Spread of total P&L from random trades

Question

A strategy makes trades as a Poisson count $N$ averaging 5 per minute. Each trade's P&L $X$ is independent with mean $0$ dollars and standard deviation $200$ dollars, independent of the number of trades. Let $S$ be the total P&L in a minute.

What are the mean and variance of $S$ ?

Accepted Answer

This is a random sum with a twist: each trade has zero expected P&L. The mean of the total is still expected count times expected size: \mathbb{E}[S] = \mathbb{E}[N]\,\mathbb{E}[X] = 5 	imes 0 = 0. But zero mean does not mean zero risk. The variance combines both layers: \operatorname{Var}(S) = \mathbb{E}[N]\operatorname{Var}(X) + \operatorname{Var}(N)\,\mathbb{E}[X]^2. For a Poisson count \operatorname{Var}(N) = 5. With \operatorname{Var}(X) = 200^2 = 40000 and \mathbb{E}[X]^2 = 0^2 = 0, the second term vanishes: \operatorname{Var}(S) = 5(40000) + 5(0) = 200000. Standard deviation \sqrt{200…

Spread of total P&L from random trades

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