Dalya invested in shares of four different companies at the beginning of the year. At the end of the year the share values had increased or decreased as follows:

For full marks your answers should be given as percents accurate to at least two decimal places.
What is the total rate of return if Dalya invests the same amount in each company?

What is the total rate of return if Dalya invests four times as much in WestCo as in the other companies?
Cola company= 9.85%
XYZ Co.= -5.03%
Westco= 4.88%
Cantu Excavating= -5.08%

To determine the total rate of return, we will calculate the weighted average of the returns based on the given investment scenarios.

### Scenario 1: Equal Investment in Each Company

When Dalya invests the same amount in each company, the total rate of return is the average of the individual returns.

Let's denote the returns as follows:
- \( R_1 = 9.85\% \) (Cola Company)
- \( R_2 = -5.03\% \) (XYZ Co.)
- \( R_3 = 4.88\% \) (WestCo)
- \( R_4 = -5.08\% \) (Cantu Excavating)

The average return \( R_{avg} \) is calculated by:
\[ R_{avg} = \frac{R_1 + R_2 + R_3 + R_4}{4} \]

Substituting the given values:
\[ R_{avg} = \frac{9.85 + (-5.03) + 4.88 + (-5.08)}{4} \]
\[ R_{avg} = \frac{9.85 - 5.03 + 4.88 - 5.08}{4} \]
\[ R_{avg} = \frac{4.62}{4} \]
\[ R_{avg} = 1.155\% \]

So, the total rate of return with equal investments in each company is:
\[ 1.155\% \]

### Scenario 2: Four Times Investment in WestCo

When Dalya invests four times as much in WestCo as in the other companies, we need to calculate the weighted average of the returns, where the weight for WestCo is four times that of the other companies.

Let's denote the weights as follows:
- Weight for Cola Company, XYZ Co., and Cantu Excavating: \( w \)
- Weight for WestCo: \( 4w \)

The total weight is:
\[ w + w + w + 4w = 7w \]

The weighted average return \( R_{weighted} \) is calculated by:
\[ R_{weighted} = \frac{wR_1 + wR_2 + wR_3 + 4wR_4}{7w} \]
\[ R_{weighted} = \frac{R_1 + R_2 + R_3 + 4R_4}{7} \]

Substituting the given values:
\[ R_{weighted} = \frac{9.85 + (-5.03) + 4.88 + 4 \cdot (-5.08)}{7} \]
\[ R_{weighted} = \frac{9.85 - 5.03 + 4.88 - 20.32}{7} \]
\[ R_{weighted} = \frac{9.85 - 5.03 + 4.88 - 20.32}{7} \]
\[ R_{weighted} = \frac{-10.62}{7} \]
\[ R_{weighted} = -1.5171\% \]

So, the total rate of return with four times the investment in WestCo is:
\[ -1.52\% \]

### Summary
- The total rate of return with equal investments is \( 1.16\% \).
- The total rate of return with four times the investment in WestCo is \( -1.52\% \).

Would you like more details or have any questions?

Here are some related questions:

1. What is the significance of calculating the weighted average in investment returns?
2. How can investment diversification affect the overall rate of return?
3. What other factors should be considered when evaluating the performance of an investment portfolio?
4. How does the standard deviation of returns impact investment decisions?
5. What is the impact of compounding on investment returns over multiple periods?
6. How would the total rate of return change if one of the companies had a significantly different return?
7. What are some strategies to mitigate investment risks?
8. How does the time horizon of an investment impact the expected rate of return?

**Tip:** Always consider the risk associated with higher returns and diversify your investments to minimize potential losses.

Learn how to calculate the total rate of return in investment scenarios using weighted averages. Explore scenarios of equal investment and differential investments in companies like Cola Company, XYZ Co., WestCo, and Cantu Excavating.

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