Question

Question 1 A
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’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 the riskier investment. Innis’s client also specified that at least $300,000 be invested in
the money market fund. ( work out the per unit annual return using rate given for LHS –annual income
and money markets units constraint RHS using unit cost )
a. Formulate a linear programming model and determine how many units of each fund Innis should
purchase for the client to minimize the total risk index for the portfolio
b. Graph the feasible region.
c. Determine the coordinates of each extreme point.
d. What is the optimal solution.
e. Solve in excel using the template provided.

Answer 1

Ans (a)	Linear programming
	Linear programming is a technique for solving problems of profit maximization or cost minimization and resource allocation. In the given scenario, we would use such technique to determine how many units of each fund should purchase to minimize the risk index for the portfolio
		Amount in $
	Relevant information
	Amount authorized to invest for the new client	1200000
	Stock fund
	Cast =$ 50
	Annural rate of return = 10%
	Risk index = 8
	Money market fund
	Cast =$ 100
	Annural rate of return = 4%
	Risk index = 3
	Amount to be invested in money market fund =$ 300000
	Working
	a) Calculation of per unit of annual return	Return per unit
	Money market fund (4%*100)	4
	Stock fund (10%*50	5
	Total annual return per unit	9
	b) money market unit constraint = 300000/100    =3000 units
	Requied Annual income from investment = $ 60000
	Required return = 60000*100/1200000 = 5%
	Linear programming model
Step-1	Defining the variables
	Let X the no. of units of Stock investment
	Let Y the no. of units of Money market investment
Step- 2	Establish the constraints
	4X + 5Y = 60000
	Since the risk associated with money market investment is less than stock investment, the company should invest in the money market which has constraints of 3000 units ( maximum investment limit) for obtaining the benefit of diversification for reducing the overall risk.
	So, where we purchase 3000 units of money market the no. of units to be purchased for the stock fund are :
	4X + 5*3000 =60000
	4X = 60000-15000
	X = 45000/4
	X= 11250
	Hence, Innis should purchase 3000 units of Money market fund and 11250 units of stock fund to achieve the required return of $ 60000
Ans (b)
	for graph the feasible reason, we should determine the following variables
	Where, X =0, the Y =60000/5 =12000 units
	Where, Y =0, the X =60000/4 =15000 units
	Maximum units of X (money market can be bought ) = 3000

Ans (c)	Coordinates of each extreme point.
	Where, X =0, the Y =60000/5 =12000 units
	Where, Y =0, the X =60000/4 =15000 units
Ans (d)	The optimum solution should be :
	Where no restriction on investment in money market fund
	15000 units of money market fund if there is no restriction on maximum investment in money market fund which is $ 300000 (3000 units @ 100 each )
	Where restriction on investment in money market fund - Maximum investment $ 300000 (3000 units @ 100)
	3000 units of money market fund and 11250 units of the stock fund if there is the restriction on maximum investment in money market fund which is $ 300000 (3000 units @ 100 each )