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] An investment manager wants to invest $1million dollars across multiple funds. Each investment fund has a different unit price and exposure to (or percentage of fund value invested in) shares, property and the bond market.

The investment manager is required to have an average exposure of 50% to shares, 30% to property and 20% to
bonds. Here exposure is again calculated as the percentage of the total dollar amount that is ultimately invested in
each type of asset.
Write these equations in matrix form. Explain, based on the application, why one of the equations
might be redundant (not needed).

The table provides information about four investment funds and their exposure percentages to shares, property, and bonds. The goal is to distribute $1 million across the funds in such a way that the total exposure to shares, property, and bonds meets specific target percentages.

### Variables:
- Let $ x_1 $, $ x_2 $, $ x_3 $, and $ x_4 $ represent the dollar amounts invested in Funds A, B, C, and D, respectively.

### Constraints:
The total investment must equal $1 million:
\[
x_1 + x_2 + x_3 + x_4 = 1,000,000
\]
Additionally, the exposure to shares, property, and bonds must meet the required averages:
- Exposure to shares: 50% 
- Exposure to property: 30%
- Exposure to bonds: 20%

The given table tells us the percentage of each fund invested in shares, property, and bonds.

- Fund A: $ 20\% $ shares, $ 75\% $ property, $ 5\% $ bonds
- Fund B: $ 35\% $ shares, $ 30\% $ property, $ 35\% $ bonds
- Fund C: $ 45\% $ shares, $ 50\% $ property, $ 5\% $ bonds
- Fund D: $ 75\% $ shares, $ 10\% $ property, $ 15\% $ bonds

The exposure conditions lead to the following system of equations:

1. **Shares exposure**: 
   \[
   0.2x_1 + 0.35x_2 + 0.45x_3 + 0.75x_4 = 0.50 \times 1,000,000 = 500,000
   \]

2. **Property exposure**: 
   \[
   0.75x_1 + 0.30x_2 + 0.50x_3 + 0.10x_4 = 0.30 \times 1,000,000 = 300,000
   \]

3. **Bonds exposure**: 
   \[
   0.05x_1 + 0.35x_2 + 0.05x_3 + 0.15x_4 = 0.20 \times 1,000,000 = 200,000
   \]

### Matrix Form:
This system of equations can be written in matrix form as:

\[
\begin{bmatrix}
1 & 1 & 1 & 1 \\
0.20 & 0.35 & 0.45 & 0.75 \\
0.75 & 0.30 & 0.50 & 0.10 \\
0.05 & 0.35 & 0.05 & 0.15
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
1,000,000 \\
500,000 \\
300,000 \\
200,000
\end{bmatrix}
\]

### Redundancy Explanation:
Notice that the total exposure to shares, property, and bonds must sum to 100% of the investment (50% + 30% + 20% = 100%). Therefore, the sum of these percentages provides no new information. One of the equations is redundant because the three exposure equations are related to each other. For example, if you know the exposure to shares and property, the bond exposure is automatically determined since the total must always sum to the total investment. Thus, any two of the exposure equations are sufficient, and the third is not needed to solve the system.

Would you like further details on this or assistance with a specific part of the problem?

### Follow-up Questions:
1. What is the exact solution for the amounts $x_1, x_2, x_3, x_4$ invested in each fund?
2. Can you explain how to solve this system of equations?
3. How would the matrix change if the target exposures were different?
4. What happens if the total investment is less than $1 million?
5. How does the unit price of each fund affect the overall strategy?

### Tip:
You can simplify a system of linear equations by removing any dependent equations to reduce computational effort.

An investment manager wants to invest $1 million dollars across multiple funds. Each investment fund has a different unit price and exposure to (or percentage of fund value invested in) shares, property and the bond market. The investment manager is required to have an average exposure of 50% to shares, 30% to property and 20% to bonds. Here exposure is again calculated as the percentage of the total dollar amount that is ultimately invested in each type of asset. Write these equations in matrix form. Explain, based on the application, why one of the equations might be redundant (not needed).

Explore how to distribute $1 million across investment funds to meet specific exposure targets using matrix operations and linear algebra. This advanced mathematical guide includes formulas and concepts related to linear systems and dependencies among equations. Suitable for university-level students and professionals in finance or mathematics.

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