Financial Risk Manager Part 1
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A portfolio manager's bonus depends on the return generated by the fund. The different bonus bands are listed below:
| Band | Bonus % |
|---|---|
| Return > 5% | 2% |
| Return > 8% | 4% |
| Return > 12% | 10% |
| Return > 20% | 14% |
| Return > 25% | 20% |
The mean return and the standard deviation of the fund managed by the portfolio manager stood at 8% and 2%, respectively. Assuming that mutual fund returns are normally distributed, what is the probability that the portfolio manager earns a bonus of between 4% to 10% this year?
Explanation:
Explanation
The bonus of 4% corresponds to the fund return of greater than 8%, and the bonus of 10% corresponds to the fund return of greater than 12%. Therefore, the task is to calculate the probability of the return being between 8% and 12%.
To calculate this probability, we need to convert these returns into z-scores using the z-score formula:
Where:
- μ = 8% (mean return)
- σ = 2% (standard deviation)
Step 1: Calculate z-scores
- For 8%: z = (8 - 8)/2 = 0
- For 12%: z = (12 - 8)/2 = 2
Step 2: Find probabilities from z-table
- P(Z ≤ 0) = 0.5
- P(Z ≤ 2) = 0.9772
Step 3: Calculate probability between 8% and 12% P(8% < Return < 12%) = P(0 < Z < 2) = P(Z ≤ 2) - P(Z ≤ 0) = 0.9772 - 0.5 = 0.4772
Therefore, the probability that the portfolio manager earns a bonus between 4% to 10% is 0.4772.