of 59
CHAPTER 16--RISK ANALYSIS
Student: ___________________________________________________________________________
9. A project with a 75% chance of earning $4,000 in profit and a 25% chance of earning $12,000 in profit has
12. If profits are normally distributed with a mean of $12 and a standard deviation of $4, there is a 50/50 chance
16. If you are indifferent between $1 and a lottery ticket that gives you a 0.001 chance of winning $1,000 you
17. To justify an investment that involves an out-of-pocket cost of $100 and a 50/50 chance of payoffs of $0 or
18. A valuation model that explicitly accounts for risk can be written as:
26. Risk Attitudes. Identify each of the following as consistent with risk-averse, risk-neutral, or risk-seeking
behavior in investment project selection:
27. Certainty Equivalents. The certainty equivalent concept can be widely employed in the analysis of
personal and business decision making. Identify each of the following statements as true or false and explain
why.
28. Expected Return Analysis. William Mays offers free investment seminars to local PTA groups. On
29. Expected Return Analysis. Alex P. Keaton has just accepted a job as a broker at a major NYSE member
30. Expected Return Analysis. Barry Bonds offers free fixed-income investment seminars to local YMCA
31. Expected Return Analysis. Dr. John Carter offers health seminars to local PTA groups. On average, Carter
32. Probability Analysis. WD-50, Inc. has just completed development of a new spray lubricant. Preliminary
market research indicates two feasible marketing strategies: developing general consumer acceptance through
media advertising, or developing distributor acceptance through intensive personal selling by company
representatives. The marketing manager has developed the following estimates for sales under each alternative:
Media Advertising Strategy
Personal Selling Strategy
Probability
Sales
Probability
Sales
0.1
$125,000
0.3
$250,000
0.4
375,000
0.4
375,000
0.4
625,000
0.3
500,000
0.1
875,000
33. Probability Analysis. Ceramic Tile, Inc. wishes to adopt one of two feasible marketing strategies:
developing general consumer acceptance through media advertising, or developing distributor acceptance
through intensive personal selling by company representatives. The marketing manager has developed the
following estimates for sales under each alternative:
Media Advertising Strategy
Personal Selling Strategy
Probability
Sales
Probability
Sales
0.3
$ 2,500,000
0.2
$ 5,000,000
0.4
10,000,000
0.6
7,500,000
0.3
17,500,000
0.2
10,000,000
34. Probability Analysis. The Dental Clinic, Inc. is contemplating replacing an obsolete word processing
system with one of two innovative lines of equipment. Alternative 1 requires a current investment outlay of
$26,022, whereas alternative 2 requires an outlay of $31,048. The following cash flows (cost savings) will be
generated each year over the five-year useful lives of the new systems.
Probability
Alternative 1
0.32
0.36
0.32
Alternative 2
0.18
0.64
0.18
35. Probability Analysis. The Medical Centre is considering taking on a new lease for additional office space
in alternative suburban shopping areas. Alternative 1 requires a current investment outlay of $50,000;
alternative 2 requires an outlay of $75,000. The following cash flows will be generated each year over an initial
five-year lease period.
Probability
Alternative 1
0.245
0.510
0.245
Alternative 2
0.18
0.64
0.18
36. Probability Analysis. The Seattle HMO, Inc. is considering entering into a data processing contract with a
leading consulting firm. Entering into such an agreement would require a current investment outlay of
$200,000. The following net cash flows (cost savings) will be generated each year over the ten-year life of the
management contract:
37. Certainty Equivalent Method. Saddie Hawkins, a management analyst with Mobile Telephone Services,
Inc., has collected the following information about three investment projects undertaken by the firm during the
past six month period. Hawkins wishes to use this information as a backdrop against which to evaluate the
attractiveness of a recent investment proposal put forth by the quality control department. In that proposal,
dubbed Project X, the quality control department proposes to spend $100,000 to modify transmission equipment
at the Colorado Springs, Colorado facility. Annual expected cost savings of $25,000 per year over the 10-year
2005-2014 period have been projected, and verified as reasonable by Hawkins.
Expected Cash Flows Per Year
Year
Project X
Project Y
Project Z
2008
$ 25,000
$ 50,000
$ 7,500
2009
25,000
45,000
12,500
2010
25,000
40,000
17,500
2011
25,000
35,000
22,500
2012
25,000
30,000
27,500
2013
25,000
25,000
32,500
38. Probability Analysis. Tex-Mex, Inc. is a rapidly growing chain of Mexican-food restaurants. The company
has a limited amount of capital for expansion, and must carefully weigh available alternatives. Currently, the
company is considering opening restaurants in Phoenix and/or Tucson, Arizona. Projections for the two
potential outlets are:
City
Outcome
Annual Profit Contribution
Probability
Phoenix
Failure
$200,000
0.5
Success
$300,000
0.5
Tucson
Failure
$100,000
0.5
Success
$500,000
0.5
Each restaurant would involve a capital expenditure of $1.5 million, plus land acquisition costs of $500,000 for Phoenix and $1,050,000 for Tucson.
The company uses the 10% yield on riskless U.S. Treasury bills to calculate the risk-free annual opportunity cost of investment capital.
39. Certainty Equivalents. Rabbit Food, Inc., is a rapidly growing chain of health food restaurants. The
company has a limited amount of capital for expansion, and must carefully weigh available alternatives.
Currently, the company is considering opening restaurants in Fresno and/or Pasadena, California. Projections
for the two potential outlets are:
City
Outcome
Annual Profit Contribution
Probability
Fresno
Failure
$300,000
0.5
Success
$500,000
0.5
Pasadena
Failure
$250,000
0.5
Success
$750,000
0.5
Each restaurant would involve a capital expenditure of $2.5 million, plus land acquisition costs of $500,000 for Fresno and $1.5 million for Pasadena.
The company uses the 10% an estimate of a minimal risk-free annual opportunity cost of investment capital.
40. Certainty Equivalents. Pier-4, Inc. is a rapidly growing chain of sea-food restaurants. The company has a
limited amount of capital for expansion, and must carefully weigh available alternatives. Currently, the
company is considering opening restaurants in Providence, Rhode Island and/or Gloucester, Massachusetts.
Projections for the two potential outlets are:
City
Outcome
Annual Profit Contribution
Probability
Providence
Failure
$500,000
0.5
Success
$750,000
0.5
Gloucester
Failure
$500,000
0.5
Success
$1,000,000
0.5
41. Certainty Equivalents. Rajun Cajun's, Ltd., is a rapidly growing chain of Cajun-style cuisine restaurants.
The company has a limited amount of capital for expansion, and must carefully weigh available alternatives.
Currently, the company is considering opening restaurants in Montgomery, Alabama and/or Pensacola Beach,
Florida. Projections for the two potential outlets are:
City
Outcome
Annual Profit Contribution
Probability
Montgomery
Failure
$150,000
0.5
Success
$350,000
0.5
Pensacola Beach
Failure
$200,000
0.5
Success
$400,000
0.5
42. Decision Trees. Atlanta Corporation has been supplying Raleigh Manufacturing, Inc. with electronic
control systems, and Raleigh is satisfied with their performance. However, Raleigh has just received a
competing bid from Brahmin, Inc., a firm that is aggressively marketing its products. Brahmin has offered to
supply systems for a price of $237,500, or $12,500 below the $250,000 price for the Atlanta system. In addition
to an attractive price, Brahmin offers a money-back guarantee. That is, if Brahmin's systems do not match
Atlanta's quality, Raleigh can reject them and return them for a full refund. However, if it must reject the
machines and return them to Brahmin, Raleigh will suffer a manufacturing delay costing the firm $50,000.
43. Decision Trees. Arnie Becker, an attorney with Dewey, Cheetum & Howe in Los Angeles, California, must
44. Standard Normal. A leading company in the freight forwarding business offers Overnight Letter delivery
service with a record of on-time delivery for 99% of shipped parcels. The price of this service is $15. Express
Mail, offered by a leading competitor for $10, has an on-time delivery record of 95%.
A.
Calculate the cost incurred due to late delivery that would make shippers indifferent to these deliver service alternatives.
B.
Which delivery alternative is preferred if a $100 cost would be incurred due to late delivery?
45. Standard Normal. Personal Business Cards, Inc. supplies customized business cards to commercial and
individual customers. Although paper, ink, and other costs cannot be determined precisely, Personal anticipates
that costs will be normally distributed around a mean of $15 per unit (each 500-card order) with a standard
deviation of $2 per unit.
46. Standard Normal. Chip Technologies, Inc. supplies wafer-thin computer chips to industrial customers.
Although labor and material costs cannot be determined precisely, CTI anticipates that costs will be normally
distributed around a mean of $3 per unit with a standard deviation of 20¢ per unit.
47. Standard Normal. University Savings, Inc offers personal checking accounts to commercial and individual
customers. Although unit costs cannot be determined precisely, University anticipates that monthly costs will be
normally distributed around a mean of $5 per unit with a standard deviation of $1 per unit.
48. Game Theory. Catskill Mountain Bike, Inc. is a producer and wholesaler of rugged bicycles designed for
mountain touring. The company is considering upgrading its current line by making standard high-grade
Chromalloy frames. Of course, the market response to this upgrade in product quality would depend on the
competitor response, if any. The company's comptroller projects the following annual profits (payoffs)
following resolution of the upgrade decision.
States of Nature
Catskill's Decision Alternatives
Competitor Upgrade
No Competitor Upgrade
Upgrade
$2,500,000
$3,500,000
Don't upgrade
$1,500,000
$5,000,000
49. Game Theory. Jessica's, a local retailer of women's clothing is considering adoption of new Sunday hours
between 12:00 PM and 6:00 PM. Jessica's is closed on Sundays. Of course, the consumer response to this
extension in hours depends on the competitor response, if any. The following annual profit contributions
(payoffs) are expected:
States of Nature
Jessica's Decision Alternatives
Competitor Open
Competitor Closed
Open Sundays
$75,000
$100,000
Closed Sundays
$50,000
$150,000
A.
Which decision alternative would Jessica's choose given a secure strategy criterion? Explain.
B.
Calculate the opportunity loss (or regret) matrix.
C.
Which decision alternative would Jessica's choose if the company seeks to minimize opportunity cost? Explain.
50. Game Theory. F&M Manufacturing, Inc., a diversified manufacturer of packaging products, is considering
upgrading its current line by making available a new line of coated paper products. Of course, the market
response to this upgrade in product quality would depend on the competitor response, if any. The company's
comptroller projects the following annual profits (payoffs) following resolution of the upgrade decision.
States of Nature
F&M's Decision Alternatives
Competitor Upgrade
No Competitor Upgrade
Upgrade
$5 million
$7.5 million
Don't upgrade
$4 million
$8 million
A.
Which decision alternative would F&M choose given a secure strategy criterion? Explain.
B.
Calculate the opportunity loss or regret matrix.
C.
Which decision alternative would F&M choose if the company seeks to minimize opportunity cost? Explain.
CHAPTER 16--RISK ANALYSIS Key
9. A project with a 75% chance of earning $4,000 in profit and a 25% chance of earning $12,000 in profit has
12. If profits are normally distributed with a mean of $12 and a standard deviation of $4, there is a 50/50 chance
16. If you are indifferent between $1 and a lottery ticket that gives you a 0.001 chance of winning $1,000 you
17. To justify an investment that involves an out-of-pocket cost of $100 and a 50/50 chance of payoffs of $0 or
18. A valuation model that explicitly accounts for risk can be written as:
26. Risk Attitudes. Identify each of the following as consistent with risk-averse, risk-neutral, or risk-seeking
behavior in investment project selection:
A.
Ignoring risk levels of investment alternatives.
B.
Larger risk premiums for riskier projects.
C.
Valuing equally certain sums and expected risky sums of equal dollar amounts.
D.
Increasing marginal utility of money.
E.
Preference for larger, as opposed to smaller, coefficients of variation.
27. Certainty Equivalents. The certainty equivalent concept can be widely employed in the analysis of
personal and business decision making. Identify each of the following statements as true or false and explain
why.
A.
If previously accepted projects with similar risk have a's in a range from a = 0.4 to a = 0.5, an investment with an expected return of
$200,000 is acceptable at a cost of $125,000.
B.
A project for which NPV < 0 using a risk-adjusted discount rate will have an implied a factor that is too large to allow project
acceptance.
C.
The appropriate certainty equivalent adjustment factor a indicates the minimum price in certain dollars that an individual should be
willing to pay per risky dollar of expected return.
D.
State lotteries that pay out 40% of the revenues they generate require players who place at least a certain $2.50 value on each $1 of
expected risky return.
28. Expected Return Analysis. William Mays offers free investment seminars to local PTA groups. On
average, Mays expects 1% of seminar participants to purchase $25,000 each in tax sheltered investments, and
2% to purchase $10,000 each in stocks and bonds. Mays earns a 4% net commission on tax shelters, and 1% on
stocks and bonds.
A.
Calculate Mays' expected net commissions per seminar if attendance averages twenty-five persons.
A.
Expected net commission will be the sum of net commissions on tax shelters (TS) and stocks and bonds (S&B).
E(NCTS)
= Expected Sales ´ Commission Rate
= (0.01)($25,000)(25) ´(0.04)
29. Expected Return Analysis. Alex P. Keaton has just accepted a job as a broker at a major NYSE member
firm, and has been asked to develop a list of customers by telephoning medical doctors and other professionals
located in the metropolitan area. On average, Keaton expects 1% of those called to purchase $15,000 each in
mutual fund investments, and 3% to purchase $10,000 each in stocks and bonds. Keaton earns a 2% net
commission on mutual funds and 1% on stocks and bonds.
A.
Calculate Keaton's expected net commissions if he calls an average of twenty-five persons per day.
A.
Expected net commission will be the sum of net commissions on mutual funds (MF) and stocks and bonds (S&B).
E(NCMF)
= Expected Sales ´ Commission Rate
= (0.01)($15,000)(25) ´ (0.02)
30. Expected Return Analysis. Barry Bonds offers free fixed-income investment seminars to local YMCA
groups. On average, Bonds expects 10% of seminar participants to purchase customized financial planning
services priced at $500 each, and 25% to purchase an investment option priced at $100.
A.
Calculate Bonds' expected gross return per seminar if attendance averages ten persons.
A.
The expected gross return will be the sum of fees generated from estate planning (EP) and will writing (W) services.
E(Fee)
= E(FeeEP) + E(FeeW)
= 0.1(10)($500) + 0.25(10)($100)
= $750
31. Expected Return Analysis. Dr. John Carter offers health seminars to local PTA groups. On average, Carter
expects 2% of seminar participants to become patients of his HMO organization at a gross billing of $2,500 per
patient per year.
A.
Calculate Carter's expected net return per dollar of gross patient billings if attendance averages fifty persons per seminar, and a
first-year net return of $100 must be earned to justify Carter's time and effort per seminar.
A.
The net return per seminar is found as:
32. Probability Analysis. WD-50, Inc. has just completed development of a new spray lubricant. Preliminary
market research indicates two feasible marketing strategies: developing general consumer acceptance through
media advertising, or developing distributor acceptance through intensive personal selling by company
representatives. The marketing manager has developed the following estimates for sales under each alternative:
Media Advertising Strategy
Personal Selling Strategy
Probability
Sales
Probability
Sales
0.1
$125,000
0.3
$250,000
0.4
375,000
0.4
375,000
0.4
625,000
0.3
500,000
0.1
875,000
A.
Assume that the company has a 20% profit margin on sales. Calculate expected profits for each plan.
B.
Construct a simple bar graph of the possible profit outcomes for each plan. Which plan appears to be more risky?
C.
Assume that management's utility function resembles the one illustrated below. Which strategy should the marketing manager
recommend?
0.1
$125,000
$ 25,000
$ 2,500
0.4
375,000
75,000
30,000
0.4
625,000
125,000
50,000
0.1
875,000
175,000
17,500
1.0
E(pMA) = $100,000
0.3
$250,000
$ 50,000
$15,000
0.4
375,000
75,000
30,000
0.3
500,000
100,000
30,000
1.0
E(pPS) = $75,000
B.
The anticipated distribution of profits for each strategy appear as follows:
0.1
$ 25,000
200
20
0.4
75,000
330
132
0.4
125,000
380
152
0.1
175,000
400
40
1.0
E(U) = 344
0.3
$ 50,000
280
84
0.4
75,000
330
132
0.3
100,000
360
108
1.0
E(U) = 324
33. Probability Analysis. Ceramic Tile, Inc. wishes to adopt one of two feasible marketing strategies:
developing general consumer acceptance through media advertising, or developing distributor acceptance
through intensive personal selling by company representatives. The marketing manager has developed the
following estimates for sales under each alternative:
Media Advertising Strategy
Personal Selling Strategy
Probability
Sales
Probability
Sales
0.3
$ 2,500,000
0.2
$ 5,000,000
0.4
10,000,000
0.6
7,500,000
0.3
17,500,000
0.2
10,000,000
A.
Assume that the company has a 10% net profit margin on sales. Calculate expected profits for each plan.
B.
Construct a simple bar graph of the possible profit outcomes for each plan. Which plan appears to be more risky?
C.
Assume that management's utility function resembles the one illustrated below. Which strategy should the marketing manager
recommend?
0.3
$ 2,500,000
$ 250,000
$ 75,000
0.4
10,000,000
1,000,000
400,000
0.3
17,500,000
1,750,000
525,000
1.0
E(pMA) = $1,000,000
0.2
$ 5,000,000
$ 500,000
$100,000
0.6
7,500,000
750,000
450,000
0.2
10,000,000
1,000,000
200,000
1.0
E(pPS) = $750,000
B.
The anticipated distribution of profits for each strategy are:
0.3
$ 250
150
45
0.4
1,000
325
130
0.3
1,750
435
130.5
1.0
E(U) = 305.5
0.2
$ 500
250
50
0.6
750
325
195
0.2
1,000
375
75
1.0
E(U) = 320
34. Probability Analysis. The Dental Clinic, Inc. is contemplating replacing an obsolete word processing
system with one of two innovative lines of equipment. Alternative 1 requires a current investment outlay of
$26,022, whereas alternative 2 requires an outlay of $31,048. The following cash flows (cost savings) will be
generated each year over the five-year useful lives of the new systems.
Probability
Alternative 1
0.32
0.36
0.32
Alternative 2
0.18
0.64
0.18
A.
Calculate the expected cash flow for each investment alternative.
B.
Calculate the standard deviation of cash flows (risk) for each investment alternative.
C.
The firm will use a discount rate of 15% for the cash flows with a higher degree of dispersion and a 12% rate for the less risky cash
flows, calculate the expected net present value for each investment. Which alternative should be chosen?
Alternative 1
0.32
$ 7,000
$ 2,240
0.36
10,000
3,600
0.32
13,000
4,160
Alternative 2
0.18
7,500
$ 1,350
0.64
10,000
6,400
0.18
12,500
2,250
E(CF2) = $10,000
Alternative 1
0.32
-$3,000
$9 ´ 106
$2,880,000
0.36
0
0
0
0.32
3,000
9 ´ 106
2,880,000
= $5,760,000
Alternative 2
0.18
-$2,500
$6.25 ´ 106
$1,125,000
0.64
0
0
0
0.18
2,500
6.25 ´ 106
1,125,000
= $2,250,000
35. Probability Analysis. The Medical Centre is considering taking on a new lease for additional office space
in alternative suburban shopping areas. Alternative 1 requires a current investment outlay of $50,000;
alternative 2 requires an outlay of $75,000. The following cash flows will be generated each year over an initial
five-year lease period.
Probability
Alternative 1
0.245
0.510
0.245
Alternative 2
0.18
0.64
0.18
A.
Calculate the expected cash flow for each investment alternative.
B.
Calculate the standard deviation and coefficient of variation of cash flows (risk) for each investment alternative.
C.
The firm will use a discount rate of 15% for the cash flows with higher degree of dispersion and a 12% rate for the less risky cash flows,
calculate the expected net present value for each investment. Which alternative should be chosen?
Alternative 1
0.245
$10,000
$ 2,450
0.510
15,000
7,650
0.245
20,000
4,900
Alternative 2
0.18
15,000
$ 2,700
0.64
25,000
16,000
0.18
35,000
6,300
E(CF2) = $25,000
Alternative 1
0.245
-$5,000
$25 ´ 106
$6,125,000
0.510
0
0
0
0.245
5,000
25 ´ 106
6,125,000
= $12,250,000
Alternative 2
0.18
-$10,000
$1.0 ´ 108
$1.8 ´ 107
0.64
0
0
0
0.18
10,000
1.0 ´ 108
1.8 ´ 107
= $36,000,000
36. Probability Analysis. The Seattle HMO, Inc. is considering entering into a data processing contract with a
leading consulting firm. Entering into such an agreement would require a current investment outlay of
$200,000. The following net cash flows (cost savings) will be generated each year over the ten-year life of the
management contract:
A.
Calculate the expected cash flow.
B.
Calculate the standard deviation and coefficient of variation of cash flows (risk).
C.
Calculate the expected net present value for the investment if the firm uses a discount rate of 20%. Should the investment be
undertaken?
0.32
$25,000
0.36
50,000
0.32
75,000
E(CF) = $50,000
0.32
-$25,000
$6.25 ´ 108
$2.0 ´ 108
0.36
0
0
0
0.32
25,000
6.25 ´ 108
2.0 ´ 108
s2 = $4.0 ´ 108
s
37. Certainty Equivalent Method. Saddie Hawkins, a management analyst with Mobile Telephone Services,
Inc., has collected the following information about three investment projects undertaken by the firm during the
past six month period. Hawkins wishes to use this information as a backdrop against which to evaluate the
attractiveness of a recent investment proposal put forth by the quality control department. In that proposal,
dubbed Project X, the quality control department proposes to spend $100,000 to modify transmission equipment
at the Colorado Springs, Colorado facility. Annual expected cost savings of $25,000 per year over the 10-year
2005-2014 period have been projected, and verified as reasonable by Hawkins.
Expected Cash Flows Per Year
Year
Project X
Project Y
Project Z
2008
$ 25,000
$ 50,000
$ 7,500
2009
25,000
45,000
12,500
2010
25,000
40,000
17,500
2011
25,000
35,000
22,500
2012
25,000
30,000
27,500
2013
25,000
25,000
32,500
2014
25,000
20,000
37,500
2015
25,000
15,000
42,500
2016
25,000
10,000
47,500
2017
25,000
5,000
52,500
PV of Cash Flow @ 8%
?
$205,620
$180,210
2007 Investment:
$100,000
$100,000
$100,000
A.
Calculate the present value of anticipated cost savings using an 8% discount rate as a reasonable estimate of the risk-free cost of capital.
B.
In light of the $100,000 investment required for each of these projects, and the discounted present value of future benefits, calculate the
certainty equivalent adjustment factor a implicit in the decision to fund each of these investment projects.
C.
Assume that the a's implicit in the decisions to fund projects Y and Z represent the upper limits for investment projects of this type.
Would a decision to fund project X be consistent or inconsistent with the firm's decision to fund projects Y and Z?
A.
The present value of anticipated benefits from project X can be calculated using the Present Value of an Annuity interest factor table
from the Appendix. Using an 8% discount rate as a reasonable estimate of the risk-free cost of capital plus a 10-year horizon, the
PVIFA.08,10 = .67101.
Therefore, the discounted present
value of future benefits anticipated
from project X is:
PV of Benefits
= $25,000 ´ PVIFA (i = 0.08, n = 10),
38. Probability Analysis. Tex-Mex, Inc. is a rapidly growing chain of Mexican-food restaurants. The company
has a limited amount of capital for expansion, and must carefully weigh available alternatives. Currently, the
company is considering opening restaurants in Phoenix and/or Tucson, Arizona. Projections for the two
potential outlets are:
City
Outcome
Annual Profit Contribution
Probability
Phoenix
Failure
$200,000
0.5
Success
$300,000
0.5
Tucson
Failure
$100,000
0.5
Success
$500,000
0.5
Each restaurant would involve a capital expenditure of $1.5 million, plus land acquisition costs of $500,000 for Phoenix and $1,050,000 for Tucson.
The company uses the 10% yield on riskless U.S. Treasury bills to calculate the risk-free annual opportunity cost of investment capital.
A.
Calculate the expected value, standard deviation, and coefficient of variation for each outlet's profit contribution.
B.
Calculate the minimum certainty equivalent adjustment factor for each restaurant's cash flows that would justify investment in each
outlet.
C.
Assuming the management of Tex-Mex is risk averse, and uses the certainty equivalent method in decision making, which is the more
attractive outlet? Why?
39. Certainty Equivalents. Rabbit Food, Inc., is a rapidly growing chain of health food restaurants. The
company has a limited amount of capital for expansion, and must carefully weigh available alternatives.
Currently, the company is considering opening restaurants in Fresno and/or Pasadena, California. Projections
for the two potential outlets are:
City
Outcome
Annual Profit Contribution
Probability
Fresno
Failure
$300,000
0.5
Success
$500,000
0.5
Pasadena
Failure
$250,000
0.5
Success
$750,000
0.5
Each restaurant would involve a capital expenditure of $2.5 million, plus land acquisition costs of $500,000 for Fresno and $1.5 million for Pasadena.
The company uses the 10% an estimate of a minimal risk-free annual opportunity cost of investment capital.
A.
Calculate the expected value, standard deviation, and coefficient of variation for each outlet's profit contribution.
B.
Calculate the minimum certainty equivalent adjustment factor for each restaurant's cash flows that would justify investment in each
outlet.
C.
Assuming the management of Rabbit Food is risk averse, and uses the certainty equivalent method in decision making, which is the
more attractive outlet? Why?
40. Certainty Equivalents. Pier-4, Inc. is a rapidly growing chain of sea-food restaurants. The company has a
limited amount of capital for expansion, and must carefully weigh available alternatives. Currently, the
company is considering opening restaurants in Providence, Rhode Island and/or Gloucester, Massachusetts.
Projections for the two potential outlets are:
City
Outcome
Annual Profit Contribution
Probability
Providence
Failure
$500,000
0.5
Success
$750,000
0.5
Gloucester
Failure
$500,000
0.5
Success
$1,000,000
0.5
Each restaurant would involve a capital expenditure of $2.5 million, and the company uses the 10% yield on risk-free U.S. Treasury bills to calculate
the risk-free annual opportunity cost of investment capital.
A.
Calculate the expected value, standard deviation, and coefficient of variation for each outlet's profit contribution.
B.
Calculate the minimum certainty equivalent adjustment factor for each restaurant's cash flows that would justify investment in each
outlet.
41. Certainty Equivalents. Rajun Cajun's, Ltd., is a rapidly growing chain of Cajun-style cuisine restaurants.
The company has a limited amount of capital for expansion, and must carefully weigh available alternatives.
Currently, the company is considering opening restaurants in Montgomery, Alabama and/or Pensacola Beach,
Florida. Projections for the two potential outlets are:
City
Outcome
Annual Profit Contribution
Probability
Montgomery
Failure
$150,000
0.5
Success
$350,000
0.5
Pensacola Beach
Failure
$200,000
0.5
Success
$400,000
0.5
Each restaurant would involve a capital expenditure of $1.5 million, and the company uses the 10% yield on risk-free U.S. Treasury bills to calculate
the risk-free annual opportunity cost of investment capital.
A.
Calculate the expected value, standard deviation, and coefficient of variation for each outlet's profit contribution.
B.
Calculate the minimum certainty equivalent adjustment factor for each restaurant's cash flows that would justify investment in each
outlet.
A.
Montgomery
E(pM)
= $150,000(0.5) + $350,000(0.5) = $250,000
sM
= $100,000
VM
= sM/E(pM) = 0.4
Pensacola Beach
E(pp)
= $200,000(0.5) + $400,000(0.5) = $300,000
sP
= $100,000
VP
= sP/E(pP) = 0.3
B.
To justify each investment alternative, the company must have a certainty equivalent adjustment factor of at least a = 0.6 for the
Montgomery outlet and a = 0.5 for the Pensacola Beach outlet because:
a
42. Decision Trees. Atlanta Corporation has been supplying Raleigh Manufacturing, Inc. with electronic
control systems, and Raleigh is satisfied with their performance. However, Raleigh has just received a
competing bid from Brahmin, Inc., a firm that is aggressively marketing its products. Brahmin has offered to
supply systems for a price of $237,500, or $12,500 below the $250,000 price for the Atlanta system. In addition
to an attractive price, Brahmin offers a money-back guarantee. That is, if Brahmin's systems do not match
Atlanta's quality, Raleigh can reject them and return them for a full refund. However, if it must reject the
machines and return them to Brahmin, Raleigh will suffer a manufacturing delay costing the firm $50,000.
A.
Construct a decision tree for this problem and determine the maximum probability Raleigh can assign to rejection of the Brahmin
system before it would reject the offer, assuming it decides on the basis of minimizing expected costs.
B.
Assume that Raleigh assigns a 40% probability of rejection of the Brahmin controls. Would Raleigh be willing to pay $5,000 for an
assurance bond that would cover manufacturing delay costs if the Brahmin controls fail the quality check? (Use the same objective as in
part A above.) Explain.
A.
The decision tree for this situation is as follows:
The maximum probability of rejection that could be assigned to the
Brahmin control system is the probability that makes the expected cost
equal for the two alternatives.
Expected Brahmin Cost
= Atlanta Cost
(1 - a)$237,500 + a($300,000)
= $250,000
$237,500 - $237,500a + $300,000a
= $250,000
$62,500a
= $12,500
a
= 0.2
43. Decision Trees. Arnie Becker, an attorney with Dewey, Cheetum & Howe in Los Angeles, California, must
serve a subpoena to an individual in New York, New York by 10:00 a.m. tomorrow morning. If the subpoena is
delivered late, Becker stands to lose $5,000 in fees. The subpoena can be delivered by mail at a cost of $25, or
by courier at a cost of $225. Based on passed experience, Becker assigns a 99% change of on-time delivery
using the courier service. Because Express Mail is a relatively new service, Becker does not know the
probability of on-time delivery using this service.
A.
Construct a decision tree for this problem and calculate the minimum probability of on-time delivery for Express Mail that would make
Becker indifferent to the two delivery services.
A.
The decision tree for this problem is as follows:
The minimum acceptable probability of on-time arrival by
Express Mail is the probability that makes the expected cost
equal for the two alternatives.
44. Standard Normal. A leading company in the freight forwarding business offers Overnight Letter delivery
service with a record of on-time delivery for 99% of shipped parcels. The price of this service is $15. Express
Mail, offered by a leading competitor for $10, has an on-time delivery record of 95%.
A.
Calculate the cost incurred due to late delivery that would make shippers indifferent to these deliver service alternatives.
B.
Which delivery alternative is preferred if a $100 cost would be incurred due to late delivery?
A.
Shippers will be indifferent to the delivery alternatives if the expected cost is equal for the two alternatives:
EC(Overnight Letter)
= EC(Express Mail)
0.99($15) + 0.01($15 + X)
= 0.95($10) + 0.05($10 + X)
0.04X
= 5
X
= $125
45. Standard Normal. Personal Business Cards, Inc. supplies customized business cards to commercial and
individual customers. Although paper, ink, and other costs cannot be determined precisely, Personal anticipates
that costs will be normally distributed around a mean of $15 per unit (each 500-card order) with a standard
deviation of $2 per unit.
A.
What is the probability that Personal would make a profit at a price of $15 per unit?
B.
Calculate the unit price necessary to give Personal a 95% chance of making a profit on the order.
C.
If Personal submits a successful bid of $18.20 per unit, what is the probability that it will make a profit?
z
1.645
46. Standard Normal. Chip Technologies, Inc. supplies wafer-thin computer chips to industrial customers.
Although labor and material costs cannot be determined precisely, CTI anticipates that costs will be normally
distributed around a mean of $3 per unit with a standard deviation of 20¢ per unit.
A.
What is the probability that CTI would make a profit at a price of $3 per unit?
B.
Calculate the unit price necessary to give CTI a 95% chance of making a profit on the order.
C.
If CTI signs a contract to supply chips at a price of $3.20 per unit, what is the probability that it will make a profit?
z
1.645
$0.329
= P - $3
P
= $3.329
47. Standard Normal. University Savings, Inc offers personal checking accounts to commercial and individual
customers. Although unit costs cannot be determined precisely, University anticipates that monthly costs will be
normally distributed around a mean of $5 per unit with a standard deviation of $1 per unit.
A.
What is the probability that University would make a profit at a checking price of $5 per unit?
B.
Calculate the unit price necessary to give University a 90% chance of making a profit on an individual checking account.
C.
If University offers its accounts at a price of $6, what is the probability that it will make a profit?
z
1.232
0.3413 = 84.13% probability that University will be able to make a profit at a price of $6 per unit.
48. Game Theory. Catskill Mountain Bike, Inc. is a producer and wholesaler of rugged bicycles designed for
mountain touring. The company is considering upgrading its current line by making standard high-grade
Chromalloy frames. Of course, the market response to this upgrade in product quality would depend on the
competitor response, if any. The company's comptroller projects the following annual profits (payoffs)
following resolution of the upgrade decision.
States of Nature
Catskill's Decision Alternatives
Competitor Upgrade
No Competitor Upgrade
Upgrade
$2,500,000
$3,500,000
Don't upgrade
$1,500,000
$5,000,000
Decision Alternatives
1. Competitor Upgrade
2. No Competitor Upgrade
49. Game Theory. Jessica's, a local retailer of women's clothing is considering adoption of new Sunday hours
between 12:00 PM and 6:00 PM. Jessica's is closed on Sundays. Of course, the consumer response to this
extension in hours depends on the competitor response, if any. The following annual profit contributions
(payoffs) are expected:
States of Nature
Jessica's Decision Alternatives
Competitor Open
Competitor Closed
Open Sundays
$75,000
$100,000
Closed Sundays
$50,000
$150,000
Decision Alternatives
1. Competitor Open
2. No Competitor Closed
50. Game Theory. F&M Manufacturing, Inc., a diversified manufacturer of packaging products, is considering
upgrading its current line by making available a new line of coated paper products. Of course, the market
response to this upgrade in product quality would depend on the competitor response, if any. The company's
comptroller projects the following annual profits (payoffs) following resolution of the upgrade decision.
States of Nature
F&M's Decision Alternatives
Competitor Upgrade
No Competitor Upgrade
Upgrade
$5 million
$7.5 million
Don't upgrade
$4 million
$8 million
Decision Alternatives
1. Competitor Upgrade
2. No Competitor Upgrade