Betty and Bob maintain a portfolio of three stocks, A, B, and C (short positions are allowed). Each year they adjust the proportions or weightings to reflect what they hope to accomplish for the portfolio. This year the reward (rate of return) and the risk (beta) are as per
A	         B	          C	    Portfolio
Weight	         x	         y	          z	            1
Return	      .20	      .05	        .35	         .15
Beta	       1.5	        .5	         2.0	          1.2

The portfolio return is a weighted average of the returns of A, B, and C.
The portfolio Beta is a weighted average of the betas of A, B, and C.
Clearly each portfolio characteristic is a linear combination of the characteristics of A, B, and C.

Write the information in the table as a system of three linear equations in three unknowns.
Solve the system. Interpret your results. If they have $100,000 assigned for use in the portfolio then what amount of money would be allocated to each stock? Be sure to indicate whether the stock position is long or short.

### Step 1: Write the Information as a System of Linear Equations

Given the weights of the three stocks $x$, $y$, and $z$, the equations for the portfolio return and beta can be written as follows:

1. **Portfolio Weights:**
   \[
   x + y + z = 1
   \]

2. **Portfolio Return:**
   \[
   0.20x + 0.05y + 0.35z = 0.15
   \]

3. **Portfolio Beta:**
   \[
   1.5x + 0.5y + 2.0z = 1.2
   \]

This gives us the following system of linear equations:
\[
\begin{aligned}
x + y + z &= 1 \quad \text{(Equation 1: Weight constraint)} \\
0.20x + 0.05y + 0.35z &= 0.15 \quad \text{(Equation 2: Return constraint)} \\
1.5x + 0.5y + 2.0z &= 1.2 \quad \text{(Equation 3: Beta constraint)}
\end{aligned}
\]

### Step 2: Solve the System of Linear Equations

We can solve this system using substitution or elimination. Let's use the elimination method.

Start with Equation 1:
\[
x + y + z = 1
\]
We can express $y$ in terms of $x$ and $z$:
\[
y = 1 - x - z
\]

Substitute $y = 1 - x - z$ into Equations 2 and 3:

**Substituting into Equation 2:**
\[
0.20x + 0.05(1 - x - z) + 0.35z = 0.15
\]
\[
0.20x + 0.05 - 0.05x - 0.05z + 0.35z = 0.15
\]
\[
0.15x + 0.30z + 0.05 = 0.15
\]
\[
0.15x + 0.30z = 0.10 \quad \text{(Equation 4)}
\]

**Substituting into Equation 3:**
\[
1.5x + 0.5(1 - x - z) + 2.0z = 1.2
\]
\[
1.5x + 0.5 - 0.5x - 0.5z + 2.0z = 1.2
\]
\[
1.0x + 1.5z + 0.5 = 1.2
\]
\[
1.0x + 1.5z = 0.7 \quad \text{(Equation 5)}
\]

Now we have the following two equations:
\[
\begin{aligned}
0.15x + 0.30z &= 0.10 \quad \text{(Equation 4)} \\
1.00x + 1.50z &= 0.70 \quad \text{(Equation 5)}
\end{aligned}
\]

To eliminate $x$, multiply Equation 4 by 10 and Equation 5 by 3 to align the coefficients of $x$:

\[
\begin{aligned}
1.5x + 3.0z &= 1.0 \quad \text{(Equation 6)} \\
3.0x + 4.5z &= 2.1 \quad \text{(Equation 7)}
\end{aligned}
\]

Subtract Equation 6 from Equation 7 to eliminate $x$:

\[
(3.0x + 4.5z) - (1.5x + 3.0z) = 2.1 - 1.0
\]
\[
1.5x + 1.5z = 1.1
\]
\[
x + z = \frac{11}{15} \quad \text{(Equation 8)}
\]

### Step 3: Back-substitute to find $x$, $y$, and $z$

Substitute $z$ from Equation 8 into Equation 4:

Let's rearrange and solve:
\[
x + z = \frac{11}{15}
\]
Thus, $y = 1 - \frac{11}{15} = \frac{4}{15}$

Substitute $z$ into Equation 4:
\[
z = \frac{4}{5}
\]

Finally, the weights are:
\[
x = \frac{1}{3}, \, y = \frac{1}{15}, \, z = \frac{11}{15}
\]

### Step 4: Interpret the Results

Since the weights $x$, $y$, and $z$ are positive, all positions are long. The fractions represent the proportions of the $100,000 that should be allocated to each stock:

\[
\text{Stock A: } \frac{1}{3} \times 100,000 = \$33,333.33 \, \text{(long position)}
\]
\[
\text{Stock B: } \frac{1}{15} \times 100,000 = \$6,666.67 \, \text{(long position)}
\]
\[
\text{Stock C: } \frac{11}{15} \times 100,000 = \$73,333.33 \, \text{(long position)}
\]

### Summary
- Stock A: $33,333.33 (long)
- Stock B: $6,666.67 (long)
- Stock C: $73,333.33 (long)

All positions are long.

### Additional questions:
1. How would the results change if the portfolio return was targeted at 20%?
2. What would be the new allocation if Stock C’s beta increased to 2.5?
3. How does the portfolio beta change if Stock B’s weight is doubled?
4. What is the impact on the portfolio if short positions were allowed for Stock A?
5. Can you construct a new portfolio with the same return but reduced risk?

**Tip**: Ensure that the sum of weights equals 1 when constructing a portfolio.

Learn how to optimize a portfolio with stocks A, B, and C using linear equations and matrix algebra. Calculate the optimal allocation of $100,000 based on desired portfolio return and beta characteristics. Understand how to solve systems of linear equations to achieve financial portfolio goals.

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