2. A portfolio manager tracks the return on investment (ROI) for three different assets as 6%,

8%, and 10%.

- Compute the average ROI.

- Calculate the sample variance, variance, and standard deviation of the ROI.

To solve the problem, let's break it down into two parts: calculating the average ROI and then computing the variance, sample variance, and standard deviation.

### Part 1: Compute the Average ROI
The average ROI is calculated by taking the sum of all the ROIs and dividing by the number of assets.

Given ROIs: 6%, 8%, 10%

\[
\text{Average ROI} = \frac{6\% + 8\% + 10\%}{3} = \frac{24\%}{3} = 8\%
\]

### Part 2: Calculate the Variance, Sample Variance, and Standard Deviation
Let's first calculate the variance, then the sample variance, and finally the standard deviation.

#### 1. Variance
Variance (\(\sigma^2\)) measures the dispersion of the ROIs from the mean.

1. **Calculate the deviations from the mean**:
   - \(6\% - 8\% = -2\%\)
   - \(8\% - 8\% = 0\%\)
   - \(10\% - 8\% = 2\%\)

2. **Square the deviations**:
   - \((-2\%)^2 = 4\%\)
   - \((0\%)^2 = 0\%\)
   - \((2\%)^2 = 4\%\)

3. **Calculate the average of the squared deviations**:
   \[
   \text{Variance} = \frac{4\% + 0\% + 4\%}{3} = \frac{8\%}{3} \approx 2.67\%
   \]

#### 2. Sample Variance
Sample variance (\(s^2\)) is calculated similarly to variance, but we divide by \(n-1\) instead of \(n\), where \(n\) is the number of observations.

\[
\text{Sample Variance} = \frac{4\% + 0\% + 4\%}{2} = \frac{8\%}{2} = 4\%
\]

#### 3. Standard Deviation
Standard deviation (\(\sigma\)) is the square root of the variance.

\[
\text{Standard Deviation} = \sqrt{2.67\%} \approx 1.63\%
\]

For the sample standard deviation, take the square root of the sample variance:

\[
\text{Sample Standard Deviation} = \sqrt{4\%} = 2\%
\]

### Summary of Results
- **Average ROI**: 8%
- **Variance**: 2.67%
- **Sample Variance**: 4%
- **Standard Deviation**: 1.63%
- **Sample Standard Deviation**: 2%

Would you like any further details or have any questions?

Here are five related questions you might find helpful:
1. How does the standard deviation reflect the risk in an investment portfolio?
2. What is the difference between variance and sample variance?
3. How would the ROI statistics change if another asset with a different ROI was added?
4. Why is the sample variance larger than the population variance?
5. How can the standard deviation be used to assess the stability of asset returns?

**Tip:** When calculating sample variance, always remember to divide by \(n-1\) to account for the smaller sample size, providing an unbiased estimate of the population variance.

Learn how to calculate the average return on investment (ROI), variance, sample variance, and standard deviation for a portfolio with specific ROIs. This detailed solution covers statistical concepts and formulas, providing insights into investment analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation