Variance of a two-regime return
With probability a return is drawn from , and otherwise from .
Find the variance of the return.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Law of total variance: , where is the regime.
The regime mean is with equal chances.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Split the variance into the noise within each regime and the spread between the two regime means.
Step 1: within regimes. Each regime has variance 1, so .
Step 2: between regimes. The regime mean is or with probability each, so it has mean 0 and variance 100.
Step 3.
The trap. Answering 1, the variance of each regime. A mixture is not a weighted average of variables: the uncertainty about which regime you are in usually dominates. In general, with weights and , the between-regime part is .