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 fund 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 income from the investment be at least $60,000. According to Innis’ 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 the riskier investment. Innis’s client also specified that at least $300,000 be invested in the money market fund.

Let   S = units purchased in the stock fund

        M = units purchased in the money market fund

Min	8S	+ 3M
s.t.
	50S	+ 100M	<=	1,200,000
	5S	+ 4M	>=	60,000
		M	>=	3,000
		S, M	>=	0

The computer solution is as follows

1. What is the optimal solution, and what is the minimum total risk?
2. Specify the objective coefficient ranges.
3. How much annual income will be earned by the portfolio?
4. What is the rate of return for the portfolio?
5. What is the dual value for the funds available constraint?
6. What is the marginal rate of return on extra funds added to the portfolio?
7. Suppose the risk index for the stock fund (the value of ) increases from its current value of 8 to 12. How does the optimal solution change, if at all?
8. Suppose the risk index for the money market fund (the value of ) increases from its current value of 3 to 3.5. How does the optimal solution change, if at all?
9. Suppose increases to 12 and increases to 3.5. How does the optimal solution change, if at all?

Answer 1

a.         Let S = number of units purchased in the stock fund

                 M = number of units purchased in the money market fund

Min	8S	+	3M
s.t.
	50S	+	100M	≤	1,200,000    Funds available
	5S	+	4M	≥	60,000   Annual income
			M	≥	3,000 Minimum units in money market

                                      S, M, ≥ 0

Optimal Solution:  S = 4000, M = 10000, value = 62000

b.         Annual income = 5(4000) + 4(10000) = 60,000

c.         Invest everything in the stock fund.