Question 1207673: The American Association of Individual Investors publishes an annual guide to the top mutual funds (The Individual Investor’s Guide to the Top Mutual Funds, 22e, American Association of Individual Investors, 2003).
The total risk ratings for 29 categories of mutual funds are as follows.
Total Risk Number of Fund Categories
Low 7
Below Average 6
Average 3
Above Average 6
High 7
a. Let x ‹ 1 for low risk up through x ‹ 5 for high risk, and develop a probability distribution for level of risk.
b. What are the expected value and variance for total risk?
c. It turns out that 11 of the fund categories were bond funds. For the bond funds, seven categories were rated low and four were rated below average.Compare the total risk of the bond funds with the 18 categories of stock funds.
Answer by asinus(45)   (Show Source):
You can put this solution on YOUR website! To analyze the total risk ratings for mutual funds, we will follow the steps outlined in the question.
### Step 1: Develop a Probability Distribution for Level of Risk
#### a) Assign Values and Calculate Probabilities
We will assign values to the risk categories as follows:
- Low risk: $ x_1 = 1 $
- Below Average risk: $ x_2 = 2 $
- Average risk: $ x_3 = 3 $
- Above Average risk: $ x_4 = 4 $
- High risk: $ x_5 = 5 $
Next, we will calculate the total number of fund categories and the probability for each risk level.
**Total Number of Fund Categories**:
$$
\text{Total} = 7 + 6 + 3 + 6 + 7 = 29
$$
**Probability Distribution**:
$$
P(x_1) = \frac{7}{29}, \quad P(x_2) = \frac{6}{29}, \quad P(x_3) = \frac{3}{29}, \quad P(x_4) = \frac{6}{29}, \quad P(x_5) = \frac{7}{29}
$$
The probability distribution can be summarized as follows:
$$
\begin{array}{|c|c|c|}
\hline
\text{Risk Level} & \text{Value (x)} & \text{Probability (P)} \\
\hline
\text{Low} & 1 & \frac{7}{29} \\
\text{Below Average} & 2 & \frac{6}{29} \\
\text{Average} & 3 & \frac{3}{29} \\
\text{Above Average} & 4 & \frac{6}{29} \\
\text{High} & 5 & \frac{7}{29} \\
\hline
\end{array}
$$
### Step 2: Calculate the Expected Value and Variance
#### b) Expected Value $ E(X) $
The expected value $ E(X) $ is calculated as follows:
$$
E(X) = \sum (x_i \cdot P(x_i))
$$
Calculating each term:
$$
E(X) = 1 \cdot \frac{7}{29} + 2 \cdot \frac{6}{29} + 3 \cdot \frac{3}{29} + 4 \cdot \frac{6}{29} + 5 \cdot \frac{7}{29}
$$
Calculating each term:
$$
E(X) = \frac{7}{29} + \frac{12}{29} + \frac{9}{29} + \frac{24}{29} + \frac{35}{29}
$$
Summing these values:
$$
E(X) = \frac{7 + 12 + 9 + 24 + 35}{29} = \frac{87}{29} \approx 3
$$
#### c) Variance $ Var(X) $
The variance $ Var(X) $ is calculated using the formula:
$$
Var(X) = E(X^2) - (E(X))^2
$$
First, we need to calculate $ E(X^2) $:
$$
E(X^2) = \sum (x_i^2 \cdot P(x_i))
$$
Calculating each term:
$$
E(X^2) = 1^2 \cdot \frac{7}{29} + 2^2 \cdot \frac{6}{29} + 3^2 \cdot \frac{3}{29} + 4^2 \cdot \frac{6}{29} + 5^2 \cdot \frac{7}{29}
$$
Calculating each term:
$$
E(X^2) = \frac{7}{29} + \frac{24}{29} + \frac{27}{29} + \frac{96}{29} + \frac{175}{29}
$$
Summing these values:
$$
E(X^2) = \frac{7 + 24 + 27 + 96 + 175}{29} = \frac{329}{29} \approx 11.34
$$
Now, we can calculate the variance:
$$
Var(X) = E(X^2) - (E(X))^2 = \frac{329}{29} - \left(\frac{87}{29}\right)^2
$$
Calculating $ (E(X))^2 $:
$$
(E(X))^2 = \left(\frac{87}{29}\right)^2 = \frac{7569}{841}
$$
Now, we need to convert $ E(X^2) $ to have a common denominator:
$$
E(X^2) = \frac{329}{29} = \frac{329 \times 29}{29 \times 29} = \frac{9541}{841}
$$
Now we can calculate the variance:
$$
Var(X) = \frac{9541}{841} - \frac{7569}{841} = \frac{1972}{841} \approx 2.34
$$
### Step 3: Compare Total Risk of Bond Funds with Stock Funds
#### c) Bond Funds Analysis
From the information provided:
- Total bond fund categories: 11
- Low risk bond funds: 7
- Below average bond funds: 4
**Risk Distribution for Bond Funds**:
- Low: 7
- Below Average: 4
- Average: 0
- Above Average: 0
- High: 0
**Probability Distribution for Bond Funds**:
$$
\begin{array}{|c|c|c|}
\hline
\text{Risk Level} & \text{Value (x)} & \text{Probability (P)} \\
\hline
\text{Low} & 1 & \frac{7}{11} \\
\text{Below Average} & 2 & \frac{4}{11} \\
\text{Average} & 3 & 0 \\
\text{Above Average} & 4 & 0 \\
\text{High} & 5 & 0 \\
\hline
\end{array}
$$
**Risk Distribution for Stock Funds**:
- Total stock fund categories: $ 29 - 11 = 18 $
- Low: $ 7 $
- Below Average: $ 6 $
- Average: $ 3 $
- Above Average: $ 6 $
- High: $ 7 $
**Probability Distribution for Stock Funds**:
$$
\begin{array}{|c|c|c|}
\hline
\text{Risk Level} & \text{Value (x)} & \text{Probability (P)} \\
\hline
\text{Low} & 1 & \frac{7}{18} \\
\text{Below Average} & 2 & \frac{6}{18} \\
\text{Average} & 3 & \frac{3}{18} \\
\text{Above Average} & 4 & \frac{6}{18} \\
\text{High} & 5 & \frac{7}{18} \\
\hline
\end{array}
$$
### Summary of Findings
- **Expected Value for Total Risk**: $ E(X) \approx 3 $
- **Variance for Total Risk**: $ Var(X) \approx 2.34 $
- **Bond Funds**: Higher concentration in low risk (7 out of 11) compared to stock funds.
- **Stock Funds**: More diverse risk distribution across categories.
This analysis shows that bond funds tend to be rated lower in risk compared to stock funds, which have a wider range of risk ratings.
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