M12 · Problem 1 ★ Check

Single-day AR variance

For the market model with $X$ having rows $(1,R_{m\tau})$ , invert $X'X$ explicitly and derive (12.7):

\operatorname{Var}(AR_{i\tau})=\sigma^2\left[1+\frac1{L_1}+\frac{(R_{m\tau}-\bar R_m)^2}{\sum_s(R_{ms}-\bar R_m)^2}\right].

On a day when the market moves four standard deviations of its estimation-window distribution, by how much is the AR variance inflated relative to $\sigma^2(1+1/L_1)$ , with $L_1=240$ ?

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