Answer-first summary for fast verification
Answer: 5.0%
Since the returns of the two funds are **independent** and **normally distributed**, the combined (merged) portfolio has: ### Combined Expected Return (μ) The portfolio weights are: - Prudent Fund: $50M / $250M = 0.20$ (20%) - Aggressive Fund: $200M / $250M = 0.80$ (80%) $$ \mu = 0.20 \times 3\% + 0.80 \times 7\% = 0.6\% + 5.6\% = 6.2\% $$ ### Combined Volatility (σ) Because the returns are independent, the portfolio variance is the weighted sum of the individual variances (no covariance term): $$ \sigma = \sqrt{(0.20)^2 (0.07)^2 + (0.80)^2 (0.15)^2} $$ $$ = \sqrt{0.04 \times 0.0049 + 0.64 \times 0.0225} $$ $$ = \sqrt{0.000196 + 0.0144} = \sqrt{0.014596} \approx 0.1208 \approx 12.1\% $$ ### Z-score for 26% return We want $P(R > 26\%)$: $$ Z = \frac{26\% - 6.2\%}{12.1\%} = \frac{19.8\%}{12.1\%} \approx 1.64 $$ ### Probability From standard normal tables (or calculator): $$ P(Z > 1.64) = 1 - P(Z \leq 1.64) \approx 1 - 0.9495 = 0.0505 \approx 5.0\% $$ **Final estimate: ≈ 5.0%**
Author: LeetQuiz Editorial Team
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Prudent Fund (USD 50 million AUM) has recently underperformed, prompting the institutional sales team to recommend merging it with Aggressive Fund (USD 200 million AUM). Prudent Fund’s returns are normally distributed with a mean of 3% and standard deviation of 7%. Aggressive Fund’s returns are normally distributed with a mean of 7% and standard deviation of 15%. Assuming the returns of the two funds are independent, senior management has asked an analyst to estimate the probability that the combined (merged) portfolio will generate a return exceeding 26%. The analyst’s estimate for this probability is closest to:
A
1.0%
B
2.5%
C
5.0%
D
10.0%
