In the APT model, what is the nonsystematic standard deviation of an equally-weighted portfolio that has an average value of $\sigma$(e_i) equal to 20% and 20 securities? A)12.5% B)625% C)4.47% D)3.54%

Question

Quizplus · Accepted Answer

The formula for the nonsystematic standard deviation of an equally-weighted portfolio in the APT model is:
σe = √(w^2[1]σ^2[1] + w^2[2]σ^2[2] + ... + w^2[n]σ^2[n])
where n is the number of securities in the portfolio, w is the weight of each security (in this case, w = 1/20 for an equally-weighted portfolio), and σ is the standard deviation of each security's nonsystematic risk (in this case, an average of 20%).
Substituting these values into the formula, we get:
σe = √[(1/20)^2(0.20)^2 + (1/20)^2(0.20)^2 + ... + (1/20)^2(0.20)^2]
σe = √(20)(0.20)^2
σe = √0.08
σe = 0.2828
Therefore, the nonsystematic standard deviation of the portfolio is 28.28%, which is equivalent to 4.47% per security (28.28% / √20). Thus, the correct answer is (C) 4.47%.

In the APT Model, What Is the Nonsystematic Standard Deviation σ\sigmaσ

In the APT Model, What Is the Nonsystematic Standard Deviation $\sigma$