A. The client needs to decide if he should invest on the stocks of a food delivery company, called Pippo. The annual profit per 100 invested for Pippo has been estimated that will follow the probability distribution in the following table:

Pippo
Profit per 100	Probabilities
160	0.10
120	0.25
700	0.20
200	0.15
-300	0.20
-800	0.10

Use an appropriate approach to make the decision of whether to invest in this particular stock. [5 marks]

B. A second stock, of the rival food delivery company called Paperino has the following estimated profit per 100 invested and associated probabilities:

Paperino
Profit	Probabilities
85	0.10
60	0.25
45	0.20
25	0.15
15	0.20
10	0.10

Which investment between Pippo and Paperino should the company prefer? Clearly support your claim with appropriate measures. [5 marks]

C. If the company adopts the same stocks portfolio of last year, the rate of return is estimated that will follow a normal distribution with a mean of 15% and a variance of 4%. What is the probability that the rate of return of the portfolio will be included between 16% and 20%? [5 marks]

D. The company considers very good investments the ones characterized by a rate of return above the 18%. What percentage of the stocks included in the portfolio will have a rate of return above the 18%? [5 marks]

The client is also investigating the possibility to invest in stocks related to delivery companies and wants to estimate the average rate of return for the stocks of a delivery companies. It has been estimated that standard deviation for delivery companies stocks is approximately 20%. The client wants to estimate a 95% confidence interval for the average rate of return that must extend no more than 2% on each side of the sample mean.

E. The client will conduct this analysis only if the sample observations needed are less than 100. Can you suggest if he needs more or less than 100 observations? [5 marks]

In the past, the client invested in two different types of stocks: high tech stocks and pharmaceutical stocks. He needs to decide how many units of each type he should keep this year. He has a budget to invest up to 1.5 million. He has decided that, independently from his decision on the other stocks, he will invest 100000 in the most profitable stocks between Pippo and Paperino. On the other end, each unit of the high tech stocks costs 60 and provides an annual rate of return of 10%; each unit of pharmaceutical stocks 100 and provides an annual rate of return of 4%.

The client wants to minimise the diversification of the portfolio. He requires that the annual income from the investments will be at least 70000. He has calculated a diversification investment measure. Each unit invested in high tech stocks has a diversification index of 7. Each unit invested in the pharmaceutical stocks has a diversification index of 3, because it has a smaller range of assets. The client has also specified that at least 300000 should be invested in the high tech stocks.

F. The client wants to know how many units of each type of investments he should purchase to minimise the total diversity index for the portfolio. Formulate a linear programming model for this problem [20 marks]

G. Produce an appropriate graphical representation of the feasible region and determine how much annual income the optimal portfolio that minimise the total diversity index will generate. [15 marks]

H. Solve with the Excel optimization solver the linear programming problem formulated for question F. [15 marks]

I. Suggest what the increase in diversity for the portfolio is if the company wants to invest an additional amount of 60 in high tech stocks. [10 marks]

J. The client has never heard of linear programming techniques. Use the scientific literature to support your decision of adopting linear programming techniques. Find at least two different applications where linear programming techniques have been used in portfolio investment decision and briefly describe the approach that the authors have used. You must use appropriate sources and you must include a reference list and need to follow the APA 7th edition. [15 marks]