Example 3 Financial Model Rate of Return Summation Variable/Constraint Linear Programming Example 3 Financial Model Rate of Return Summation Variable/Constraint

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

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 .08X1 + .06X2 + .05X3 + .07X4 + .04X5 + .02X6

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 ≥

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

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

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 X’s ≥ 0

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

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 = 6050/100000 = 6.05%

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

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.