Answer-first summary for fast verification
Answer: 0.4772
**Explanation:** To earn a 4% bonus, the return must be greater than 8% but less than or equal to 12% (since the next bonus tier of 10% starts at returns > 12%). Given: - Mean (μ) = 8% - Standard deviation (σ) = 2% - Returns are normally distributed **Step 1: Calculate z-scores** 1. For return = 8%: z₁ = (8% - 8%) / 2% = 0 2. For return = 12%: z₂ = (12% - 8%) / 2% = 2 **Step 2: Find probabilities from z-table** - P(return < 8%) = P(z < 0) = 0.5 (50%) - P(return < 12%) = P(z < 2) = 0.9772 (97.72%) **Step 3: Calculate probability for 4% bonus** P(8% < return < 12%) = P(return < 12%) - P(return < 8%) = 0.9772 - 0.5 = 0.4772 **Step 4: Verify bonus conditions** - Bonus of 4% applies when return > 8% - But when return > 12%, the bonus becomes 10% - Therefore, the 4% bonus applies specifically for returns between 8% and 12% Thus, the probability of earning a 4% bonus is 0.4772 or 47.72%.
Author: Nikitesh Somanthe
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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 4% this year?
Click here to view the normal distribution table.
A
0.6737
B
0.5
C
0.4226
D
0.4772
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