M04 · Problem 3 ★★

Sign rules under Gaussianity

Let returns be zero-mean, jointly Gaussian and stationary with autocovariances $\gamma_k$ . Let $x_t=\sum_{i=0}^{L-1}w_ir_{t-i}$ and consider the sign rule with position $\operatorname{sign}(x_t)$ over day $t+1$ . Prove

\mathbb E[\operatorname{sign}(x_t)\,r_{t+1}]=\sqrt{\tfrac{2}{\pi}}\;\frac{\sum_iw_i\gamma_{i+1}}{\sigma_x},\qquad\sigma_x^2=\sum_{i,j}w_iw_j\gamma_{|i-j|}.

Conclude that the sign rule has positive expected return iff the linear rule does.

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