1. Example 3 Financial Model Rate of Return Summation Variable/Constraint
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 Total Expected Annual Return
MAX .08X X X X X X6

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 =
X1 + X2 + X3 – X7 = 0 or
Amount in Intel
Must be
At least
25% of tech total X3
.25 X7 ≥
X3 – .25 X7 ≥ 0 or

6. Amount in Low Return Investments
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
X4 + X5 – X7 ≥ 0 or
Amount in Low Return Investments
Cannot
Exceed
25000
X3 + X5 + X6 ≤
25000

7. Complete Model MAX .08X1 + .06X2 + .05X3 + .07X4 + .04X5 + .02X6
s.t. X X X X4 + X X ≤100000
X ≥
X X X X7 = 0
X X7 ≥ 0
X4 + X X7 ≥ 0
X X X ≤ 25000
ALL X’s ≥ 0

8. =SUMPRODUCT($C$3:$I$3,C5:I5)
Drag down

9. Total Expected Return on Investment = $6050 All $100,000 is invested.
$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/ = 6.05%

10. before it is worthwhile
Sensitivity Report
These low numbers
imply that Cisco and
Bank of America are
the most sensitive
to change. Thus
we should double-
check the ROI
estimates.
Expected returns for
both Microsoft and
First Bank must
increase to 8%
before it is worthwhile
to invest in them.
Future Investment
Yields a 7% return
True for all
future investment

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 ROI’s are the most sensitive to affecting changes to the optimal solution.