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Risk Attitude

Dr. Yan LiuDepartment of Biomedical, Industrial & Human Factors Engineering

Wright State University

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Introduction

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.

Which game would you choose, game 1 or game

2?

Game 1

(0.5) (0.5)

Payoff

$30-$1

(0.5) (0.5)

$2,000-$1,900

Game 2

EMV=$14.5

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.

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Utility Function

Utility functions are models of an individual’s attitude toward risk Utility functions translate dollars into utility units

Graph Table Mathematical expression

xxxx )(U),log()(U

Utility

Dollarsx

U(x)A utility function that displays risk-aversion

(upward sloping and concave)

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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

xxx )(U )U(Δ)U()Δ(U xxxx (continuous) (discrete)

xxx )U( (continuous) (discrete

))(ΔU)(U)Δ(U xxxx

Risk-Seeking: Willing to Accept More Risk Would play a state lottery U(x) is a convex curve

Risk-Neutral: An EMV Decision Maker Maximizing utility is the same as maximizing EMV U(x) is a straight line

xx

)U(

(discrete))U(Δ)U()Δ(U xxxx is constant (continuous)

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Risk Attitude (Cont.)

Dollars

Utility

Risk-Averse

Risk-Neutral

Risk-Seeking

Shapes of Utility Functions of Three Different Risk Attitudes

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Some Terminologies

Expected Utility (EU) 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.

The risk premium of the lottery is: (0.3*200+0.7*(-10))-30 =53-30=$23

Your certainty equivalent for this lottery is $30

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Certainty Equivalent

(CE)

EMV

Risk Premium

Utility Curve

Expected Utility (EU)

Utility

Dollar

U(CE) = EU

For a risk-seeking person, CE would be on the right side of EMV on the horizontal axis

Graphical Representation of Expected Utility, Certainty Equivalent, and Risk Premium

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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

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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

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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

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.

A (0.5)

(0.5)$100$10

B CE

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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)

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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

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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

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Risk Tolerance and Exponential Utility Function

Exponential Utility Function

Rx

ex

1)(UR is risk tolerance, showing how risk-adverse the function is. Larger R means less risk-adversion and makes the utility function flatter

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 10 11 12 12 14 15

R=1

R=3

R=5

Exponential Utility Functions with Three Different Risk Tolerances

x ↑ => U(x) →1x =0 => U(x) = 0

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Risk Tolerance and Exponential Utility Function (Cont.)

Assess Risk Tolerance R

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 900/1)(U xex

E

(0.5)

(0.5)

$Y

– $Y/2F $0

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

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Risk Tolerance and Exponential Utility Function (Cont.)

2

R

25.0CE

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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=$1114.71

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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)

(0.5)$x+15

$x –15

$ in Pocket

Probability Tree of the Bet

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$x EV ($) CE ($) Risk Premium ($)

25 25 21.88 3.12

35 35 31.88 3.12

45 45 41.88 3.12

55 55 51.88 3.12

When x=$25EU=0.5∙U($10)+0.5∙U($40)=0.5∙(1-e-10/35)+0.5∙(1-e-40/35)=0.4648EU=U(CE) 0.4648=1-e-CE/35CE=$21.88EMV= 0.5∙10+0.5∙40=$25Risk Premium=EMV-CE=25-21.88=$3.12

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Constant and Decreasing Risk-Aversion (Cont.)

Decreasing Risk-Aversion 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

)ln()(U xx

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$x EV ($) CE ($) Risk Premium ($)

25 25 20 5.00

35 35 31.62 3.38

45 45 42.43 2.57

55 55 52.92 2.08

When x=$25EU=0.5∙U($10)+0.5∙U($40)=0.5∙ln(10)+0.5∙ln(40)=2.9957EU=U(CE) 2.9957=ln(CE)CE=$20EMV= 0.5∙10+0.5∙40=$25Risk Premium=EMV-CE=25-20=$5

In the previous example of betting, suppose your utility function can be modeled as a logarithmic function

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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

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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.

a.What should the investor do?

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Invest

$10,000

success (0.5)

10,000-5,000= $5,000

Don’t

Invest

10,000-5,000+15,000 =$20,000Failure

(0.5)

Total Wealth

- $5,000

$15,000

a.

EU(invest) = 0.5∙U($20,000)+0.5∙U($5,000)=0.5∙ln($20,000)+0.5∙ln($5000) = 9.21EU(Don’t invest) = U($10,000) = ln($10,000) = 9.21Therefore, the investor is indifferent between the two alternatives

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?

Invest

Success (0.5)

10,000+1,000-5,000= $6,000

Don’t Bet

10,000+1,000-5,000+15,000 =$21,000

Failure (0.5)

Total Wealth

-$1,000

$15,000

Bet

Win (0.5)

Lose (0.5)

$1,000

-$5,000

Don’t

Invest

10,000+1,000 = $11,000

Invest 10,000-1,000-

5,000= $4,000

10,000-1,000-5,000+15,000 =$19,000

$15,000-$5,000

Don’t

Invest

10,000-1,000 = $9,000

Invest 10,000-5,000=

$5,000

10,000-5,000+15,000 =$20,000

$15,000

-$5,000

Don’t

Invest

$10,000

Success (0.5)Failure (0.5)

Success (0.5)Failure (0.5)

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b.

EU(Invest|Win) = 9.326

EU(Don’t Invest|Win) = 9.306

EU(Invest|Lose) = 9.073

EU(Don’t Invest|Lose) = 9.105

EU(Bet) = 9.216

EU(Don’t Bet) = 9.21

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If he wins the bet:

EU(Invest) = 0.5∙ln($21,000) + 0.5∙ln($6,000) = 9.326EU(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) = 9.073EU(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) = 9.216EU(Don’t Bet) = 9.21 (from part a)

Therefore, he should bet