Between-pod and within-pod PnL variance

Question

A fund has three pods of equal size (so a randomly chosen trader is equally likely to be from any pod). The pods' average daily PnLs are $+2$ , $0$ , and $-2$ , and each pod has the same within-pod PnL variance of $6$ .

Use the law of total variance to find the total variance of a random trader's daily PnL, and the fraction explained by which pod they belong to.

Accepted Answer

Let X be the pod (uniform over the three) and Y the trader's PnL. The law of total variance is \operatorname{Var}(Y) = \mathbb{E}[\operatorname{Var}(Y \mid X)] + \operatorname{Var}(\mathbb{E}[Y \mid X]). Within-pod part. All three within-pod variances equal 6, so their average is \mathbb{E}[\operatorname{Var}(Y \mid X)] = 6. Between-pod part. The pod means are +2, 0, -2, with overall mean 	frac{2 + 0 - 2}{3} = 0. Their variance is \operatorname{Var}(\mathbb{E}[Y \mid X]) = 	frac13\left[(2)^2 + (0)^2 + (-2)^2ight] = 	frac13(4 + 0 + 4) = 	frac{8}{3} \approx 2.67. Total. \operatorname{Var}…

Between-pod and within-pod PnL variance

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