Financial Risk Manager Part 1
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A portfolio has an expected return of 9% with a standard deviation of 7%. If the returns are normally distributed, then what is the probability that the return will be greater than 16%?
Explanation:
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.