Question

Innis Investments manages funds for a number of companies and wealthy clients.The investment strategy is tailored to each client's needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: A stock fund and a money market fund. Each unit of the stock fun costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%.The client wants to minimize risk subject to the requirement that the annual incomes from the investment be at least $60,000. According to Innis's risk measurement system, each unit invested in the stock fund has a risk index of 8,and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is riskier investment. Innis's client has also specified that at least $300,000 be invested in the money market fund. The client wants to minimize the total risk index for the portfolio.

Let

S = units purchased in the stock fund, and

M = units purchased in the money market fund.

This leads to the following formulations:

Min 8 S + 3 M;

s.t.

50 S + 100 M <= 1,200,000 (Funds available);

5 S + 4 M >= 60,000 (Annual income);

M >= 3000 (Units in money market);

S, M >= 0.

(a)What is the minimum total risk?

(b): Refer to the Innis Investment question above and answer, specify the range of optimality for the STOCK FUND's objective coefficient.

(c): Refer to the Innis Investment question above. How much annual income will be earned by the portfolio?

(d): Refer to the Innis Investment question above. What is the rate of return for the portfolio?

(e): Refer to the Innis Investment question above. What is the shadow price for the funds available constraint?

g)Suppose the risk index for the stock fund (Cs) increases from its current value of 8 to 12. How does the optimal solution change, if at all?

(h) Suppose the risk index for the money market fund (Cm) increases from its current value of 3 to 3.5. How does the optimal solution change, if at all?

Answer 1

Using the funds constraint we have:

50*S + 100*M <= 1200000 or S+2M <= 24000

and annual income constraint is: 5S + 4M>=60000

solving the two we get: 3S = 12000 or S = 4000

which gives M = 10000

For M = 3000 we have: S = 18000 and S = 9600

For each of the above combinations we have the risk of the portfolio as:

62000, 153000, 85800

The minimum risk corresponds to the combination of S= 4000 and M = 10000

The minimum total risk is 62000 units.

b) The range of stock funds coefficient is (4000, 18000)

c) The annual income earned by the portfolio will be $60000.

d) The rate of return on the portfolio will be 60000/1200000 = 5% p.a.

P.S.: we can only do 4 parts at a time in a question.