Question

Sharon has a total of $230,000 to invest in three types of mutual funds: growth, balanced, and income funds. Growth funds have a rate of return of 12%/year, balanced funds have a rate of return of 10%/year, and income funds have a return of 6%/year. The growth, balanced, and income mutual funds are assigned risk factors of 0.1, 0.06, and 0.02, respectively. Sharon has decided that at least 50% of her total portfolio is to be in income funds and at least 25% of it in balanced funds. She has also decided that the average risk factor for her investment should not exceed 0.05. How much should Sharon invest in each type of fund in order to realize a maximum return on her investment? Hint: The constraint for the average risk factor for the investment is given by
0.1x + 0.06y + 0.02z ≤ 0.05(x + y + z).

growth funds $:

balanced funds $:

income funds $:


What is the maximum return?

Answer 1

The linear programming model is:
max 0.12x+0.1y+0.06z
The risk constraint is: 0.1x+0.06y+0.02z<=0.05(x+y+z)
The total investment is: x+y+z=230000
The minimum investment in income funds is: z>=0.5*230000
The minimum investment in balanced funds is: y>=0.25*230000

Solving this using a software like LINGO gives a value of x = 57500 (growth funds), y = 57500 (balanced funds), and z = 115000 (income funds). This gives a maximum return of 19550.

Solving this systematically by hand is not advisable because it entails drawing in 3 dimensions, with planes cutting the search space and numerous test points to verify. The LINGO code is:
max = 0.12x+0.1y+0.06z;
0.1x+0.06y+0.02z <= 0.05(x+y+z);
x+y+z = 230000;
z >= 0.5*230000;
y >= 0.25*230000;

Answer 2

Answer: 1)Sharon invested in growth funds=$57500,in balanced funds=$57500 and in income funds=$115000 .

2)The maximum return=$19550 .

Step-by-step explanation:

Total amount  that Sharon has =$230,000

Rate of return by growth funds(x) per year =12%=0.12

Rate of return by balanced funds(y) per year =10%=0.1

Rate of return by income funds (z) per year=6%=0.06

So the total investment = x+y+z and

the total return to Sharon = 0.12x+0.1y+0.06z

Now the amount she invested in each fund is given by

z(income funds)=50% of total investment=0.50(230000)=$115000

y(balanced funds)=25% of total investment=0.25(230000)=$57500

x(growth funds)=230000-y-z=230000-115000-57500=$57500

Now,the growth, balanced, and income mutual funds are assigned risk factors of 0.1, 0.06, and 0.02, respectively and she has decided that the average risk factor for her investment should not exceed 0.05.

Therefore,the constraint for the average risk factor for the investment is given by  

0.1x + 0.06y + 0.02z ≤ 0.05(x + y + z)........(1)

by putting the values of x,y and z in *(1), we got our values satisfied.

So the maximum return = 0.12x+0.1y+0.06z =0.12(57500)+0.1(57500)+0.06(115000)

=$19550.