
Explanation:
Using Grinold's Fundamental Law of Active Management, the expected standard deviation (volatility) of alpha is driven by the Information Coefficient (IC) and the residual risk ().
Standard deviation of alpha () = .
Given and , the standard deviation of alpha is or $1.8%$.
Assuming alphas are normally distributed with a mean of 0%, we want to find the proportion of stocks with an alpha greater than 3.6% or less than -3.6%.
A value of 3.6% is exactly 2 standard deviations away from the mean ().
In a normal distribution, approximately 5% of observations lie outside standard deviations from the mean (using the empirical rule).
Number of stocks = $5% \times 500 = 25$.
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Q.62 An analyst regresses the returns of 500 stocks against the returns of the S&P 500 index. The resulting pool of 500 alphas has a residual risk of 18% and an information coefficient of 10%. Assuming that the alphas are normally distributed with a mean of 0%, roughly how many stocks have an alpha greater than 3.6% or less than -3.6%?
A
5
B
250
C
475
D
25