You are considering the risk-return profile of two mutual funds for investment. The relatively risky fund promises an expected return of 12.9% with a standard deviation of 16.7%. The relatively less risky fund promises an expected return and standard deviation of 4% and 6.7%, respectively. Assume that the returns are approximately normally distributed.

**a-1.** Calculate the probability of earning a negative return for each fund.

**Note: Round your final answers to 4 decimal places.**
Calculate the probability of earning a return above 8.2% for each fund.

To calculate the probabilities associated with each fund, we will use the properties of the normal distribution. For a normally distributed return \( X \) with an expected return \( \mu \) and standard deviation \( \sigma \), we can calculate probabilities by converting the return into a Z-score using the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

We will calculate two probabilities for each fund:

1. The probability of earning a negative return (\( P(X < 0) \)).
2. The probability of earning a return above 8.2% (\( P(X > 8.2) \)).

We’ll compute this for both the risky fund and the less risky fund.

### **1. Risky Fund**:
- Expected return \( \mu_1 = 12.9\% \)
- Standard deviation \( \sigma_1 = 16.7\% \)

#### (a) Probability of earning a negative return (\( P(X_1 < 0) \)):

\[
Z_1 = \frac{0 - 12.9}{16.7} = \frac{-12.9}{16.7}
\]

#### (b) Probability of earning a return above 8.2% (\( P(X_1 > 8.2) \)):

\[
Z_2 = \frac{8.2 - 12.9}{16.7} = \frac{-4.7}{16.7}
\]

### **2. Less Risky Fund**:
- Expected return \( \mu_2 = 4\% \)
- Standard deviation \( \sigma_2 = 6.7\% \)

#### (a) Probability of earning a negative return (\( P(X_2 < 0) \)):

\[
Z_3 = \frac{0 - 4}{6.7} = \frac{-4}{6.7}
\]

#### (b) Probability of earning a return above 8.2% (\( P(X_2 > 8.2) \)):

\[
Z_4 = \frac{8.2 - 4}{6.7} = \frac{4.2}{6.7}
\]

Now, I will calculate these Z-scores and use the standard normal distribution to find the corresponding probabilities.Here are the calculated probabilities:

### **1. Risky Fund:**
- The probability of earning a negative return is approximately **0.2199** (21.99%).
- The probability of earning a return above 8.2% is approximately **0.6108** (61.08%).

### **2. Less Risky Fund:**
- The probability of earning a negative return is approximately **0.2752** (27.52%).
- The probability of earning a return above 8.2% is approximately **0.2654** (26.54%).

Would you like any further details or explanations?

### Related Questions:
1. How do changes in the standard deviation impact these probabilities?
2. What would happen to the probabilities if the expected return increased by 2% for both funds?
3. How would you calculate the probability of earning between 5% and 10% for each fund?
4. What is the 90th percentile return for each of these funds?
5. How does compounding affect long-term returns for these mutual funds?

### Tip:
To reduce risk in your investment portfolio, diversifying across funds with different risk-return profiles can smooth out extreme outcomes over time.

You are considering the risk-return profile of two mutual funds for investment. The relatively risky fund promises an expected return of 12.9% with a standard deviation of 16.7%. The relatively less risky fund promises an expected return and standard deviation of 4% and 6.7%, respectively. Assume that the returns are approximately normally distributed.

**a-1.** Calculate the probability of earning a negative return for each fund.

**Note: Round your final answers to 4 decimal places.**
Calculate the probability of earning a return above 8.2% for each fund.

Calculate the probability of earning a negative return or a return above 8.2% for two mutual funds using normal distribution concepts. This includes Z-score calculations for a risky fund with an expected return of 12.9% and standard deviation of 16.7%, and a less risky fund with a 4% expected return and 6.7% standard deviation. Suitable for finance and statistics students at the college or university level.

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