'Optimal' Probabilistic Predictions for Financial Returns

We examine the `relative optimality' of sign predictions for financial returns, extending the work of Christoffersen and Diebold (2006) on volatility dynamics and sign predictability. We show that there is a more general decomposition of financial returns than that implied by the sign decomposition and which depends on the choice of the threshold that defines direction. We then show that the choice of the threshold matters and that a threshold of zero (leading to sign predictions) is not necessarily `optimal'. We provide explicit conditions that allow for the choice of a threshold that has maximum responsiveness to changes in volatility dynamics and thus leads to `optimal' probabilistic predictions. Finally, we connect the evolution of volatility to probabilistic predictions and show that the volatility ratio is the crucial variable in this context. Our work strengthens the arguments in favor of accurate volatility measurement and prediction, as volatility dynamics are integrated into the `optimal' threshold. We provide an empirical illustration of our findings using monthly returns and realized volatility for the S&P500 index.