Strategy actual expectancy uncertainty

Despite that any trading strategy, even with a high Expected Value and awesome looking historical equity curves, possible losses within a small sample (e.g. a few hundred trades) are expected. Of course there's also the chance that it does not really make money, and that the back test data simply got lucky. The same thing could happen with a conventional business failures where they may have started with awesome concepts, but with no empirical evidence.

So how do we deal with it? As Nassim Taleb has mentioned in Models on Models, practically profitable trading strategies often hold robust theory and have little dependence on probability estimate.

Purely statistics based trading strategies are dangerous.


 Pure number crunching does not reflect how markets evolve, and that financial distributions have consistently implied a pathological nature. This means we can NOT assume that bigger sample back test results converge to population statistics, i.e. great historical performance are really meaningless without robust principle explaining the inefficiency exploited.

Here's a simple applet from HQuotes showing possible equity curves with customized expected values, for a small number of bars(trades). We can see that unless you have a significant edge, short term performance really can not be trusted!

Kelly bet sizing, equity growth probability, and stuff

(THE KELLY CRITERION IN BLACKJACK SPORTS BETTING, AND THE STOCK MARKET1

 

 

where:

• f* is the fraction of the current bankroll to wager;
• b is the net odds received on the wager ("b to 1"); that is, you could win $b (plus the $1 wagered) for a $1 bet
• p is the probability of winning;
• q is the probability of losing, which is 1 − p.

 These are the guys who created the original theory of card counting in blackjack, way before the MIT team. Some important points off the above linked research paper, keep in mind everything's proved numerically

1) Kelly's Criterion for optimal bet sizing

2) Probability estimate for reaching specific future equity levels with respect to n trials(bets)

3) Estimate of trials needed to reach specific future equity levels

4) Actual examples applied to black jack card counting, sports betting, and of course financial trading

This was the result of a 3 month sports betting strategy, applying the Kelly Ratio for each bet size. They placed 5-15 bets per day to allow for the Law of Large Numbers to kick in sooner than later. With an initial bankroll of $50k, we can see that the actual profits beat the expected a bit and ended close to 100% for the period.

Fortune's Formula book review

This has got to be one of the most practical and entertaining books I've gone through. It involves crazy smart mathematicians like Claude Shannon, John Kelly, Van Thorp working with gangsters like Longy Zwillman.

A few key concepts worth noting and still practical in today's markets,

1) Statistical arbitrage

Thorp started with Warrents (still traded over the ASX today), convertible bonds against underlying stocks. The average return of these guys in the 80s BEAT Warren Buffet's track record. There still exists plenty of other opportunities today.


2) The Kelly Criterion

where:

• f* is the fraction of the current bankroll to wager;
• b is the net odds received on the wager (that is, odds are usually quoted as "b to 1")
• p is the probability of winning;
• q is the probability of losing, which is 1 − p.

While making returns more volatile, with a positive expectancy this formula guarantees the highest possible return over time while keeping risk of ruin minimal.


3) Shannon's Demon
Claude Shannon was a freaking genius, and this is his version of a "balanced portfolio", where as long as the traded instrument IS indeed stochastic, a positive expectancy is guaranteed. So it can fit into the category of statistical arbitrage.