Question

The standard deviation of a portfolio:

Multiple Choice

• is a weighted average of the standard deviations of the individual securities held in the portfolio.

• is an arithmetic average of the standard deviations of the individual securities which comprise the portfolio.

• can never be less than the standard deviation of the most risky security in the portfolio.

• can be less than the standard deviation of the least risky security in the portfolio.

• must be equal to or greater than the lowest standard deviation of any single security held in the portfolio.

Answer 1

Answer: The standard deviation of a portfolio: "can be less than the standard deviation of the least risky security in the portfolio".

Explanation:
A portfolio will consist of multiple securities.
Example of a 2 asset portfolio:
Standard deviation of the portfolio=[((Wa)^2)((Standard deviation of a)^2) +((Wb)^2)((Standard deviation of b)^2) +2*Wa*Wb*(Standard deviation of a)*(Standard deviation of b)*(Correlation between a and b)]^(1/2)
Here, Wa and Wb refers to the weight of stock a and b.
Now, suppose the correlation between a and b is -1.

Then the equation becomes:
[((Wa)^2)((Standard deviation of a)^2) +((Wb)^2)((Standard deviation of b)^2) - 2*Wa*Wb*(Standard deviation of a)*(Standard deviation of b)]^(1/2)

[{(Wa)(Standard deviation of a) - (Wb)(Standard deviation of b)}^2]^(1/2)
=(Wa)(Standard deviation of a) - (Wb)(Standard deviation of b)
Suppose weight of each stock be 1. Then,
Portfolio standard deviation=Standard deviation of a - Standard deviation of b
Therefore, as the standard deviations are subtracted in this case, the portfolio standard deviation can be less than the standard deviation of the least risky security in the portfolio.