2.5.3 Directional Trading System

2.5.3 Directional Trading System

Forecasters’ objectives reflect their utility and action rules. As a third example, consider the decisions of a risk-neutral market timer whose utility is linear in the payoff, $U(\delta (f_t),y_{t+1})={\delta }_t{y}_{t+1}$ , where $y_{t+1}$ is the return on the market portfolio in excess of a risk-free rate at time $t+1$ and ${\delta }_t$ is the investor’s holding of the market portfolio which depends on his forecast of stock returns as of time t,

$$U(\delta (f_t),y_{t+1})={\delta }_t{y}_{t+1}.\text{\hspace{1em}(2.50)}$$

Again, we can think of the market timer’s loss as the negative of (2.50). Moreover, assume that the investor follows the decision rule, $\delta (f_t)$ , of going “long” one unit in the risky asset if a positive return is predicted $(f_t>0)$ , otherwise going short one unit. In this case the investor’s decision, $\delta (f_t)$ , depends only on the predicted sign of $y_{t+1}:$

$$\delta (f_t)=\{\begin{array}{ll}1& \mathrm{if}{f}_t\ge 0,\\ -1& \mathrm{if}{f}_t<0.\end{array}\text{\hspace{1em}(2.51)}$$

Trading profits depend on the sign of $y_{t+1}$ and $f_t$ as well as on the magnitude of $y_{t+1}$ . To see this, let 1 $(f_t>0)$ be an indicator function that equals 1 if $f_t>0$ and otherwise equals 0. Then the return from the trading strategy in (2.51) becomes

$$U(\delta (f_t),y_{t+1})=(21(f_t>0)-1)y_{t+1}.\text{\hspace{1em}(2.52)}$$

As one would expect from (2.51), both the sign (in relation to that of the forecast, $f_t)$ and magnitude of excess returns, $y_{t+1}$ , matters to the trader’s utility, while only the sign of the forecast enters into the utility function. Note that large forecast errors for forecasts with the correct sign lead to smaller loss than small forecast errors for forecasts with the wrong sign. This example also raises the issue of which forecast approach would be best suited given the directional trading rule. Since the trader ignores information about the magnitude of the forecast, an approach that focuses on predicting only the sign of the excess return could make sense.

How the forecaster maps predictions into actions may thus be helpful in explaining properties of the observed forecasts. Leitch and Tanner (1991) studied forecasts of Treasury bill futures contracts and found that professional forecasters reported predictions with higher MSE than those from simple time-series models. At first, this seems puzzling since the time-series models presumably incorporate far less information than the professional forecasts. When measured either by their ability to generate profits or to correctly forecast the direction of future interest rate movements the professional forecasters did better than the time-series models, however. A natural conclusion to draw from this is that the professional forecasters’ objectives are poorly approximated by the MSE loss function and are closer to a directional or “sign” loss function. This would make sense if investors’ decision rule is to go long if an asset’s excess payoff is predicted to be positive, and otherwise go short, i.e., sell the asset.