An approach to portfolio optimization is the scenario approa

An approach to portfolio optimization is the scenario approach. In this, we identify a few scenarios (less than 10, say) that might occur during the next year. For each scenario we estimate the return on each investment. Then, we estimate the expected return and risk of the portfolio The Sentinel Finance Company, a small firm, wishes to invest in four stocks. The cost of each stock (S per share) and the forecasts of the return (S per share) for each stock made by the company\'s five analysts are given in the following table Stock 1 Stock2Stock3Stock 4 $53.00 Cost $30.00 $45.00 $27.00 Forecast 1 Forecast 2 Forecast 3 Forecast 4 Forecast 5 3.00 1.00 2.75 4.50 3.25 13.00 4.50 1.75 5.00 2.75 4.00 .60 2.75 1.90 3.75 25.00 15.00 20.00 5.00 35.00 5.40 20.00 Expected return S/share 2.90 2.60 Assume that all forecasts are equally likely. Additionally, the finance company would like to invest no more than $100,000. Sentinel has the following requirements for its investment portfolio I. Achieve an expected return of at least 10% of total amount invested 2. Achieve a minimum risk [as measured by the absolute deviation from the expected return (a surrogate for variance) 3. Invest at least 10% of the total investment in stock 4. Formulate the above portfolio optimization problem as a linear program.

Solution

Calculate Risk of each stock

Stock 1 risk = (ABS(2.9-3)+ABS(2.9-1)+ABS(2.9-2.75)+ABS(2.9-4.5)+ABS(2.9-3.25))/5 = 0.82

Stock 2 risk = (ABS(5.4-13)+ABS(5.4-4.5)+ABS(5.4-1.75)+ABS(5.4-5)+ABS(5.4-2.75))/5 = 3.04

Stock 3 risk = (ABS(2.6-4)+ABS(2.6-0.6)+ABS(2.6-2.75)+ABS(2.6-1.9)+ABS(2.6-3.75))/5 = 1.08

Stock 4 risk = (ABS(20-25)+ABS(20-15)+ABS(20-20)+ABS(20-5)+ABS(20-35))/5 = 8.00

LP model

Decision variables: Xi = Amount ($) invested in stock i

Objective: Min 0.82X1 + 3.04X2 + 1.08X3 + 8X4

s.t.

X1 + X2 + X3 + X4 <= 100000

2.9X1 + 5.4X2 + 2.6X3 + 20X4 >= 0.10*(X1+X2+X3+X4)

X4 >= 0.1*(X1+X2+X3+X4)

X1, X2, X3, X4 >= 0