# Predict Log Stock Return Direction and Trading Strategy

The $$k$$ period log return is defined as $$r_{t}(k)=log(S_{t}/S_{t-k}),$$ Where $$S_{t}$$ is the stock closing price at time $$t$$. For argument sake, assume that by time I mean a stock trading day and historical stock closing prices are known and denoted by $$S_{1}, S_{2}, S_{3}, ...., S_{T-1}$$. Here are my questions:

1. I have seen people trying to predict the change in stock price direction e.g. one period log return i.e. $$r_{T}(1)=log(S_{T}/S_{T-1})$$ is positive or negative based on past historical data. The argument is that they buy if they expect $$r_{T}(1)>0$$ and they sell if $$r_{T}(1)<0$$.

• (a) Could anyone please tell me in this case that at which point of time during trading day $$T$$ exactly we should buy the stock and at which point of time we should sell?
• (b) Should we buy in the morning of day $$T$$ and then wait until the end of day $$T$$ to sell it?
• (c) Can we put a sell order exactly at closing time on day T?
• (d) Given $$k=1$$, does this sort of trading strategy fall in the form of "day trading"?
2. Suppose one predicts $$r_{T}(5)=log(S_{T}/S_{T-5})>0$$.

• (a) Does this mean that investor should by the stock at time $$T-5$$ i.e. 5 days earlier and hold on to it for the next five days and the sell it in trading day $$T$$?