Law of total variance, decomposing PnL risk

Question

State and prove the law of total variance, then use it to split a trader's daily PnL variance into a "within-regime" part and a "between-regime" part.

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Statement. For any random variables X, Y with finite variance, \operatorname{Var}(Y) = \underbrace{\mathbb{E}\big[\operatorname{Var}(Y \mid X)\big]}{	ext{within (unexplained)}} + \underbrace{\operatorname{Var}\big(\mathbb{E}[Y \mid X]\big)}{	ext{between (explained)}}. Proof. Use \operatorname{Var}(Y) = \mathbb{E}[Y^2] - (\mathbb{E}[Y])^2 and the tower property \mathbb{E}[Y] = \mathbb{E}[\mathbb{E}[Y\mid X]]. By the definition of conditional variance, \mathbb{E}[Y^2 \mid X] = \operatorname{Var}(Y \mid X) + (\mathbb{E}[Y\mid X])^2. Take expectations: \mathbb{E}[Y^2] = \mathbb{E}\big[\operator…

Law of total variance, decomposing PnL risk

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