Ashley has earmarked at most $25 0,000 for investment in three mutual funds: a m

Question

Ashley has earmarked at most $25 0,000 for investment in three mutual funds: a money market fund, an international equity fund, and a growth and return income fund. The money market fund has a rate of return of 6%/year, the international equity fund had a rate of return of 10% and the growth-and-income fund has a rate of return of 15%/year. Ashley has stipulated that no more than 25% of her total portfolio should be in the growth-and-income fund and that no more than 50% of her total portfolio should be in the international equity fund. To maximize the return on her investment, how much should Ashley invest in each type of fund? What is the maximum return?

Explanation / Answer

(1) Find an equation for Ashley's total return on her investments.
What is Ashley's return on each investment?
[Amount invested]*[Rate of return]
What is Ashley's total return?
[Amount invested in money market]*[Rate of return for money market] + [Amount invested in equity fund]*[Rate of return for equity fund] + [Amount invested in growth-and-income fund]*[Rate of return of growth-and-income fund]
Using the labels and values given above, we get [Total Return] = .06x + .10y + .15z

(2) Identify your decision variables.
What factors have an impact on Ashley's total return?
From the equation above, the amount Ashley invests in each option and the rate of return for each option.
Are any of these factors constant?
Yes, each rate of return is a constant.
Can Ashley control any of the factors?
Yes, she can control x, y, and z, so these will be her decision variables.

(3) Identify your constraints.
From reading the problem, we get the following:
[Budget Constraint]: Ashley's total investment cannot exceed $250,000.
[G&I Limit Constraint]: Her investment in growth-and-income cannot be more than 25% of her total investment.
[Int. Eq. Constraint]: Her investment in international equity cannot be more than 50% of her total investment.
To rewrite these constraints as mathematical expressions, we need an expression for Ashley's total investment.
[Total investment] = [investment in money market] + [investment in equity] + [investment in growth-and-income].
These amounts have already been labeled as x, y, and z respectively, so
[Total investment] = x + y + z.

We can rewrite these constraints mathematically:
[Budget Constraint]: x + y + z <= 250000
[G&I Limit Constraint]: z <= .25(x + y + z)
[Int. Eq. Constraint]: y <= .50(x + y + z)

We're done, right?
Kind of. Generally, we want to write our constraints so that all of the variables are on one side of the <= sign, with a fixed constant on the other side.
Using a little algebra, we get
[G&I]: -.25x + -.25y + .75z <= 0
[Int. Eq.]: -.50x +.50y -.50z <= 0

Putting everything together, we get

maximize .06x + .10y + .15z
subject to
x + y + z <= 250000
-.25x + -.25y + .75z <= 0
-.50x +.50y -.50z <= 0

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