Question

A winner of the Texas Lotto has decided to invest P,000 per year in the stock market. Under consideration are stocks for a petrochemical firm and a public utility. Although a long range goal is to get the highest possible return, some consideration is given to the risk involved with the stocks. A risk index on a scale of 1 - 10 (with 10 being the most risky) is assigned to the two stocks. The total risk of the portfolio is found by multiplying the risk of each stock by the dollars invested in that stock. The following table provides a summary of return and risk:

STOCK       ESTIMATED RETURN                      RISK INDEX

PETROCHEMICAL      12%                                          9

UTILITY                           6%                                           4

The investor would like to maximize the return on the investment, but the average risk index of the investment should not be higher than 6. How much should be invested in each stock? What is the average risk for this investment? What is the estimated return for this investment?

trI saw a lot of answers on Chegg, but they all seem to skip the small steps. I need to study this for a final. I will like to understand how people get the optimal solution of X1= 20000 and X2= 30000? I have been stuck on trying to understand this question. Thank you in advance!

Answer 1

Decision Variables:

Let the amount invested in Petrochemical and utility stock be x and y respectively.

Objective: To Maximize the expected return of the portfolio of two stocks.

Objective Function: Max(Return) = 0.12x + 0.06y

Subject to the constraints:

Constraint 1:

x + y = 50,000 (Amount to be invested in stock market)

Constraint 2: Average risk index of portfolio should not be greater than 6

Average risk of Petrochemical stock = Risk of stock / Total Investment = 9x / (x+y)

Average risk of Utility stock = Risk of utility stock / Total Investment = 4y / (x+y)

Average risk of portfolio = Sum ( Average risk of Both Stocks) = (9x + 4y) / (x+y)

(9x + 4y) / (x+y) <= 6

(Average Risk of the portfolio of the two stocks should be less than or equal to 6)

Simplifying, we get:

9x + 4y <= 6x + 6y

3x - 2y <= 0

Constraints 3 & 4:

x >= 0

y >= 0 (Non- Negative constraint to denote $ values in positive)

Solution (Excel Solver Screenshot):

Thus, we get the required optimal solution, which states investing $20,000 in petrochemical stocks and $30,000 in utility stocks.