Zoe 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 Zoe invests the same amount in each company?

What is the total rate of return if Zoe invests three times as much in SouthCo as in the other companies?

westco= 1.86%
northco= -9.62%
classic coverings= -3.39%
southco= -8.42%

To determine the total rate of return for Zoe's investments under the two different scenarios, we'll use the weighted average return formula. 

### Scenario 1: Investing the Same Amount in Each Company

If Zoe invests the same amount in each company, the total rate of return is simply the arithmetic average of the returns from each company.

Given the returns:
- WestCo: \(1.86\%\)
- NorthCo: \(-9.62\%\)
- Classic Coverings: \(-3.39\%\)
- SouthCo: \(-8.42\%\)

The total rate of return \(R_{\text{equal}}\) is calculated as follows:

\[ R_{\text{equal}} = \frac{1.86\% + (-9.62\%) + (-3.39\%) + (-8.42\%)}{4} \]

\[ R_{\text{equal}} = \frac{1.86 - 9.62 - 3.39 - 8.42}{4} \]

\[ R_{\text{equal}} = \frac{-19.57}{4} \]

\[ R_{\text{equal}} = -4.8925\% \]

So, the total rate of return if Zoe invests the same amount in each company is \(-4.89\%\).

### Scenario 2: Investing Three Times as Much in SouthCo as in the Other Companies

In this scenario, let \(x\) be the amount invested in each of WestCo, NorthCo, and Classic Coverings, and \(3x\) be the amount invested in SouthCo.

The weighted return \(R_{\text{weighted}}\) is calculated by:

\[ R_{\text{weighted}} = \frac{x \cdot 1.86\% + x \cdot (-9.62\%) + x \cdot (-3.39\%) + 3x \cdot (-8.42\%)}{x + x + x + 3x} \]

\[ R_{\text{weighted}} = \frac{1.86x - 9.62x - 3.39x - 25.26x}{6x} \]

\[ R_{\text{weighted}} = \frac{1.86 - 9.62 - 3.39 - 25.26}{6} \]

\[ R_{\text{weighted}} = \frac{-36.41}{6} \]

\[ R_{\text{weighted}} = -6.0683\% \]

So, the total rate of return if Zoe invests three times as much in SouthCo as in the other companies is \(-6.07\%\).

### Summary

1. Total rate of return (equal investment in each company): \(-4.89\%\)
2. Total rate of return (three times as much in SouthCo): \(-6.07\%\)

Do you want more details or have any questions?

Here are some related questions you might find interesting:
1. How does the rate of return change if Zoe invests twice as much in SouthCo instead of three times?
2. What would be the rate of return if NorthCo's performance was 0%?
3. How does the variance in the rates of return affect Zoe's overall investment risk?
4. What would happen to the total return if Classic Coverings had a positive return instead of negative?
5. How does diversifying investments impact the total rate of return?
6. What if Zoe had invested equally in five companies instead of four?
7. Can you calculate the overall return if one company is removed from the portfolio?
8. How would an increase in the return of the best-performing company affect the overall return?

**Tip:** Always consider the weight of each investment in your portfolio to understand its impact on the total return.

Learn how to calculate the total rate of return for Zoe's investments in two scenarios: equal investment in each company and three times as much in SouthCo. Understand the arithmetic and weighted average return formulas with detailed examples. Suitable for undergraduate students and investors interested in portfolio management.

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