Profit Maximizers, Risk Averse Traders and Separable Models

In trying to predict a market, there are essentially three classes of forecasting model we can attempt to build: directional models, which forecast only the future direction of the market; point models, which forecast a conditional mean of the future distribution of returns for some horizon from any given decision point; and, distributional models that attempt to describe the entire conditional probability density function (or maybe the lower moments of it) from which the next period(s) realized return will be drawn.

(Note, in the following we asume that the forecasting models are all actually valid out-of-sample.)

In the first circumstance, we do not have much sophistication. The actual return that occurs on any given day can be modelled as an i.i.d. random number plus a constant times our directional forecast. That constant is the mean return conditioned on our forcast indicator (+1, 0, -1).

Now, if that mean return is less than the costs of trading (slippage, brokerage fees, etc.) then a rational profit maximizer would not ever trade. If it exceeded the costs of trading then a rational profit maximizer would always trade in the direction of the forecast.

With the second style of forecast, a point forecast which changes dynamically, then a rational profit maximizer could compare the forecast on any given day with his knowledge of transaction costs (slippage as a function of position size, fixed trading fees etc.) and decide dynamically whether to trade on any given forecast. Since this makes a conditional decision, which should take the trader out of the market on some occasions when the first scenario would have us in the market, we expect that this strategy would perform better than the first (it may fail to for a particular realization). With a point forecast a profit maximizer would trade only on a forecast with a net positive expectation (i.e. forecast return exceeds all trading costs) and they will trade on every forecast with a net positive expectation.

However, we are not pure profit maximizers. Most of us are risk averse to some extent, and few of us would go any buy 10,000 S&P futures contracts if our net expected profit was just $1 on the whole trade (to give a cartoon example). A risk averse trader, which describes anybody who seeks to maximize a risk adjusted returns metric such as the Sharpe Ratio, does not do every trade that has a net positive expectation. They choose to veto the set of trades in which the expected profit does not pay them sufficiently for the risk they are taking in entering the trade.
This is equivalent to adding an additional cost to the trade: I will trade when my point forecast exceeds my trading costs and my risk penalty.

Thus a risk averse trader will outperform a profit maximizing trader in expectation on a risk adjusted basis. This is because they will not do some of the trades that the profit maximizer would enter.

Perhaps suprisingly, we should take this all into account when choosing our basic modelling paradigm. The reason is the Law of Large Numbers, which tells us that when we make a measurement with a large sample size we get a more accurate estimate of the true value of a parameter than when we make a measurement with a small sample size.

When we build a trading system which is optimized by examing a backtest of the trades, we are dealing with a sub-sample of the data available --- the data for the trades actually entered. When we build a forecasting system which is optimized by following a statistical estimation procedure on the returns data, we are dealing with the full dataset available. Thus the accuracy of our parameter estimates, by the Law of Large Numbers, is as high as possible. Furthermore, we are not embedding assumptions about our risk aversion or transaction costs etc. into our estimation procedure.