utility function can be described by the log utility function U(x) =ln(x), where x is the total wealth.
What should the investor do?

24. a. Total Wealth success (0.5) 10,000-5,000+15,000 =$20,000 $15,000
Invest
$10,000
success (0.5)
10,000-5,000= $5,000
Don’t Invest
10,000-5,000+15,000 =$20,000
Failure (0.5)
Total Wealth
- $5,000
$15,000
EU(invest) = 0.5∙U($20,000)+0.5∙U($5,000)=0.5∙ln($20,000)+0.5∙ln($5000) = 9.21
EU(Don’t invest) = U($10,000) = ln($10,000) = 9.21
Therefore, the investor is indifferent between the two alternatives

25. b. Suppose the investor places a bet with a friend before making the investment decision. The bet is for $1,000; if a fair coin lands heads up, the investor wins $1,000, but if it lands tails up, the investor pays $1,000 to his friend. Only after the bet has been resolved will the investor decide whether or not to invest in the venture. If he wins the bet, should he invest? What if he loses the bet? Should he toss the coin in the first place?

26. b. EU(Invest|Win) = 9.326 Total Wealth Success (0.5)
10,000+1,000-5,000+15,000 =$21,000
$15,000
Invest
Failure (0.5)
Win (0.5)
10,000+1,000-5,000= $6,000
-$5,000
$1,000
Don’t Invest
EU(Don’t Invest|Win) = 9.306
10,000+1,000 = $11,000
Success (0.5)
EU(Bet) = 9.216
EU(Invest|Lose) = 9.073
10,000-1,000-5,000+15,000 =$19,000
$15,000
Invest
Bet
Failure (0.5)
Lose (0.5)
10,000-1,000-5,000= $4,000
-$5,000
-$1,000
Don’t Invest
EU(Don’t Invest|Lose) = 9.105
10,000-1,000 = $9,000
Don’t Bet
Success (0.5)
10,000-5,000+15,000 =$20,000
$15,000
EU(Don’t Bet) = 9.21
Invest
Failure (0.5)
10,000-5,000= $5,000
-$5,000
Don’t Invest
$10,000

27. If he wins the bet:
EU(Invest) = 0.5∙ln($21,000) + 0.5∙ln($6,000) = 9.326
EU(Don’t Invest) = ln($11,000) = 9.306
Therefore, if he wins the bet, he should invest the venture
If he loses the bet:
EU(Invest) = 0.5∙ln($19,000) + 0.5∙ln($4,000) = 9.073
EU(Don’t Invest) = ln($9,000) = 9.105
Therefore, if he losses the bet, he should not invest the venture
EU(Bet) = 0.5∙EU(Invest|win) + 0.5∙EU(Don’t Invest |lose)
= 0.5(9.326)+0.5(9.105) = 9.216
EU(Don’t Bet) = 9.21 (from part a)
Therefore, he should bet