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.