Stocks & Commodities V. 31:13 (8-13): Expectancy Risks In Trading by Norman J. Brown

Expectancy Risks In Trading by Norman J. Brown

The Kelly Criteria

There are always some unexpected trading profits and associated risks involved with expectancy and with the use of the Kelly criteria for betting. Here’s a clarification of two important issues.

There is some uncertainty in understanding trading using expectancy and the Kelly criteria (the optimum bet, FO); this article should clarify those two issues. To calculate expectancy (E), the equation for such is simple enough:

Expectancy (E) = B * R – (1 – B) = B * (1 + R) –1

(See sidebar, “Expectancy Glossary,” for definition of terms.) If the expectancy is greater than zero, it gives you an edge in your trading. This makes sense because a positive expectancy leads to positive (enhanced profit) trading, while a zero or negative expectancy means you should not be trading at all.

Basically, there are two types of trading: fixed-dollar trading usually associated with casino gambling and fixed-fraction (FF) trading usually associated with stock market trading. For example, in roulette gambling, we usually bet a fixed-dollar amount and repeat this in an iterative, noncompounding, approach. It turns out that roulette trading is a losing game to the gambler as E = ‑0.0526.

In the long run, the gambler will lose his money (of course, there is always the exception where the lucky gambler beats the house). Because no compounding (generally) is involved, the gambler loses $2 for every 38 spins of the wheel (betting $1 per turn), resulting in a linear loss at a ‑5.26% rate, increasing directly as the number of bets made (on average). Thus, the resulting loss of equity on average is given by:

EE = E * N * Amount bet

The FF investing in the marketplace is different as the returns and losses compound at an exponential rate, given by the following equity trading formula:

Equity = 1 + Profit = (1 + R * Bet)BN * (1 – Bet) [(1 – B) * N]

Here, the resulting equity varies at a compounded, nonlinear rate, with up and down surges. In the case of fixed-dollar gambling, the ending equity is predictable since you know the values of E, N (number of trades), and the dollar amount bet. This is not true in FF trading as the compounded results eventually turn down, as you will soon find out.