Optimal Sample Size Before a Directional Bet

An asset's one-step return is $R \sim N(\mu, \sigma^2)$ where $\mu$ is unknown and $\sigma^2$ is known. Before placing a trade, you choose a position $w \in \{-1, 0, +1\}$ (short, flat, or long) with payoff $wR$. Before choosing $w$, you may pay a cost $c$ to observe $n$ i.i.d. signals $Y_1, \ldots, Y_n$ where $Y_i \sim N(\mu, \tau^2)$ (noise level $\tau^2$ is known). You then pick $w$ based on the sample mean $\bar{Y}_n$. 1. Given that you have sampled $n$ signals, derive the optimal decision rule for $w$. 2. Compute the expected profit from sampling $n$ signals, net of the cost $nc$. 3. Find the optimal sample size $n^{*}(c)$ that maximizes net expected profit. 4. Find the threshold $c^{*}$ below which sampling at all is worthwhile.