An individual has 25000 to invest. As their financial consultant, you recommend investing in Treasury Bills earning 7%, Corporate Bonds earning 9% and Junk Bonds earning 11%. Find all the investment alternatives that will earn $2000 per year.
To find all the investment alternatives that will earn $2000 per year, we can set up equations based on the given interest rates and the total amount to be invested.
Let's denote the amount invested in Treasury Bills as T, in Corporate Bonds as C, and in Junk Bonds as J.
According to the information provided, the total amount invested is $25,000, so we have the equation: T + C + J = $25,000.
Now let's calculate the annual earnings from each investment:
- Treasury Bills: The interest rate is 7%, so the earnings from T will be 7% of T, which is 0.07T.
- Corporate Bonds: The interest rate is 9%, so the earnings from C will be 9% of C, which is 0.09C.
- Junk Bonds: The interest rate is 11%, so the earnings from J will be 11% of J, which is 0.11J.
To earn $2000 per year, the sum of the earnings from each investment should equal $2000, so we have the equation: 0.07T + 0.09C + 0.11J = $2000.
Now we can solve the two equations simultaneously to find the investment alternatives:
1. T + C + J = $25,000
2. 0.07T + 0.09C + 0.11J = $2000
There are various methods to solve these equations, such as substitution or elimination. For simplicity, let's use substitution:
From equation 1, we can express T as T = $25,000 - C - J.
Now substitute T in equation 2 with $25,000 - C - J:
0.07($25,000 - C - J) + 0.09C + 0.11J = $2000.
Simplify the equation:
$1,750 - 0.07C - 0.07J + 0.09C + 0.11J = $2000.
Combine like terms:
$1,750 + 0.02C + 0.04J = $2000.
Rearrange the equation by moving $1,750 to the right side:
0.02C + 0.04J = $250.
Now we have a new equation that relates the amounts invested specifically in Corporate and Junk Bonds, which will earn $2000 per year.
To find the investment alternatives, we can plug in different values for C (remembering that T + C + J = $25,000) and solve for J. For example, let's consider a few scenarios:
1. If we invest $10,000 in Treasury Bills (T = $10,000), then C + J = $15,000.
Plugging in the values, we have:
0.02C + 0.04J = $250
0.02($15,000 - J) + 0.04J = $250
$300 - 0.02J + 0.04J = $250
0.02J = $50
J = $2,500.
Therefore, if $10,000 is invested in Treasury Bills, $2,500 should be invested in Junk Bonds, and the remaining $12,500 can be invested in Corporate Bonds to earn $2000 per year.
2. Another scenario could be investing $5,000 in Treasury Bills (T = $5,000).
Plugging in the values:
0.02C + 0.04J = $250
0.02($20,000 - C - J) + 0.04J = $250
$400 - 0.02C - 0.02J + 0.04J = $250
-0.02C + 0.02J = -$150
J - C = $7,500.
In this case, there are many possible combinations of Corporate and Junk Bond investments that will satisfy the equation. For example, if $7,000 is invested in Corporate Bonds (C = $7,000), then $14,500 should be invested in Junk Bonds (J = $14,500). Alternatively, if $5,000 is invested in Corporate Bonds (C = $5,000), then $12,500 should be invested in Junk Bonds (J = $12,500), and so on.
By applying different values for T and calculating the corresponding values for C and J, you can find all the investment alternatives that will earn $2000 per year.