A pension fund manager decides to invest a total of at most $35 million in U.S. Treasury bonds paying 5% annual interest and in mutual funds paying 8% annual interest. He plans to invest at least $5 million in bonds and at least $15 million in mutual funds. Bonds have an initial fee of $100 per million dollars, while the fee for mutual funds is $200 per million. The fund manager is alllowed to spend no more than $6000 on fees. How much should be invested in each to maximize annual interest?
Let x be the amount in millions dollar invested in bonds paying 5% annual interest. And let y be the amount in million dollars invested in mutual funds paying 8% annual interest.
Then, determine the constraints for each given conditions.
For the first condtion, the manager decides to invest a total of at most $35 millions. So, its equivalent equation is:
`x + y lt= 35`
Next, he plans to invest at least $5 million in bonds and at least $15 millions in mutual funds. So, the equivalent equations are:
`x gt= 5`
Also, the manager is allowed to spend no more that $6000 on fees. Since the initial fees for bonds is $100 per million dollars and for mutual funds $200 per million, then its equation is:
`100x + 200y lt= 6000`
And this simplifies to:
`x + 2y lt=60`
Hence, we have four constraints. These are:
EQ1: `x + y lt= 35`
EQ2: `x gt= 5`
EQ4: `x + 2y lt= 60`
Next, let's determine the objective function. Since we have to maximize the annual interest for the two investments, apply the formula for simple interest.
where P is the principal amount (amount invested), r is the annual rate and t is the number of years.
So, the interest earned in each investment for one year is:
`I_x=x*0.05*1 = 0.05x`
And, the annual interest earned for the two investments is:
`I_y+I_x=0.05x + 0.08y`
Hence, our objective function is:
`z=0.05x + 0.08y`
To determine the values of x and y that would maximize our z, graph the four inequality equations. Then, consider only the region that satisfy the four inequalities.
Hence, is the region that satisfy the four inequalities is:
(Note: Green - boundary line of EQ1. Red - boundary line of EQ2.
Orange - boundary line of EQ3 Blue - boundary line of EQ4.)
In our bounded region, the intersection of the boundary lines are (5,15) , (20,15), (10,25) and (5,27.5).
Plug-in these intersection point to the objective function
`z= 0.05x + 0.08y`
For (5,15), the value of z will be:
`z=0.05(5) + 0.08(15) = 1.45`
For (20,15), the value of z will be:
For (10,25), the value of z will be:
And for (5,27.5), the value of z will be:
`z=0.05(5) + 0.08(27.5)=2.45`
Hence, the values of x and y that will maximize z are x=10 and y=25
Therefore, to maximize the annual interest earned, the manager should invest $10,000,000 in bonds paying 5% annual interest and $25,000,000 in mutual funds paying 8% annual interest.