Linear Programming

1
Linear Programming

• Example 3
• Financial Model
• Rate of Return
• Summation Variable/Constraint

2
The Problem

• An investor has 100,000 to invest in the stock
  market and is considering investments in six
  stocks with the following current projected rates
  of return
• Tech Stocks Cisco (8), Microsoft (6) and
  Intel (5), Bank Stocks B of A (7), First Bank
  (4)
• Money Market -- ING(2)
• Constraints
• At least 20,000 is to be invested in ING
• At least 25 of the funds in tech stocks must be
  in Intel
• At least as much in bank stocks as tech stocks
• No more than 25,000 in investments with
  projected returns of 5 or less

3
Decision Variables/Objective

• X1 invested in Cisco
• X2 invested in Microsoft
• X3 invested in Intel
• X4 invested in Bank of America
• X5 invested in First Bank
• X6 invested in ING

MAX Total Expected Annual Return
MAX .08X1 .06X2 .05X3 .07X4 .04X5 .02X6
MAX Total Expected Annual Return
4
Constraints

• Cannot invest more than 100,000
• At least 20,000 invested in ING

Total Amount Invested
Cannot Exceed
100000
X1 X2 X3 X4 X5 X6
100000

Amount Invested in ING
Must be At least
20000
X6
20000

5
Constraints

• Summation constraint for total in tech
• At least 25 of tech should be in Intel

Total Amount In Tech
IS
X7
X1 X2 X3
X7

Amount in Intel
Must be At least
25 of tech total
X3
.25 X7

6
Constraints

• Invest at least as much in banking as tech
• At most 25,000 in investments with returns of 5
  or less

Amount In Banking
Must be At least
Amount in Tech
X4 X5
X7

Amount in Low Return Investments
Cannot Exceed
25000
X3 X5 X6
25000

7
Complete Model

• MAX .08X1 .06X2 .05X3 .07X4 .04X5 .02X6
• s.t. X1 X2 X3 X4 X5
  X6 100000
• X6 20000
• X1 X2 X3 - X7
  0
• X3 -.25X7 0
• X4 X5 - X7 0
• X3 X5 X6
  25000
• ALL Xs 0

8
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9
Invest 15,000 in Cisco 5,000 in
Intel 60,000 in B of A 20,000 in ING
Total Expected Return on Investment 6050 All
100,000 is invested. So return on investment
6050/100000 6.05
10
Sensitivity Report
11
Review

• Financial situations can be modeled by linear
  programs.
• How to estimate the return on investment (ROI) of
  a portfolio.
• How to estimate the ROI for future investments.
• Determining how much the estimate on a ROI must
  increase before it is profitable to invest in the
  stock.
• Determining which ROIs are the most sensitive to
  affecting changes to the optimal solution.