Levered portfolio return after a fee

Question

An asset's return is R \sim N(	ext{mean } 0.08,\ 	ext{variance } 0.04). A fund takes 2x leverage on it and charges a flat fee of 0.02, giving investor return Y = 2R - 0.02. What are the mean and variance of Y?

Accepted Answer

Apply the linear-transform rules to Y = aR + b with a = 2 (the leverage) and b = -0.02 (the fee): - Mean: \mathbb{E}[Y] = 2(0.08) - 0.02 = 0.16 - 0.02 = 0.14. - Variance: \operatorname{Var}(Y) = 2^2 	imes 0.04 = 4 	imes 0.04 = 0.16. Standard deviation \sqrt{0.16} = 0.4, double the original sd of \sqrt{0.04} = 0.2. Leverage doubles the spread of returns (and the standard deviation), while the flat fee only shifts the mean down. Takeaway. Leverage a = 2 scales variance by a^2 = 4 (to 0.16) and the fee only lowers the mean; risk grows with the square of leverage.

Levered portfolio return after a fee

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