Chapter 15Risk Analysis

Frequency definition of probability

Given a situation in which a number of possible outcomes might occur, the probability of an outcome is the proportion of times that it occurs if the situation exists repeatedly.

P(A) = r/R is the probability of outcome A, given r A’s in R trials.

Subjective definition of probability

Given a situation in which a number of possible outcomes might occur, the probability of an outcome reflects the degree of confidence that the decision maker has that this particular outcome will occur.

Probability distribution

Profit Probability

$1,000,000 .6

$ - 600,000 .4

Expected value of profit:

($1,000,000).6 + ($-600,000).4 = $360,000

Constructing a decision tree

Decision fork: a juncture representing a choice where the decision maker is in control of the outcome

Chance fork: a juncture where “chance” controls the outcome

Constructing a decision tree

.5$800,000$100,000

$200,000

Do not increase price

Increase price

Successful

Unsuccessful.5

-$600,000

Tomco Oil Corporation

- $90,000$56,000

$0

Do not drill well

Drill well

No oil = .6

10,000 B = .15$100,00020,000 B = .15

$300,000

30,000 B = .10

$500,000

Expected utility

The sum of the utility if each outcome occurs times the probability of occurrence of the outcome

Using a utility functionTomco Oil Corporation

Utility

Profit0

-90

10

20

30

40

50

0 100 300 500

Expected utility for Tomco Oil

E(U) = .6U($-90) + .15U(100) + .15U(300) + .10 U(500)

=.6 (0) + .15(20) + .15(40) + .10(50)

= 14

Attitudes toward risk

Utility

Profit0

Risk Seeking

Risk Neutral

Risk Averse

Measures of risk

Standard deviation:

= [ni=1i - E(i))2 Pi]1/2

Coefficient of variation

V = / E()

Adjusting the valuation model for risk

Certainty equivalent approachRisk-adjusted discount rates

Winner’s curse

In an auction situation, the highest bidder is likely to pay more for the good than it is worth

The highest bid, by definition, must be greater than the average bid (unless all bids are equal)

Risk vs. uncertainty

Risk occurs when the outcome is not certain, but the probabilities of all outcomes are known or estimable

Uncertainty refers to a situation where some or all probabilities are unknown

Problem 1

1. a. E(Px) = .2(20) + .3(8) + .4(10) + .1(3) = $10.7 million.

sX = [(20 – 10.7)2(.2) + (8 – 10.7)2(.3) + (10 –

10.7)2(.4) + (3 – 10.7)2(.1)]1/2

= 5.0606.

VX = sX /E(Px) = 5.0606/10.7 = .4729.

b. E(Py) = .1(12) + .3(9) + .1(16) + .5(11) = 11.

sY = [(12 – 11)2(.1) + (9 – 11)2(.3) + (16 – 11)2(.1) +

(11 – 11)2(.5)]1/2 = 1.97.

VY = sY /E(Py) = 1.97/11 = .18.

Problem 1 (cont.)

1. c. VX = VY X is riskier.

d. Y, since Y is both less risky and has a higher expected value than X, characteristics valued by Martin’s president.

Problem 2

2.a. 18 percent – 6 percent = 12 percent.

b. 6 percent.

c. 18 percent.

d. 18 percent.

Problem 3

3.a. No, 0 > –$8 million, so the worst outcome is least bad if Zodiac does nothing.

b. It assigns 100 percent of the probability to the worst possible case for each option and therefore might lead you to choose an action with a much lower expected value than available alternatives.

c. No, it would be 50P if P is the chance of success for the university research, but this is unknown.

Problem 4

4. a. E(P) = $20(100,000) – $1,000,000

= $1,000,000.

b. s = 10,000($20) = $200,000.

c. V = $200,000/$1,000,000 = .2.

Problem 5

5. U(X) = X/100,000 – 1.

a. U(400) = 4 – 1 = 3.

b. U(40) = .4 – 1 = –.6.

c. U(–20) = –.2 – 1 = –1.2.

Problem 6

6. b. She is risk neutral since the indifference curve is a straight line.

Problem 7

7. b. Only one: To buy or not to buy the firm.

c. Only one: The firm is effective or it is not.

d. Yes, $50,000 > $0.

e. (1) Yes, the “won’t work” branch has another chance fork attached with .2 leading to $100,000 and .8 leading to –$400,000.

(2) Yes, (a) the project works, (b) the project is resold, and (c) the project loses money.

(3) (a) .5, (b) .5(.2) = .1, (c) .5(.8) = .4.

(4) (a) $500,000, (b) $100,000, (c) –$400,000.

Problem 7 (cont.)

7. f. E(P) = 500,000(.5) + 100,000(.1) – 400,000(.4) = $100,000. Buy the firm.

g. (1) E(P) = (500,000 – X) x (.5) – 400,000(.5) = 0 implies X = $100,000.

(2) E(P) = (500,000 – X) x (.5) + 100,000(.1) – 400,000(.4) = 0 implies X = $200,000.

Problem 8

8. Given the low risk factor NASA used, the attractiveness of a launch appeared greater than an unbiased appraisal would suggest.

Problem 9

9. b. The first fork is a decision fork to publish or not to publish. The second fork is a chance fork, where publishing either succeeds or fails.

Problem 10

10. b. No, there are no probabilities given.

c. 1/4(800) – 3/4(200 = 50 > 0, so a person who is risk neutral would drill. However, if very risk averse, the person would not want to drill.

d. Yes, since the project has both a positive expected value and contains risk, Mr. Lamb will be doubly pleased.

e. Yes, Mr. Lamb cares only about expected value, which is positive for this project.