Readings
Readings
Chapter 5
Advanced Linear Programming Applications
BA 452 Lesson A.10 Advanced Finance Applications
1
Overview
Overview
BA 452 Lesson A.10 Advanced Finance Applications
2
Overview
Portfolio Selection Problems with Asset Classification help financial managers
restrict the maximum investment in any single class (equities, debt, real estate,
…) to ensure diversification.
Portfolio Selection Problems with Planning Scenarios help financial managers
predict returns based on past performance. Hence, managers can compute
minimum returns and expected returns.
Portfolio Selection Problems with a Maximin (Maximize a Minimum) Objective
help compute minimum returns for risk-averse (pessimistic) investors
concerned with the worst-case scenario.
Production Scheduling Problems with Adjustment Costs help find an efficient
low-cost production schedule when there are costs of increasing or decreasing
production or storage.
BA 452 Lesson A.10 Advanced Finance Applications
3
Tool Summary
Tool Summary
 Do not make integer restrictions, and maybe the solution are integers.
• Second Example: IMD = number of Indianapolis-Memphis-Discount
seats
 Use compound variables:
• Second Example: IMD = number of Indianapolis-Memphis-Discount
seats
 Constrain a weighted average with a linear constraint:
• Third Example: Constrain the weighted average risk factor to be no
greater than 55: 60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 +
10X9 < 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
 Interpret the unrealistic assumptions needed for a linear formulation:
• First Example: Assume the set of alternative inputs and outputs for a
high school is convex. Thus if 100 teaching hours gets 20 more
students admitted to college, then 10 teaching hours gets at least 2
admitted.
• Second Example: Assume demand has only two values.
• Third Example: Assume risk is a linear function of investment shares.
BA 452 Lesson A.10 Advanced Finance Applications
4
Tool Summary
Tool Summary
 Critique the unrealistic or approximate variables used because data on
better variables may not be available:
• First Example: High school output is measured by only three variables.
• Average SAT Scores, even though maximizing those scores is not
the same as maximizing learning.
• The number of High School Graduates, even though there is no
accounting for learning beyond a minimal level.
• The number of College Admissions, even though there is no
accounting for the quality or selectivity of the colleges.
• First Example: High school input is measured by only three variables.
• Senior Faculty
• Budget ($100,000's)
• Senior Enrollments, even though there is no measure for the quality
of those students before their senior year.
BA 452 Lesson A.10 Advanced Finance Applications
5
Portfolio Selection with Asset Classification
Portfolio Selection with Asset
Classification
BA 452 Lesson A.10 Advanced Finance Applications
6
Portfolio Selection with Asset Classification
Overview
Portfolio Selection Problems with Asset Classification help
financial managers restrict the maximum investment in any
single class (equities, debt, real estate, …) to ensure
diversification, and so to reduce risk.
BA 452 Lesson A.10 Advanced Finance Applications
7
Portfolio Selection with Asset Classification
Question: John Sweeney is an investment advisor
who is attempting to construct an optimal portfolio
for a client who has $400,000 cash to invest. There
are ten different investments, falling into four broad
categories that John and his client have identified as
potential candidates for this portfolio.
The investments and their important
characteristics are listed in the table on the next
slide. Note that Unidyde Corp. under Equities and
Unidyde Corp. under Debt are two separate
investments, whereas First General REIT is a single
investment that is considered both an equities and a
real estate investment.
BA 452 Lesson A.10 Advanced Finance Applications
8
Portfolio Selection with Asset Classification
Exp. Annual
After Tax Liquidity Risk
Return
Factor Factor
Category
Investment
Equities
(Stocks)
Unidyde Corp.
CC’s Restaurants
First General REIT
15.0%
17.0%
17.5%
100
100
100
60
70
75
Debt
(Bonds)
Metropolis Electric
Unidyde Corp.
Lewisville Transit
11.8%
12.2%
12.0%
95
92
79
20
30
22
Real Estate
Realty Partners
First General REIT
22.0%
0
50
( --- See above --- )
Money
T-Bill Account
Money Mkt. Fund
Saver's Certificate
9.6%
10.5%
12.6%
80
100
0
BA 452 Lesson A.10 Advanced Finance Applications
0
10
0
9
Portfolio Selection with Asset Classification
Construct a portfolio that maximizes John's client's total
expected after-tax return over the next year, subject to the
following constraints placed upon him by the client for the portfolio.
1. The weighted average liquidity factor for the portfolio must be
at least 65.
2. The weighted average risk factor for the portfolio must be no
greater than 55. (Using a weighted average or any linear function is an
inaccurate risk measure.)
3. No more than $60,000 is to be invested in Unidyde stocks or bonds.
4. No more than 40% of the investment can be in any one
category except the money category.
5. No more than 20% of the total investment can be in
any one investment except the money market fund.
6. At least $1,000 must be invested in the Money Market fund.
7. The maximum investment in Saver's Certificates is $15,000.
8. The minimum investment desired for debt is $90,000.
9. At least $10,000 must be placed in a T-Bill account.
BA 452 Lesson A.10 Advanced Finance Applications
10
Portfolio Selection with Asset Classification
Answer:

Define the decision variables
X1 = $ amount invested in Unidyde Corp. (Equities)
X2 = $ amount invested in CC’s Restaurants (Equities)
X3 = $ amount invested in First General REIT (Equities, Real Estate)
X4 = $ amount invested in Metropolis Electric (Debt)
X5 = $ amount invested in Unidyde Corp. (Debt)
X6 = $ amount invested in Lewisville Transit (Debt)
X7 = $ amount invested in Realty Partners (Real Estate)
X8 = $ amount invested in T-Bill Account
X9 = $ amount invested in Money Mkt. Fund
X10 = $ amount invested in Saver's Certificate
BA 452 Lesson A.10 Advanced Finance Applications
11
Portfolio Selection with Asset Classification
Define the objective function
Maximize the total expected after-tax return over the next year:
Max .15X1 + .17X2 + .175X3 + .118X4 + .122X5
+ .12X6 + .22X7 + .096X8 + .105X9 + .126X10

Define the constraint that total investment is at most $400,000:
(1) X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 < 400,000

BA 452 Lesson A.10 Advanced Finance Applications
12
Portfolio Selection with Asset Classification

(2)

(3)

(4)

(5)
(6)
(7)
Constrain the weighted-average-liquidity factor to be at least 65:
100X1 + 100X2 + 100X3 + 95X4 + 92X5 + 79X6 + 80X8 + 100X9 >
65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
Constrain the weighted average risk factor to be no greater than 55:
60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 <
55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
Constrain that no more than $60,000 be invested in Unidyde Corp:
X1 + X5 < 60,000
Constrain that no more than 40% of the total investment be in any one
category except the money category:
X1 + X2 + X3 < 0.40 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 +
X10)
X4 + X5 + X6 < 0.40 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 +
X10)
X3 + X7 < 0.40 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
BA 452 Lesson A.10 Advanced Finance Applications
13
Portfolio Selection with Asset Classification

Constrain that no more than 20% of the investment can be in any one
investment, except the money market:
(8) X1 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
(9) X2 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
(10)X3 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
(11) X4 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
(12) X5 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
(13) X6 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
(14) X7 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
(15) X8 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
(16) X10 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
BA 452 Lesson A.10 Advanced Finance Applications
14
Portfolio Selection with Asset Classification
Constrain that at least $1,000 be in the Money Market:
(17) X9 > 1,000

Constrain that the maximum investment in Saver's Certificates is
$15,000:
(18) X10 < 15,000

Constrain that the minimum investment in the Debt category is
$90,000:
(19) X4 + X5 + X6 > 90,000

Constrain that at least $10,000 be in a T-Bill account:
(20) X8 > 10,000

BA 452 Lesson A.10 Advanced Finance Applications
15
Portfolio Selection with Asset Classification
Interpretation:
Total expected after-tax return = $64,355
X1 = $0 invested in Unidyde Corp. (Equities)
X2 = $80,000 invested in CC’s Restaurants
X3 = $80,000 invested in First General REIT
X4 = $0 invested in Metropolis Electric
X5 = $60,000 invested in Unidyde Corp. (Debt)
X6 = $74,000 invested in Lewisville Transit
X7 = $80,000 invested in Realty Partners
X8 = $10,000 invested in T-Bill Account
X9 = $1,000 invested in Money Mkt. Fund
X10 = $15,000 invested in Saver's Certificate
BA 452 Lesson A.10 Advanced Finance Applications
16
Portfolio Selection with Planning Scenarios
Portfolio Selection with Planning
Scenarios
BA 452 Lesson A.10 Advanced Finance Applications
17
Portfolio Selection with Planning Scenarios
Overview
Portfolio Selection Problems with Planning Scenarios help
financial managers predict returns based on past
performance. From predicted returns, managers can
compute minimum returns for risk-averse investors, and
expected returns for risk-taking investors.
BA 452 Lesson A.10 Advanced Finance Applications
18
Portfolio Selection with Planning Scenarios
Question: Portfolio manager Max Gaines needs to develop
an investment portfolio for his clients who are willing to
accept a moderate amount of risk. His task is to determine
the proportion of the portfolio to invest in each of the five
mutual funds listed below so that the portfolio provides an
annual return of no less than 3%, and it maximizes the
average annual return assuming that each planning
scenario is equally likely.
BA 452 Lesson A.10 Advanced Finance Applications
19
Portfolio Selection with Planning Scenarios
Formulate and solve an appropriate linear program based
on the following past performance.
Mutual Fund
International
Stock
Large-Cap
Blend
Mid-Cap
Blend
Small-Cap
Blend
Intermediate
Bond
Annual Returns (Planning Scenarios)
Year 1
Year 2
Year 3
Year 4
22.00
26.00
6.00
-3.00
14.00
18.00
10.00
5.00
19.00
18.00
5.00
1.00
13.00
11.00
2.00
6.00
7.00
8.00
9.00
-3.00
BA 452 Lesson A.10 Advanced Finance Applications
20
Portfolio Selection with Planning Scenarios
Answer: Max R1 + R2 + R3 + R4
R1 = 22 IS + 14 LC + 19 MC + 13 SC + 7 IB
R1 > 3
(Scenario 1)
R2 = 26 IS + 18 LC + 18 MC + 11 SC + 8 IB
R2 > 3
(Scenario 2)
R3 = 6 IS + 10 LC + 5 MC + 2 SC + 9 IB
R3 > 3
(Scenario 3)
R4 = -3 IS + 5 LC + 1 MC + 6 SC – 3 IB
R4 > 3
(Scenario 4)
IS + LC + MC + SC + IB = 1
(Total proportion must equal 1)
IS, LC, MC, SC, IB > 0
(Non-negativity)
BA 452 Lesson A.10 Advanced Finance Applications
21
Portfolio Selection with Planning Scenarios
BA 452 Lesson A.10 Advanced Finance Applications
22
Portfolio Selection with Maximin
Portfolio Selection with Maximin
BA 452 Lesson A.10 Advanced Finance Applications
23
Portfolio Selection with Maximin
Overview
Portfolio Selection Problems with a Maximin (Maximize a
Minimum) Objective help financial managers compute
minimum returns for risk-averse investors. Risk-averse
investors are pessimists, and want to maximize their
investment return in the worst-case (minimum return)
scenario.
BA 452 Lesson A.10 Advanced Finance Applications
24
Portfolio Selection with Maximin
Question: Portfolio manager Max Gaines needs to develop
an investment portfolio for his conservative clients. His
task is to determine the proportion of the portfolio to invest
in each of the four mutual funds listed below so that the
portfolio provides the best return possible with a minimum
risk.
Mutual Fund
Annual Returns (Planning
Scenarios)
Year 1
Year 2
Year 3
International
Stock
Large-Cap
Blend
Mid-Cap Blend
12.00
16.00
6.00
14.00
18.00
10.00
9.00
18.00
5.00
Small-Cap
Blend
3.00
11.00
-2.00
BA 452 Lesson A.10 Advanced Finance Applications
25
Portfolio Selection with Maximin
Answer: Formulating this problem depends on the definition
of “minimum risk”. Risk is different from variability. Risk is
only concerned with the potential for loss. Hence, the best
return possible with a minimum risk means to maximize
investment return in the worst-case (minimum return)
scenario.
BA 452 Lesson A.10 Advanced Finance Applications
26
Portfolio Selection with Maximin
Let IS = proportion of portfolio invested in International
Stock, and so on.
Let M = guaranteed minimum return (with zero risk).
Max M
s.t.
- M + 12 IS + 14 LC + 9 MC + 3 SC > 0 (Scenario 1)
- M + 16 IS + 18 LC + 18 MC + 11 SC > 0 (Scenario 2)
- M + 6 IS + 10 LC + 5 MC - 2 SC
> 0 (Scenario 3)
IS + LC + MC + SC = 1
(Total proportion must equal 1)
M, IS, LC, MC, SC > 0
(Non-negativity)
(If the optimal minimal return M turns out to be negative, then the
problem above is infeasible. In that case, methods are needed that are
more advanced than we cover in this class.)
BA 452 Lesson A.10 Advanced Finance Applications
27
Portfolio Selection with Maximin
Solution:
BA 452 Lesson A.10 Advanced Finance Applications
28
Portfolio Selection with Maximin
Solution: The optimum is
remarkable because the
rate of return in planning
scenarios 2 and 3 exceed
the guaranteed minimum
return.
BA 452 Lesson A.10 Advanced Finance Applications
29
Production Scheduling with Adjustment Costs
Production Scheduling with
Adjustment Costs
BA 452 Lesson A.10 Advanced Finance Applications
30
Production Scheduling with Adjustment Costs
Overview
Production Scheduling Problems with Adjustment Costs
help managers find an efficient low-cost production
schedule for one or more products over several periods in
the future (weeks or months) when there are costs of
increasing or decreasing production or storage from one
period to the next.
BA 452 Lesson A.10 Advanced Finance Applications
31
Production Scheduling with Adjustment Costs
Question: Anderson makes replacement windows. In
January, the company produced 15,000 windows and
ended the month with 9,000 windows in inventory.
Anderson’s management team would like to develop a
production schedule for the next two months. A smooth
production schedule is desirable because it maintains the
current workforce. However, given sales forecasts, the
management team does not think a smooth schedule is
possible.
BA 452 Lesson A.10 Advanced Finance Applications
32
Production Scheduling with Adjustment Costs
Anderson’s accounting department estimates that
increasing production from one month to the next will cost
$4 per unit increase, and decreasing production from one
month to the next will cost $2 per unit decrease. Assume
there is no cost of changing storage from one month to the
next.
February
March
Sales forecast
15,000
16,000
Production
capacity
14,000
15,000
Storage capacity
6,000
7,000
BA 452 Lesson A.10 Advanced Finance Applications
33
Production Scheduling with Adjustment Costs
Formulate a linear programming model to minimize the cost
of adjusting production so that you meet your sales
forecasts in each month.
BA 452 Lesson A.10 Advanced Finance Applications
34
Production Scheduling with Adjustment Costs
Answer: The essential data are: A) Increasing production
from one month to the next will cost $4 per unit increase,
and decreasing production from one month to the next will
cost $2 per unit decrease. B) January production and
storage is 15,000 and 9,000. C) Data for February and
March are:
February
March
Sales forecast
15,000
16,000
Production
capacity
14,000
15,000
Storage capacity
6,000
7,000
BA 452 Lesson A.10 Advanced Finance Applications
35
Production Scheduling with Adjustment Costs
Let
F = number of windows manufactured in February
M = number of windows manufactured in March
Im = increase in production level necessary during month m
Dm = decrease in production level necessary during month m
sm = ending inventory in month m
Min
4I1 + 4I2 + 2D1 + 2D2
s.t.
9000 + F - s1 = 15,000
or
(1) F - s1 = 6000
February Demand
(2) s1 + M - s2 = 16,000
March Demand
BA 452 Lesson A.10 Advanced Finance Applications
36
Production Scheduling with Adjustment Costs
F - 15,000 = I1 - D1
or
(3) F - I1 + D1 = 15,000
M - F = I2 - D2
or
(4) M - F - I2 + D2 = 0
Change in February Production
Change in March Production
(5) F  14,000
February Production Capacity
(6) M  15,000
March Production Capacity
(7) s1  6,000
February Storage Capacity
(8) s2  7,000
March Storage Capacity
BA 452 Lesson A.10 Advanced Finance Applications
37
Production Scheduling with Adjustment Costs
BA 452 Lesson A.10 Advanced Finance Applications
38
Production Scheduling with Adjustment Costs
Solution: The third and
fourth constraints
F - 15,000 = I1 - D1
M - F = I2 - D2
for the changes in
production (on the lefthand side of those
equations) decompose
those changes into either
an increase or a
decrease but not both.
For example, the
February change in
production is a decrease
of D1 = 3000.
BA 452 Lesson A.10 Advanced Finance Applications
39
BA 452
Quantitative Analysis
End of Lesson A.10
BA 452 Lesson A.10 Advanced Finance Applications
40