Financial Risk Manager Part 1

Explanation

To solve this problem, we need to calculate the probability that the portfolio return exceeds 16% given a normal distribution with:

• Mean (μ) = 9%
• Standard deviation (σ) = 7%

Step 1: Calculate the Z-score

The Z-score measures how many standard deviations a value is from the mean:

[ Z = \frac{X - \mu}{\sigma} = \frac{16 - 9}{7} = \frac{7}{7} = 1 ]

Step 2: Find the probability

We want P(X > 16%), which is equivalent to P(Z > 1).

From the standard normal distribution table:

• P(Z ≤ 1) = 0.8413
• Therefore, P(Z > 1) = 1 - P(Z ≤ 1) = 1 - 0.8413 = 0.1587

Step 3: Interpretation

This means there is a 15.87% probability that the portfolio return will exceed 16%.

Key Concept: For a normal distribution, approximately:

• 68% of values fall within ±1σ of the mean
• 95% of values fall within ±2σ of the mean
• 99.7% of values fall within ±3σ of the mean

Since 16% is exactly 1 standard deviation above the mean, the probability of exceeding this value is the complement of the cumulative probability up to Z=1.