A stock broker has $200,000 to invest in stocks and bonds. She wants to invest at least $100,000 in stocks and at least $50,000 in bonds. If stocks have an annual yield of 9% and bonds have an annual yield of 7% how much should she invest in each to maximize her income? Find the maximum return.
you want to
maximize p = .09x + .07y subject to
x+y <= 200000
x >= 100000
y >= 50000
A little linear programming reveals
x = 150000, y = 50000 p=17000
To find the maximum return, we need to determine the amount of money the stock broker should invest in stocks and bonds. Let's assume she invests an amount x in stocks and (200,000 - x) in bonds.
To maximize her income, we will create an equation based on the annual yields of stocks and bonds.
The income from stocks can be calculated by multiplying the investment in stocks (x) by the annual yield of 9%:
Income from stocks = 0.09x
Similarly, the income from bonds is determined by multiplying the investment in bonds (200,000 - x) by the annual yield of 7%:
Income from bonds = 0.07(200,000 - x)
Since the total income is the sum of the income from stocks and bonds, the equation becomes:
Total Income = Income from stocks + Income from bonds
= 0.09x + 0.07(200,000 - x)
To find the maximum return, we need to find the value of x that maximizes the Total Income. We can start by simplifying the equation:
Total Income = 0.09x + 0.07(200,000 - x)
= 0.09x + 0.07(200,000) - 0.07x
= 0.09x + 14,000 - 0.07x
= 0.02x + 14,000
Now, we can take the derivative of the Total Income with respect to x and set it equal to 0 to find the critical point where the maximum occurs:
d(Total Income)/dx = 0.02 = 0
2 = 0 (since 0.02x becomes 0)
However, this equation has no solution since 2 cannot equal 0. Since there is no critical point, we need to consider the endpoints of the investment range.
Given that the stock broker wants to invest at least $100,000 in stocks and at least $50,000 in bonds, we can set up the following inequalities:
x ≥ 100,000 (investment in stocks)
200,000 - x ≥ 50,000 (investment in bonds)
Simplifying the second inequality, we get:
x ≤ 150,000
Taking these inequalities into account, we find the valid range for x is between 100,000 and 150,000.
Now we evaluate the Total Income at the endpoints:
For x = 100,000:
Total Income = 0.02(100,000) + 14,000
= 2,000 + 14,000
= 16,000
For x = 150,000:
Total Income = 0.02(150,000) + 14,000
= 3,000 + 14,000
= 17,000
Therefore, the maximum return is achieved when she invests $150,000 in stocks and $50,000 in bonds, resulting in a maximum annual income of $17,000.