A winner of the Florida Lotto has decided to
invest $500,000 per year. Two possible
considerations are an international stock with an
estimated return of 12% and a mutual fund with an
estimated return of 6%. The estimated risk index for
the international fund is 9 while the mutual fund
risk index is only 4. The total risk of the portfolio is
found by multiplying the risk of each account by the
dollars invested in that option. The investor would
like to maximize the return on the investment, but
the average risk index of the portfolio should not be
higher than 6 based on the estimated retirement
date. How much should be invested in each option?
What is the average risk for this investment? What
is the estimated return for the investment?
To determine how much should be invested in each option, we'll set up a system of equations.
Let's assume the amount invested in the international stock is x dollars, and the amount invested in the mutual fund is y dollars.
Since the investor wants to invest $500,000 per year, we can write the equation:
x + y = 500,000 ------ Equation 1
Next, let's consider the return on investment. The estimated return for the international stock is 12%, which translates to 0.12, and the estimated return for the mutual fund is 6%, which is 0.06. The total return on the portfolio can be calculated by multiplying the return of each account by the dollars invested in that option and summing them up:
0.12x + 0.06y = Total return
Now, we know that the investor wants to maximize the return on the investment. So we need to find the maximum value for the total return.
To find the maximum value, we introduce a new constraint:
The average risk index of the portfolio should not be higher than 6.
The risk index is found by multiplying the risk of each account by the dollars invested in that option and summing them up.
So, the risk index for the international stock is 9x, and the risk index for the mutual fund is 4y.
The average risk index of the portfolio is given by:
(9x + 4y) / (x + y) ≤ 6
Now, we have two equations and two unknowns:
x + y = 500,000 ------ Equation 1
(9x + 4y) / (x + y) ≤ 6
To solve the system of equations, we can substitute the value of y in Equation 1:
x + (500,000 - x) ≤ 500,000 / 6 - 9x + 6y
Simplifying the equation gives:
9x - 6y ≤ 3,000,000 - 6x
15x ≤ 3,000,000
x ≤ 200,000
Using this maximum value for x, we can solve for y:
y = 500,000 - x
y = 500,000 - 200,000
y = 300,000
So, the investor should invest $200,000 in the international stock and $300,000 in the mutual fund.
To calculate the average risk index:
Average risk index = (9x + 4y) / (x + y)
Average risk index = (9(200,000) + 4(300,000)) / (200,000 + 300,000)
Average risk index = (1,800,000 + 1,200,000) / 500,000
Average risk index = 3,000,000 / 500,000
Average risk index = 6
The average risk index of the portfolio is 6, which satisfies the constraint of not exceeding 6.
Finally, to calculate the estimated return for the investment:
Total return = 0.12x + 0.06y
Total return = 0.12(200,000) + 0.06(300,000)
Total return = 24,000 + 18,000
Total return = 42,000
The estimated return for the investment is $42,000.