Trading Tradeoffs With Risk Vs. Reward by Norman J. Brown
Three variations of risk measures can be applied to evaluate the trading of mutual funds, equities, and commodities.
There’s a well-known principle in trading that it is essential to have an edge, or expectancy (E), that is positive in order to create a winning strategy where your trading equity is increasing over time. The expectancy formula, as explained in the sidebar “Calculating Standard Deviation,” is:
E = Bet [RW - (1-W)] = Bet [W(1 + R) - 1]
In trading, if you have a positive edge even though you have a low winning ratio (W), you can succeed as long as it is compensated with a high winning ratio (R). As an example, E = 0.20 can be obtained with two widely differing parameters: (1) R = 5, W = 0.20 or (2) R = 0.5, W = 0.80. If you set E = 0, you get this simple formula:
W = 1/(1 + R)
It clearly indicates the level of W required for a given R-value in order to achieve a winning level of trading. For example, R = 1.5 requires W > 0.40, whereas R = 4 only requires W > 0.20 to achieve a positive equity return.
There is far more to trading than expectancy, of course. In particular, you must be aware of your risks. There are two measures of risk that are widely used: volatility as measured by standard deviation (Std) and maximum drawdown (Mdd). I have not seen an expression for standard deviation using the trading expectancy I use here, so I derived an equation for it (equation 3 in the sidebar, “Calculating Standard Deviation”). The data shows its influence on risk. A parameter that reflects trading risk is Mdd (which measures the maximum drawdown of equity, in percent, over the trading interval) with data that clearly indicates that trading with low values of W results in high values of Mdd. This is one of the most interesting conclusions resulting from the study.