Spread between two independent stock returns

Question

Stock A's daily return (in percent) is A \sim N(	ext{mean } 0.5,\ 	ext{variance } 4); stock B's is B \sim N(	ext{mean } 0.2,\ 	ext{variance } 5), independent of A. A long-short trader holds the spread D = A - B. What are the mean and variance of D, and what is its distribution?

Accepted Answer

A difference of independent normals is still normal. The mean subtracts, but the key trap is the variance: - Mean: \mathbb{E}[D] = 0.5 - 0.2 = 0.3. - Variance: for a difference of independent variables, variances still add, they do not subtract, because subtracting B is the same as adding -B, and flipping a sign does not change spread. So \operatorname{Var}(D) = 4 + 5 = 9. Thus D \sim N(0.3,\ 9), standard deviation \sqrt{9} = 3. Takeaway. For A - B with independence, means subtract (0.3) but variances add (9); a spread is more variable than either leg suggests at first glance.

Spread between two independent stock returns

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